-A
Comprehensiue Introduction
to .
'
DIFFERENTIAL GEOMETRY
VOLUME ONE
Third Edition
MICHAEL SPIVAK
PUBLISH OR PERISH, INC.
Houston, Texas 1999
ACKNOWLEDGEMENTS
1 am greatly indebted to
without his encouragement
these volumes would have remained
a short set of mimeographed notes
and
without his TE/X program they would
never have become typeset books
PREFACE
T
he preface to the first edition, reprinted on the succeeding pages, excused
this book's deficiencies on grounds that can hardly be justified now that
these "notes" tmily have become a book.
At one time I had optimisticaliy planned to completely revise ali this material
for the momentous occasion, but I soon realized the futility of such an undertaking. As I examined these five volumes, written so many years ago, I could
scarcely believe that I had once had the energy to learn so much material, or
even recall how I had unearthed some of it.
So I have contented myself with the correction of errors brought to my attention by diligent readers, together with a few expository ameliorations; among
these is the inclusion of a translation of Gauss' paper in Volume 2.
Aside from that, this third and final edition diffiersfrom the previous ones only
in being typeset, and with figures redrawn. I have merely endeavored to typeset
these books in a manner befitting a subject of such importante and beauty
As a final note, it should be pointed out that since the first volumes of this
series made their appearance in 1970, references in the text to "recent" results
should be placed in context.
HOW THESE NOTES CAME TO BE
m d how they d i d not come t o be a book
For many years 1 have wanted t o w r i t e t h e Great American D i f f e r e n t i a l
Geometry book. Today a d i l e m a c o n f r o n t s any one i n t e n t on p e n e t r a t í n g t h e mysteries of d i f f e r e n t i a l geometry. On t h e one hand, one can
consult numerous c l a s s i c a l t r e a t m e n t s of t h e s u b j e c t i n an attempt t o
f orm some i d e a how t h e concepts w i t h i n it developed. Unf o r t u n a t e l y ,
a modern mathemat i c a l education t e n d s t o make c l a s s i c a l mathematical
works i n a c c e s s i b l e , p a r t i c u l a r l y t h o s e i n d i f f e r e n t i a l g e o m e t r y . Onthe
o t h e r hand, one can now f i n d t e x t s a s modern i n s p i r i t , and a s c l e a n i n
e x p o s i t i o n , a s Bourbaki's Algebra.
But a thorough study of t h e s e books
u s u a l l y l e a v e s one unprepared t o c o n s u l t c l a s s i c a l works, and e n t i r e l y
ignorant of t h e r e l a t i o n s h i p between e l e g a n t modern c o n s t r u c t i o n s and
t h e i r c l a s s i c a l c o u n t e r p a r t s . Most s t u d e n t s e v e n t u a l l y f i n d t h a t t h i s
ignorance of t h e r o o t s of t h e s u b j e c t has i t s p r i c e -- no one d e n i e s t h a t
modern d e f i n i t i o n s a r e c l e a r , e l e g a n t , a n d p r e c i s e ; i t ' s j u s t t h a t i t ' s
impossible t o comprehend how any one e v e r thought of them, And even a f t e r
one does master a modern treatment of d i f f e r e n t i a l geometry , o t h e r modern
t r e a t m e n t s o f t e n appear simply t o b e a b o u t t o t a l l y d i f f e r e n t s u b j e c t s .
Of course, t h e s e remarks merely mean t h a t no matter how w e l l some of t h e
present day t e x t s achieve t h e i r o b j e c t i v e , 1n e v e r t h e l e s s f e e l t h a t an
i n t r o d u c t i o n t o d i f f e r e n t i a l g e o m e t r y ought t o have q u i t e d i f f e r e n t aims.
There a r e two main premises on which t h e s e n o t e s a r e based. The f i r s t
premise i s t h a t i t i s a b s u r d l y i n e f f i c i e n t t o eschewthe modern language
of manifolds, bundles, forms, e t c . , whichwas d e v e l o p e d p r e c i s e l y i n
order t o r i g o r i z e t h e concepts of classicaldifferentialgeometry.
Rephrasing every t h i n g i n more element a r y t e m s involves i n c r e d i b l e
c o n t o r t i o n s which a r e not only unnecessary , but misleading. The work
of Gauss , f o r example, which uses i n f i n i t e s i m a l s throughout , i s most
n a t u r a l l y rephrased i n terms of d i f f e r e n t i a l s , even i f it is p o s s i b l e
t o r e w r i t e i t i n terms of d e r i v a t i v e s . For t h i s reason, t h e e n t i r e
f i r s t volume of t h e s e notes is devoted t o t h e theory of d i f f e r e n t i a b l e
manifolds, t h e basic language of modern d i f f e r e n t i a l geometry. This
language i s compared whenever possible with t h e c l a s s i c a l l a n p a g e , so
t h a t c l a s s i c a l works can then be read.
The second premise f o r t h e s e notes i s t h a t i n order f o r a n i n t r o d u c t i o n
t o d i f f e r e n t i a l geometry t o expose t h e geometric aspect of t h e s u b j e c t ,
an h i s t o r i c a l approach is necessary; t h e r e i s no point i n introducing
t h e curvature t e n s o r without explaining how i t was invented and what it
has t o do with curvature. 1 personally f e l t t h a t 1 could never acquire
a s a t i s f a c t o r y understanding of d i f f e r e n t i a b l e geometry u n t i l 1 read
t h e o r i g i n a l works. The second volume of t h e s e notes g i v e s a d e t a i l e d
exposition of t h e fundamental papers of Gauss and Riemann. Gauss' work
i s now a v a i l a b l e i n E n g l i s h (General I n v e s t i g a t i o n s of Curved Surfaces;
Raven P r e s s ) . There a r e a l s o two E n g l i s h t r a n s l a t i o n s of Riemann's work,
but 1 have provided a (very f r e e ) t r a n s l a t i o n i n t h e second volume .
Of course, 1 do not t h i n k t h a t one should f ollow a l 1 t h e i n t r i c a c i e s of
t h e h i s t o r i c a l process, with its i n e v i t a b l e d u p l i c a t i o n s and f a l c e l e a d s
What i s intended, r a t h e r , is a p r e s e n t a t i o n of t h e s u b j e c t along t h e
l i n e s which i t s development might have followed; a s Bernard Morin s a i d
t o me, t h e r e is no reason, i n mathematics any more than i n biology, why
ontogeny must r e c a p i t u l a t e phylogeny. When modern terminology f i n a l l y
i s introduced, i t should be a s an outgrowth of t h i s (mythical) h i s t o r i c a l
development, And a l 1 t h e major approaches have t o be presented, f o r they
were a l 1 r e l a t e d t o each o t h e r , and a l 1 s t i l l play an important r o l e .
A t t h i s point 1 am reminded of a paper described i n Littlewood' S
Mathematician' S M i s c e l l a n y . The paper began "The aim of t h i s paper i s
t o prove .
.." and it t r a n s p i r e d only much l a t e r t h a t t h i s aim was not
achieved (the author h a d n ' t claimed t h a t it was) . What 1 have outlined
above i s t h e content of a book t h e r e a l i z a t i o n of whose b a s i c p l a n and t h e
incorporation of whose d e t a i l s would perhaps be impossible ; what 1 have
w r i t t e n i s a second or t h i r d d r a f t of a preliminary version of t h i s book,
1 have had t o r e s t r i c t myself t o what 1 could w r i t e and l e a r n about within
t h e present academic year, and a l l r e v i s i o n s and c o r r e c t i o n s have h a d t o
be made within t h i s same period of time. Although 1 may some day be a b l e
t o devote t o i t s completionthe time which s u c h a n u n d e r t a k i n g deserves,
a t p r s s e n t 1 have no plans f or t h i s . Consequent l y , 1 would l i k e t o make
t h e s e n o t e s a v a i l a b l e now, d e s p i t e t h e i r d e f i c i e n c i e s , a n d w i t h a l 1 t h e
compromises 1 learned t o make i n t h e e a r l y hours of t h e morning.
These notes were w r i t t e n while 1 was teaching a year course i n d i f f e r e n t i a l g e o m e t r y a t Brandeis University, during t h e academic year
1969-70. The course was taken by s i x juniors and s e n i o r s , and audited by
a few graduate s t u d e n t s . Most of them were f a m i l i a r with t h e m a t e r i a l i n
Calculus on Manifolds, which i s e s s e n t i a l l y regarded a s a p r e r e q u i s i t e .
More p r e c i s e l y , t h e complete p r e r e q u i s i t e s a r e advanced c a l c u l u s using
l i n e a r algebra and a b a s i c knowledge of metric spaces. An acquaintance
with t o p o l o g i c a l spaces i s even b e t t e r , s i n c e it allows one t o avoid t h e
t e c h n i c a l t r o u b l e s which a r e sometimes r e l e g a t e d t o t h e Problems, but 1
t r i e d hard t o make everything work without i t .
The m a t e r i a l i n t h e p r e s e n t v o l u m e was covered i n t h e f i r s t t e r m , except
f o r Chapter 10, which occupied t h e f irst couple of weeks of t h e second
term, and Chapter 1 1 , which was not covered i n c l a s s a t a l l . We f ound it
necessary t o t ake r e s t cures of n e a r l y a week af t e r completing Caapters 2 ,
3, and 7 . The same m a t e r i a l could e a s i l y be expanded t o a f u11 year course
i n manifold theory with a pace t h a t few would describe a s excessively
leisurely
. 1 am gratef u1 t o t h e c l a s s f ú r keeping up w i t h my accelerated
pace, f o r otherwise the second half of these notes would not have been
w r i t t e n . 1 am a l s o extremely g r a t e f u l t o Bichard P a l a i s , whose expert
knowledge saved me innumerable hours of labor.
TABLE OF
CONTENTS
Although the chapters are not divided into sections.
the listing for each chapter @ves some indication
which topics are treated. and on what pages.
CHAPTER 1 . MANIFOLDS
Elementary properties of manifolds . . . . . . . . . . . . . . . . 1
Exarnples of manifolds . . . . . . . . . . . . . . . . . . . . . 6
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 20
CHAPTER 2 . DIFFERENTI AL STRUCTURES
Cm structures . . . . . . . . . . . . . . . . . . . . . . . . 27
Cm functions . . . . . . . . . . . . . . . . . . . . . . . . 31
Partial derivatives . . . . . . . . . . . . . . . . . . . . . . 35
Critica1 points . . . . . . . . . . . . . . . . . . . . . . . . 40
Immersion theorems . . . . . . . . . . . . . . . . . . . . . 42
Partitions of unity . . . . . . . . . . . . . . . . . . . . . . 50
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 53
CHAPTER 3. T H E TANGENT BUNDLE
The tangent space of IRn . . . . . . . . .
Tlle tangent space of an imbedded manifold .
Vector bundles . . . . . . . . . . . . . .
The tangent bundle of a manifold . . . . .
Equivalence classes of curves. and derivations
Vector fields . . . . . . . . . . . . . . .
Orieil tation . . . . . . . . . . . . . . .
Addendum . Equivalence of Tangent Bundles
Problems . . . . . . . . . . . . . . . .
xiu
CHAPTER 4. TENSORS
The dual bundle . . . . . . . . . . . . . . . . . . . . . . . 107
The difkrential of a function . . . . . . . . . . . . . . . . . 109
Classical versus modern terminology . . . . . . . . . . . . . . 1 1 1
Mul tilinear functions . . . . . . . . . . . . . . . . . . . . . 115
Covariant and contravariant tensors . . . . . . . . . . . . . . 117
Mixed tensors. and contraction . . . . . . . . . . . . . . . . 121
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 127
CHAPTER 5. VECTOR FIELDS AND DIFFERENTIAL EQUATIONS
Integral curves . . . . . . . . . . . . . . . . . . . . . . . . 135
Existence and uniqueness theorems . . . . . . . . . . . . . . . 139
The local flow . . . . . . . . . . . . . . . . . . . . . . . . 143
One-parameter groups of difkomorphisms . . . . . . . . . . . 148
Lie derivatives . . . . . . . . . . . . . . . . . . . . . . . . 150
Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Addendum 1. Dserential Equations . . . . . . . . . . . . . . 164
Addendum 2. Parameter Curves in Two Dimensions . . . . . . . 167
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 169
CHAPTER 6. INTEGRAL MANIFOLDS
Prologue; classical integrability theorems . . . . . . . . . . . . 179
Local Theory; Frobenius integrability theorem . . . . . . . . . . 190
Global Theory . . . . . . . . . . . . . . . . . . . . . . . . 194
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 198
CHAPTER 7 . DIFFERENTIAL FORMS
Alternating functions . . . . . . . . . . . . . . . . . . . . . 201
The wedge product . . . . . . . . . . . . . . . . . . . . . . 203
Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . .207
DSerential of a form . . . . . . . . . . . . . . . . . . . . . 210
Frobenius integrability theorem (secondversion) . . . . . . . . . 215
Closed and exact forms . . . . . . . . . . . . . . . . . . . . 218
Tlle Poincaré Lemma . . . . . . . . . . . . . . . . . . . . . 225
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 227
CHAPTER 8. INTEGRATION
Classical line and surface integrals
Integrals over singular k-cubes . .
The boundary of a chain . . . .
Stokes' Theorem . . . . . . . .
Integrals over manifolds . . . . .
Volume elements . . . . . . . .
Stokes' Theorem . . . . . . . .
de Rliarn cohomology . . . . .
Problems . . . . . . . . . . .
CHAPTER 9. RIEMANNIAN METRICS
Inner products . . . . . . . . . . . . . . . . . . . . . . . . 301
Riemannian metrics . . . . . . . . . . . . . . . . . . . . . 308
LRngth of curves . . . . . . . . . . . . . . . . . . . . . . . 312
The calculus of variations . . . . . . . . . . . . . . . . . . . 316
The First Variation Formula and geodesics . . . . . . . . . . . 323
The exponential map . . . . . . . . . . . . . . . . . . . . . 334
Geodesic com ple teness . . . . . . . . . . . . . . . . . . . . 341
Addendum . Tubular Neighborhoods . . . . . . . . . . . . . . 344
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 348
CHAPTER 1 O . LIE GROUPS
Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 371
L f t invariant vector fields . . . . . . . . . . . . . . . . . . . 374
Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . 376
Subgroups and su balgebras . . . . . . . . . . . . . . . . . . 379
Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 380
One-parame ter subgroups . . . . . . . . . . . . . . . . . . . 382
The exponential map . . . . . . . . . . . . . . . . . . . . . 384
Closed subgroups . . . . . . . . . . . . . . . . . . . . . . 391
LRft invariant forms . . . . . . . . . . . . . . . . . . . . . 394
Bi-invariant metrics . . . . . . . . . . . . . . . . . . . . . . 400
The equations of structure . . . . . . . . . . . . . . . . . . . 402
Probleins . . . . . . . . . . . . . . . . . . . . . . . . .
406
Conlents
xuz
CHAPTER 1 1. EXCURSION IN T H E REALM OF ALGEBRAIC
TOPOLOGY
Complexes and exact sequences . . . . . . . . . . . . . . . . 419
The Mayer-Vietoris sequence . . . . . . . . . . . . . . . . . 424
Triangulations . . . . . . . . . . . . . . . . . . . . . . . . 426
The Euler characteristic . . . . . . . . . . . . . . . . . . . .428
Mayer-Vietoris sequence for compact supports . . . . . . . . . . 430
The exact sequence of a pair . . . . . . . . . . . . . . . . . 432
Poincaré Duality . . . . . . . . . . . . . . . . . . . . . . . 439
The Thom class . . . . . . . . . . . . . . . . . . . . . . . 442
Index oí' a \lector field . . . . . . . . . . . . . . . . . . . . .446
Poincaré-Hopf Theorem . . . . . . . . . . . . . . . . . . . 450
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 453
APPENDIX A
To Chapter I . . . . . . . . . . . . . . . . . . . . . . . . 459
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 467
To Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . 471
Problems . . . . . . . . . . . . . . . . . . . . . . . . . - 4 7 2
To Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . 473
To Chapters 7.9. 10 . . . . . . . . . . . . . . . . . . . . . 474
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . - 4 7 5
NOTATlON INDEX . . . . . . . . . . . . . . . . . . . . . . . 477
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . 481
A
Comprehemive Introduction
to
DIFFERENTIAL GEOMETRY
VOLUME ONE
CHAPTER 1
MANIFOLDS
T
he nices t example of a metric space is Euclidean n-space R", consisting of
al1 n-tuples x = ( x l , . . . ,Y) with each x' E IR, where R is the set of real
numbers. Whenever we speak of R" as a metric space, we s h d assume that it
has the "usual metric"
unless anotlier metric is explicitly suggested. For n = O we will interpret R0 as
the single point O E R.
A manifold is supposed to be "locally" like one of these exemplary metric
spaces R". To be precise, a manifold is a metric space M with the following
proper ty:
If x E M, then there is some neighborhood U of x and some integer
n 2 O such that U is homeomorphic to R".
The simplest example of a manifold is, of course, just Rn itself; for each
x E R" we can take U to be ali of R". Clearly, R" supplied with an equivalent metric (one which makes it homeomorphic to R" with the usual metric),
is also a manifold. Indeed, a hasty recollection of the definition shows that
anything l-iomeomorphic to a manifold is also a manifold-the specific metric with which M is endowed plays almost no role, and we s h d alrnost never
mention it.
[If you know anything about topological spaces, you can replace "metric
space7'by "topological space" in our definition; this new definition allows some
pathological creatures which are not metrizable and which fad to have other
properties one might carelessly assume must be possessed by spaces which are
locally so nice. Appendix A contains remarks, supplementing various chapters,
which should be consulted if one allows a manifold to be non-metrizable.]
The second simplest example of a manifold is an open bali in R"; in this
case we can take U to be the entire open ball since an open ball in R" is
homeomorphic to R". This example immediately suggests the next: any open
subset V of R" is a manifold-for each x E V we can choose U to be some open
ball with x E U c V. Exercising a mathematician's penchant for generalization,
we immediately announce a proposition whose proof is left to the reader: An
open subset of a manifold is also a manifold (calIed, quite naturally, an open
submanifold of the original manifold).
The open subsets of R" already provide many dXerent examples of manifolds
Cjusi 11ow maiiy is the subject of Problem 24), though by no means all. Before
proceeding to examine other examples, which constitute most of this chapter,
some preliminary remarks need to be made.
If x is a point of a manifold M, and U is a neighborhood of x (U contains
some open set V with x E V) which is homeomorphic to RR by a homeomorphism #: U + R",then #(V) c R" is an open set containing #(x). Conse-
quently, there is an open ball W with #(x) E W c $(V). Thus x E #-'(w) c
V c U. Since # : V + R" is continuous, the set #-'(W) is open in V, and thus
open in M; it is, of course, homeomorphic to W, and thus to R". This complicated little argumentjust shows tliat we can always choose the neighborhood U
in our definition to be an open neighborhood.
Witli a little tliought, it begns to appear that, ii-i fact, U mtwf be open. But
to prove this, we need the following theorem, staied I-iere without proof.*
1. THEOREM. If U c Rn is open and f : U + Rn is one-one and continuous, then f (U) c IR" is open. (It foUows that f (V) is open for any open V c U,
so f
-' is continuous, and f is a homeomorphism.)
Theorem 1 is called "Invariance of Domain", for it implies that the property of
being a "domain" (a connected open set) is invariant under one-one continuous
maps into R". The p o o f that the neighborhood U in our definition must be
open is a simple deduction from Invariance of Domain, left to the reader as ail
easy exercise (it is also easy to see that if Theorem 1 were false, then there would
be an example where the U in our definition was not open).
We next turn our attention to the integer n appearing in our definition. Notice
that n may depend on tlie point x. For example, if M c IR3 is
ihen we can clioose n = 2 for points in MI and ri = 1 for points in M2. Tliis
example, by the way, is an unnecessarily complicated device for ~roducingone
manifold from two. In general, given Mi and Ms, with metrics di and d2, we
can firsi replace each di witli an equivalent metric di such that d i ( x , y ) .E 1 for
al1 X, y E Mi; for example, we can define
* Al1 proofs require some amount of macliiner)' The quickest routes arc probably provided by Vick, Hont o1o.g TJleory and Massey, Singular Hont ology Theoy. An old-fashioned ,
but pleasanily geometric, treátment may be found in N e w a n , %pologp ofPlenr Se&.
Then we can define a metric d on M = Mi U M2 by
d(x,v) =
di(x, y)
if therc is some i such that x , y E Mi
o therwise
(we assume that Mi and M2 are disjoint; if not, they can be replaced by new sets
which are). In tlie new space M, both Mi and M2 are open sets. If MI and M2
are manifolds, M is clearly a manifold also. This coiistruction can be applied
to any number of spaces-even uncountably many; the resulting metric space is
called the disjoint union of the metric spaces Mi. A disjoint union of manifolds
is a manifold. In particular, since a space with one point is a manifold, so is any
discrete space M, defined by the metric
Although ddfeient 11's may be required at ddferent points of a manifold M,
it would seem tliat only one n can work at a giveii point x E M. For the proof
of this ii-ituitivelyobvious assertion we llave recourse once again to Invariance
of Domain. As a first step, we note that R" is not liomeomorphic to R m wlien
11 # 111, for if 11 > 111, t1-iei-i iliei-e is a one-oiie continuous map from R m into
a iloii-open subset of R". The furtlier deduction, tliai the n of our definition
is uilique at each s E M, is lefi to the reader. This unique n is called the
dimension of M at x. A manifold has dimension 11 01- is n-dimensional or is an
a-manifold if it lias dimension i~ at each point. It is convenient to refer to tlie
inaiiifold M as M" wlieii we want to indicaie tliai M has dimensioii n.
Consider once more a discrete space, whicli is a O-dimensional maiiifold.
The oiily compact subsets of such a space are finite subsets. Consequently, an
ui-icountablediscrete space is not o-compact (it cannot be written as a countable
uiiioi~of com11act sul~sets).Tlie same plieiiomenon occurs with higher-dime~isional manifolds, as we see by taking a disjoint uilion of uncountably many
manifolds liomeomorpl~icto R". In tliese examples, however, h e manifold is
not coi~i-iected.Mie \vil1 often ileed to know that tliis is the only way in wliicl-i
o-compactness can fail to hold.
2. THEOREM. If X is a connecied, Iocally coinpaci metric space, tl-ien X is
o -compact.
PR001;. Foi- each 1- E X cconsider tl-iose numbers r > O such that the closed
ball
(y E X : d(x-,y) 5 i ' )
is a compact set (there is at least one such r > O, since X is locally compact).
The set of al1 such I. > O is an interval. If, for soine A-, tliis set includes all r > 0 ,
then X is o-compact, since
If not, then for each x E X define r ( x ) to be one-half the least upper bound of
al1 su& r.
The triangle inequality implies that
so that
which implies that
Jnterchanging xi and x2 gives
so tlle function r : X + IR is continuous. This has tlle following important
consequence. Suppose A c X is compact. Let A' be the union of all closed
balls of radius r (y) aiid cei-iter y, for all 11E A . Then A' is also compact. The
poof is as follows.
Let z1,22,23,. . . be a sequence in A'. For eacli i there is a yi E A such
that zi is in the ball of radius r(yi)with center yi. Since A is compact, some
subsequence of the y i , which we might as well assume is the sequence itself,
converges to some point y E A . Now the closed ball B of radius
and
center y is compact. Since yi :
iy and since the function r is continuous,
eventually the closed b d s
are contained in B. So the sequence ri is eventually in the compact set B, and
consequently some subsequence converges. Moreover, the limit point is actually
in the closed ball of radius r(y) and center y (Problem 10). Thus A' is compact.
Now let AO E X and consider the compact sets
Their union A is clearly open. It is also closed. To see this, suppose that x is
a point in the dosure of A . Tlicn there is sorne y E A with d(x, y) c i r ( x ) .
This shows tliat jf y E A,, tlien x E A,', so x E A .
Siiice X is coi-ii-iected,and A # 0 is operi a ~ i dclosed, it must be tliat X = A ,
wliich is o-compact. 6
After this liassle with point-set topology?we present the long-promised exarnples of manifolds. Tlie oiily connected i-manifolds are tlie line IR and the circle,
or I -dimensional sphere, S', defined by
The function f : (0,2ir) + S' defined by f (O) = (cos O, sin O) is a homeomorphism; it is even continuous, though not one-one, on [0,2n]. We will
orten denote the ~ o i n (cos
t O, sin O) E S' simply by O E [O, 2x1. (Of course, it
is always necessary to check that use of this notation is valid.) The function
g : (-s, s ) + S', defined by the same formula, is also a h o m e o m ~ r ~ h i s m ;
together with f it shows that S' is indeed a manifold.
There is another way to prove this, better suited to generalization. The projection P from the point (O, 1) onto the line IR x (-1) c R x R, illustrated in
the above diagram, is a homeomorphism of S' - ((0, 1)) onto R x (- 1): this is
poved most simply by calculating P : S' - ((O,])) + R x (- 1 ) explicitly. The
point (O, 1) may be taken care of similarly, by projecting onto R x (I), or it suffices to note that S' is "homogeneous"-there is a homeomorphism taking any
point into any other (namely, an appropriate rotation of IR2). Considerations
similar to these now show that the n-sphere
is an n-maiiifold. The 2-spliere S2,commonly known as "the sphere", is our
first example of a compact 2-manifold or surface.
From these few manifolds we can already construct many others by noting
tl-iat if M iare manifolds of dimension ir j (i = 1,2),then M I x M2 is an (ni +n2)manifold. In particular
n times
is called the n-torus, while S' x S' is commonly called "the torus". 1t is obviously liomeomorphic to a subset of IR4, and it is also liomeomorphic to a
certain subset of IR3 which is what most people have in mind when they speak of
"the torus": This subset may be obtained by revolving the circle
{(O,y,a) E IR3 : (JJ - 1)2 + z2 = 1/41
around the z-axis. The same construction may be applied to any 1 -manifold
coiitaiiied ili {(O,jp,r) E IR3 : 1):> O). Tlie resul tiiig surface, called a surface of
revolution, lias coinpoiients lioineomorpl~iceither to tlie toi-us or to the cyliiideiSi x R, tlie latter of which is also homeomorphic to the annulus, the region of
tlie plai-ie contail-ied be tweeii two concentric circles.
l'hc i-iestsimplest compact 2-maiiifold is tlie 2-lioled torus. To provide a more
explicit description of tlie 2-lioled torus, it is easiest to begin witli a "l-iandle", a
space liomeomorphic to a torus witli a hole cut out; more precisely, we tlirow
away d the points on one side of a certain circle, wbich remains in our llandle,
and which will be referred to as the boundary of the handle. The 2-holed
torus may be obtaiiied by piecing two of tliese together; it is also described as
the disjoint uiiion of two handles witli corresponding points on the boundaries
"identified".
Tlie >I-holedtorus may be obtained by re~eatedapplications of this procedure. It is homeomorpliic to iIie space obtained by starting with the disjoint
union of i 7 liandlcs and a spliere with n holes, and then identif9ng points on
the boundary of the ithIiandle with corresponding points on the ithboundary
piece of the sphere.
There is one 2-manifold of which most budding mathematicians make the
acquaintance wheli tliey still know more aboui paper and paste than about
metric spaces-ihe famous Mobim S@, which you "make" by giving a strip
of paper a half twist before pasting its ends together. This can be described
analytically as the image in R3 of the function f : [O, 2x1 x (- 1,l) + R3 defined
by
e c o s B ,2sinB+icosIsinB,
e
f(B,i)= (2cosB+icos~
If we define: f on [O, 2x1 x [- I,1] instead, we obtain the Mobius strip with
a boundary; as iiivestigation of the paper model will show, this boundary is
homeomorphic to a circle, not to two disjoint circles. With our recently introduced terminology, tlie Mobius strip caii also be described as [O, I] x (- 1,l)
with (O, i) and ( 1, -1) "identified".
We have not yet liad to make precise this notion of ccidentification'y,but our
next example will force tlie issue. We xvisli to identify eacli point x E s2witli
its aiitipodal poiiit -x E S2.The space whicli results, the projecúve plane, IP2,
is a lot harder io visualize than previous examples; indeed, there is no subset
of IR3 which represents it adequately.
The precise definition of IP2 uses the same trick that mathematicians always
use when they want two things wliich are not equal to be equal. The p0in.hof P 2 are defilied to be the sets ( p , - p ) for p E s2.We wilI denote this set
by [ p ] E IP2, so tliat [ - p ] = [ p ] . We thus have a map f : S2 + IP2 @ven
by f ( p ) = [ p ] , for whicl-i f ( p ) = f ( q ) implies p = fq . We will postpone
for a while the problem of defining the metric giving the distance between two
points [ p ] and [ q ] , but we can easily say wliat the open sets will turn out to be
(and this is all you need to know iil order to check that P 2 is a surface). A subset
U c p2 will be open if aiid only if f ( U ) c S2is open. This just means that
the open sets of P 2 are of tlie form / ( V ) where V c S2 is an open set with
the addiiional important property that if it contains p it also contains - p .
-'
In exacily ihe same wa): we could have defined ihe poinis of the Mohius
sirip M io be
al1 points (s,f) E (O, 1) x (-1,I)
iogeiher wiih
allseis {(O,r),(l,-111,
denotedby[(O,f)Ior[(I,-111.
(S, f ) if s # O, 1
[(s,f)] i f s = O or 1,
-'
and U c M is open if and only if f (U) c [O, 11 x (- 1 , l ) is open, so tliat
tl-ie open seis of M are of the form f ( V ) ~rliereV is open and coniains (S, -1)
wlienevcr it coniains (S, f ) for s = O or 1.
To gei an idea of wliat p 2looks Iike, we can make tliings easier for ounelves
by first ilirowiiig away all poiiiis of s2helow tlie (x, y)-plane, siiice iliey are
ideniified with poinis ahmre ihe (x, y)-plane anyway. This leaves the upper
I-icniisplier-e(including tlie bounding ci~-clc), wliicli is Iiomeoinor~~li
ic io ilie disc
aiid wc must idcntify eacli p E Si wiili - p E S'. Squaring tliiiigs o K a
bit, tliis is ilie same as identifying points on ilie sides of a square according
to tlie sclienie shown below (poinis on sides wiil-i ihe same lahel are ideniified
in sucli a way iliat ille heads of tlie arrows are ideniified wiih each otlier). Tlie
dotied liiies iii ihis piciui-e are h e key io undersiaiiding IP2. If we disiori ilie
region beiween tl-iem a bit we see tliat the froni pari of B followed by t l ~ e
I~ackpart of A, ai the upper left, is to be ideniified wiih the carne thing ai the
10wer righi, in reverse direction; in other wods, we obtain a Mobius sirip with
a boundary (narnely, ihe doited line, which is a single circle). If this h4obius
strip is removed, \ve are lefi wiih two pieces \vI~ichcan be rearranged to form
something homeomorphic to a disc. The projectiw plane is thus oh<ained from
tl-ie disjoini union of a disc and a Mobius sirip with a boundary, by identifying
points on the boundary and poinis on the boundary of the disc, both of which
are circles. Thus io make a model of iP2 we jusi have to sew a circular piece of
cloth and a clotlr Mobius strip iogci1ie1-along ifieir cdges. Unforiunately a litile
experimentaiion will convince you ilrai ihis cannot be done (without having tlle
iwo pieces of doth pass through each other}.
The subsei of IR3 obiained as the union of ilie Mobius sirip and a disc, alihough noi hoiiicoiiioipliic to iP2, can siill be described mathematically in terms
of IP2. There is clearly a coiitinuous funciion / : iP2 + IR' wliose image is this
subsei; moreovei; althougli f is not one-one, ii is locally one-one, that is, every
point p E IP2 has a neigliborhood U on whicli f is one-one. Sucli a function f
is called a topological immersion (the single word "immersion" has a more v e cialized meaiiiiig, explained in Chapter 2). IVe can thus say thai P2 can be
topologicaIIy immersed in W ', al though noi topologically imbedded (ihere is no
liomeomoi-phism f fiom PZ LO a subset of W3), In W4, liowever, wiih an extra
dimension to play around ivith, the disc can be added so as noi io interseci the
Mobius strip.
Another topological immersioii of P2in W3 can be obtained by first immersing ihe Mobius sirip so iflat its boundary cirde lies iri a plane; ibis can be done
in the follo~vingway. The figures below show thai the Mobius sirip may be obtairled from an annulus by identifying opposite points of the inner circle. (This
is also obvious from ihe fact thai ihe A4obius sirip is the projective plane wiih a
disc removed.} TIlis inner circle can l ~ replaced
e
by a quadrilateral. When ihe
resuliing figure is di.a\vn up irito 3-space and the appropriatc ideniifications are
niade we ol~tairiibe "cross-cap". The cross-cap iogether wiih the disc ai tlie
bo~tomis a iopologically immersed P2.
'llic onc gap in ihe precedii-ig discussion is ilre definiiion of a metric for P2.
'l'lir niissiiig nleir-ic can he supplied by an appeal to Problem 3-1, whicl~will
I;itri. l ~ used
e
quite often, and which the reader should peruse sometime hefoi-e
1-i.;icliiigCliapter 3. Rouglily speaking, ii sliows ihai iliiiigs llke P Z ,whidi ougli r
t i ) be inanifolds, are. (Those who know ahout iopological spaces will recognize
ii 21s a disguised case of ille Urysohn Metrizaiion Theorem.) For the preseiii,
Iio~*rver,
we will obtain our metric by a irick thai simultaneously provides an
iiiibidding of P 2 in JR4. Coiisider ihe fuiiciion f : s2+ JR4 defined by
f(x, y, 5 ) = (yz, xz, xy, x 2 + 2y2 + 3 2 ) .
f(p) = f(-p). We mainiain ihai f(p) = f(q) implies ihat p = fy.
'li) p~-ovetl~is~
suppose ihai f(xi y , ~ =
) f(ai b, c). We have, firsi of ail
( :lr.asly
xy = ab.
I1' a , b ,c # 0, ihis leads to
r4
Now
lience
(3)
Using (2), ihis gives
so s = f a . Similarly, Mre obiain y = f b , z = f b , wiih ihe same sign (which
comes from (3))holding for dl three equations. In this case we have proved our
conteniion ~riihouteven using the fourth coordinate of f. Now suppose a = 0.
If x # 0; then (1) would immediately give y = z = O, so that
(x, y, z ) = (fl , O , O).
But y = ;= O implies (by (1) again) that bc = O, so b = O or c = O and
These equations clearly con tradict
Thus x = O aiso, and Mre have
But (6) implies ihat
2). 2
+ 3z2 = zY2 + 3(1 - y2)
= 3 - y ,2
and similarly for b and c, so (5) gives
Now (4) $ves
(tlris holds eveil if y = b = 0, since then 1,c = f 1). Clearly, (4) also shows thai
thc same sign holds in (7) arid (8).which completes the proof.
Sinre f(p) = f(q) precisely wlieii p = f q , we can define j : P2 + R4 b!,
This nlap is ont-oi~eand wc can LISC it to define the meti-ic in IP2:
Then one can check that the open sets are indeed the ones described above.
By ihe way, the map g : P
' + W3 defined by tlie firsi 3 componenis of j;
is a topologicai immersion of P' in W 3 . The image in W3 is Steiner's "Romaii
surface".
Wiih the new surface P2 at our disposal, we can create other surfaces in
t he same way as the n-holed torus. For example, to a handle we can attacli
a projective space 14th a hole cut out, or, what amounts to the same thing, a
h4obius strip. The closesi we can come io picturing ihis is by drawing a crosscap sticking on a iorus. We can dso join together a pair of projective planes
with holes cut out, whicll amounts to sewing two Mobius strips together along
tlieir boundary. Although this can be pictured as two cross-caps joined together,
it has a nicer, and famous, representation. Consider the surface obtained from
ilre square with identifications indicated below; it may aiso be obtained from
tlie cylinder [O, l] x S' by identifying (O, x) E [O, l] x S' with (1, x ' ) , where x'
is ihe reflection of x through a fixed diameter of ihe circle. Notice that ihe
identifications on the square force PI, P2, P3, and f i to be identified, so that
the set {Pl,P2, P 3 ,P 4 ) is a single point of our new space. The doited Iines
beloy dividing the sides into thirds, form a single circle, which scparates the
surface into two parts, one of which is shaded.
Rearrangemeni of ihe iwo parts shows thai ihis surface is precisely iwo
Mobius strips witb corresponding points o11 iheir boundary ideniified. The
description iti terms of [O, 11 x S' immediaiely suggests an immersion of the
surface. Turriing one e i ~ dof the cylinder around and pushing ii through itself
orients ihe left-hand boundary so that (O, x) is direcily opposite (1, x') , io which
it can then be joined, forming the "Klein bottle".
1;xamples of higher-dimensional manifolds ~ ~ inoi
l l be treated in nearly such
ifvtail, I~ui!in addition to the famiiy of 11-manifolds S", we wiIl mention ihe
i.i.lated fa mil^ of "projeciive spaces". Projective 11-space P" is defined as ihe
i.olleciion of al1 sets (p, -p) for p E S". T h e description of the open sets in P"
is ~x.eciselyanalogous to tlie descripiion for P2. Alihough these spaces seem to
li)i-ina family as regular as the family S", we ~cillsee Iater that the spaces P"
liii. cven n dXer in a very important way from the same spaces for odd n.
One further definition is needed io complete this introduction to manifolds.
\%. have dready discussed some spaces which are noi manifolds only because
tliey have a "boundarf"' for example, the h4obius strip and the disc. Points
r i i i these "boundaries" do not have neighborhoods homeomorphic to IRn, but
t l rey do llave neighhorhoods homeomorphic io a n important subse t of IR". Tlie
(closed)hdf-space W" is defined by
W" = { ( x t ,...,x") E IRn : x" 2 O}.
!'Imanifold-with-boundary is a metric space M with the following property:
If x E M , ihen there is some neighborhood U of x and some integer
rr > O such ihai U is homeomorphic to either R" or W".
A poini in a manifold-with-boundary canriot have a neighborhood liomeomor-phic to both IR" and W" (Invariance of Domain again); we can therefore
distirlguish ihose poinis x E M having a neighborhood homeomorphic to Hn.
.]'he sei of al1 such x is cdled ihe boundary of M arid is denoted by aM. If M is
;ictually a manifold, tlien 8M = 0. Notice that if M is a subset of IR", then aM
is noi necessarily the same as the boundary of M in the old sense (defined for
iiny subset of IRn); indeed, if M is a manifold-wiih-boundary of dimension < n,
tlien ali poinis of M wiil be boundary points of M.
If manifolds-~lith-boundaryare studied as frequently as manifolds, it becomes
I~oibersomeio use tl-iis long designaiion. Ofien, ihe word "manifold" is used
ior "manifold-~~ith-11oundary"-A manifold in our sense is then cdled "nonI~ounded";a non-l~oundedcompact mailifold is cailed a "closed manifold". We
\vil1 stick to the other termiiiology, hui ~ l i l sometimes
l
use "bounded manifold"
insiead of "manifold-with-boundary".
PRQBLEMS
1. Slio\v that if d is a metric on X, then both d = d/(l + d ) and d =
min(1,d) are d s o metrics and ihai tliey are equivdent to d (Le., the identiv
map 1 : (X, d ) + (X, d) is a homeomorphism).
2. Jf (Xi, di) are metric spaces, for i E 1, with meirics di < 1, and Xi n Xj = 0
for i # j , tl~en(X, d ) is a meiric space, where X = U iXj, and d(x, y) =
di(x, 1.) if A+, y E Xi for some i, while d(x, y) = 1 otherwise. Each Xi is an
open subsei of X, and Y is homeomorphic to X if and only if Y = U iYi
where the Yiare disjoint open sets and Yi is homeomorphic to Xi for each i.
T l ~ space
e
(X, d ) (or any space homeomorphic io it) is cdled the disjoint union
of ihe spaces Xi.
3. (a) Exre177 manifold is locally compaci.
(b) Every manifold is locdly pathwise connecied, and a connected manifold is
pathwise connected.
(c) A connecied manifold is arcwise connected. (A path is a continuous image
of [O, 11: but an arc is a une-ui~econtinuous image. A difficult theorem states
that every path contains an arc between iis end points, but a direct proof of
arcwise-conneciedness can be @ven for manifol ds.)
4. A space X is cdled locdl)~connected if for eacll x E X ii is ihe case that
every neighborhood of x contains a connected neighborhood.
(a) Conncctedness does iiot imply local conneciedness.
01) An open subset of a locally coi~nectedspace is locally connected.
(c) X is locally connected if and only if components of open sets are open, so
evcry iieighboi-liood of a poini in a Iocally connected space coniains an open
connected neigh borhood.
(d) A locally connecied space is homcomorphic to the disjoint union of its componen ts.
(e) Every manifold is locally connected, and consequently homeornorpliic io
the disjoin t unioii of its componeii ts, wliich are open submanifolds.
5. (a) Tlie neigliborhood U in oui. definition of a manifold is always open.
(b) The integer n in our definition is unique for each x.
6. (a) A subset of aii n-manifold is an n-manifold if and only if it is open.
(h) lf M is connected, ihen ihe dimension of M at x is the same for al1 x E M.
7. (a) lf U c IR is an intenlal and f : U + IR is contiriuous and one-one,
ihen .f is eiiher increasing or decreasing.
(11) 'l'lie image f ( U )
(f.)
is open.
'l'he inap f is a homeomorpl~ism.
8. Ikr t his problem, assume
(1) (Tlie Generalized Jordan Cunle Theorem) If A c Rn is homeomorphic
to S"-', then IR" - A has 2 components, and A is the houndary of eacli.
(2) If B c IR" is homeomorphic to D" = (x E R" : d(x,O) 5 l), then
R" - B is connecied.
One componeni of R" - A (the "outside of A") is unhounded, and the other
(ihc "inside of A") is bounded.
(h) If U c W" is open, A c U is homeomorphic to SR-' and f : U +
Ri" is oi~e-oneaiid coniinuous (so ihai f is a honieomorphism on A), thei~
-/'(iiiside of A ) = inside of f (A). (First prove c .)
(f.) Pi-ove Invariance of Domain.
(a)
9. (a) Give an elenientary proof tliat IRi is not liomeomorphic to R" for n > 1.
(h) l'rove directly from the Generalized Jordan Cunre Theorem thai IRm is no1
Iiorneomorphic to IR" for nr # n .
10. In ihe proof of TI~eoirin2, sliow that the limii of a convergent subsequence
oi'the zi is aciudly i i i the closed ball of radius r(y) and cenier y.
11. Every connecied maiiifold (which is a metr-ic space) has a couniahle 11asc
li~i-its iopology, and a countahle dense subset.
P
12. (a) Compute tlie composiiion f = S' - ((0, 1)) + Ri x (- 1 ) + Ri
i:spIiciily for ihe map P on page 7, and show that it is a homeomorphism.
(h) Do the same for f : S"-' - ((O,. . .,O, 1)) + IRn-'.
13. (a) Tlie text describes tlie open subsets of iP2 as seis of ihe form f(V);
\\*liei-eV c S' is open aiid coiitaiiis - p wlienever it contains p. Show tliat this
last condition is aciually unnecessary.
(II) The andogous condiUon Zr necessary for the Mobius sirip, which is discussed
iinmediately afterwards. Explain how ihe two cases differ.
14. (a) Check tliat tlie inetric defined for IP2 gises tlie open sets described in
ihe texi.
(II) Check that P* is a surface.
15, (a) S h o ~that
l P' is homeomorphic to S ' .
(b) Since we can consider S"-' c S " , and since aniipodal points in S"-' are
siill antipodal when considered as points in S", we can consider P"-' C P" in
an obvious way. Show that P" - iPn-' is Iiomeomorphic to interior D" = (x E
R" : d(x,O) -= 1).
16. A classical tlieorem of topology states that every compact surface otl~er
~liaiiS 2 is obiained by gluiiig iogether a certain number of ton and projective
spaces, and that all compact surfaces-~4th-boundaryare ohtained from these
by cutting out a finite number of discs. To whích of these "standard" surfaces
are the following homeomorphic?
a hole in
a hole in
a hole
17. Lei C c R c IR2 be the Cantor sei. Show thai W 2 - C is homeomorpliir
to tlie surface shown at the top of the next page.
illis uii.clc is tiol in tlic sur fa c.^
these circles are
18. A locdly compact (but non-compact) space X "has one end" if for every
voinpact C c X there is a compact K such that C c K c X and X - K is
c-onnected.
(a) R" Iias one end if n > 1, but not if n = 1.
(13)
Rn - (O) does not "have one end" so Rn - ( O ) is not homeomorphic to Rm.
l 9. This proaem is a seque1 to the previous one; it will be used in Problem 24.
Aii end of X is a function E which assigns io each compaci subset C c X a
iion-empty component E(C) of X - C, in sudi a way that Cl c C2 hplies
~ ( C ZC) E(C1).
(a) If C c R is compact, ihen R - C has exactly 2 unbounded components, the
"left" component coiltaining dl iiumbers < some N, the "right" one containing
al1 ilumbers > some N. If E is an end of R, show thai E(C) is either dways tlie
"lefi" componeni of IR - C, o r dways the "rhiglli" one. Thus R has 2 ends.
(11) Show that R" has only one end E for n > 1. More generdly, X has exactly
oiie end E if and o~llyif X "has one end" in tlie sense of Problem 18.
(c) This part requires some knowledge of topological spaces. Let E(X ) be the
set of d ends of a coi-ii-iected, locally connected, locdly compact Hausdofl
sl~aceX. Define a topology on X U E(X) by choosing as neighborhoods N c (EO)
of an end EO die sets
for- dl compact C. Sl-iow ihat X U G(X) is a compact HausdorK space. Whai
is R U E(R), and R" U E(Rn) for n > l ?
20. Coilsider ihe folloiving- tliree surfaces.
(A) The infiniie-lioled torus:
(B) The doubly infiniie-holed torus:
(C) The infinite jail cell window:
_ k l l-~ : ~-~Ji!+
lli ~J\t
-
e
*
*
-
-
-
e
*
.
(a) Surfaces (A) and (C) liavc oiie end, wllile surface (B) does not.
(b) Surfaces (A) and (C) are homeomorphic! Hi?zt: Tl-ie region cut out by ihe
lines in tlie piciure below is a cylinder, wliicl-i occurs ai tlle left of (A). Now draw
in two more liiies enclosirlg more holes, aild consider ihe region between the
iwo pairs.
21. (a) The three open subsets of RZ shown below are Iiomeomorphic.
(b) The points kside the three surfaces of Problem 20 are homeomorphic.
22, (a) Every open subset of ES is homeomorphic to the disjoint union ofintervals.
(b) Ther-e are only countably many non-homeomorphic open subsets of R.
23. For the purposes of this problem we will use a consequence of the Urysohn
Metrization Theorem, that for any connected manifold M , there is a homeomorphism f from M to a subset of the countable product R x R x - .
(a) If M is a connected non-compact manifold, then there is a continuous
function f :M + R such that f "goes to ca at m", i-e., if (x,)is a sequence
which is eventually in the complement of every compact set, then f (x,) + m.
(Compare with Problem 2-30.)
(b) Given a homeomorphism f : M + R x R x - - - and a g : M + R which
goes to ca at m, define f : M + R x ( R x R x - - - )by f ( x ) = ( g ( x ) ,f ( x ) ) .
Show that f ( ~ is)closed.
(c) Thei-e are ai most c non-liomeomorphic coniiected manifolds (where c = ZN"
is the cardinality of ES).
24. (a) It is posible for IRZ - A and IR2 - B to be homeomorphic even though A
and B are non-liomeomorphic closed subsets.
(b) If A c IRz is closed arid totally disconnected (the only components of A are
points), then E(RZ- A) is liomeomorphic to A. Hence IR2 - A and IRZ - B are
non- homeomorph ic if A and B are non-homeomorphic closed to tally disconnected sets.
(c) The derived set A' of A is the set of al1 non-isolated points. We define A(")
inductively by A ( ' ) = A' and A ( ~ + ' =
) (A("))'.For each n there is a subset A,
of R such that A,(") consists of one poini.
*(d) There are c non-homeomorphic closed totally disconnected subsets of IRZ.
Hint: Let C be the Cantor set, and ci C: cz C: c3 C: . -. a sequence of points
in C. For each sequence nl r m r - - - , one can add a set A,, such that its
n jth derived set is (ci).
(e) Tliere are c non-homeomorphic connected open subsets of IRZ.
25. (a) A manifold-with-boundary could be defined as a metric space M with
the property tliat for each x E M there is a neighborhood U of x and an
inieger n 2 O such thai U is homeomorphic to an open subset of W".
(h) If M is a manifold-with-boundary, then aM is a closed subset of M and
aM and M - aM are manifolds.
(c) If Cj, i E 2 are the components of aM, and Ir c I , then M - UiEl,
Ci is
a manifold-with-boundary.
26. If M c Rn is a closed sei and an 11-dimensional manifold-with-boundary,
ilien the iopological boundary of M , as a subsei of IR", is aM. This is not
iiecessarily true if M is not a closed subset.
27. (a) Every poini (a, b, c ) oii Steiner's surface satisfies bZc2+ ozcZ+ 02b2 =
abc.
@) If (a, b,c) saiisfies tliis equation and O # D = JbZcZ + oZcZ + oZbZ,then
(a, b, c) is on Sieiner's suríiace. Hint: LRt x = bclD, etc.
(c) The sei {(a, b, e) E IR3 : bZcZ+ ozcz + azbz = abc) is the union of tlie
Steiner surface and of the portions (-m, - 1 12) and (1 12, m) of each axis.
CHAPTER 2
DIFFERENTIABLE STRUCTURES
e are now ready to apply analysis to the study of manifolds. The necessary tools of "adva~lcedcalculus", which the reader should bring along
fieshly sharpened, are contained in Chapters 2 and 3 of Cizku1u.r on A4an8Ms.
We will use freely the notation and results of these chapters, including some
11roblems, notably 2-9, 2-15, 2-25, 2-26, 2-29, 3-32, and 3-35; however, we wül
denote the identity map from W" to Wn by 1, rather than by n (which will be
used oRen enough iii other contexts), so that l i ( x ) = xi.
Ori a general manifold M the notion of a continuous function f : M + W
inakes sense, but tlie notiori of a differentiable function f : M + W does not.
This is the case despite the fact that M is locally like IR", where differentiability of funclions can be defined. If U c M is ari open set and we choose
a homeomorpliism #: U + IRn, it would seem reasonable to define f to be
differentiable on U if f o #-' : R" + W is differeiitiable. Unfortunately, if
q : V + W" is anotl~erhomeomorphism, and U n V # 0, then it is not
necessarily true that f o SI-': W" + W is also dfirentiable. Indeed, since
we can expect f o VI-' to be differentiable for al1 f wliich make f o#-' diñereiitiable only if # o q-' : Wn + W" is differeiitiable. This is certainly not always
tlie case; for example, one need merely choose # to be h o@,where h : R" + R"
is a liomeomorphism that is not differentiable.
If we insist on defining differentiable functions on any manifold, there is no
way out of this impasse. It is necessary to adorn our manifolds with a little
additional structure, the precise nature of which is suggested by the previous
discussion.
Among all possible homeomorphisms from U c M onto Rn,we wish to select
a certain collection witli tlie properry that # o @ - ' is differentiable whenever #, @
are in the collection. This is precisely whai we shall do, but a few refinements
will be introduced along the way.
First of all, we M i l 1 be interested almost exclusively in functions f : R" + R"
wliich are C m (that is, each component function f i possesses continuous partial
derivatives of all orders); sometimes we will use the words "dflerentiable" or
c'smoothy'to mean Cm.
Moreover, insiead of considering h o m e o m ~ r ~ h i s mfrom
s
open subsets U
of M orlto R" ,it will sufice to consider Iiomeomorphisms x : U + x ( U ) c R"
onto open subsets of R".
The use of tlie letters x, 11,etc., for tliese liomeoniorphisms, henceforth adIiered to almost religiously is meant to encourage the casual confusion of a point
p E M with x ( p ) E R n : wliich has "coordinates" x' ( p ) , .. . ,x n ( p ) . The only
time tliis notation will be confusing (and it will be) is when we are referring to
ihe maiiifold R", whcre ii is hard not to lapse back into the practice of denoting
points by x and y. IVe will often mention tlie pair (x, U), instead of x aloile,
just to provide a coii\rei~ieniname for the domain of x .
If U and V are opeii siibsets of M, two homeomorphisms x : U + x ( U ) c
R" arid 11:V + j c ( V ) c R" are called Cm-related if the maps
are Cm. This make seiise, since x ( U nV ) and y(U nV ) are open subsets of R".
Also, it makes sense, aild is automatically true, if U n V = 0.
A family of mutually Cm-related homeomorpliisms whose domains cwer M
is called an atlas for M. A particular member ( x , U ) of an atlas 36 is called
a chart (for tlie atlas 36):. or a coordinate system on U, for the obvious reason
iliai it provides a way of assigiiing "coordinaies" to points on U , namely, tlic
coordinates x' ( p ) , . .. ,x n ( p ) to the point p E U .
I4Je can even imagine a mesh of coordinate lines on U , by considerirlg the
l)2jintz/iable Structure~
iiiverse images uiider x of lines in Rn parallel to one of the axes.
. . . . .........
. . . .
..........
.. .. .. .. .. .. .. .. ..
. . . . -.......
. . -.....
............
.. .. .. .. ,........
. . . .
.............
. . . . . .. .. ,.. ...
..m..
L.......
......s......
....................
. . . . . . . . .
. .. .. .. .. .. .. .. .. ..
d...
.. .. .. .,...........
. . . . .
.........
.. . . ., -.,
.. .. .. .. .-........
.. .. . . . .
............
. . . . . . . .-...
The simplest example of a manifold together ~ r i t han atlas consists of Rn with
ari atlas A of only one map, the identity 1: R" + Rn. We can easily make the
atlas bigger; if U a ~ i dV are llomeomorphic operi subsets of R", we can adjoin
any homeomorphism s: U + V with the property that x and x-l. are Cm.
Indeed, we can adjoiii as many such x's as we like-it is easy to check that
they are al1 Cm-relaied io each other. Tlie advantage of this bigger atlas U
is ihai the single word "cliart", wlieri applied to this atlas, denotes something
which must be described in cumbersome lariguage if one can refer only to 36.
Aside from this, U differ-s only superficially from 36; one can easily construct U
from 36 (and one would be foolish not to do so once and for all). What has just
been said for ihe atlas (1)applies to any atlas:
1. LEMMA. If 36 is an atlas of Cm-related cliarts on M, then 36 is contained
in a unique maximal atlas 36' for M .
PR001;. Let A' be il-ic set of al1 cliarts y wliicll are Cm-related to al1 charts
x E 36. It is easy to check that al1 charts in 36' are Cm-related, so 36' is an atlas,
and it is clearly tlie uriique maximal atlas coiitaining 36. *:*
Mle now dcfine a C m manifold (or differentiable manifold, or smooth manifold)
to be a pair (M, A): wliere A is a niam'rna/ atlas for M. Thus, about the simplest
example of a Cm manifold is (R", U), wliere U (the "usual Cm-structure for
Rn") is tlie maxi~naladas containing (1).Arioiher example is (R, V) where V
contains the l~~rneornor~liism
x H x3, wliosc inverse is no/ C m , together with
al1 charts Cm-relaied io it. Although (R, U) and (R, V) are not tlie same, tliere
is a one-one onto function f : R + R such thai
A-
E U
ifand only if x o f E V,
namely, the obvious map f( x ) = x Thus (R, U) aiid (8, V) are the sort
of structures one would want to cal1 "isomorphic". Tlie term actually used is
"diffe~mor~hic":two C* manifolds (M, A) and (N, a) are diffeomorphic if
there is a one-one onto function f:M + N such that
x E J? if and only if A- o f E A.
-'
Tl-ie map f is called a diffeomorphism, and f is clearly a difeomorphism
also. If we had not required our atlases to be maximal, the definition of diffeomor.pliism would have had to be more complicated.
Normally, of course, we will suppress mention of tlie atlas for a dfireritiable
mariifold, and speak elliptically of "the difirentiable manifold M"; the atlas
for M is sometimes referred io as the dzjierentiabk structure for M. Ii wdl always
be undersiood that IR" refers to the pair (IR", U).
Ii is easy to see that a difkomor-phism must be coritinuous. Consequently,
its inverse musi also be coniinuous, so ihat a difeomorpliism is automatically
a homeomorpl~ism. This r-aises the natural question whetlier, conversely, two
liomeomorpliic maiiifolds are necessarily diffeomor-phic. Later (Pi.oblem 9-24}
we will be able to prove easily that IR with any ailas is difkomorphic to (R, U).
A proof of tlie corresporiding assertion for
is much harder, the proof for IR3
would ccrtainly be ioo difficult for- iriclusion llere, aiid the proof of the essential
uniqueness of C* structures on IR" for n 2 5 requires very difficuli techniques
from topology,
In ihe case of spl-ieres,tlie projections Pl and P2 from the points (0,. . .,O, 1)
and (0, . .. ,O, - 1) of S"-' are easily seen to be Cm-related. They iherefore
clctei-minean ailz5-tl-ie "usual C* structure for S"-'". T1-iis atlas may also bc
described in terms of the 2n homeomorpliisms
defined by J(s)= gi(x) = ( x r , . . ., s i - ' , x i + ' , .. . ,xn),wbicli are Cm-related
to P1and P2.Tlrei-e are, up io diffeoniorpl-iism,unique differentiable struciures
on S" for 17 5 6. But tl-iere are 28 difeomorpl-iism classes of differeniiable
structures on S7. aiid over- 16 million on S3'. However, we shall not come
close to proving iliesc asser-tioi~s,whicli are pari of tlie field called "differential
topology", rather then diferential geometry. (hrhaps most astonishing of al1
is ilie quite receiit discovery that IR4 lias a difeirritiable structure that is not
cliffromorphic io tlie usual diffei-entiablesiructure!)
Otl-ier examples of differentiable manifolds will be @ven soon, but we cari
already describe a diffcrentiable structure A' 011 any open submarlifold N of
a differentiable manifold ( M , A); tlie atlas A' consists of all (x, U) in 36 witli
U c N.
Just as dfiomor-phisms are analogues for C* manifolds of homeomorphisms, there are analogues of continuous maps. A function f : M + N is
called differentiable if for every coordinate system (x, U) for M and (y, V)
for N, the map y o f o x-' : IRn + IRm is differentiable. More particularly, f
is called differentiable at p E M if y o f o x-' is differentiable at x(p) for
coordinate systems (x, U) and (y, V) with p E U and f (p) E V. If this is
true for one pair of coordinate systems, it is easily seen to be true for any otlier
pair. We can thus define differentiability of f on any open subset M' C M ;
as one would suspect, tliis coincides with differentiabilit~of the restricted map
f 1 M ' : M' + N. Clearly, a differentiable map is continuous.
A differeniiable function f : M + IR refers, of course, to the usual d f i r e n tiable structure oii IR, and lience .f is differentiable if and only if f o x-' is
diflerentiable foi- cach chart x. It is easy to see that
(1) a function f : IR" + IR" is diflerentiable as a map between C* manifolds
if and only if it is difler-entiable in tlie usual sense;
(2) a fuiiction f : M + IRm is differentiable ifand only if each f i : M + W m
is diferen tiable;
(3) a coordinate system (x, U) is a diffeomorphism from U to x(U);
(4) a fuiiction f : M + N is diffei-eiitiable if and only if each y' o f is
difirentiable for each coordinate system y of N;
(5) a difler-eritiable function f : M + N is a dfleomorphism if and only if
f is one-o~ieonto and f U 1 : N + M is differentiable.
Tlie differe~itiablesiructui-es on inaiiy manifolds are designed to make ccrtain
fuilciions differ-eniiable. Consider first the product MI x M2 of two dfleren-
iial~lemanifolds Mi, arid the two "projections" ni : M I x M2 + Mi defined
by ni(pl, p2) == pi. It is easy to define a differentiable structure on MI x M2
which makes eac1-i ni difirentiable. For each pair (xi, Ui) of coordinate systems
on Mi, we constmct the homeomorphism
defined by
Tlien we cxiend tliis atlas to a maximal one.
Similarly, ihere is a difirentiable structure on Pn which makes the map
. f : Sn + Pn (defiried by f (p) = [p] = (p,-p)) difirentiable. Consider
any coordinate system (x, U) for S",where U does not contain - p if it contair~sp, so tllat f 1 U is o~i,e-one.Tlie map x o (f 1 U)-" is a homeomorp1~ism
on J'(U) c Pn,and any two such are Cm-related. The collectiori, of tliese
lioineoi~iorpliismscan then be extended to a maximal atlas.
To obtain differentiable structures on other surfaces, we first note that a C m
manifold-witli-bour~darycan be defined in a n obvious way. It is only necessary
to know wlien a map f : W n + Rn is to be considered differentiable; we cal1 f
differentiable d e n it can be extended to a dflerentiable function on a n open
i-ieiglibor-hoodof W".A "Iiandle" is then a Cm manifold-with-boundary.
A diffei.eriiiable structure on tlie 2-holed torus can be obtained by "matcliiiig"
tRe dflereiitiable structure on two l-iandles. Tl-ie details involved in tliis process
are reserved for Pro blem 14.
To deal witl-i C m fuilctions effectively,one needs to know that there are lots of
ilicm. Tlie esistcrice of COQfurictions oii a manifold depends oii tlie existence of
C m funciions on R" ~ I i i c lare
i O ouiside of a compact set. We briefly recall liere
ilie necessat-y facis aboui such Cm fuiictions (c.f. Cakulus on MaliifoMs, pg. 29).
D@ben/iabL Slructu res
(1) The function h : R + R defined by
is Cm, and h(")(0) = O for al1 n.
(2) The function j: IR + R defined by
e-(x-r)-2
e-(x+~)-2 x E (-1, 1)
=
{
o
x
(-41)
-1
I
I
is Cm.
Similarly, there is a Cm function k : W + R which is positive on (0,S) and O
elsewhere.
I
S
(3) The fuiiction I : R + R defined by
is Cm; ii is O for x 5 O, increasing on (O, S), and I for x 2 S.
(4)Tlie function g : R" + R defined by
is Cm; ii is positive on (-E, E) x . - - x (-E, E) and O elsewliere.
O n a Cm manifold M we can now produce many non-constant Cm functions. The closure (x : f(m) # O) is called tlie support of f,and denoted simply
by support f (or sometimes supp f).
2. LEMMA. Let C c U c M witli C conipact aild U open. Then there
is a Cm function f : M + [O, 11 such tliat f = 1 on C and support f c U.
(Compare h e 2 of the proof of Theorem 15,)
PROOF. For each p E C, choose a coordinate system (x, V) with c U and
x(p) = O. Then x(V) 3 (-E,&) x - - - x (-E,&) for some E 3 O. T h e function
g o x (wliere g is defined in (4))is CDgon V. Clearly it remains Cm if we extend
it to be O outside of V. IRt Sp be the extended function. The function fp can be
coiisir~uctedfor each p, aiid is positive on a neighborhood of p whose closure is
coniained iii U. Since C is compact, finitely many such neighborhoods cover C,
aiid the sum, fp, + - - + -fp,, of the correcponding functions has support c U.
0
1
1C it is positive, so on C it is 2 S for some 8 3 O. Let f = Io (fp, + - . -+fpt,),
where I is defined in (3).e:*
By tlie way, we could have defined C r rnanifolds for each 1. 1, not just for
i- = m". (A function f : IRn + IR is C r if it Iias continuous partid derivatives
up to order i-). A "COfunction" is just a continuous function, so a CO manifold is
jusi a maiiifold in tlie sense of Chapter l. We caii also define analytic manifolds
(a function f : W n + W is analytic at u E IRn if f can be expressed as a
power series in the ( x i - í i i ) which converges in some neighborhood of U ) . Tlie
symbol C Wstaiids for analytic, and it is conveiiient to agree that r .:w c: o
for each iiiteger i . > O. If a < B, tlien tlie cliarts of a maximal CB atlas are al1
Ca-related, bui ihis atlas can always be extended to a btggu atlas of Ca-related
cliaris, as in Lemma 1. Tlius, a CB structure on M can always be extended
io a C" structure i i ~a uiiique way; tlie smaller siructui-e is tlie "stronger" one,
tlie CO structure (coiisistiiig of al1 homeomoi~pliismsx: U + W " ) beiiig tlie
lüigest. The coiiverse of this trivial remark is a hard tlieorem: For a 2 1,
ewry C" structurc roiitains a CB structurc for eacli /3 3 a;it is not unique, of
c,ourse, but it is ui~iqueup to diffeomorpliism. Tliis will not be proved liere.*
In fact, C" maiiifolds for a # w will hardly ever be mentioned again. One
irinark is iii ordei- 1 3 0 ~ tlie
~ ; proof of h m m a 2 produces an appropriate C"
fiinciion f oii a C" inaiiifold, for O 5 a 5 w. Of course, for a = o the proof
6c
* hi.a p1-00fsee h4unkrcs, EIentefztay Ditpereniial %/~olog)l.
fails completely (aiid ihe result is fahe-an analytic function which is O on a n
open set is O everywhere).
With difirentiable functions now at our disposal, it is fitting that we begin
differentiating them. What we S hall define are the partid derivatives of a differentiable function f : M + R, with respeci to a coordinate system (x, U). At
this point classical notation for partid derivatives is systematically introduced,
so it is wortli recalling a logicai notation for the partial derivatives of a function
f : R" + R . We denote by Di f (u) the number
lim
f ( u ' y * . * y u i + h , - . - , u "-) f(u)
I?
h-O
The Chaiii Rule states that if g : Rm + R" and f : R" + R, then
n
Di f (g(u)) - ~j gi (u).
Dj (f 0 g) (u) =
i=21
Now, for a function f : M + R and a coordinate system (x, U ) we define
a f = D i(f o x-') o r, as an equation behveen functions). If we
(or simply axl
define tlie curve ci : (-E, E) + M by
then this partial derivative is just
so it measures the rate change of f along tlie curve c;;in fact it is just (f oci)'(0).
Notice tliat
ax3
1 ifi= j
-(p) = 6; -
a A-
J
1f x liappens to be the identity map of Rn, then Dt f (p) = 8f/axi(p), whidi
is the classical synibol for this partid derivative.
Anot her classical instance of this notation, often not completely clarified, is
thc use of the syrnbols a/ar and a/a9 in connection with "polar coordinates".
On the subset A of JRZ defined by
by'
we can iii troduce a "coordinate system" P : A + IW
where r ( x , y) =
m
and 9(x, y) is the unique numbcr in (O, 2rr) with
Tliis really is a coordinate system on A in our sense, with its image being the
set (r : r > O) x (0, k).(Of course, the polar coordinate system is often
not restricted to tlie set A . One can delete any ray other than L if 9(x, 1))
is restricted to lie in the appropriate interval (60, 90 + 2 ~ ) many
;
results are
essentially independent of which line is deleted, and this sometimes justifies the
slopl~inessinvolved in the definition of the polar coordinate system.)
IVe liavc 1-eallydefined P as an inverse function, whose inverse P-' is defined
simply by
P-'(r,9) = (r cos9,r sino).
From this formula we can compute f i a r explicitly:
(f o P-')(r, O) = f (r cos O, i- sin O),
af
-(x?
ar
Y) = Di (f 0 p-'1 (P(x9 Y))
= DI f (P-' ( ~ ( xY))
, Di [P-'11 (P(x, Y))
+ ~ z (p-'(p(x,
f
Y)) - Di [ P - ' I ~ ( P ( XY))
~
by the Chain Rule
= DI f (x, y) tos O(x, y) Dzf (x, Y) sino(x, Y).
+
Tliis formula just gives tlie value of the directional derivative of f at (x, JI),
along a uiiit \lector v = (cos O (x, y), sin O (x, y)) poin ting outwards from the
origin to (x, y). Tliis is to be expected, because cl, the inverse image under P
sin B(x,y)
of a culve along t l ~ e"1.-axis", is just a line in tliis direction.
A similar computation gives
to v , and thus tlle
The lector w = (- siii O(x, y), cos 0(x, y)) is
direction, at tlie point (A-, y), of the curve cz ivliicli is tlie inverse image under P
of a cunfe along tlle "O-axis". Tlie factoi- i. (x, y) appears because this cuive
goes around a circle of that radius as 8 goes from O to 2n, so it is going i-(x, y)
times as fast as it should go in order to be used to compute the directional
derivative of f in the direction w , Note that af/aO is independent of which
line is deleted from the plai~ein order to define the function 8 unarnbiguously.
Using the notation af/ax for Di f, etc., and suppressing the argument (x, 11)
evenutliere (thus writing an equation about functions), we can write the above
equations as
af
a f sin 8
a f cos 8 + -
- c = -
ay
ax
8.f = ?f
ae ax
-(-)e
ay
sin 8) + -ra.f cos 8.
ay
111 particular, these formulas also te11 us what ax/ai- etc., are, where (x, y)
rlrnotes ilie identit~coordinate system of RZ.We have ax/ar = cose, etc., so
oui- formulas can be put in the form
af = -afax
+-al-
a~-al-
aya~
In classical notation, the Cl~ainRule would always be written in this way. It is
a pleasui-e to repoi-t thai Iienceforih this may always be done:
3 . PROPOSITION. If ( S , U ) and (y, V) are coordinate systems on M, and
f : M + R is differentiable, then on U n V we have
PROOF. It's tlie Chaii~Rule, of course, if you just keep your cool:
At this point we could introduce the "Einstein summation convention". Notice that the summation in this formula occurs for the index j, which appears
both "above" fin a x j / a y i ) and "below" (in a f / a x j ) . There are scads of formulas in which this happens, often with hoards of indices being summed over,
and the convention is to omit the C sign completely-double indices (wliicli
by luck, tlie nature of t hings, and felicitous choice of notation, almost always
occur above and below) being summed over. 1 won't use this notation because
wheiiever 1 do, 1 sooii forget I'm supposed to be summing, and because by doing things "riglit", one can avoid what Élie Cartan lias called the "debauch of
indices".
We will often write formula (1) in the form
here alay' is considered as aii operator taking the function f to a f / a y i . The
operator taking f to af /ayi ( p ) is denoted by
For later use we record a property of f = a/axil,: it is a "point-derivation".
4. PROPOSITION. For any diflerentiable f , g : M + IR,and any coordinate
system (A-,U ) with p E U, the operator l = a/axilPsatisfies
K f g ) = f (p)f(g)+f(f )g(p).
PROOF. Lft to the reader.
If (A-,U) and (x', U ' ) are two coordinate systems on M, the n x n matrix
is just the Jacobian matrix of x1 o A--'
inverse is clearly
at x(p). It is non-singular; in fact, its
Now if f : M" + N m is Cm and (y, V ) is a coordinate system around f(p),
t he rank of the m x 11 matrix
clearly does not depeiid on the coordinate system (x, U ) or (y, V ) . It is called
tlie rank of f at p. T h e point p is called a critical point of f if the rank of f
at p is 4 i?7 (the dimension of the image N); if p is not a critical point of f,
it is called a regular point of f. If p is a critical point of f,the value f (p) is
called a cñtical value of f. Other points in N are regular values; thus y E N
is a regular value if and only if p is a regular point of f for ewry p E f (y).
Tliis is true, in particular, if q $ f (M)-a non-value of f is still a "regular
value".
If f : W + W, tlien x is a critica] point of f if and only if f t ( x ) = O. It
is ~ossiblefor al1 points of the intenral [u, b] to be critical points, although this
can happen only if f is constant on [u, b]. If f : WZ + IR has al1 points as
critical values, tlien Dl f = Dzf = O everywhere, so f is again constant. O n
tlie other hand, a function f : WZ + IRZ may have al1 points as critical points
without being constant, for example, f ( x , y) = A . In this case, Iiowever, the
iinage f (WZ)= W x { O ) c WZ is still a "small" subset of IRZ. The most importan1
theorem about critical points generalizes this fact. To state it, we will need some
terminology.
Recall tliat a set A c W" has "measure zero" if for every E > 0 tliere is a
sequeilce Bi, Bz, B 3 , . . . of (closed or open) rectangles with
-'
and
wliere u(&) is the volume of B,l. We want to define the same concept for a
subset of a manifold. To do this we need a lemma, wliich in turn depends on
a Icmma from Caku/us on Adanfolds, which we merely state.
5. LEMMA. Let A C IR" be a rectangle and let f : A + Rn be a function
such that ~ ~ ~5 fK ' onl A for i, j = 1 ,..., n. Then
for al1 x , y E A.
6. LEh4MA. If f : IR" + IR" is C' and A c IR" has measure O, then f ( A ) has
measure O.
PROOF. We can assume that A is contained in a compact set C (since IR" is a
countable union of compact sets}. Lemma 5 implies that there is some K such
t hat
I f (x) - f (r)l 5 n2Klx - yl
for al1 x , y E C. Tllus f takes rectangles of diameter d into sets of diameter
- >I' Kd. This clearly implies that f ( A ) has measure O if A does. +3
A subset A of a Cm 71-manifold M Iias measure zero if there is a sequence
of charts (xi, U i ) , with A c Ui Ui, S U C ~that each set x i ( A n U i ) c IR" has
measure O. Using L m m a 6, it is easy to see tliat if A c M has measure O, then
x(A n U ) c IR" Iias measure O for any coordinate system ( x , U). Conversely,
if this condition is satisfied and M is connected, or has only countably many
compoiieiits, then it follows easily from Theorem 1-2 that A has measure O. (But
if M is the disjoint union of uncountably many copies of IR, and A consists of
one poirit from each compoiient, then A does not have measure O). Lemma 6
thus implies another result:
7. COROLLARY. If f : M + N is a C' fu~ictionbetween two n-manifolds
and A c M has measure O, then f ( A ) c N has measure O.
PR001;. Tliere is a sequelice of charts (xi, U i ) with A c Ui U; and each set
xi(A n U i ) of measure O. If ( y , V ) is a cliart on N, then f ( A ) n V = Ui f (A n
U i ) n V . Each set
Iias measure O, by Lemma 6. Thus y( f ( A )nV ) has nieasure O. Since f (UiUi)
is contained in tlie uiiion of at most countably many components of N, it follows
that f ( A ) has measure O.
e+
8. THEOREM (SARD'S THEOREM). If f : M + N is a C i map between
n-manifolds, and M has at most countably many components, then the critical
values of f form a set of measure O in N.
PROOI;. It clearly suffices to consider the case where M and N are Rn. But
this case is just Theorem 3.14 of Calculw on M a n f o b .
Tlie stronger version of Sard's Theorem, which we will never use (except once,
in Problem 8-24), states* tliat the critical values of a C' map f : M" + N"'
are a set of measure O if k 2 1 + max(n - in,O). Theorem 8 is the easy case,
and tlie case nz > n is the trivial case (Problem 20). Although Theorem 8 will
be very important later on, for the present we are more interested in knowing
wliat tlie image of f : M + N looks like locally, in terms of the rank k of f
at p E M . More exact information can be @ven wlien f actually has rank k
i n a neighborhood of p. It should be noted that f must have rank 2 k in
some iieiglihor-hood of p, because some k x k submatrix of (a(yi o f)/axi)
has non-zero determinant at p , and hence in a neighborhood of p.
9. THEOREM. (1) If f : M" + N m has rank k at p, then tliere is some coordinate system (x, U) around p and some coordinate system (y, V) around f (p)
with y o f ox-' in the form
Moreover, given any coordinate system J?, the appropriate coordinar e syst em
ori N can be obtained merely by pei-muting the component functions of y.
(2) If f has rank k in a rieiglil~orlioodof p, theii tliere are coordinate systems
(A-, U) and (y, V ) such that
Ren~ark:Tlie special case M = Rn, N = Rm is equivalent to the ge~ieraltlieorem, wliicli @\res only local results. If y is tlie identity of R", part (1) says thai
ly first pei-formiiig a difleoniorpliism on Rn, arid tlieri permuting the coordiiiates in Rm,we can irisure that f keeps tlie first k components of a poiiit fixed.
Tliesc difTcornoiphisms on Rn and Rm are clearly necessary, since f may not
everi be oiie-oiic on Rk x { O ) c RW aad itr image could, for example, contain
oiily ~ o i n t swith first coordinate O.
* Foi- a proof, scc hqiliroi; To/)ologilh t n tlic Ditp~~'nl&inblc
I4cuy)oi)ir or Sternberg, LEcllrres ott
Dflirenliol Geonzelv.
In part (2) we must clearly allow more leeway in the choice of y , since f (IRn)
rnay not be contained in any k-dimensional subspace of IRn.
I'ROOF. (1) Choose some coordinate system u around p . By a permutation of
ilie coordinate functions u' and y' we can arrange that
Define
Condition (1) implies that
This shows thai A- = (x o u - ' ) o u is a coordinate system in some neighborI~oodof p, since (2) and the Inverse Function Theorem show that x o u-' is a
dfleomorphism in a ncighborhood of u ( p ) . Now
q = x - ' ( a ' , . . . , a n ) means
hence
1
x ( q ) = (a ,. .. , a n ) ,
x'(~=
) ai ,
SO
Y ~ f o x - l ( a l , . . . , a J=1y) o f ( q )
= ( a 1 ,..., a k ,
for q = x-l (a 1 ,. . ., a n )
).
(2) Choose coordinate systems x and v so that v o f o x-' has the form in (1).
Since rank f = k in a neigliborhood of p, the lower square in the matrix
must vanish in a neighborhood of p. Thus we can wnte
Define
ya = va
y ' = vr - $ r o ( ~ ' , . . . , V k ).
Since
has non-zero determinant, so y is a coordinate system in a neighborhood
of f (p). Moreover,
Theorem 9 acquires a special form when the rank of f is n or nz:
10. THEOREM. (1) If m 5 ,I and f : M" -t Nm lias rank m at p , then for
any coordinate sysiem (y, V ) around f (p), there is some coordinate system
(x, U) around p with
y o f ox-'(a',.. .,an) = (a',. . . , a m ) .
(2) If 11 5 m and $: M n + N"' has rank ir ai 11, tlien for any coordinate
system (x, U) around p , there is a coordinate system (y, V ) around $ ( p ) with
y o f ~ x - ~ i(, a. . . , a n ) = (a 1 ,..., a n , O ,...,0).
PROOF. (1) This is practicdly a specid case of (1) in Theorem 9; it is only
rlecessary to observe that when k = rn, it is clearly unnecessary, in the proof of
this case, to permute ihe yi in order to arrange that
a,p = 1,. . .,m;
) 0
det ( a ( y ~ u ~ f ) ( p ) f
only the u' need be permuted.
(2) Since the rank of f at any point must be 5 n, the rank of $ equals n
R"
in some neighborhood of p. It is convenient to think of the case M
and N = RJn and produce the coordinate system y for Rm when we are Siven
ihe identity coordinate system for R". Par-t (2) of Theorem 9 yields coordinate
systems # for Rn and S, for Rm such that
1
1
@ o f o#-'(a ,...,a") = (a ,..-,an,O,..., O),
-
Even if we clo rioi pei-form #-'. firsi, tl-ie map f still takes EXn into the subset
(R") wliich S, takes to R" x (O) c Rm-the points of Rn just get moved to
ihe wrong place in Rn x (O). This can be corr-ected by another map on R".
Define h Ily
h(bi ,. . . ,bJn)= (4-' (bl,. . . ,b"), b"+', . . . ,bm).
Then
h o S , o $ ( a i , ...,a n ) = h o S , o $ o # - ' ( b 1 ,..., b")
for (b', .. . ,bn) = #(a)
= h(bl,. . .,ón,0,. . .,O)
= ($-'(bl, .. ., b"),O,. . .,O)
..
= (a i , . , a R ,O,
. ,O),
so h o S, is tl-ie desired
If we are Siven a coordinate system x on R" other
than the identify, we just define
h(bl, . . .,b"') = ( ~ ( 4 - ' ( b i ,- . . ,b')), b"+I,. . . ,bm);
)J.
it is easily cliecked tliat y = h o S, is rlow the desired y. +3
Aliliough p is a regular point of f in case (1) of Theorem 10 and a critica1
point in case (2) (if n < m), it is case (2) \vhich most interests us. A dflerentiable
funciion f :M n + N m is called an immersion if the rank of f is n, the
dimension of the domain M, at al1 points of M. Of course, it is necessary that
In > n, and i i is clear from Theorem 10(2) that an immersion is locally one-one
(so it is a iopological immersion, as defiiied in Chapter 1). On the other hand,
a differentiable map f need not be an immersion ewn if it is globally one-one.
The simplest example is ihe function f :IR + IR defined by f ( x ) = x3, with
f (O) = O. Another example is
A more illun~inatingexample is the function h : IR + IR2 defined by
alihough its image is tlie graph of a non-dflerentiable function, the curve itself
manages to be differenkiable by slowing down to velocity O at the point (0,O).
One can easily define a similar curve whose image looks like the picture below.
Tliree immersions of IR in IR2 are shown below. Althougll the second and
iliii-d immei-sions Bi and B2 are one-one, iheir iinages are not homeomorphic
to R . Of course, even if tl-ie one-one immersion f : P + M is not a homeoniorpliism onto iis image, tliere is certainly some metric and some dflerei~tiable
sti-ucture on f ( P ) which makes the inclusion map i : f ( P ) + M an inlmersion. In general, a subset MI c M, with a dflerentiable structure (iiot neccssarily compatible with the metric Mi inherits as a subset of M), is called an
immersed submanifoId of M if the inclusion map i : MI + M is an immersion.
l'he follo\ving picture, indicating the image of an immersion B p : R + S' x S ] ,
rS3
first time around
second time around /
sl-iows that Mi may even be a dense subset of M.
Despite these complications, if M i is a k-dimensional immersed submanifold
of M n and U iis a neighborhood in M' of a point p E M I , tllen there is a
coordinate sysiem (y, V) of M around p , such that
tliis is an immediaie consequerice of Tlieorcm 10(2), with f = i. Tlius, if
g ; Mi + N is Cm (co~isideredas a function oii tlie manifold M i ) in a neigliborhood of a point p E M I , ihen ihere is a Cm furiction g on a rieigliboi-liood
V c M of p sucli that g = g O i on V n M i -we can define
S (4) = g (qi),
where
~ ' ~ ( q=' )~ ' " ( 4 ) a = 1, ...,k
r =k + l,.s.,iz.
On the oilier hand, even if g is Cm on al1 of M I we may not be able to
define g on M. For example, this cannot be done if g is one of the functions
B;' j j ( M ) + R.
One other complicatioi-i arises with immersed submanifolds. If M I c M is
an immersed submailifold, aild f : P + M is a Cm function with f ( P ) c Mi,
ii is not necessarily true that $ is Cm when considered as a map into MI, with its
Cm structure. The following figure shows that f might not even be continuous
as a map jnto M I . Actually, tliis is the only thing that can go wrong:
11. PROPOSITION. If MI c M is an immersed manifold, f : P + M is
a Cm function witli f (P)c M I , and f is coniinuous considered as a map
irlto M I , then f is also Cm coilsidered as a map into MI.
PROOF. Let i : MI + M be tlie inclusioil n-iap. We want to show that iP1o j'
is Cm if it is continuous. Given p E P, choose a coordiilate system (y, V) for M
around f (p) such that
is a iieigliboi-hood of f (p) in MI and (yi1 U], . .. ,y k 1 U]) is a coordiliate system
of MI on U].
By assumption, i-' o f is continuous, so
-' o i(open set) is an open set.
Since Ui is open in M I , this means that f -' (Ul) c P is open. Thus f takes
f
some neighborhood of p E P into U,. Since al1 y j o f are Cm, and y ' , . ..,y k
are a coordinate system on U1, the function f is Cm considered as a map
into MI. 43
Most of these difficulties disappear wlien we consider one-one immersions
f : P + M which are liomeomorphisms onto their image. Such an immersion
is called an imbedding ("embedding" for the English). An immersed submanifold Mi c M is called siinply a (Cm) submanifold of M if the inclusion map
i ; MI + M is an imbeddirlg; it is called a closed submanifold of M if MI is
also a closed subset of M.
Tliere is one way of getiiilg submanifolds wl-iich is very important, aild @ves
the sphei-e S"-' C W" - {O) c JR", defined as (x : 1x12 = 11, as a special case.
12. PROPOSITION. If f : M" + N lias constani railk k on a neighborhood
of f (J!), then f (y) is a closed submanifold of M of dimension n - k (or
is empty). Iii particular, if y is a regular value of f : M" + Nm, then f (y)
is an (11 - 111)-dimensionalsubmanifold of M (or is empty).
-'
-'
-'
PROOF. Left to the reader. 43
It is to be hoped that however abstract the noiion of Cm manifolds may
appear, sul~manifoldsof JRN will scem like faii-ly concrete objects. Now it turns
out tliat e u q r (connected) Cm manifold can be imbedded in some J R N , so that
manifolds caii be pictured as subsets of Euclidean space (though this picture
is iiot always die nlost uscful one). We will prove tliis fact oilly for compaci
mailifolds, but we first develop some of the macl~iileiywhich would be used in
the general case, since we xrtil1 need it later on anyway. Unfortunately, there are
many definitions and theorems involved.
If O is a cover of a space M, a cover O' of M is a refinement of O (or
"refines O ") if for every U in O' there is some V in O with U c V (the sets of O'
are "smaller" than those of O)-a subcover is a very special case of a refining
cover. A cover O is called locally finite if every p E M has a neighborhood W
which intersects only finitely many sets in O.
13. THEOREM. If O is an open cover of a manifold M, then there is an open
cover O' of M whicli is locally finite and which refines O. Moreover, we can
choose al1 members of O' to be open sets dfleomorphic to R".
PROOF. \Ve can obviously assume that M is connected. By Theorem 1.2, there
are compact seis Cl,C2,C3,. . - with M = CI U C2 U C3 U - - - - Clearly Ci has
an open neighborliood U] with compact closure. Then
U Ci has an open
neighborhood U2 ~ 4 t hcompact closure. Continuing in this way, we obtain
c Ui+i, whose union contains aH C;,
open sets Ui, witli
compact and
and lience is M. Let U-] = U. = 0.
-
Now M is ilic union for i > 1 of the "aniiular" regons A i = Ui - Ui-]. Since
eacli A i is compact, we can obviously cover Ai by a finite number of open sets,
eacli coiiiaiiied in some membcr of O, and eacli coiitained in Vi = Ui+, - Ui-2.
\Ve can also clioose these open sets to be diffeomoi.phic to RR.In this way we
obiain a cover O' wliich refines O aild wliicli is locally finite, since a point in Ui
is noi in 6 for j 3 2 + i. /f,
Notice that if O is an open locally finite cover of a space M and C C M is
compact, then C intersects o~ilyfinitely many members of O. This shows that
aii open IocaHy finite cover of a connected manifold must be countable (like the
cover constructed in the proof of Theorem 13).
14. THEOREM (THE SHRINKING LEMMA). Let O be an open locally
firiiie co\rer of a manifold M. Then it is possible to choose, for each U in O, an
open set U' wiui
c U in such a way that the collection of al1 U' is also an
open cover of M.
PROOF. 1442 can clearly assume that M is connected. Let O = (Ui, U2, U3,. . .).
Then
c1= u, - (U2UU3U . - - )
is a closed set co~itainedin U], and M = Ci U U2 U U3 U. . . L t U{ be an
open set witli Ci c U; c
c U1 Now
is a closed set contained in U2, and M = U; U C2 U U3 U - . Let U; be an
c U2. Continue in this way.
open set wiili C2 c U; c
For aliy p E M tliere is a largest n witli p E U,, because O is locally finite.
Now
p E u; UU; U
U u; U (u.+, U UN+2U . - . ) ;
-
S
it follows that
P E U/UU;Ua.*,
since replacing U,+í by Un+i cannot possibly eliminate p. 34
15. THEOREL4. Let O be an open IocaHy firiite cover of a manifold M. Then
ihere is a collection of Cm functioiis # u : M + [O,11, one for each U in O,
such tllat
(1) support $U C U for each U,
(2)
$u(/)) = 1 for al1 p E M (this sum is really a finite sum in some
U
iieigliborliood of p, by (1)).
PROOF. Cme 1. Each U in O Rar ~omjia~t
closure. Choose the U' as in Theorem
14. Apply Lemma 2 to
c U c M to obtain a C m function Jru : M + [O, 11
wliicli is 1 on
aiid has support c U. Since the U' cover M ,clearly
Define
Gerie~alc a e . Tliis case can be proved in tlie same way, provided diat
Lemma 2 is irue for C c U c M with C closed (but not necessarily compact)
and U open. But this is a consequence of Case 1:
For eacli y E C choose an open set Up c U with compact closure. C w e r
M - C with open sets V. haviiig compact closure and contained in M - C.
The open cover ( U p ,V..) lias an open locally fiilite refinement O to which Case I
applies. Let
Cme 2.
Tliis sum js Cm,silice it is a fi~iitesum iii a iieigliborliood of each point. Since
#u ( p ) = 1 for al1 p , arid #u ( p ) = O when U c V., clearly J ( p ) = 1
fol. al1 p E C. Usirig tlic faci tliat O is locally finite, it is easy to see tliat
support f c U.+:+
xu
16. COROLLARY. If O is aiiy open cover of a n~aiiifoldM , then ihere is a
collection of Cm functions # i : M + [O, I] such that
(1) tlie collection of sets ( p : # i ( p ) # O) js locally finite,
(2) t i#i( p ) = I for al1 p E M ,
(3) for eacli i tliere is a U E O such that support #i C U.
(A collccti~ri( # i : M + [O, 11: satisfying (1) and (2) is called a partition of unity;
if it satisfics (3), ii is called subordinate to O.)
It is riow fairly easy to prove tlie last theorem of tliis cliapter.
17. THE0REh.I. If M " is a conipact C m mailifold, tlien tliere is aii iml~edcliiig /: M + IRN for some N.
1'1200F. TIiere are a £nite number of coordinaie systems (xl, Ul), . .. , (xk,Uk)
witl-i M = Ul U - - - UUk. Choose U; as in Tlieorem 14, and functions @i : M +
[O, 11 whicli are 1 on
and have support c Vi. Define f : M + RN, where
N=nk+k,by
Tl-iisis ai-i inlmersion, because aiiy poiiit p is in U; for some i, and on U,', where
qfj = 1, ihe N x it Jacobian matrix
It is also one-one. For suppose that f ( p ) = f(q). There is some i such that
p E U;. T1-iei-i@i(p) = 1, SO also @i(q) = 1- This shows that we musi llave
y E Ui. Moreover,
@i ' xi (P) = @i - X i (Y),
Problem 3-33 sl-iows that, in fact, we can always choose N = 2rz + 1.
1, (a) Sliow tliat beir-ig Cm-r-elated is no.! a1.i equivalente relation.
(b) 111 the pl-oof of Leninia 1, show ihat al1 charis in A' are Cm-related, as
claimed.
2. (a) If M is a 1-ileti.i~space togeiber wiil-i a collection of homeomorphisms
x: U + Rn whose domains cover M and which are Cm-related, show that
tl-ie tt at each yoiiit is ui-iique rtrillrouf using Irivariai-ice of Domain.
(b) SI-iowsin-iilarlythai aM is well-defined for a Cm manifold-with-boundary M.
3. (a) Al1 Cm fui-ictions are continuous, ai-id il-ie coinposition of Cm functions
is Cm.
(b) A funciion f : M + N is Cm if al-id only if g o f is Cm for every Cm
func tion g : N + R .
4. How inaiiy clistii-ict Cm stl-uctures are thel-e 011 R? (Tlleie is only one up to
diffeon-iorpliisin;il-iat is i-ioi the question beil-ig asked.)
5. (a) If N c M is open and A' consists of al1 ( x , U) in A with U c N, show
thai A' is maximal for N if A is maximal for M.
(b) Show that A' can also be described as ihe set of al1 (xlV í l N, V í l N) for
(x, V) in A.
(c) SIlow ihai ihe inclusion i : N + M is Cm, and that A' is the unique atlas
with this property.
6. Clieck thai tlie iwo projections Pl and P2 on S"-] are Cm related to the 2n
l~omeomor~hisms
_I;: and gj.
7. (a) If M is a connected Cm marlifold and p,q E M, tl-ien ihere is a Cm
c u m c : [O, I ] + M with c(0) = p and c(I) = q.
(b) It is even possible to clioose c to be one-one.
8. (a) Sliow ihat (MI x M2) x M3 is dfleomorphic io MI x (M2 x M3) and
iI-iai MI x M2 is dfleomorphic to M2 x M I .
(b) The dflcrentiable structure on MI x M2 makes the "slice" maps
of M I , M2 + M1 x M2 diffel-entiablefor a11 pl E M i , p2 E M2.
(c) Pl4ore gerierally, a map f : N + MI x M2 is Cm if and only if the composiiions rrl o f : N + Mi and rr2 o f : N + M2 are Cm. Moreover, ihe Cm
structure \ve llave defined for Mi x M2 is the only one wiih this property.
(d) If fj : N + Mi are C m (i = 1,2), can one determine ihe rank of ($1,$2) : N
+ Mi x M2 (71 p in terms of the ranks of _f;: at p? For fj : Ni + Mi, show thal
Si x f2: Ni x N2 + Mi x M2, defined by ji x fi(pi, p2) = (Si(pi1,Sz(p2)),
is Cm and determine its rank in terms of the ranks of ji.
9. Let g : S" + P"be tlie map p H [p]. SIiow that f : P" + M is Cm if and
only if j' o g : S" + M is Cm. Compare the rank of f and the rank of f o g .
10. (a) If U c R" is open and f : U + R is locally Cm (every point has a
iieiglil~orlioodori wllich f is Cm), tllen f is Cm. (Obvious.)
(b) If f : W" + R is locally Cm, then f is Cm, i-e-,f can be exieilded to a
Cm function on a ileighborhood of M". (Not so obvious.)
1 1 . If f : W" + R Ilas hvo extensions g, h to Cm functions in a neighborhood
of M", illeli Qg and Q h are ihe same at points of R"-' x (O) (so we can speak
of Dif at these points).
12. If M is a Cm rnanifold-wiil-i-boundary, tlien there is a unique Cm struciure
ori aM sucll iliai il-ie inclusion map i : aM + M is an imbeddiilg.
13. (a) Let U C M" be an open set sucb ihat boundary U is an (n - 1)-dimensional (dflereiitiable) submanifold. Show tliat U is aii 11-dimensionalmanifoldwith-boundary (It is well to bear in mind the following example: if U = {x E
EX' : d ( x ,O) < 1 or 1 < d ( x ,O) < 21, then Ü is a manifold-with-boundary, but
aÜ # boundary U.)
(b) Consider the figure shown below. This figure may be extended by putting
srnaller copies of tlie two parts of S' into the regions indicated by arrows, and
ihen repeatirig tliis coristi-uction iudefinitely. The closure S of the final resulting
figure is known as Alexnnder's Houied SSplnr. Sliow tliat S is liomeomorphic to s2.
(Hinl: Tlie additional poinis in tbe closure are bomeomorphic to the Cantor
sei.) If U is tlie uiibourided component of W3 - S, ihen S = boundary U, but
Ü is not a 2-dimensional manifold-with-boundary, so part (a) is true only for
differentiable su bmanifolds.
14. (a) Therc is a map f : EX2 + EX2 S U C ~ Itliat
(1) f (x,O) = (x,O) for al1 x,
(2) f(x, y) c EI2 for y > O,
(3) f ( ~ , yc) W2 - IN2 for y < O,
(4) f restricted to the upper half-plane or the lower half-plane is Cm, but f
itself is not Cm.
(b) Suppose M and N are Cm manifolds-with-bounda~yand f : aM + aN
is a difiomorphism. Let P = M U 1 N be obtained from the disjoint union
of M and N by identifying x E aM with f (x) E aN. If (x, U) is a coordinate
system around p E a M and ( y , V) a coordinate system around f ( p ) , with
f ( U n a M ) = V n a N , and ( y o f)IU n a M = xlU n a M , we can define a
homeomorphism from U U V c P to IRn by sending U to Wn by x and V to
tlle Iower half-plane by the reflection of y. Show that this procedure does not
define a CbO structure on P.
(c) Now suppose t11at there is a neighborhood U of aM in M and a dfleomorphism a : U + 8M x [O, l ) , such that a ( p ) = (p,O) for al1 p E aM, and a
similar diffeomorphism p : V + aN x [O, 1). (M'e will be able to prove later that
such diffeomorpllisms always exist). Show that tllere is a unique CbO structure
on P sucl-i tl-iat t l ~ eii~clusionsof M and N are CbO and such that the map from
U U V to aM x (- 1,l) induced by a and B is a dfleomorphism.
(d) By usiilg two difTei-entpairs (a,p), define two dflerent Cm structures on EX2,
collsidcred as thc union of two copies of W 2 witli corresponding points on a W 2
iclei~tified.SIlow that the resulting CbO manifolds are dfleomorphic, but that
the difKeomorphism ca?~nolbe chosen arbitrai-iIy close io the identity map.
15. (a) Find a Cm structure on W' x M' which makes the úidusion into EX2
a Cm map. Can the inclusioi-i be an imbedding? Are the projections on each
factor Cm maps?
(b) If M and N are manifolds-with-boundary, construct a Cm structure on
M x N such that ali the "slice maps" (defined in Problem 8) are Cm.
16. Show that the fui-iction f : IR -+ IR defined by
is Cm (the formula e-'ix2 is used just to get a function which is > O for x < 0,
and e-'/lXl could be used just as weil).
17. Lemma 2 (as addended by the proof of Theorem 15) shows that if Cl and C2
are disjoint closed subsets of M, then there is a Cm function f : M + [O, 11
such that Ci c f -'(O) and C2 c f -'(I). Actually, we can even find f with
Cl = /-'(O) and C2 = f - ' (1). The proof turns out to be quite easy, once you
know the trick.
(a) It suffices to find, for any closed C c M, a Cm function f with C = /-'(O).
(b) Let ( U i ) be a countable cover of M - C, where each Ui is of the form
for some coordinate system x taking an open subset of M - C onto EXn. Let
Si : M + [O, 11 be a Cm function with _I;: > O on Ui and j;. = O on M - Ui.
Functions like
-
axi' axíaxk5
"'
will be called mixed partials of A, of order 1,2,. .. . Let
ai = sup of all mixed partials of
Show that
Si,. .. ,f;. of al1 orders 5 i.
f = "Cai2'
-Si
i=I
is Cm, and C = f -l (O).
18. Consider the coordinate system (y1,y2) for IR2 defined by
y' ( a ,b) = a
y2(a,b) = a
+ b.
(a) Compute f/ayr(a, b) from the definition.
(b) Also compute it from Proposition 3 (to find a I i / a y j , write each 1' in terms
of y' and y2).
Notice that af/ayl # 8/ / a l r even though
on y and i, not just on y'.
= I r ; the operator alay' depends
19. Compute the "Laplacian"
in terms of polar coordinates. (First compute a/ax in terms of alar and a/a9;
r a aa;) ae
a (,=)J.
r a
then compute a2/ax2 from this). Amwer: ;
+
20. If : M" + Nm is C1 and m > n, then /(M) has measure O (provided
that M has only countably many components).
21. The following pictures show, for 11 = 1,2, and 3, a subdivision of [O, 11 x
[O, 11 into 22n squares, An,i,. . . ,An,2b; square An,k is labeled simply k. The
numbering is determined by the following conditions:
(a) The lower left square is A,,].
(b) The upper left square is An1221f.
(c) Squares A,,k and AR,k+l have a common side.
(d) Squares An,41+1, An,41+2, An,41+3i An,41+4 are contained in A,-, ,[+l.
Define f : [O, 11 + [O, 11 x [O, 11 by the condition
f(t) E An,k
for al1
k-1
22"
k
22"
<ts-.
--
Show that f is continuous, onto [O, 11 x [O, 11, and not one-one.
22. For p/2" E [O, 11, define f (p/2") E lR2 as shown below.
(a) Show that f is uniformly continuous, so that it has a continuous extension
g: [O, 11 + lR2. Show that g is one-one, and that its image wiil not have
measure O if tlie shaded triangles are chosen correctiy.
(b) Consider tlie homeomorphic image of S' obtained by adding, below the
image of g, a semi-circle with diameter tlie line segment A B , JVhat does the
inside of this curve look like?
23. Let c : [O, 11 + Rn be continuous. For eacli partition P = {to,.. .,f k ) of
[O, 11, define
k
The curve c is rectifiable if {l(c, P)) is bounded above (with length equal to
sup{l(c, P))). Show tliat the image of a rectifiable curve has measure O.
24. (a) If M is a Cm manifold, a set M i c M can be made into a k-dirnensional submanifold of M if and only if al-ound each point in M I there is a
coordinate system ( x ,U) on M such tliat M i n U = { p : xk+'(p) =
=
x" (p) = O).
(b) Tlie subset M ican be made into a closed submanifold if and only if such
coordinate systems exist around every point of M .
25. Tlie sei {(x, 1x1); x E R) is noi tlie image of aiy immersion of R into IR2.
26. (a) If U c Rk is open and f : U -t IRn-' is Cm, then the graph of
f = {(p, f(p)) E R": p E U) is a submanifold of Rn.
(b) Every submaliifold of R" is locally of tiiis form, after renumbering coordinates. (Neitlier Theorem 9 nor 10 is quite strong enough. You wiil need
the implicit function theorem (Gilculw un Manzjbolds, pg. 44)).Theorem 10 is essentially Theorem 2-1 3 of Celculw on ManifoHs; comparison with the implicit
function theorem will show liow some information has been allowed to escape.)
27. (a) An immersion from one n-manifold to another is an open map (the
irnage of an open set is open).
(b) If M and N are n-manifolds with M compact and N connected, and
f : M + N is an immersion, then f is onto.
28. k o v e Proposition 12: If f : M" + N has constant rank k on a neighborhood of f - ' ( y ) , then f ( y ) is a (closed) submanifold of M of dimension
n - k (or is empty).
-'
29. Let f : P' + W 3 be the map
defined in Chapter 1, whose image is the Steiner surface. Show that g fails to
be an immersion at 6 points (tlie image points are the points at distance &1/2
on each axis). There js a way of immersing P2 in IR3, known as Boy's Surface.
See Hilbert and Col1n-Vossen, Geomeiy and !he Imagination, pp. 3 17-32 1.
30. A continuous f~inctionf : X + Y is proper if f - ' ( C ) is compact for
evcry compact C c Y. The limit set L ( f ) of f is the set of al1 y E Y such
ihat y = lim f(x,) for some sequence xl,x2,x3,. .. E X with no convergent
subsequence.
(a) L ( f ) = 0 if and only if f is proper.
(b) f ( X ) c Y is closed if and only if L ( f ) c f ( X ) .
(c) Tliere is a continuous f : W + IWZ with f(W) closed, but L ( f ) # 0.
(d) A onc-one continuous function f : X + Y is a homeomorphism (onto its
image) if and only if L ( f ) n f ( Y ) = 0.
(e) A submanifold Mi c M is a closed submanifold if and only if the inclusion
map i: MI + M is proper.
( f ) If M is a manifold, there is a proper map f : M + IW; the function f can
be made Cm if M is a Cm manifold.
31. (a) Fiiid a cover of [O, 11 which is not locally finite but wliich is "pointfinite": every point of [O, 11 is in only finitely many members of the cover.
(b) Prove the Shrinking Lemma when the cover O is point-finite and countable
(notice that local-finiteness is not really used).
(c) Prove the Shrinking Lemma when O is a (not necessarily countable) pointfinite cover of any space. (You will need Zorn's Lemma; consider collections C
of ~ a i r (U,
s U') where U E O, U' c U, and t he union of all U' for (U, U') E C,
togetlier with all other U E O covers the space.)
32. (a) If MI c M is a closed submanifold, U 2 Mi is any neighborhood,
aiid f : Mi + R is Cm, then there is a Cm function f:M + W with / = f
on M i , and with support J c U.
(b) This is false if M = W and MI = (0,l).
(c) This is false if R is replaced by a disconnected manifold N.
Remark: It is also false X M = RZ,MI = N = S', and f = identity; in fact, in
tliis case, f has no contiiiuous extension to a map from R2 to S', but the proof
requires some topology. However, f can always be extended to a CbO function
in a neighborhood of MI (extend locally, and use partitions of unity).
33. (a) The set of aH non-siiigular n x n matrices with real entries is called
GL(i1, R), tlie general linear group. It is a CbO manifold, since it is an open
2
subset of W" . The special linear group SL(n, R), or unimodular group, is the
subgroup of al1 matrices witli det = 1. Usirig tlic formula for D(det) in CahuEw
on A4aa@ds, pg. 24, show tliat SL(n, R) is a closed submanifold of GL(n, R) of
dimension n2 - 1.
(b) The symmetric n x 11 matrices may be thought of as R"("+')/~. Define
$: GL(n, W) + (symmetric matrices) by $(A) = A . A', wliere A' is tlie
traiispose of A . Tke subgroup $-'(I) of GL(J~,W)is called the orthogonal
group O(ri). Sliow tliat A E O(n) if arid only if tlie rows [or columns] of A are
ort honormal.
(c) Show that O(n) is compact,
(d) For any A E GL(JI,R), define RA: GL(!r, R) + GL(n, R) by RA(B) = BA.
Show tliat RA is a dfleomorphism, aiid ihat $ o RA = @ for ali A E O(n). By
applying tlie chain rule, show that for A E O(n) the matrix
a@ij
lias the same rank as (-(1))
axkl
.
(Here xk' are the coordinate functions in WnL, and @'j the n(n+ 1)/2 component
functions of @.) Conclude from Proposition 12 that O(n) is a submanifold of
GL(n, R).
(e) Using the formula
62
show that
\O
otherwise.
Show that the rank of this matrix is n(n + 1)/2 at I (and hence at A for al1
A E O(n).) Conclude that O(n) has dimension n (n - 1)/2.
(fj Show that det A & 1 for al1 A E O@), The group O(n) nSL(n, R) is called
the special orthogonal group SO@), or the rotation p u p R(n).
-
34. Let M(nr, 11) denote the set of al1 n.i x n matrices, and M(n2,n; k) the set
of al1 n? x n matrices of rank k.
(a) For every Xo E M(m,n; k) there are permutation matrices P and Q such
that
PXoQ =
co
Bo) ,
Do
where A. is k x k and non-singular.
(b) There is some E > O S U C ~that A is non-singular whenever al1 entries of
A - A . are cl E .
(4 If
A B
PXQ=(C
D)
wbere the entries of A - A. are < E , then X has rank k if and only if D =
CA-'B. Hinl: If Ik denotes the k x k identity matrix, then
(d) M (112,n; k) c M (m,n) is a submanifold of dimension k (m + n - k) for a11
k 5 in,n.
CHAPTER 3
THE TANGENT BUNDLE
A
point v E R" is frequentiy pictured as an arrow from O to V. But there are
many situations where we would like to picture this same arrow as starting
at a dXerent poini p E Rn:
For example, suppose c : EX + R" is a differentiable curve. Then cr(t) =
(cl'(t), . . - ,cnr(t))is just a point of R", but the line bemeen c ( t ) and c ( t ) + cr(t)
is tangent to the curve, aiid the "velocity vector" or "tangent vector" c r ( i ) of
the curve c is customarily pictured as the arrow from c ( i ) to c ( t ) + cr(t).
To give tliis picture mathematical substance, we simply describe the "arrow"
from p to p + v by the pair (p, u). The set of al1 such pairs is just R" x R",
wliich we will also denote by T R", the "tangent space of R" "; elements of T R"
are called "tangent vectors" of R". We will often denote (p, u) E TR" by U,
("the vector v at p"); in conformjty with this notation, we will denote the set
of al1 (p,v) for v E R" by Rnp. At times, it is more convenient to denote a
member of TR" by a single letter, like v. To recover the first member of a pair
2i E TRn, we define tlie "projection" map rr : R" x IR" + R" by ~ ( ab), = a .
For any tangent vector u, the point X(V)is "where it's at".
T h e set rr-l ( p ) may be pictured as al1 arrows starting at p. Alternately,
ii can be pictur-ed more geometi-ically as a particular subset of R" x R", the
oiie visualizable case occurring when 11 = l. This picture gives rise to some
tei.minology-we call rr-'(p) ihe fibre over p. Tliis fibre can be made into a
The Tazgeni Bundle
vector space in an obvious way: we define
(p, u)
(p, w ) = (PY u + w)
a (p, u) = (p,a u)-
(The operations $ and * should really be thought of as defined on
(
)
x
~
(
~
)and ,
R x T R",
res~ectively.
,€E"
Usually we will just use ordinary + and . instead of $ and .)
If f : Rrl + Rm is a differentiable map, and p E Rn, then the linear transformation Df(p): Rn + Rm may be used to produce a linear map from
Rnp + Rni (,, defined by
This map, whose appareiitly anomalous features will soon be justified, is denoted by S,,; tlle symbol
denotes tlie map f* : TR" + TRm which is the
union of al1 f*,. Sincc j;,(u) is defined to be a vector E Rmftpl, the followiiig diagram "commutes" (the two possible compositions from TR" to Rm are
equal),
TR" f* TRm
-
J
IR"
Rm
Tlius, $, has the map f, as well as al1 maps Df(p), built into it.
This is noi the only reason for defining f* in this particular way, however.
Suppose that g : Rm + IRk is another dzerentiable function, so that, by the
chain rule,
By our definition,
This looks horribly complicated, but, using (11, it can be written
thus we have
g*of* = (g of)*.
Thjs relation would clearly fall apart completely if f*(vp) were not in R T ( p i ;
with our present definition of f*, it is merely an eiegant restatement of the
chain rule.
Henceforth, we will state almost al1 concepts about Jacobian matrices, like
rank or singularity, in terms of f,, rather than Df. T h e "tangent vector" of a
curve c: EX + R" can be defined in terms of this concept, also. T h e tangent
vector of c at t may be defined as
[If c happens to be of the form
then
tllis \lector lies along the tangent line to tlie graph of f at (1, f (l)).] Notice
thai the tangeni vector of c at 1 is the same as
c*(l*)= [Dc(t)(UIc(t) = (cl/(t), . - ,cns(l))c(*),
wliere l t = (1,I) is iIie "unit" tangent vector of R at 1.
If g : Rn + EXm is differentiable, then g o c is a curve in Rm. The tangent
T h 7angent Bundle
vector of g o c at 1 is
( g O c)*(lr) = g*(c*(lr))
= g,(tangent vector of c at t ) .
Consider now an n-dimensional manifold M and an imbedding i : M + W N .
Suppose we take a coordinate system ( x , U ) around p. Then i o x-' is a map
from EXn to W N with rank n. Consequently, (i o x - ' ) , (W\(,))
is an n-dimensional subspace of W N i ( , ) . This subspace doesn't depend on the coordinate
system x., for if y is another coordinate system, then
(i o y-'), = ( i O x-' o x O y - 1 ) ,
= (i o x - ' ) , o
( X O y-1)*
and
(x 0 y-')*y(,) :
+
is an isomorphism (with inverse ( y o x - ' ) , ~ ( , ) ) .
Wnx(P)
There is another way to see this, which justifies the picture we have drawn.
If c: ( - & , E ) + W" is a curve with c(0) = x(p), then a = i o x-' o c is a
curve in EXN which lics in i(M), and every dxerentiable curve in i(M) is of this
form (Proof?). Now
ar(lO) = (i O x-')*
o
lo),
so the tangent vector of every a is in (i ox-'), (EXn,(,)).
Moreover, every vector
in this subspace is the tangent vector of some a, since every vector in EXn,(,)
is
tlie tangent vector of some curve c. Thus, our n-dimensional subspace is just
the set of al1 tangent vectors at i(p) to dxerentiable curves in i(M). We will
denote this n-dimensional subspace by (M, i),.
We now want to look at the (disjointj union
We can define a "projection" map
As in the case of TWn, each "fibre" x-'(p) has a vector space structure also.
Beyond this we have to look a little more carefully at some specific examples.
Coiicider first the maiiifold M = S' and tlie inclusion i : S' + EX2. Tlie
curve c(O) = (cos @,sinO) passes through every point of S', and
cr(0) = (- sin O, cos O) # O.
For eacli p = (cos O, sin O) E S', let up = (- sin O, cos O), (it clearly doesn't
matter wliich of the infinitely many possible 0's we choose). Then (S', i), con-
The Ta?gel2tBundle
69
sists of al1 multiples of the vector u,,. We can therefore define a homeomorphism
h : T(S1,i) + S' x IR1 by h (hu,,) = ( p , A), which makes the following diagram commu te.
If we define tlie "Abres" of n' to be the sets n'-' (p), then each fibre has a vector
space structure in a natural way. Commutativity of the diagram means that $i
takes fibres into fibres; clearly f ] restricted to a fibre is a linear isomorphism
onto the image.
Now consider the manifold M = s2and tlie inclusion i : s2c R3. In thjs
case tliere is «o map f2 : T ( s ~ ,i) + s2x R2 with the properties of the map h.
If there were, then, for a fixed vector u # O in R2, the set of vectors
would be a collection of non-zero tangent vectors, one at each point of s 2 ,
which varied coiiiiriuously. It is a well-known (hard) theorem of topology that
this is impossible (you can't comb the hair on a sphere).
There is another example where we can prove that no appropriate homeomorphism T ( M , i) -t M x EX2 exists, without appealing to a hard theorem of
topology. The rnap i will just be the inclusion M +- where M is a Mobius
strip, to be precise, the particular subset of IR3 defined in Chapter 1-M is the
image of the map f : [O, 2x1 x (- 1 , l ) + R3 defined by
At each point p = (2 cos 8 , 2 sin 8 , O ) of M , the vector
u,, = (-2 sin 8,2 cos 0, O)p - f*((lf0)(e,0))
is a tangent vector. T h e same is true for al1 multiples of f,((O, I ) ( ~ , ~shown
)),
as dashed arrows in the picture. Notice that
while
This means tliat we can never pick non-zero dashed vectors continuowEy on the
set of al1 points (2 cos 8,2 sin @,O); If we could, then each vector would be
for some continuous function A: [0,2rr] + R. This function would have to be
non-zero everywhere and also satisfy h(2rr) == -h(O), which it can't (by an easy
theorem of topology). The impossibility of choosing non-zero dashed vectors
contiiiuously clearly sliows that there is no way to map T ( M ,i), fibre by fibre,
The 7 a z g e ~Bundle
71.
homeomorphically onto M x IR2. We thus have another case where T (M, i)
does not "look like" a product M x Rn.
For any imbedding i ; M + R N , however, the s tmc ture of T (M, i) is always
simple locab: if ( x ,U ) is a coordinate system on M , then n - ' ( U ) , the part
of T ( M ,i) over U, can always be mapped, fibre by fibre, homeomorphically
onto U x Rn. In fact, for each p E U, the fibre
( M ,i ) ,
equals
(i o x - ' ) * ~ ( , )(WnX(,)) = mp (Rnx(,)) ,
where the abbreviation 172, lias been introduced temporarily; we can therefore
define
/: n - ' ( U )+ U x R"
by
/ (m, (U,(,)))
= ( p ,u ) -
In standard jargon, T(M, i) is "locally trivial". This additional feature qualifies
T ( M ,i) to be included among an extremely important class of structures:
An ~r-dimensionalvector bundle (or n-plane bundle) is a five-tuple
where
(1) E and 3 are spaces (tlie "total space" and "base space" of 6,
respec tively),
(2) n ; E + B is a continuous map onlo B,
(3) $ and O are maps
with $ ( n - ' ( p ) x n - ' ( p ) ) c n - ' ( p ) and @ ( W x n - ' ( p ) ) c
n-' (p), which make each fibre ir-' ( p ) into an n-dimensional
vector space over IR,
such that the following "local triviality" condition is satisfied:
Foi. each p E B , there is a neighborhood U of p and a homeomorphism t : n ( U ) + U x 8" which is a vec tor space isomorphism
from each n - ' ( q ) onto q x Rn, for al1 q E U.
-'
Because tliis local triviality condition really is a local condition, each bundle
6 = ( E , n , B, $,O) automatically gives rise to a bundle 6lA over any subset
A c B; to be precise,
Notation as cumbersome as al1 tliis invites abuse, and we shall usually refer
simply to a bundle rr ; E + B , or aren denote the bundle by E alone. For
vectors u, w E n-' (p) and a E R, we wiil denote $(u, w) and @(a, v) by v + w,
and a . v or av, respectively.
The simplest example of an n-plane bundle is just X x Rn with x : X x R" +
X the projection on tlie first factor, arid tlie obvious vector space structure on
each fibre. This is called the trivial H-plane bundle over X and will be denoted
by E"(X). Tlie "tangent bundle" TR" is just E"(R").
The bundle T(S' ,i) considered before is equivalent to E' (S'). Equivalente
is here a tecl~iiicalte1.m: Two vector bundles 61 = X I : El + B and 6 2 =
n2 : E2 + B are equivalent (tih., C2) if there is a homeomorphism h : El + E2
which takes each fibre ni-' (p) isomorphically onto n2-' (p). T h e map h is
called an equivalence. A bundle cquivalent to E" (B) is called trivial. (The
local triviality condition for a bundle 6 just says that ,$[Uis trivial for some
neighborhood U of p.)
T h e bundles T ( s ~ ,i) aiid T(M, i) are not trivial, but tliere is an even simpler
example of a non-trivial buiidle, T h e Mobius strip itsey (not T(M, i)) can be
i.,ir~sideredas a 1-dimeiisional vector bundle over S', for M can be obtained
1iom [O, I] x R by identifying (0,a) with ( 1 , -a), wliile S' can be obtained from
73
The Tangen.?Bundle
[O, 11 by identifying O with 1; the map rr is defined by x(1.a) = t for O cr r cr 1
aild ~ ( ( ( 0a),
, (1, -a))) = {O, 1). The diagram above illustrates local triviality
iiear the point {O, 1) of S'. SupPose that S : S' + M is a continuous function
with rr o S = identity of M (such a function is called a section). Such a map
corresponds to a continuous function S : [O, 11 + R with S(0) = -S(l). Since S
must be O soinewliere, t he section s must be O somewliere (that is, S(@)E rr-l(8)
must be the O vector for some 0 E S'1. This surely sliows that M is not a trivial
bundle.
An equivalente is obviously the analogue of an isomorphism. T h e analogue
of a homomorphism is the following.* A bundle map from to
is a pair of
continuous maps
f), with : El + E2 and f : Bl + Bz, such that
(7,
f
c1 c2
(1) the following diagram commutes
(2) /:nl-'(p) + rrz-' (f (p)) is a linear map.
Tlie pair (f., f ) is a bundle map kom T R k to TR' for any dxerentiable
f : IRk + IR'. If M W c Rk and N m c IR1 are submanifolds. i : .M + IRk and
j : N + IR1 are tlie inclusions, and the rnap f satisfies f (M) C N, then f,
* Tliere are actually severa1 possjble choices, depending o11 whether one is considering al1 buridles at once, fixed bundles over various spaces, or a fwed base space with
varyiiig buiidles. T h u s f may be restricted to be an isomorphism on fibres and f to
be tlie identity or a Iion~eomorphism.The relations between some of these cases are
considered in the problems.
takes T ( M , i ) to T ( N ,j ) ; to see this, just remember that v E T ( M , i ) p is the
tangent vector of a curve c in M , so & ( u ) is the tangent vector of the c u m
f o c in N , and consequently f+( u ) E T ( N ,j ) . In this way we obtain a bundle
map from T ( M ,i) to T ( N ,j ) . Actually, it would have sufficed to begin with
a Cm function f : M + N, since f can be extended to IRk locally. In fact,
this construction could be generalized much further, to the case where i and j
are merely imbeddings of two abstract manifolds M and N , and f:M + N
is Cm; we just consider the function j o f o i-' : i ( M ) + i ( N ) and extend it
locally to 3.
The case which we want to examine most carefully is the simplest:
where M = N and f is the identity, while i and j are two imbeddings of M
in IRk and dR',
respectively. Elements of T ( M ,i)pare of the form (i o x-' ).
(tu)
for w E dnxnx(p),
while elements of T ( M ,j ) p are of the form ( j x-'). ( w ) for
w E IRnx(,). If we map
Q
(i O x-'), ( w ) H (jo X-'), ( w )
we obtain a bundle map from T ( M ,i)lU to T ( M ,j)lU, which is obviously an
equivalence. The map ( M ,i)p+ ( M ,j ) p iiiduced on fibres is independent of
the coordinate system x , for if ( y , V) is another coordinate system, then
We can therefore put al1 tliese maps together, and obtain an equivaEence from
T ( M ,í) to T ( M ,j ) . In otlier words, the dependence of T ( M ,i) on i is almost illusory; we could abbreviate T ( M ,i) to T M , if we agreed that T M really
denotes an equivalence class of bundles, rather than one bundle. That is the
The 7angenl Bundle
75
sort of thing an algebraist might do, and it is undoubtedly ugly. What we would
like to do is to get a single bundle for each M, in some natural way, which has
al1 the properties any one of these particular bundles T ( M , i) has. Can we do
ihis? Yes, we can. When we do, T R n will be dflerent from our old definition (namely, &"(IRn)),and so will f, for f : Rn + Rm, so in stating our result
precisely we will write "old f," when necessary.
f . THEOREM. It is possible to assign to each n-manifold M an n-plane bundle TM over M, and to each CbOrnap f : M + N a bundle map (f,, f ) , such
that:
(1) If 1 : M + M is the identity then 1,: TM + T M is the identity. If
g : N + P,then (g o f), = g, o f,.
(2) There are equivalentes t" : T Rn + E" (R") such that for every Cm function f : Rn + Rm the following commutes.
(3) If U C M is an open submanifold, then T U is equivalent to (TM) 1 U,
and for f : M + N the map (f IU), : TU + T N is just the restriction
of f,. More precisely, there is an equivalence T U 2 (TM)IU such that
the following diagrams commute, where i : U + M is the inclusion.*
PR00F. T h e construction of T M is an iilgenious, tllough quite natural, subterfuge. We will obtain a single bundle for TM, but the e h e n t s of T M will eacli
be large equivalence classes.
* M7henusiiig the notation $*,it must be understood that the symbol 'y"realiy refers to
a triple (f, M, N) wliere f : M + N. The identitymap 1 of U to itself and the inclusion
map i : U + M have to be considered as different, since the maps 1, : TU + TU and
i, : T U + TM are certainly dfleren t (they map T U into two different sets).
The construction is much easier to understand if we first imagine that we already had our bundles T M . Then if ( x , U ) is a coordinate system, we would
have a map x , : T U + T ( x ( U ) ) ,and this would be an equivalence (with
inverse (x-').). Since T U should be essentially (TM)IU3and T ( x (U)) should
essentially be x ( U ) x R n , a point e E n - ' ( p ) would be taken by x. to some
( x ( p ) ,u). Here v is just an element of Rn (and every v would occur, since x*
maps a-' ( p ) isomorphically onto { p )x R"). If y is another coordinate system,
then y,(e) would be ( y ( p ) ,w ) for some w E Rn. 'We can easily figure out what
tlle relationship between v and w would be; since ( x ( p ) ,u ) is taken to ( y @ ) ,w )
by y, o x,-' = ( y o x - ' ) , , and ( y o x - ' ) , is supposed to be the old ( y o x-').,
we would have
This condition makes perfect sense without any mention of bundles. It is the
clue wliicli enables us to now define T M .
If x and y are coordinate systems whose domains contain p, and u , w E R",
we define
if (a) is satisfied.
(x, U ) ;
( y ,w )
-
It is easy to check (using the cbain rule) that
is an equivalence relation; the
P
equivalence class of ( x , u ) will be denoted by [ x ,u], . These equivalence classes
will be called tangent vectors at p3 and T M is defined to be the set of al1 tangent
vectors at al1 points p E M ; the map n takes equivalence classes to p . We
P
define a vector space stmcture on n-' ( p ) by the formulas
tliis definition is independent of the particular coordinate system x or y , because
D ( y o x - ' ) ( x ( p ) ) is an isomorphism from RR to R".
Our definition of T M provides a one-one onto map
(b>
tx : n-' ( U ) -t U x Y , namely
[ x , v J qH ( q ,u).
We want tliis to be a homeomorphism, so we want tx-' ( A ) to be open for every
open A C U x RR3and thus we want any union of such sets to be open. There
is a metric witli exactly these sets as open sets, but it is a little ticklish to produce,
so wc leave this one part of the proof to Problem 1.
We now have a bundle n : T M + M . We will denote the fibre n - ' ( p )
by M p , iii conformity with the notation Rnp, tbough T M p might be better. If
The Znge~dBundle
77
J': M + N, and (x, U) and (y, V ) are coordinate systems around p arid f (p),
r-espectively, we define
Of course, it must be cbecked that this definition is independent of x and y
(the chain rule again),
Condition (1) of our theorem is obvious.
To prove (2), we define t n to be t l , where I is the identity map of R" and t,
is defined in (b); it is trivial, though perhaps confusing to the novice, to prove
commutativity of the diagram.
Condition (3) is pi4acticallyobvious also. In fact, the fibre of TU over p E U
is almost exactly the same as the fibre of TM over p ; the only difirence is that
each equivalence class for M contains some extra members, since in M there
are more coordinate systems around p than there are in U c M. *3
Henceforth, the bundle rr : TM + M will be called the tangent bundle of M.
If i : M + Rk is an imbedding, then TM is equivalent to T ( M , i). In fact, if
(x,U) is a coordinate system around p, and I is tbe identity coordinate system
of IRk, then
the composition tnic is easily seen to be an equivalence. But T (M, i) will play
no further rolc in this story-the abstract substitute T M will always be used
instead.
Having succeeded in producing a bundle over each M, which is equivalent to
T(M, i), we next ask how fortuitous this was. Can one find other bundles with
the same properties? Tlie answer is yes, and we proceed to define two different
such bundles.
For tbe first example, we consider curves c: (-&,E) + M, each defined
on some interval around 0, witb c(0) = p. If (x, U) is a coordinate system
around p , we define
cl 7 c2
if and only if:
x ocl and x oc2, mapping IR to IR",
have the same derivative at O.
The equivalence classes, for al1 p E M, will be the elements of our new bundle, T'M. For f :M + N tliere is a map S# taking the %
equivalence class
P
Jz)
of c to the
equivalence class of f o c. Without bothering to check details,
we can already see that this example is "redly the same" as T M [x ,v J p corresponds to:
the 7 equivdence class of x-' o y,
where y is a c u m in Rn with y'(0) = u ;
under this correspondence, $# corresponds to f,.
In the second example, things are not so simple. We define a tangent vector
at p to be a linear operator which operates on al1 CbO functions f and which
is a "derivation at p":
We have already seen that the operators = a/axilp have this property For
these operators, clearly t ( f ) = l ( g ) if f = g in a neighborhood of p. This
condition is actually tme for any derivation t. For, suppose that f = O in a
neighborhood of p. There is a Cm function h : M + R with h ( p ) = 1 and
support h c f - ' (O). Then
Thus, if f = g in a neighborhood of O, then O = e(f - g ) = e(f ) - l ( g ) . If f
is defined only in a neighborhood of p, we may use this trick to define f ) :
choose h to be 1 on a neigliborhood of p, with support h c f -l ( O ) , and define
[ ( S )as [ ( f h ) .
Tlie set of al1 sucli operators is a vector space, but it is not a jjriori clear what
its dimension is. This comes out of the following.
e(
2. LEMMA. Let f be a CbOfunction in a convex open neighborhood U of O
in IRn, with f ( 0 ) = O. Then there are CbO functions gi : U + R with
(Tlie second condition actually follows from the first.)
PROOF. For x E U, let Ax ( t ) = f ( t x ) ; this is defined for O 5 t 5 1 , since U is
convex. Then
Therefore we can let g ( x )= J: D i f ( t x ) d t . 9
The Tangeni Bundle
79
3. THEOREM. The set of al1 linear derivations at p E M" is an n-dimensional vector space. In fact, if (x,U ) is a coordinate system around p, then
span tbis vector space, and any derivation l can be written
(so e is determined by the numbers l ( x i ) ) .
PROOE Notice that
so l ( 1 ) = O. Hence l ( c ) = c l(1)= O for any constant function c on U.
Consider the case where M = R" and p = 0. Assume U is convex. Given f
on U, choose gi as in Lemma 2, for the function f - f (O). Then
n
(lidenotes the ith
coordinate func tion)
l(f)=e(f -f(o))=e
This sliows tliat a/a liiospan the vector space; they are clearly linearly independent. It is a simple exercise to use the coordinate system x to transfer this
result fmm R" to M. 4 3
Froni Tbeorcm 3 we caii see tbat, once again, a bundle constructed from al1
derivations at al1 points of M is "really tlie same" as TM. We can let
the formula
a
P
j=]
derived in Chapter 2, shows that
-1
a
x a i
axi
i= 1
n
n
=b
i = ~
i
- 1a
ayj
ifand only d b j =
ayi p
i =l
axi
-,
and this is precisely the equation which says that ( x , a ) (y,b). It is easily
checked that under this correspondence, the map which corresponds to f, can
be defined as foilows:
Ef* ( 0 1 ( g ) e ( g o f
I=
n
. -a l ,
Notice that if x denotes the identity coordinate system on Rn, then
a'
axl
i=l
corresponds to ap when we identi@ T R n with E" (Rn).
Wc will usually make no distinction whatsoever between a tangent vector
v E M P and the linear derivation it corresponds to, that is, between Ex, a], and
consequently, we will not hesitate to write u (f ) for a differentiable function f
defined in a neighborhood of p. In fact, a tangent vector is often most easiiy
described by tclliiig what derivation it corresponds to, and the map f, is often
most easiiy analyzed from the relation
11 is customary io denote the identity coordinate system on IR' by r, and to
write
ior
dt 2,
1
1;
at 1,
this is a basis for Rz,,. If c : R + M is a differentiable curve, then
is called tlie tangent vector to c at lo. We will denote it by the suggestive symbol
The 7 a n g n l Bundle
81
This symbol will be subjected to the standard abuses one finds (unexplained) in
calculus textbooks: the symbol
dC will often stand for
dt
1
dt l ,
the subscript "t" now denoting a particular number t E R, as well as the identity
coordinate system.
As you might well expect, it is no accident that our second and third exarnples
turned out to be "really the same" as TM. There is a general theorem that a11
"reasonable" examples will have this property, but it is a littie delicate to state,
and quite a mes5 to prove, so it has been quarantined in an Addendum to this
chapter.
The tangent bundle TM of a Cm manifold has a little more stmcture than
an arbitrary n-plane bundle. Since TM locally looks like U x Rn, clearly TM
is itself a manifold; there is, moreover, a natural way to put a Cm stmcture
on TM. If x : U + IRn is a chart on M, then every element u E (TM)IU is
uniquely of the form
Let us denote a' by X i(u). Then the map
U H (,Y1 (H (U)),
. . .,x n (ir (U)),x1 (U),. .. ,Xn (U)) E IR2"
is a homeomorphism from (TM)IU to x (U) x R". This map, ( x o x, X), is
simply the map x, when we identify TU with U x Rn in the standard way If
(y, V) is anotlier coordinate system, and
then, as we have already seen,
This shows that if (t,a) = (zl,. . .,t n , a l , . . . , a n ) E IR2", then
Y* 0 (x*)-I (1, a)
= ( y o x - 1 (t),
xFZlai Di(y1 x-l) (t)>- - - ,
0
This expression sliows that y* o (x*)-' is Cm.
ai ~i ( Y" 0 x-l) (t))-
\/Ve thus have a collection of Cm-related charts on T M , which can be extended to a maximal atlas.
With this Cm structure, the local trivializations x, are Cm. In general, a
vector bundle rr: E + B is called a Cm vector bundle if E and B are Cm
manifolds and there are Cm local trivializations in a neighborhood of each
point. It follows that rr : E + B is Cm.
Recall that a section of a bundle rr : E + B is a continuous function s : B + E
such that rr O S = identity of B; for Cm \lector bundles we can also speak of Cm
sections. A section of T M is called a vector field on M ; for submanifolds M
of Rn, a vector field may be pictured as a continuous selection of arrows tangent
to M. The theorem that you can't comb the hair on a sphere just states that
there is no vector field on s2which is werywhere non-zero. We have shown that
there do not exist two vector fields on the Mobius strip which are everywhere
linearly independent.
Vector fields are customarily denoted by symbols like X, Y, or 2, and the
\rector X(p) is often denoted by X, (sometimes X may be used to denote a
single vector, i ~some
i
M,). If we think of T M as the set of derivations, then for
any coordinate system (x, U), we have
n
X(P) = C ~ ' ( P )
for al1 p E U.
i=l
The functions a' are continuous or Cm if and only if X : U + T M is continuous or Cm.
If X and Y are two vector fields, we define a new vector field X + Y by
Similarly, if f : M + R, we define the vector field JX by
The 7angeni Bundle
83
Clearly X + Y and f X are Cm if X, Y, and f are Cm. On U we can write
the symbol a/axi now denoting the vector field
If f : M + R is a Cm function, and X is a \lector field, then we can define
a new fwidion 2 ( f ): M + B by letting X operate on f at each point:
It is not hard to check that if X is a Cm vector field, then f ( f ) is Cm for
every Cm function f ; indeed, iflocally
then
which is a sum of products of Cm functions. Conversely, if y(f ) is Cm for
e u q Cm function f , then X is a Cm vector field (since 2 ( x i ) = a').
Let F denote the set of al1 Cm functions on M. We have just seen that a Cm
vector field X @ves rise to a func tion 9 : P + P. Clearly,
tlius 2 is a "derivation" of the ring F. Ofien, a Cm vector field X is identified
witli the derivation f . Tlie reason for this is that if A : P + 7 is any derivation, then A = 2 for a unique Cm vector field X. In fact, we clearly rnust
define
X,(f> = A ( f )(P),
and the operator XP thus defined is a derivation at p.
The tangent bundle is the tme beginning of the study of dflerentiable manifold~,and you should not read further until you grok it.* The next few chapters
constitute a detailed study of this bundle. One basic theme in all these chapters is that any structure one can put on a vector space leads to a structure on
any vector bundle, in particular on the tangent bundle of a manifold. For the
present, we will discussjust one new concept about manifolds, which arises in
this very way from the notion of "orientation" in a vector space.
The non-singular linear maps f : V + V from a finite dimensional vector
space to itself fall into two groups, those with det f > 0, and those with det f < 0;
linear transformations in t he first group are called orientation preserving and
the others are called dentation reversing. A simple example of the latter is
the map f : Rn + R" defined by f (x) = (x1,. . .,xn-l, -xn) (reRection in the
llyperplane xn = O). There is no way to pass continuously between these two
groups: if we identify linear maps Rn + R" with n x n matrices, and thus
with IRn2, then the orientation presewing and orien tation reversing maps are
disjoint open subsets of the set of al1 non-singular maps (those with det # 0).
The terniinology "orientation presewing" is a bit strange, since we have not yet
defined anything called "orientation", which is being preserved. T h e problem
becomes more acute if we want to define orientation preseMng isomorphisms
between two dflerent (but isomorphic) vector spaces V and W; this clearly
makes no sense unless we supply V and W with more structure.
To provide tliis extra structure, we note that two ordered bases (vi,. . ., u,)
and (vf1,.. . , vfn)for V determine an isomorphism f : V + V with f(vi) = vfi;
the matrix A = (aij) of f is given by the equations
We cal1 (vi,. .. ,u,) and (vfl,.. .,u',) equally oriented if det A > O (i.e., if f is
orientation preserving) and oppositely oriented if det A < 0.
Tlle relation of being equally oriented is clearly an equivalence relation, dividing tlie collection of al1 ordered bases into just two equivalence classes. Either of
iliese two equivalence classes is called an orientation for V. The class to which
(vi,. . .,un) belongs will be denoted by [vi,. . .,vn], so that if p is an orientation
of V, then (vi,... ,un) E ,u if and only if [vi,. . .,un] = p. If ,u denotes one
* A cult word of the sixties, "grok" was coined, purportedly as a word from the Martian
language, by Robert A. Heirilein in his pop ccience fiction novel Shnger in a Slrange
Lnttd. lts sense is nicely coiiveyed by the defiriitjoii in The Anzen'can H d h g e Diclio?tag*:
"To understand profoundly through intuition or empathy".
The 7angeni Bundle
Examples of equally oriented ordered bases in R, IR2, and IR3.
orientation of V, tbe other will be denoted by - p , and the orientation [ e ] , ., . ,e,]
for R" will be called the "standard orientation".
Now if (V, p ) aild (W, u ) are two ri-dimensional vector spaces, together with
orientations, an isomorphism f : V + W is called orientation preserving (with
respect to p and u) if [f ( v i ) ,- .-,f ( v , ) ] == v whenever [ V I , . . ., u,] = p; if this
holds for any one ( v i ,. . . ,u,), it clearly holds for all.
For tlie trivial bundle P ( X ) = X x R" we can put the "standard orientation"
[ ( x ,el),.. . , ( x ,e,)] on each fibre {x)x Rn. If f : e n ( X )+ e n ( X )is an equivalente, aiid X is connected, tlien f is either orientation presewing or orientation
reversing on eacli fibre, for if we define the functions aij : X + R by
then det ( a i j ): X + R is coniinuous and never O . If rr : E + B is a nontrivial 11-plane bundle, an orientation p of E is defined to be a collection of
condition"
orientatioiis pp for rr-I ( p ) wllicll satisfy the following "~ompatibilit~
for any open connected set U C B:
lf t : x-' ( U ) + U x IRn is an equivalence, and the fibres of U x IR" are
given tlie standard orientation, tlien t is either orientation presewing
or orientation reversing on all fibres.
Notice tliat if tliis condition is satisfied for a certain t, and t' : n-' ( U ) + U x Rn
is another equivalence, then t' automatically satisfies the same condition, since
t'o t
-' : U x 1"+ U x 1"is an equivalente. This shows that the orientations pp
define an orientation of E if the compatibility condition holds for a collection
of sets U which cover B.
If a bundle E has orientation p = {,up), it has another orientation -p =
{-,up), but not every bundle has an orientation. For example, the Mobius
strip, considered as a 1-dimensional bundle over S ' , has no orientation. For,
although the Mobius strip has no non-zero section, we can pick two vectors
from each fibre so that the totality A looks like two sections. For example,
we can let A be [O, i ] x {- 1, 1) with (O, a ) identified with (1, -a); then A just
looks like the boundary of the Mobius strip obtained from [O, i] x [- 1 , 11. If
we had compatible orientations p,,, we could define a section S : S' -, M by
choosing s ( p ) to be the unique vector s ( p ) E A nn-'(p) with [s(p)] = pp.
A bundle is called orientable if it has an orientation, and non-orientable otherwise; an oriented bundle is just a pair (6, p ) where p is an orientation for 6.
Tliis definition can be applied, in particular, to the tangent bundle TM of a
Cm manifold M. In this case, we cal1 M itself orientable or non-orientable depending on whether TM is orientable or non-orientable; an orientation of TM
is also called an orientation of M, and an oriented manifold is a pair ( M , p )
where p is an orientation for TM.
Tlle manifold Rn is orientable, since T W 2 E" (Rn), on which we have the
standard orientation. The sphere S"-' c Rn is also orientable. To see this we
note that for each p E S"-' the vector w = pp E en(Rn) 2 TR" is nol in
.i (Sn-',,) E T Rn,, (Roblem 2 l), so for V I ,.. . ,v.- 1 E Sn-lp we can define
(vi,. .. ,u n - ' ) E ,up if and only if (w, i,(vi), . .. ,i,
is in the standard
orientation of Rnp. The orientation p = {/L, : p E S"-' ) thus defined is called
the "standard orientation" of S"-'.
The iorus S' x S' is aiiotlier example of an orientable manifold. This can
be seen by noting that for any two manifolds Mi and M2 the fibre (Ml x M2)p
of T (MI x M2) can be written as VI $ V2p where (ni)*
: K,,+ (Mi),, is an
isomorphism and the subspaces Vi, vary continuously (Problem 26). Since Ts'
is trivial, this shows that T (S' x S') is also trivial, and consequently orientable.
Any n-holed torus is also orientable-the proof is presented in Problem 16,
which also discusses the tangent bundle of a manifold-with-boundary.
The Mobius strip M is the simplest example of a non-orientable 2-manifold.
For tlie imbedding of M considered previously we have aiready seen that on the
subset S = ((2cos 8 , 2 sin 8, O)) c M there are continuously varying vectors u,,,
but tliat it is impossiWe io clioose continuously from among the dashed vectors
wp = f*((O, I ) ( ~ , ~and
) ) their negatives. If we had orientations /l, for p E S,
then we could simply choose wp if [up, wp] = pp and -wp otherwise.
T h e projective plane IP2 must be non-oiieniable also, since it contains the
Mobius strip (for any orientable bundle 6 = rr : E + By the restriction 6IB'
to aiiy subset B' c B is also orientable). Non-orientability of IP2 can be seen
in aiiother way, by considering the "antipodal map" A : S2 -t S2 defined by
A(p) = -p. This rnap is just the restriction of a linear map A: R3 + R3
defined by tlie same formula. T h e rnap A * : S2,, -t s ~ is
~just
( (~
p ,)~ H
)
A(v)), wlien S2,, is identified with a subspace of { p ) x R3. T h e map A
is orientation reversing, so if vi = (p, ui) E SZP,the bases
(ui, u2, p)
and
(A(ui), A(u2), A(P))
are oppositely oriented. Tliis shows that if ,u is the standard orientation of S2
and [ul, vi] E ,up, then [A,v1, A,v2] E -/,LA(,,).
Thus the map A : S2+ s2is
"orientation reversing" (the notion of an orientation preserving or orientation
reversing map f : M + N makes sense for any imbedding f of one oriented
manifold into another oriented manifold of the same dimension). From this fact
it follows easily that IP2 is not orientable: If IP2 had an orientation v = {v[,l)
and g : s2 -t IP2 is the map p H [ p ] , then we could define an orientation
{ji,) on S" by requiring g to be orientation preserving; the map A would then
be orientation preserving with respect to fi, which is impossible, since k. = p
or -p.
For projective 3-space IP3 the situation is just tlle opposite. In this case, the
antipodal map A : S' + s3ir orientation presening. If g : S 3 + IP3 is the
map p H [P],we obviously can define orientations v, for IP3 by requiring g
to be orientation presewing. In general, these same arguments show that IP" is
orientable for n odd and non-orientable for n even.
There is a more "elementary" definition of orientability which does not use
the tangent bundle of M at all. According to this definition, M is orientable if
there is a subset A' of the atlas A for M such that
(1) tlle domains of al1 (x,U) E A' cover M,
(2) for al1 (x, U) and (y, V ) E A',
An orientation p of TM allows us to distinguish the subset A' as the collection
of al1 (x, U) for which x, : TM lU + T(x(U)) i x(U) x IRn is orientation
preserving (when x ( U ) x IR" is given the standard orientation). Condition (2)
liolds, because it is just the condition that (y o x-'),: T(x(U)) + T(x(U))
is orientation preserving. Conversely, given A' we can orient the fibres of
TM IU in such a way that x, is orientation preserving, and obtain an orientation
of TM. Although our original definition is easier to picture geometrically, the
determinant condition will be very important later on.
The 7angeni Bundle
ADDENDUM
EQUIVALEMCE OF T A N G E N T B U N D L E S
The fact that al1 reasonable candidates for the tangent bundle of M turn out
to be essentially the same is stated precisely as foilows.
4. THEOREM*. If we have a bundle T'M over M for ea& M, and a bundle
map (h,f ) for each Cm map f : M + N satisfying
(1) of Theorem 1,
(2) of Theorem 1, for certain equivalences t'",
(3) of Theorem 1, for certain equivalences T'U h: (TfM)IU,
then there are equivalences
e M : TM + T'M
such that the following diagram commutes for every Cm map f : M + N.
PROOF. The details of this proof are so horrible that you should probably skip
it (and you should definitely quit when you get bogged down); the welcome
symbol 4 4 occurs quite a ways on. Nevertheless, the idea behind the proof is
simple eilough. If (x, U) is a chart on M, tllen both (TM)I U and (T'M) 1 U
"look like" x ( U ) x Rn,so there ought to be a map taking the fibres of one to
the fibres of the other. Wliat we have to hope is that our conditions on TM and
T'M make them "look alike" in a sufficiently strong way for this idea to really
work out. Those who have been through this sort of rigamarole before know
(i-e.,have faith) that it's going to work out; those for whom this sort of proof is
a new expei-ienccshould find it painful and instructive,
* Functorites will iiotice that Theorems 1 and 4 say that there is, up to natural equivaleiice, a uiiique functor from the category of Cm manifolds and Cm maps to the
category of bundles and bundle maps which is naturally equi\~alentto (E", old f,) on
Euclidean spaces, and to the restriction of the functor on open submanifolds.
Let ( x , U ) be a coordinate system on M . Then we have the following string
of equivalences. Two of them, which are denoted by the same symbol 2 ,are
the equivalences mentioned in condition (3). Let a, denote the composition
a, = ( f n l x ( u )0) 0 x* 0 (-)-l.
Similarly, using equivalence 2' for T I , we can define p,.
Then
Bx-' o ax : (TM)IU -t (T'M)IU
is an equivalence, so it takes tlie fibre of T M over p isomorphically to the fibre
of T I M over p for ea& p E U . Our main task is to show that this isomorphism
between the fibres over p is independent of the coordinate system ( x , U ) . This
will be done in three stages.
(1) Suppose V c U is open and y = xlV. We will need to name all the inclusion
maps
To compare a, and ay,consider the following diagram.
TIze 7angeni Bundle
91
O,
Each of the four squares in this diagam commutes. To see this for square
we enlarge it, as sllown below. The two triangles on the left commute by condition (3) for TM, and the one on the right commutes because i o j = r .
Square @ commutes because k o y = x o j. Square @ commutes for the
same reason as square
- @; tlie inclusions x ( U ) + Rn and y(V) + IRn come
into play. Square (4) ob~iouslycommutes. Chasing through diagram (1) now
shows tllat the follo-ving commutes.
This means that for p E V, the isomorphism ay bemeen the fibres over p
is the same as a,. Clearly the same is true for B, and By, since our proof
used only properties (l), (21, and (3), not the explicit construction of TM. Thus
By-' o (Y,, = B x - l O ax on the fibres over p, for every p E V.
(11) We now need a Lcmma which applics to both TM and TIM. Again, it will
be proved for TM (where it is actually obvious), using only properties (l), (2),
and (3), so that it is also true for T'M.
LEMMA. If A c IRn and B c IRm are open, and f : A + B is Cm, then the
following diagram commutes.
PROOF. Case 1. %e
ir a map /:Wn + Rm with 7 = f on A . Consider the
following diagram, where i : A + IRn and j : B + IRm are the incIusion maps.
Everythirlg in tliis diagram obviously commutes. This implies tliat the two
compositions
-
tnlA
C
old /;
T A 2( T R")I A d&"(Rn)IA dE" (IRn)
.em(Rm)
Ev
and
are equal and this proves the Lemma in Case 1, since the maps "old
"old f," are equal on A.
T*'' and
&e 2. General' cae. For each p E A , we want to show that m o maps are the
same on the fibre over p. NOWthere is a map j : IRn + IRm with j' = f on an
open set A', wl~erep E A' C A. We tben have tlie following diagram, wliere
every i comes from the fact tliat some set is an open submanifold of another,
The Gngeni Bundle
and i: A' + A is the inciusion map.
0,
Boxes @,
and @ obviously commute, and
commutes by &e l . To
see that square @ (whidi has a triangle witliin it) commutes, we imbed it in
a larger diagram, in which j: A + R" is the incIusion map, and other maps
have also been named, for ease of reference.
T Rn
(u)
h
/
TA
E
(TRn)IA
i*]
E
+(TP)IAr
TA1
To prove that h o i, = p o K, it suffices to prove that
since v is one-one. Thus it suffices to prove j* o i* = v o p o o, which amounts
to proving commutativi ty of the foilowing diagram.
Since j o i is just the inclusion of A' in Rn, this does commute.
Commutativity of diagram (2) shows that the composition
coincides, on the subsel (TA)IA', with the composition
and on A' we can replace "old /," by "old f,". In other words, the two compositions are equal in a neighborhood of any p E A, and are thus equal, which
proves the h m m a .
(111) Now suppose ( x , U) and (y, V) are any two coordinate systems with p E
U n V. To prove that By-' 0 a, and Bx-' o a, induce the same isomorphism
un the fibre of TM at p , we can assume without loss of generality that U = V,
because part (1) applies to x and xl U n V, as well as to y and y1 U n V.
Assuming U = V, we have the following diagram.
ry
a.-
(3) (TM)lU 4
TU
old ( y o x-'),
(y 0 h.-1)*
O-R")ly(U)
Tlle triangle obviously commutes, and tlle rectangle commutes by part (11).
Diagram (3) thus shows tbat
Exactly the same result llolds for T':
The desired result B,-' o a, = B,-' o a, follows immediately.
Now that we have a well-defined bundle map TM + T'M (the union of al1
Bx-' 0 a,), it is dearly an equilraience e M . The proof that eni o f, = ffi o e M is
left as a masochisiic exercise for the reader. *:*
n l e Zngeni Bundle
PROBLEMS
1. Let M be any set, and { ( x i ,Ui)) a sequence ofone-one functions xi : Ui + Rn
with Ui c M and x (Ui) open in Rn, su& that each
is continuous. It would seem that M ought to have a metric which makes
each Ui open and each xi a homeomorphism. Actually, this is not quite true:
(a) Let M = R U {s), where s 6 R. Let U] = R and x l : Ul + R be the
identity, and let U2 = R - {O) U {s), with x2: U2 + R defined by
Show that there is no metric on M of the required sort, by showing that every
neighborhood of O would have to intersect every neighborhood of s. Nevertheless, we can find on M a pseudometric p (a function p : M x M + R with
a11 properties for a metric except that p (p, q) may be O for p S; q) such that p
is a metric on each Ui and each x j is a homeomorphism:
(b) If A C IRn is open, then there is a sequence A l , A2, A j, .. . of open subsets
of A such that every open subset of A is a union of certain Ai's.
(c) There is a sequence of continuous functions 1;.: A + [O, l], with support J
c A, which "separates points and closed sets": if C is closed and p E A - C,
then there is some j;: with fi (p) 6 (A nC). Hint: First arrange in a sequence
all pairs (Ai, Aj) of part (b) with
C Al.
(d) Let Alj, j = 1,2,3,. .. be such a sequence for each open set xi (Ui)- Define
gi,j : M + [o, 11 by
Arrarlge a11 gil] in a single sequence Gl, G2, G3,. .. , let d be a bounded metric
on R, and define p on M by
Show that p is the required pseudometric.
(e) Suppose that for every p,q E M there is a Ui and Uj wit.h p E Ui and
q E Uj and open sets Bi c xi(Ui) and Bj c xj(Uj) SO that p E xi-'(Bi),
q E xj-'(l?j), and xi-'(Bi) n xj-'(Bj) = 0. Sliow that p is actually a metric
on M.
Tle Tangeni Bundle
97
9. (a) Show that the correspondence between T M and equivalence classes of
curves under which [ x ,u], corresponds to the 7 equivalence clau of x d l o y,
for y a curve in R" with y'(0) = u, makes f, correspond to $#.
(b) Show that under the correspondence [ x , a ] ,n Ciaia/axilp,the map f,
can be defined by
Ef*re)l(g)= [ ( g f ) .
10. If V is a finite dimensional vector space over R, define a Cm structure on V
and a homeomorphism from V x V to T V which is independent of choice of
bases. As in the case of R", for u, w E V we will denote by u, E Vw the vector
corresponding to ( w , u).
1 1. If g ; R + R is Cm show that
for some Cm function h : R + R.
12, (a) Let F
, be the set of al Cm functions f : M + R with f ( p ) = 0, and
1et
F, + R be a linear operator with f g ) = O for al1 f,g E F,. Show
that l has a unique extension to a derivation.
(b) Let W be the vector subspace of 3, generated by al1 products fg for f , g E
F,. Show that the \rector space of al1 derivations at p is isomorphic to the dual
space (.7;p / W )*.
(c) Since (F,/ W)* has dimension n = dimension of M, the same must be true
of F,/ W. If x is a coordinate system with x ( p ) = O, show that x1 + W , . . .,
x" + W is a basis for F,/ W (use Lemma 2). The situation is quite different for
C1 functions, as the next problem shows.
e:
e(
13, (a) Let V be the \lector space ofall C1 functions f : R + R with f ( 0 ) = 0,
and let W be the subspace generated by ail products. Show that lim f ( x ) / x 2
x-o
exists for al1 f E W.
(b) Foro < E < 1, jet
Show that al1 $, are in V, and that they rcpresent linearly independent elements
of V / W.
(c) Conclude that ( V / W)* has dimension cC = 2'.
14, If f ; M + N aild f, is the O map on each fibre, then f is constant on
each component of M .
15, (a) A map f : M + N is an immersion if and only if f* is one-one on
each fibre of T M . More generally, the rank of $ at p E M is the rank of the
linear transformation f , : Mp + N f ( p l .
@) If f o g = f , where g is a difkomorphism, then the rank of f o g at a
equals the rank of f at g ( a ) . (Compare with Problem 2-33(d).)
16. (a) If M is a manifold-with-boundary, the tangent bundle T M is defined
exactly as for M ; elements of M p are y equivalente classes of pain ( x ,u).
Although x takes a neighborhood of p E aM onto W n , rather than R", the
ueciors v still run through IR", so Mp still has tangeilt vectors "pointing in ail
directions". If p E aM and x : U + W" is a coordinate system around p, then
x,-' (Rn-',(,)) c M, is a subspace. Show that tliis subspace does not depend
on the choice of x;in fact, it is i,(aM),,, where i : aM + M is the inclusion.
(b) Let a E IRn-' x { O ) c W n . A tangent vector in W", is said to point "inward" if, under the identification of T W n with &"(Mn), the vector is ( a , u) where
un > O. A vector v E Mp which is not in i,(aM)p is said to point "inward" if
x*( u ) E Wnqp1 points inward. Show that this definition does not depend on
the coordinate system x.
(c) Show tllat if M has an orientation p, then aM has a unique orientation
a p such that [ v i ,. .- ,
= (ap),, if and only if [ w ,i,vi,. .. ,i , ~ , - ~ ]= pp for
every outward pointing w E M p .
(d) If p is the usual orientation of W",show that a,u is (- l ) n times the usual
orientation of IRn-' = aWn. (The reason for this choice will become clear in
Chapter 8.)
(e) Suppose we are in the setup of Problem 2-14, Define g : 8M x [O, 1 ) +
aN x [O, 1 ) by g ( p , t) = ( $ ( p ) , 1 ) . Show that T P is obtained from T M U T N
Tlze Z q m i Bundle
(f) If M and N have orientations p and u and f : ( a M , a p ) + (aN,au) is
orientation-reuer~rg,show that P has an orientation which agrees with p and u
on M c P and N c P.
(g) Suppose M is s2 with two holes cut out, and N is [O, 11 x S ' . Let f be
a dfiomorphism from M to N which is orientation preserving on one copy
of S' aiid orientation reversing on the other. Wliat is the resulting manifold P?
17. Sliow that T P is~ homeomorphic to the space obtained from T ( s ~ ,by
~)
identifying ( p , u) E (s2,
i), with ( - p , - u ) E (s2,
i)-,.
18. Although there is no everywhere non-zero \rector field on s2,there is one
on s2- { ( O , 0, l ) ) , which is dfiomorphic to IR2. Show that such a vector field
can be picked so that near ( O , 0 , l ) the vector field looks like the following picture
(a "magnetic dipole"):
19. Suppose we have a "multiplication" map ( a ,b) H a - b from IR" x R" to R"
that inakes R" into a (non-associative)division algebra. That is,
(al + a2) b = al b + a2 b
a (bi +b2) = a .bl + a .b2
h(a - b ) = (ha) b = a (hb) for h E R
a (l,O,. ,.,O) = a
and there are no zero divisors:
a,b#O
ab#O.
(For example, for 12 = 1, we can use ordinary multiplication, and for n = 2
we can use "complex multiplication", (a, b) (c, d) = (ac - bd, a d + bc).) Let
e l , . ,e, be the standard basis of Rn,
..
(a) Every point in S"-' is a - el for a unique a E Rn.
@) If a # O, then a el, . . ,, a en are linearl~independent.
(c) If p = a el E S"-', then the projection of a - ez,. . , a . e, on (Sn-',i),
are linearly independent.
(d) Mu~tiplicationby a is continuous,
(e) TSn-' is trivial.
(f) TP"-' is trivial.
The tangent bundles TS3 and TS' are both trivial. Multiplications with
the required properties on R4 and R8 are provided by the "quaternions" and
"Cayley numbers", respectively; the quaternions are not commutative and the
Cayley numbers are not even associative, It is a classical theorem that the
reals, complexes, and quaternions are the only associative examples. For a
simple proof, see R. S, Palais, TLv Classi;cation oJReal DiuisiDn Akebrdi, Amer,
Math. Monthly 75 (1968), 366-368. J. E Adams has proved, using methods of
algebraic topology, that n = 1, 2, 4, or 8.
pncidentally, non-existence of zero divisors immediately implies that for a # O
there is some b with a b = (1,0,. . ,O) and b' with b'a = (1,0,. . ,O). If the
multiplication is associative it follows easily that b = b', so that we always have
multiplicative inverses. Conversely, tliis condition implies that there are no zero
divisors if the multiplication is associative; otherwise it suffices to assume the
existence of a unique b with a b = b a = (1, O,, . ,,O).]
.
+
.
.
20. (a) Consider the space obtained from [O, 11 x Rfl by identifying (0, u) with
(1, Tu), where T : R" + Rn is a vector space isomorphism. Show that this can
be made into the total space of a vector bundle over S' (a generalized Mobius
strip),
@) S11ow that the resulting bundle is orientable if and only if T is orientation
preserving.
21. Show that for p E s 2 , the \rector pp E R3, is not in i.(s2,) by showing that
the inner product (p, c'(0)) = O for al1 curves c with c(0) = p and Ic(l)l = 1
for al1 t. (Recall that
wliere denotes the traiispose; see C a h b on Man@Idr, pg. 23-1
22, Let M be a Cm manifold, Suppose that (TM)I A is trivial whenever A c M
is homeomorphic to S'. Show that M is orientable. Hinl: An arc c from
The h g m t Bundle
101
po E M to p E M is contained in some cuch A so (TM)lc is trivial. Thus one
can ('transportJithe orientation of Mpo to Mp. It must be checked that this is
independent of the choice of c. First consider pairs c, c' which meet only at p0
and p, T h e general, possibly quite messy, case can be treated by breaking up c
into s m d pieces contained in coordinate neighborhoods,
Remark: Using results from the Addendum to Chapter 9, together with Problem 29, we can conclude that a neighborhood of some S' c M is non-orientable
if M is non-orientable,
The next two problems deal with important constructions associated with
vector bundles.
-
23. (a) Suppose 6
x : E + X is a bundle and f : Y + X is a continuous map. Let E' c Y x E be the set of al1 ( y , e ) with f ( y ) = x ( e ) , define
n': E' + Y by x r ( y , e )= y, and define f:E' + E by j ( y , e ) = e. A vector
space structure can be defined on
by using the vector space structure on x-'( f ( y ) ) . Show that x': E' + Y is a
bundle, aiid ( j ,f ) a bundle map which is an isomorphism on each fibre. This
bundle is denoted by f *(C), and is called the bundle induced (from 6 ) by f .
(b) Suppose we have anotlier bundle 6" = n": E" + Y and a bundle map
-
(Y,f ) from 6" to 6 wliich is an isomorphism
- on each fibre. Show that 6"
' =f *
2
) Hid: Map e E E" to (+"'e), ?(e)) E E'.
(c) If g : + y, then ( f og)*(6) 8 * ( f * ( 6 ) ) .
(d) If A C X and i : A + X is the inclusion map, then i*(6) 2 6 1 A.
(e) If 6 is orientable, then f
is also orientable.
(f) Give an example where 6 is non-orientable, but f *(6) is orientable.
(g) Let = n : E + B be a vector bundle, Since n : E + 3 is a continuous
map from a space to the base space B of 6 , the symbol n * ( ( ) makes sense,
SIiow that if 6 is not orientable, then x*(6)is not orientable.
*(e)
e
24, (a) Given an n-plane bundle 6 = x : E + B and an m-plane bundle 7 =
rr': E' + 3, 1et E" C E x E' be the set of al1 pairs (e,ef)with x ( e ) = x1(e').
Let x"(e,el) = x ( e ) = nl(e'). Show that n": E" + B is an (n m)-plane
+
I~undle.It js called the Whitncy sum 6 $ 7 of 6 and 7; the fibre of 6 $ 7 over p
is the direct sum n-' ( p )$ n/-' ( p ) .
(b) If f : Y + 3 , show that f 4 ( 6 $ 7 ) 2r S*(.$)$ f 4 ( 7 ) .
(c) Given bundles ti = ni: Ei + Bi, define X : El x E2 + B1 x B2 by
rr(el,e2)= (rr1(ei),rr2(e2)).Show that this is a bundle x over Bl x B2.
(d) If A: B + B x B is the "diagonal map", A(x) = (x,x), show that
EZ
A*(6 ZJ).
(e) If 6 and q are orientable, show that $ z~ is orientable.
(f) If 6 is orientable, and ZJ is non-orientable, show that $ ZJ is also nonorientable.
(g) Define a "natural" orientation on V $ V for any vector space V, and use
this to show that 6 $ 6 is always orientable.
(11) If X is a "figure eight" (c.L Problem 5), find two non-orientable 1-plane
bundles 6 and q over X such that 6 $ q js also non-orientable.
c2
e$z~
e
25, (a) If rr r E + M is a Cm vector bundle, then rr, has maximal rank at
ixch point, and each fibre n-' (p) is a Cm submanifold of E.
(b) T h e O-section of E is a submanifold, carried dfiomorphically onto B by rr.
26. (a) If M and N are Cm manifolds, and l i [or
~ rrN] : M x N + M [or N]
is the projection 017 M [or N], then T ( M x N) h. XM*(TM)$ rrhr*(TN).
(b) If M and N are orientable, then M x N is orientable.
(c) If M x N is orientable, then both M and N are orientable,
27. Show that the Jacobian matrix of y, o (x.)-'
is of the form
Tliis shows that the manifold TM is alz~rqsork~ttabk,i,e., the bundle T(TM) is
orientable. (Here is a more coilceptual formulation: for v E TM, the orientation
for (TM), can be defined as
thc form of y, o (x,)-' shows that this 01-ientation is iildependeni of the choice
of A-.) A differeilt proof that TM is orieiltable is $ven in Problem 29.
28, (a) Let (x, U ) be a coordinate system on M with x(p) = O and let v E Mp
he Cy=la' a/axi . Consider the curve c in TM defined by
lp
Tlze 7ageni Bundle
Show that
(b) Find a curve whose tangent vector at O is a/a(xi o rr)I,.
29. This problem requires some familiarity with the
- notion of exact sequences
f
2
(c.f. Chapter 11). A sequence ofbundle maps El -+ E2 -+ E3With $ = g =
identity of B is exact if at each fibre it is exact as a sequence of vector space
maps.
(a) If ,$ = rr : E + B is a CbO vector bundle, show that there is an exact
sequence
O + rr*(.$) + TE + rr*(TB) + O.
f i t : (1) An element of the total space of ~ " ( 6 is
) a pair of points in the same
fibre, which determines a tangent vector of the fibre. (2) Map X E (TE)e to
(e, GX).
(b) If O + El + Ez + E3 + O is exact, then each bundle Ei is orientable if
the other two are.
(c) T(TM) is always orientable,
(d) If rr r E + M is not orientable, then the manifold E is not orientable. (This
is why the proof that the Mobius strip is a non-oriei-itable manifold is so similar
to the proof that the Mobius bundle over S' is not orientable.)
T h e ilext two Problems contain more information about the groups introduced in Problem 2-33. 111addition to being used in Problem 32, this information wiH al1 be important in Chapter 10,
.
30. (a) Let po E S"-' be the point (0,. .,O, 1). For n 3 2 define / : SO(n) +
S"-' by / ( A ) = A(po). Show that / is continuous and open. Show that
(po) is homeomorphic to SO(n - I), and then show that j-'(p) is homeomorphic to SO(n - 1) for ali p E S"-'.
(b) SO(1) is a point, so it is connected. Using part (a), and induction on n,
prove that SO(n) is connected for ali n > 1 .
(c) Show that O(tz) has exactly two components.
/-'
31, (a) If T : IRn + IR" is a linear transformation, T*: IRn + Rn, the adjoint
of T, is defined by (T*v, w) = (u, Tw) (for each u, the map w H (u, Tw) is
linear, so it is w H (T*v, w) for a unique T*v). If A is the matrix of T with
respect to the usual basis, show that the matrh of T* is the transpose A'.
(b) A linear transformation T : R" + Rn is self-adjoint if T = T*, so that
(Tu, w ) = (u, Tw) for a l u, w E R". If A is the matrix of T d t h respect to the
standard basis, then T is self-adjoint if and only if A is symmetric, A~ = A. It
is a standard theorem that a symmetric A can be written as CDC-1 for some
diagonal matiix D (for an analy& proof, see CaEculw on Man@ldr, pg. 122).
Show that C can be chosen orthogonal, by showing that eigenvectors for distinc t
eigenvalues are orthogonal.
(c) A self-adjoint T (or the corresponding symmetric A) is called positive semid d n i t e if (Tu, u) > O for al1 u E Rn, and positive definite if (Tu, u) > O for al1
v # O. Show that a positive definite A is non-singular. Hznt: Use the Schwarz
inequality.
(d) Show that A' - A is always positive semi-definite.
(e) Sliow that a positive semi-definite A can be written as A = B2 for some B.
(Remember that A is symmetrjc.)
(f) Show tliat every A E GL(n, R) can be written uniquely as A = A l .A2 where
A i E O(n) and A 2 is positive definite. Hht: Consider A' . A, and use part (e).
(g) The matrices A l and A2 are continuous functions of A. Hint: If A(") + A
and A(") = A(")] ~ ( " 1 2 ,then some subsequence of ( A ( " ) ~converges.
)
) GL(i1, R) is homeomorpliic to O(n) x R"("("*')/~
and has exactly two components, ( A : det A > 0) and ( A : det A < O). (Notice that this also gives us
anotller way of fiilding the dimension of O(n).)
32, Two continuous fuilctioils So,5 : X + Y are called homotopic if there is
a coiltjiluous function H : X x [O, i] + Y such that
The functions H; : X + Y defined by Hl (x) = H(x, t ) may be thought of as a
path of fuilctions fi-om Ho = fa to HI = J. The map H is called a homotopy
between fo and J .
The i-iotation f : (X, A) + (Y, B), for A c X and B c Y, means that
f : X + Y and f (A) c B. We cal1 f l ~ ,J : (X, A ) + (Y, 3)homotopic (as maps
from (X, A ) to (Y, B)) if there is an H as above such that each Hr : (X, A ) +
(Y, B)(a) If A : [O, 11 + GL(n, R) is continuous aild H : R" x [O, 11 + R" is defiiled by
H(x, 1 ) = A(r)(x),show that H is continuous, so tllat Ijg and Hl are homotopic
as nlaps from (R" ,R" - (O)) lo (Rn,R" - (O)). Conclude that a non-singular
linear trailsformation T : (R", R" - (0)) + (R", R" - (O)) with det T > O is
homotopic to ihe identity map.
(b) Suppose f : R" + R" is Cm and f(0) = 0, while f (R" - (0)) c R" - (O).
If Df(0) is non-singular, sl-iow that f : (R", Rn - (O)) + (R", R" - (O)) is
homotopic to Df (O): (Rn,Rn - (0)) + (Rn,Rn - (O)). Hint; Define H ( x , I ) =
f(tx) for O < 1 5 1 and H(x, O) = Df (O) (x). To prove continuity at points
(x, O), use Lemma 2.
(c) Let U be a neighborhood of O E R" and f:U -4 Rn a homeomorphism
with f(O) = O. Let B, C V be the open ball with center O and radius r, and let
h : Rn + B, be the homeomorphism
then
f 0 , z : (R",rw" - (O)) + (Rn,Rn
- (O)).
MTe will say that f is orientation preserving at O if f o h is homotopic to the
identity map 1 : (Rn,Rn - (0)) + (Rn,Rn - (O)). Check that this does not
depend on the choice of B, c V.
(d) For p E R", Jet Tp: R" + Rn be T'(q) = p + q . If f :U 4 V is a
l-iomeomorphism, where U, V c Rn are open, we will say that f is orientation
preserving at p if T-J(p) O f O T,, is orientation preserving at O. Show that if M
is orientable, then there is a collection C of charts ~ l h o s edomains cover M such
that for every (x, U) and (y, V) in e, the map y ox-' is orientation presening
at x ( ~ for
) a11 p E U n V.
(e) Noiice that the condition on y o xd' in part (d) makes sense even if y ox-'
is not djffereniiable. Thuus, if M is any (not necessarily dxerentiable) manifold,
we can define M to be orientable if there is a collection C of homeomorphisms
x : U + R" whose domains cover M, such that C satisfies the condition in
part (d), To prove that this definition agrees with the old one we need a fact
from algebraic iopology: If f : Rn + Rn is a homeomorphism with f(O) = O
and T ; Rn S'> R" is ~ ( x ' .,. .,x") = ( x ' , . . . ,xn-l, -xn), then precisely one
of f and T o f is orientation presen~ii-igat O. Assuming this result, show that
if M has such a collection C of l-iomeomorphisms, then for any Cm structure
on M the tailgeiit bundle TM is orientable.
33. Let M" c RN be a Cm n-dimensional submanifold. By a chord of M we
meai-i a point of IRN of the form p - q for p , q E M.
(a) Prove tliat if N > 212 + 1, then there is a vector v E sN-lsuch that
(i) no chord of M is parallel to u,
(ii) no tangent plane M, contains v .
Hint; Coilsider ceriain n-iaps from appropriate open subsets of M x M and
TM to S"-'.
(b) Let IRN-l c IRN be the subspace perpendicular to U, and n : IRN + IRN-l
the correspoilding projection. Show that n l M is a one-one immersion. In
particular, if M is compact, then rrl M is an imbedding.
(c) Every compact Cm rl-dimensional manifold can be imbedded in IR2"+'.
Note: This is the easy case of M'hitney's classical theorem, which gives the
same result even for non-compact manifolds (H. Mfhitney, fizermtzabk man@ldí,
Ann. of Math. 37 (1935), 645-680). Proofs may be found in Auslander and
MacKenzie, Intmduction ¿o Dzffmentiabk A4un@lds and Sternberg, Latures on fiy
fmential Geotne~tIn Munkres, Eln~tentaryDflia~tialZpology, there is a dflerent
sort of argumerlt to prove that a not-necessarily-compact n-manifold M can be
imbedded in some IRN (in fact, With N = (n + I ) ~ )T. h a l we may show that M
imbeds in IR2"+' using essentiaily the argument above, together with the existence of a proper map f : M + IR, given by Problem '2-30 (compare Guillemin
,:lid Pollack, fizmnlial Topology). A much harder result of Whitn y shows that
M" can actually be imbedded in IR2" (H. Wllitney, nie selj-intersectioonroja mooU1
u-t~ranfoldin 2n-space, Ai111. of h4ath. 45 (1 944), 220-246).
CHAPTER 4
TENSORS
A
II the constructions on vector buildles carried out in this chapter have a
common feature. In each case, we replace each fibre x - ' ( p ) by soirie
other vector space, and then fit al1 these new vector spaces together to form a
new vector bundle over the same base space.
Tlie simplest case arises when we replace each fibre V by its dual space V*.
Recal1 that V* deilotes the vector space of al1 linear functions A : V + R. If
f r V + W is a linear transformation, then there is a linear transformation
f * : W* + V* defined by
It is clear that if 1 v : V + V is the ideiitity, tlien 1 v* is the identity map of V*
and if g : U + V, then (f o g)* = g* o f *. These simple remarks already
sliow tliat f * is an isomorphism if f : V + W is, for (f
o f ) * = 1 v* and
(f 0 f-l)* = 1W+.
Tlie dimensjon of V* is tlie same as that of V, fol. finite dimeilsional V. In
E V*, defined by
fact, if vi,. . .,v, is a basis for V, theii the elements
-'
are easily cliecked to be a basis for V*. The linear fuiiction v*i depeilds on tlle
eiltire set vi,. . .,v,, not just on vi alone, aiid tlle isomorphism from V to V*
obtained by sending vi to
is not indepeildeilt of the choice of basis (consider
what happens if vi is replaced by 2vl),
On the othei. hand, if v E V, we caii define u** E V** = (V*)* unambiguously by
If v**(h) = O for every h E V*, then h(v) = O for al1 h E V*, which implies that
v = O. Thus the map v w v** is an isomorphism from V io V*'. It is called
the natural isomorphism from V to V**.
(Problem 6 gives a precise meaning to the word "natural", formulated only
after the term had long been in use. Once the meaning is made precise, we can
prove that there is no natural isomorphism from V to V*.)
Now let 6 = n : E + B be any vector bundle. k t
and define the function x': E' + B to take each [n-'(p)]*to p- If U c B,
lind I : ?r-' ( U )+ U x R" is a trivialization, then we can define a function
I' : nf-'( U )+ U x (Rn)*
in the obvious way: sincc the map r restricted to a fibre,
tP : H-'(~)
+ {p)x R",
is an isomorphism, it gives us an jsomorphism
(g*)-' : [ir-'(p)]*+ {p} x (R")'
We can makc n': E' + B into a vector bundle, t l ~ ed u d bundle '6 of 6,by
requiring that al1 such i r be local trivializations. (We first pick an isomorphism
from (Rn)*to R",once and for all.)
e*
At fii-st jt n-iigllt appear tliat
2 6, since eacli n-'(p) is isomorphic to
x ( p . However, tliis is true merely because t l ~ etwo vector spaces havc tlie
same dime~ision. Tl-ie lack of a natural isomorpl-iism from V to V* prevents
us from constructing an equivalence between '6 and 6. Actually, we will see
later that in "most" cases
Zr equivalei-it to 6; for the present, readers may
ponder tl-iis question for tliemselves. In contrast, the bundle
= (t*)' is
nlu~oysequivalcnt to 6. We consti-uct the cquivalence by mapping the fibre V
of 6 over p to the fibre V** of t'* oler p by the natural isomorphism. If you
e*
e**
e*
tl-iink about how is constructed, it will appear obvious that this map is indeed
a n equivalence.
Even if 6 can be pictured geometrically (e.g., if 6 is TM), there is seldom a
geometric picture for C*. Rather, 6* operates on 6: If s is a section of 6 and o
is a section of t*,then we can define a function from B to R by
P
G(P)(S(P))
s(p)E f i(p)
o ( p ) E x'-l ( p ) =
,-' ( p ) " .
This function wjll be denoted simply by o ( s ) .
When this construction is applied to the tangent bundle TM of M, the resulting bundle, denoted by T*M, is called the cotangent bundle of M ; the fibre
of T*M over p is (M,)". Like TM, the cotangent bundle T * M is actuaiiy a
C m vector bundle: since two trivializatjons X* and y, of TM are Cm-related,
the same is clearly true for x.' and y,' (in fact, y.' o (x.')-' = y, o (x,)-').
1% can thus define Cm, as wcll as continuous, sections of T*M. If w is a C m
section of T*M and X is a C m lector field, then w ( X ) js the C m function
P l+ ~ ( P ) ( X ( P ) ) .
If f : M + R is a Cm function, then a Cm section d f of T*M can be
defined by
d f ( p ) ( X ) = X ( f ) for X E M,.
The sectio~idf is called tlie differentd of f . Suppose, in particular, that X is
dcldt 1 , where c(io) = p. Recall that
This means that
Adopting the elliptical notations
this equation takes the nice form
If (x,U) is a coordinate system, tlien tlie dxi are sections of T * M over U.
Applyiiig the definition, we see tliat
Tlius d x l ( p ) , . . . ,d x n ( p ) is just tlie basis of M', dual to tlie basis a/axl l,
a/axnIP ~f M,.
Tliis means tl~atevery section w can be expressed u~~iquely
on U as
. . .,
for certaiii functio~~s
w ; o11 U. Tlie section w is coiitiiluous or Coaif aiid only if
the functions wi are.
We can also write
if wc define sums of sections and products of functions and scctioiis in the
olwious way ("poii~rwise"additioii ai-id multiplication).
Tlie sectioii <f must liave some sucli expressioii. In fact, we obtain a classical
formula:
1. THEOREM. If (x, U ) is a coordiiiate system and f is a Cm function, then
tlien
Thus
C lassical differential geome ters (and classical aiialysts) did not liesitate to talk
about "infiiijtely small" clianges dxi of the coordinates x i ,just as Leibnitz had.
No one wanted to admit tliat tliis was nonsense, because true results were obtaincd wlien tliesc iiifinitely small quantities wcre divided into each other (provided one did it in tlie right way).
Eventually it was rcalized that the closest one can come to describing an
infinitely sinall cliange is to describe a directiori in wliicli tliis change is supposed
to occur, Le., a tangent vector. Since df is supposed to be the infinitesimal
changc of f uiidei- aii iiifiiiitesimal chaiige of tlic poirit, df must be a function
of this chaiige, whicli means that df should be a function on tangent vectors.
The dxi tliemselves tlien metamorpliosed into fuiictions, and it became clear
tliat tliey must be distinguislied from tlie tangent vectors a/dxi.
Oiice this realization came, it was only a matter of making new definitions,
which presenled tlie o k i notation, and waitiiig for everybody to catch up. In
sliort, al1 classical iiorioiis involviiig iilfinitely small quantitics became functions
on tangent vectors, likc df, eexcept for- quotieiits of infinitely smaü quantities,
which became taiigent vectors, like dc/dr.
Looking back at tlic classical works fi-om our modern vantage point, one can
usually see that, 110 niatter liow obscui*elyexpressed, this point of view was in
some sense the one always taken by classical geometers. In fact, the differential
d f was usually introduced in the following way:
CLASSICAL FORMULATION
MODERN FORMULATION
LRt f be a function of the x', . . .,x",
say f = f ( x l ,. . . ,x").
k t j be a function on M, and x a
coordinate system (so that f = f o x
for some function f on W", namely
j = f 0.-'1.
Let x i be functions of r , say x' =
x i ( l ) . Then f becomes a function
of 1, f ( t ) = f ( x l ( r ).,. . , - Y " ( ¡ ) ) .
Let c : W -. M be a curve. Then
f o c : W + R, where
We now have
We now have
df
dr
" af
-=EGd;.
i=
f O c(r) = j ( x l o c ( I ) ,. . .,xn O e(¡)).
( f O c)I(r)
n
I
=
E D;j ( x ( ~ ( t ) )()x i c ) ( ( I )
m
o
i=I
(Tlle classical notation, which
suppresses tlie curve c , is still used
by physicists, as we shall point out
once again in Chapter 7.)
=
af
E" ,(c(r))
a~
- (xi0 c)I(r)
f =I
or
d ( f ( c ( r ) ) ) " af
dr
=Ea;i(c(0).
i= 1
Multiplying by d r gives
df =
a.f d x i .
E" axl
1= 1
(Tl-iis equation signifies that true
results are obtained by dividing by
dr again, 710 mal& what ~e functions
x i ( t ) are. It is the closest approach
in classical analysis to the realization of d f as a function on tangent
vectors.)
dxi(c(r))
di
Consequently,
i=l
Since every tangent vector at c(r) is
of the form dcldr, we have
df =
af d x i .
E" i= 1
a ~ i
In preparation for our reading of Gauss and Riemann, we will continually
examine tlie classical way of expressilig al1 concepts which we introduce. After
a whiie, the "translation" of classical terminology becomes only a little more
difficult than tlie translation of the German in which it was written.
Recall that if f : M + N is Cm, then there is a map f,: TM + TN;
for each p E M, we have a map f*p : Mp + Nf ( p ) . Since f Z p is a linear
transformation between t ~ 7 ovector spaces, it gives rise to a map
Strict notational propriety would dictate that this map be denoted by (f,,)*,
but everyone denotes it simply by
Notice tliat we cannot put al1 4 together to obtain a bundle map from T*N
to T*-M;in fact, tlle same q E N may be f (pi)for more than one pi E M,
and there is no reason why fWp, should equal f,p,. On the other hand, we can
do sometl~ingwith t l ~ ecotaiigent bundle that we could not do with the tangent
bundle. Suppose w is a section of T*N. Then we can define a section rj of T*M
as follows:
V(P) = w ( f ( p ) ) 0 f*p,
(The complex symbolism tends to hide the simple idea: to operate on a vector,
we push it over to N by f*,and then operate on it by u . ) This section q is
denoted, naturally enough, by $*m. There is no corresponding way of transferring a vector field X on M w e r to a vector field on N.
Despite these differences, we can say, roughly, that a map f : M + N produces a map f* going in tlie same direction on tlle tangent bundle and a map
f * going in the opposite direction on tlle cotangent bundle. Nowadays such
situations are always distinguished by calling the things which go in the same
direction "covariant" and the tliings which go in the opposite direction "contravariant". Classical terminology used these same words, and it just happens
to llave reversed tllis: a vector field is called a contravariant vector fieId, whiie
a section of T*M is called a covariant vector fieId. And no one has had the
gall or autllority to reverse terminology so sanctified by years of usage. So
it's very easy to remember which kii-id of vector field is covariant, and which
contravariant -itJs j ust t l ~ eopposite of wliat it logically ought to be.
The rationale behind the classical terminology can be seen by considering
coordinate systems x on R" which are linear transformations. In this case, if
x(vi) = el, then
x ( a 1 vi
a%,)= (a 1 ,...,a"),
+ . v . +
so h e x coordinate system is just an "oblique Cartesian coordinate system".
If x' is aiiotlier such coordinate system, then x'j = =yzI
Clearly aij = ax"/axi, so
aijxi for certain aij.
ilib can be seeii directly from die fact that h e matrix (ax'J/axi) is the constant
inatrix D(xt o -Y-' ) = xt o x-' . Comparing (*) with
fi-om Tlieorem 1, we see tliat tlie differeiitials dxi "cliange in tlie same way" as
tlic coo~diiiatcsx t , lience tliey are "covai-iant". Coiisequently, any combination
is also called "covariant". Notice that if we also llave
o=
otidx",
tl~en\ve can express the wti in terms of the wt. Substituting
n
dx' =
E axi d x t J
j==l
into tlle first expression for w and comparing coeffrcients with the second, we
find that
1
W j =
t-1
On the otller hand, given two expressjons
for a vector field, tlie functions atj must satisfy
Tllese expressioiis can always be remembered by iioting that indices which are
summed over always appear once "above" and once "below". (Coordinate functions x ' , . . . ,x" used to be denoted by xl, . . . ,A-.. Tliis suggested subscripts wi
for covariaiit vector fields and superscripis o' íor coiiti.avariant vector fields. Mter tllis was fiimly established, the indices oli the x's were shifted upstairs again
to inake tlle summation convention work out.)
Covariar-it and contravariant vector fields, i.e., sections of T*M and TM,
respectively, are also called covariant ar-id coi1travarian t tensors (or tensor fields)
of order 1 , which is a warning that worse things are to come. We begin with
some worse algebra.
If VI, . . . , Vm are vector spaces, a function
is limar for each choice of VI,.. . ,vk-l,vk+lr.. . , u m . The set of aii such T
is clearly a vcctoi- space. If VI ,. . . , Vm = V, this vector space will be denoted
by T m ( V ) . Notice that 7' (V) = V*. If f : V + W is a linear transformation, then there is a linear transformation $" : T m( W) + Tm(V), defined
completely analogously to the case m = 1:
For T E Tk(V), and S E 7' ( v ) we can define the "tensor product" T @SE
T'+'(v) by
Of course, T @ S is not S @ T. On the otl-ier lland, (S@ T ) @ U = S @ ( T @ U),
so we can define n-fold tensor products unambiguously; this tensor product
operation is itself multilinear, (Sl S2) 8 T = SI 8 T S28 T, etc. In
particular, if V I , . . . ,vn is a basis for V and v * ~ ., .. ,v*, is the dual basis for
V* = T'(V), then the elemen ts
+
+
are easily seen to be a basis for T ~ ( V )which
,
ihus has dimension nk.
We can use this new algebraic construction to obtain a new bundle from any
\rector bundle 6 = ?r : E + B. We let
E' =
uTk(?r-l
(p)),
P@
and let
n i : E' + 3
take
T~(s-' (P)) to p.
If U c B and
r:n-I(u)+
U xwn
is a ti+i\4alization7then the isomorphisms
ip : n-I
(p) + {p} x Rn
yield isomorpllisms
(rpL)-' : Tk(n-' (p)) + {p} x 7 " ~ " ) .
If we choose an isomorphism Tk(wn)+ Rnk once and for all, these maps can
bc put together to give a map
\Ve rnake k e r r > E'
:
+ B into a vector bundle T~ (6) by requiring that al1 such 'i
1)e local trivializations. The bundle 6* is the special case k = 1.
For the case of T M , h e bundle T ~ ( T Mis) called the bundle of covariant
tensors of order k, and a section is called a cwariant tensor fidd of order k. If
( x , U) is a coordinate system, so that
is a basis for (Mp)*,tlien tlie k-fold tensor products
are a basis for T k ( M , ) . Thus, on U every covariant tensor feld A of order k
can be written
A(P)=
il
,...,ik
or simply
A =
Ai, ...i , ( p ) dxi(( p ) @ - . @ dxik ( p ) ,
E Ai, dxi1 8 - -
.@ dxfk,
wllere dxil 8 - - @ dx-'k riow denotes a section of T ~ ( T M )If. we also have
then
(tlie pi-oducts are just 01-diiiaryproducts of fuilctions). To denve this equation,
we just use equation (**) on page 114, and multilinearity of B. The section A
is continuous or Coo if and only if the functions Ai,...;, are.
A covariant tensor feld A of order k can just be tliought of as an operation 2
on k vector fields XI,. . . ,Xk which yields a function:
Notice that A is rnultilinear on h e set V of Cm vector fields:
Moreover, because A is defined 'pointwise", it is actually linear ouer h e Cm
fuízdions F; ie., if f is Coo,then
for we have
\Ve are finally ready for anotlier theorem, one that is used over and over.
is liiiear over Y,theii tliel-e is a unique tensor field A with A = A.
PXOOF. Note first that if v E Mp is any tangent \lector, then there is a vector
Geld X E V witll X(p) = v. In fact, if (x, U) is a coordinate system and
then we can define
(0
outside U,
~ ~ l i e eacli
r e af now denotes a constant function and f is a Cm function with
f (p) = 1 and support f c U.
Now if v1, .. . ,vk E Mp are extended to vector fields Xf,. . . ,Xk E V we
clearly must define
The problem is to prove tl-iat tl-iis is well-defined: If Xr(p) = Yi(p) for eacl-i i,
we claim that
d(x1,- - ,Xk)(p) = A(Y1 - ., Yk)(p)*
3
(The map A "lives at points", to use tlie in terminology.) For simplicity, take the
case k = 1 (tlle general case is exactly ailalogous). The proof that A ( X ) ( p ) =
A ( Y ) ( p )when X ( p ) = Y ( p ) is in two steps.
(1) Suppose first that X = Y in a neighborhood U of p. Let f be a Cm
function witli f ( p ) = I and support f c U. Then fX = f Y, so
evaluating at p gives
~ ( X ) ( P=)& ( Y ) ( P ) .
(2) To prove tlle result, it obviously suffrces to show that A ( X ) ( p ) = O if
X ( p ) = O. k t (x, U) be a coordinate system around p , so that on U we
can write
n
m
x = ~~~d
where al1 b ' ( p ) = O.
f-l
axi
If g is 1 i ~ai iieigliboi-liood V of p, and support g c U, tlien
is a well-defined Cm \rector field on al1 of M wliich equals X on V, so that
Now
= O,
since b f ( ~=)0. +3
Because of Tlieoi-em 2, we will ilevei- distinguisli between tlie tensor field A
and tlie ol~csationA,ilor will we use the symbol A any longer. Note that
Tlieorem 2 applies, in particular, to tlie case k = 1, wliere 7 ' ( T M ) = T'M,
tlie cotangent bui-idle: a fiiilctioii fi-om V + 9 wliicll is linear over 9 comes
fi-om a covariaiit vectoi- field u.Just as witli covariant vector fields, a Cm map
f : M + N @ves a map f* taking covariant tensor fields A of order k on N
to covariaiit tei-isor fields f * A of 01-der k on M:
Moreover, if A and B are covariani tensor fields of orders k and I , respectively,
then we can define a new covariant tensor field A @ 3 of order k + I :
( A @ B) (p) = A (p) @ B(p)
(operating on Mp x
. x Mp k + I times).
Although covariant tensor fields will be our main concern, if only for the
sake of completeness we should define coiitravariant tensor fields. Recall that
a contravariant vector field is a seciion X of TM. So each Xp E Mp. Now an
element v of a vector space V can be thought of as a linear function v : V' + R;
we just define v(h) to be h(v). A contravariant tensor field of order k is just
a section A of the bundle T ~ ( T * M ) thus,
;
each A(p) is a k-linear function
on Mp*. We could also use the notation T k ( T M ) ,if we use Tk ( V ) io denote al1
k-linear functioiis on V'. In local coordinates we can write
(remember that each a/axj 1, operates on Mp*), or simply
If we have another such expression,
tl-ien we easiiy compute that
A contravariant tensor field A of order k can be considered as an operator A
iaking k covariant vector fields wl ,. . . ,wk into a function:
Naturally, tliere is aii analogue of Tlieorem 2, proved exactly the same way, that
allows us to dispense with the notation A,and to identify contravariant tensor
fields of order k with operators on k covariant vector fields that are linear over
,!he Coofunciioni F.
Finally, we are ready to introduce "mixed" tensor fields. To make the introduction less painful, we consider a special case first. If V is a vector space, let
q' ( V ) denote al1 bilinear functions
e = a : E + B gives rise to a vector bundle q' ( e ) ,obtained by
( p ) by q' (a-' ( p ) ) . In particular, sections of
(TM)
replacing each fibre
A vector bundle
K-'
are called tensor fields, covariant of order 1 and contravariant of order 1.
There are al1 sorts of algebraic tricks one can play with 7,' ( V ) ;although they
should be kept to a minimum, certain ones are quite important. k t End(V)
denote the vector space of al1 linear transformations T : V + V ("endomorphisms" of V ) , Notice that each S E End( V ) @ves rise to a bilinear S E II;' ( V ) ,
by the formula
Moreover, the correspondence S I+ S from End(V) to q' ( V ) is linear and oneone, for S = O implies that A ( S ( v ) )= O for al1 A, which irnplies that S ( v ) = O,
for al1 v . Since both End(V) and
( V ) have dimension n 2 , this map is an
isomorphisrn. The inverse, however, is not so easy to describe. Given 3, for
each v the vector S ( v ) E V is merely determined by describing the action of a h
on it according to (*). It is not hard to check tl-iat this isomorphism of End(V)
and T 1 ( V )makes the identity map 1: V + V in End(V) correspond to the
cc
evaluation" map
e : V x V * + W in T'''(v)
given by
e ( v ,A) = A(v).
Generally speaking, our isomorpliism can be used to transfer any operation
from End(V) to ?''(V). In particular, given a bilinear
we can take the trace of the corresponding S : V + V ; this number is called
the contraction of T . If v i , . . . ,v , is a basis of V and
then we can find the matrix A = (aij) of S, defined by
in terms of the T,' ; in fact,
contraction of T =
T.:
(The term "contraction" comes from the fact that the number of indices is contracted from 2 to O by setting tl-ie upper and lower indices equal and summing.)
Tl-iese identifications and operations can be carried out, fibre by fibre, in any
fibre bundle T,' (5). Thus, a section A of 7'' (5) can just as well be considered
as a section of tlie bundle End(h), obtained by replacing each fibre a-' (p) by
~ n d ( a - ' ( p ) ) .In tliis case, each A(p) is an endomorphism of a-'@). Moreover, each section A gives rise to a function
(contraction of A) : B + R
defined by
p I+ contraction of A ( p )
= trace A ( p)
if we consider A (p) E ~ n (n-'
d @))
In particular, given a tensor field A, covariant of order 1 and contravariant
(TM), we can consider each A (p) as an
of order 1, wliich is a section of
endomorpllism of M,, and we obtain a function "contraction of A". If in a
coordinate system
tl-ien
(contraction of A ) = y A:.
Tlle general notion of a mixed tensor field is a straightfonvard generalization.
Define 7jk( V) to be the set of al1 (k + !)-linear
k times
l times
Every bundle 5 gives rise to a bundle ~ ' ( 5 ) Sections
.
of q ( T M ) are called
tensor fields, covariant of order k and contravariant of order i, or simply of type
(f), an abbreviation that also saves everybody embarrassment about the use of
the words "covariant" and "coníravarian t". Locally, a tensor field A of type (f )
can be expressed as
and if
then
Classical differentia1 geometry books are filled with monstrosities like this
equation. 111fact, the classical definition of a tensor field is: a n assignment
of nk+* functions to every coordinate system so that (*) holds between the nk+'
functions assigned to any two coordinate systems x and x'. (!) O r even, "a set
functions which changes according to (*)" . Consequently, in classical
of ,rk+'
differential geonletry, al1 important tensors are actually defined by defining the
functions A;, in terms of the coordinate system x, and then checking that (*)
holds.
Here is an important example. In every classical dxerential geometry book,
one will find the following assertion: ' T h e Kronecker delta 6; is a tensor." In
other words, it is asserted that if one cliooses the same n2 functions 6; for each
coordinate system, then (*) holds, i.e.,
this is certainly true, for
From our point of view, what this equation shows is that
A =
C6; dx'
i, j
a
@axj
is a certain tensor field, independent of the choice of the coordinate system x
To identify the mysterious map
we consider v E Mp and h E Mp* with the expressions
then
Thus A ( p ) is just tl-ie evaluation map M p x Mp* + R; considered as an ei-idomorphism of M,, it is just the identity map.
Tl-ie conti.actjon of a tensor is defined, classically, in a similar manner. Given
a tensos, Le., a collection of functions A!, one for each coordinate system, satisfying
we note that
n
axi
=EA/E-axiU a x j
i,j
a-l
so tliat tliis sum is a well-defined function. Tliis calculation tends to obscure
the one part which is really necessary-verification of the fact that the trace of
a linear transformation, defined as the sum of the diagonal entries of its matrix,
is indcpendent of the basis with respect to which the matrix is written.
Incideiitally, a teiisor of type (f) can be contracted with respect to any pair
of upper and lower indices. For example, the functions
a-l
"transform correctly" if the AaPY do. If we consider each A ( p ) E T..
( M ~ ) then
,
we are taking B(p) E q Z ( M p )to be
~ ( p ) ( vvz,~ Al,
, Az) = contraction of: (v, A) H A(p)(v, vi, vz7A l 7 hz, A).
Wlliie a corltravai-iant vector feld is classically a set of n functions which
"transforms in a certain way", a vector at a single point p is classically just an
assigiiment of n tiumbns u ' , . . . , a n to each coordinate system x , such that the
numbers a", . . . ,a'" assigned to x' satisfy
Tllis is preciseb the defi~iitionwe adopted when we defined tangent vectors as
equivalente classcs [A-,a],. T h e revolutioil iil tlle modern approach is tliat the
set of al1 vectors is made into a bundle, so tl~at\lector fields can be defined as
sections, ratller tlian as equivalente classes of sets offunctions, and that al1 other
types of tensors are constructed from this bundle. The tangent bundle itselfwas
almost a victim of the excesses of revolutionary zeal. For a long time, the party
line held tliat TM must be defined either as derivations, or as equivalence classes
of curves; the return to the old definition was influenced by the "functorial"
point of view of Theorems 3-1 and 3-4.
The modern revolt against the classical point of view has been so complete
in certain quarters that some matliematicians will give a three page proof that
avoids coordinates in preference to a three line proof that uses them. We won't
go quite that far, but we will give an "invariant" definition (one that does not
use a coordinate system) of any tensors that are defined. Unlike the "Kronecker
delta" and contractions, such invariant definitions are usually not so easy to
come by. As we shall see, invariant definitions of al1 the important tensors in
differential geometry are made by means of Theorem 2. We seldom define A ( p )
directly; instead we define a function A on vector fields, which miraculously
turns out to be linear over the Coo functions Y,and hence must come from
some A . At t l ~ eappropriate time we will discuss wliether or not this is al1 a big
cheat.
PROBLEMS
l. LRt f : M" + Nm, and suppose that (x, U ) and (y, V ) are coordinate
systems around p and f (p), respectively.
(a) If g : N + R, then
(Proposition 2-3 is tlie special case f
(b) Show that
and, more generally, express j;(C;,,
(c) Show that
= identity.)
aia/axi 1P ) in terms of the a/ayjIp.
(d) Express
in terms of the dx'.
2. If f , g: M + N are Cm, sl~owtllat
3. LRt f : M + R be Cm. For v E Mp, show tllat
4. (a) Sllow that if the ordered bases vi, . . . , v, aiid wl ,. .. ,W , for V are equaiiy
oriented, tllen tlle same is true ofthe bases v*I, . . . , v*, and w * ~. ,. . ,u*,for V * .
(b) Show that a bundle 6 is orientable if and only if is orientable.
e*
5. The following statements and problems are al1 taken from Eisenhart's classical work Riemalznian Geometgt. In each case, check them, using the classicai
methods, and then translate the problem and solution into modern terms. An
"invariant" is just a (well-defined) function. Remember that the summation
convention is always used, so h i p i means
h i p i . Hints and answers are
given at the end, after (xiii).
x7=,
(i) If the quantity h i p i is an invariant and either hi or pi are the components
of an arbitrary [covariant or coiitravariant] vector field, tlie otlier sets are components of a vector field.
(ii) If haIiare the components of n vector fields [in an n-manifold] , where i
for i = 1 , . . . , I Z indicates the component and a for a = 1 , . . .,n the vector, and
tbese vectors are independent, that is, det(hUii)# O, then any vector-field h' is
expressible in the form
h' = ~ ~ h , ~ ' ,
where the a's are invariants.
(iii) If p; are the componeiits of a given vector-field, any vector-field hi satisfyiiig h i p i = O is expressible linearly in terms of ,z - 1 independent vector fields
h,lr for a = 1 , . . . ,n - 1 wliicli satisfy the equation.
( i ) If uij = aji
for the components of a tensor field in one coordinate system,
..
then U"J = arJ' for the coordinates in any other coordinate system.
(1) If uij and bi; are compoiieii ts of a teiisor field, so are aij
b *. ~If aiJ aiid
bki are components of a tensor field, so are aijbki.
(vi) If uohihJ is an invariant for 1' an arbitiary vector, then uij aji are the
componeiits of a teiisor; in particular, if uUhihj = O, then aij aji = O.
(vii) If uijhihJ = O for al1 vectors hi sucli tliat hipi = O, wliere pi is a given
covariant vector, ir vi is defined [c.f. (iii)] by a i j h U i i ~=j 0, a = 1,. . . ,n - 1 and
pivi # 0, and by definition
+
+
+
1
tlieii ( ~ ' j 7 p i o 1 ) e i e j = O is satisfied by every vector field Pi, and consequently
(viii) If a,, are tlie componeiits of a tensor aiid b aiid c are invariants, sl~ow
tliat if bu,, ea,, = O, tlien either b = -c and u,, is symmetric, or b = c and
u,, is skew-symmetric.
(ix) By delinition tlie rnlzk of a tensor of the second order a;j is tlie rank of
the matrix ( u t j ) . Show tliat the rank is invariant under al1 transformations of
coordinates.
+
(x) Show tliat the rank of the tensor of components aibj, where ai and bj
are the components of two vectors, is one; show that for the syrnmetric tensor
aibj + ajbi the rank is two.
(xi) Show that the tensor equation aijhi = ahj, where a is an invariant, can
be written in the form (aij - aGij)hi = O. Show also that aij = Gija, if the
equation is to hold for an arbitrary vector hi.
(xii) If aijh< = a h j holds for al1 vectors hi S U C ~that pihi = O, where ,ui is a
@ven vector, then
a ij = 0161 + oi.pi.
.
(O
.
~JI...JI>
il ...i,
-
1
(-1
if j, = j~for some a #
or i, = ip for some a # fi
or if { j l , .. . ,jp} # { i l , .. . ,ip}
if j l , . . . ,jp is an even permutation of i l , . . . ,ip
if jl ,. . . ,jp is an odd permutation of ii, . . . ,ip
then 6tA.'...!'I , are tlie components of a tensor in al1 coordinate systems.
HINTS AND ANSWERS.
(i) w is determined if w (X) is known for al1 X, and uice versa.
(iii) Given w (with w(p) # O for al1 p], there are everywhere linearly independent vector fields X 1 , .. . ,X,-1 which span ker w at each point. (This is true
only locally. For example, on
x IR ihere is an w such that ker w(p, í) consists
of vectors tangent to S2x {í}.)
(vi) For T : V x V + IR, let Tt(v, w) = T(w, u). Tlien T + T' is determined by
S(v) = T(v, u). For, T(u + w, u w) = T(v, u) T(u, w) T(w, v) T(w, w).
Similarly, T(v, u) = O for al1 v implies that T + T' = 0.
(vii) Give~iw (with w ( p) # O for al1 p] , clioose Y complementary to ker w at
al1 points. If o ( 2 ) = T(Y, Z ) , then T (2,2 ) = w (Z)o(Z)/w (Y) for all vector
fields 2.
(ix) T : V x V + W corresponds to T : V + V* [where T(v)(w) = T (v, w)].
The rank of T may be defined as the rank of T (coiisider the matrix of T with
respect to bases U , , . . . , u n and v * i , . . . ,u",).
(xii) Let V = Mp*. If T : V + V and p E V* and T(v) = a v for aii u E kerp,
there is a y complementary to ker ,u such that
+
+
+
+
(Begin by clloosing yo complementary to ker ,u and writing v uniquely as vo+cyo
for u0 E ker p.)
(xiii) Define
--
S: v x . . . x v x v * x - - ~ x v * + R
p times
p times
6. (a) Let iv : V + V ** be the "natural isomorphism" i v (v)(l) = l (v). Show
that for aily linear traiisformation f : V + W, tlie following diagram commutes:
v iv ) y * *
h) Show tliat there do not exist isomorphisms i v : V + V* such that the fol-
lowing diagram always commutcs.
v-v* 2 v
Hint: There docs not even exist an isomorphism i : R + R* which makes tlie
diagram commute for al1 linear f : R + R.
7. A covariantfuílctol- from (finite dimensional) vector spaces to vector spaces is a
function F wliich assigns to every vector spacc V a vector space F( V) and to evcry linear transformation f : V + W a limar uansformation F(f ) : F(V) +
F(W), sucli that F ( l v ) = 1 ~ ( vand
) F ( g 0 f ) = F ( g ) o F(v).
(a) Tlie "identity functor", F ( V ) = V, F(f ) = f is a functor.
(b) The "double dual functor", F( V) = V **, F( f ) = f ** is a func tor.
(e) The "Tk fuiictor", F ( V ) = &(V) = T k ( v * ) ,
is a functor.
(d) If F is any fuiictor and f : V + W is an isomorphisrn, tlien F(f ) is an
isomorphism.
A contrava?iatiifutictur is defined similarly, except tliat F ( f ) : F(W) + F(V) and
F ( g o f ) = F ( f ) o F(g). Functors of moi-e tlian one argument, covariant in
some and contravariant in otliers, may also be defined.
(e) The "dual functor", F ( V ) = V*, F(f ) = f * is a contravariant functor.
(f) The "T' fuiictor", F(V) = T~(V), F ( f ) = f * is a contravariant functor.
8. (a) k t Hom( V, W) denote al1 linear transformations from V to W. Choosing a basis for V and W, we can identify Hom(V, W) with the m x n matrices,
and consequently give it the metric of Rnm. Show that a dfierent choice of
bases leads to a homeomorphic metric on Hom(V, W).
(b) A functor F @ves a map from Hom(V, W) to Hom(F(V), F(W)). Cal1 F
mntinuous if this map is always continuous [using the metric in part (a)]. Show
that if = ?r : E + B is any vector bundle, and F is continuous, then there is
a bundle F(e) = ir': E' + B for which ?ir-' (p) = ~ ( ? r " ' ( p ) ) ,and such that
to every trivialization
r : a - ' ( u ) + U x Wn
e
corresponds a trivialization
ri: ? r ~ -(U)
~ + U x F(Rn).
(c) The functor TI (V) = T1( V*) = V** is continuous. (The bundle (TM) is
just a case of the construction in (b).)
(d) Define a continuous contravariant fuiictor F, and show how to construct a
bundle F(6).
(e) The functor F(V) = V* is continuous. (The bundle T*M is a special case
of the construction in (d).)
Geiierally, tlie same construction can be used when F is a functor of several
argumeii ts. The bundles T~( M ) are al1 special cases. See the next two problems
for otlier examples, as well as an example of a functor which is no1 continuous.
9. (a) Let F be a functor from V", tlle class of tz-dimensional vector spaces,
to vk. Given A E GL(n, W) we can consider it as a map A : Rn + Rn. Then
F(A) : F(Rn) + F(RN). Cl~oose,once and for all, an isomorphism F(Rn) +
Rk. Tllen F(A) can be considered as a map h(A): Rk + IRk. Show that
zi : GL(t1, R) + GL(k,R) is a homomorphism.
(b) How does tlle llomomorpllism h depeild on tlle initial choice of the isomorphism F(R") + IRk?
(c) Let v = (VI , . . .,v,) and w = (wl, . . . , w,) be ordered bases of V and let
e = ( e l , .. . ,e,) be the standard basis of R". If e + v denotes the isomorphism
taking ei to vi, sllow that the following diagram commutes
where A = ( a i j ) is defined by
j-i
After identifying F ( R n )with R k , this means that
also commutes. Tliis suggests a way of proving the following.
THEOREM. If h : GL(n,R ) + GL(k,R ) is any homomorphism,
there is a functor Fh : V" + V' such that the homomorphism defined
in part (a) is equal to h.
(d) For q,qt E Rk, define
( v ,Y )
- (w,Y')
if q = h(A)qtwhere wi = E = ,ajivj.. Show that -- is an equivalencc relation,
and that every equivalence class contains exactly one element ( v ,y ) for a @venv.
We will denote the equivalence class of ( Y , y ) by [v,y].
(e) Show that the operations
are well-defined operations making the sct of al1 equivalcnce classcs into a
k-dimensional vector space Fh ( V ) .
(f) If V , W E V" and / : V + W, choose ordered bases v, w, define A by
/ ( v i ) = I jn =Llji
, W j , and define
Sliow that this is a well-defined liiiear transformation, tllat FA is a functor, and
that Fh(A)= h ( A ) when we identify Fh(Rn)with IRk by [e,q]H y .
(g) k t a : R + R be a non-continuous homomorpllism (compare page 380),
aiid let b : GL(n,R) + G L ( I , R ) = R be h ( A ) = a(det A). Then Fh: V" + VI
is a noil-continuous functor.
10. In classical tensor analysis there are, in addition to mixed tensor fields, other
"quantities" which are defined as sets of functions which transform according
to yet other rules. These new rules are of the form
A' = A operated on by h
(S)
For exarnple, assignments of a single function a to each coordinate system Asuch that the function a' assigned to x' satisfies
are called (even) scalar densities; assignments for which
Il ) ; : (
a ' = det - - a
are called odd scaIar densities. T h e Theorem in Problem 9 allows us to construct a bundle whose sections correspond to these classical entities (later we
will have a more illuminating way):
(a) Let h : GL(n7B) + GL(1, R) take A into multiplication by det A. Let Fh
be tlie functor @ven by tlie Theorem, and consider the 1-dimensional bundle
Fh ( T M ) obtained by replacing each fibre Mp witl~F],( M p ) . If (x, U ) is a
coordinate system, then
is non-zero, so every section on U can be expressed as a a, for a unique
function a. If xf is anotller coordinate system and a a, = a f axl, show that
) . - ( axi
o' = det
u.
(b) If, instead, 7i takes A into multiplication by 1 det Al, show that the corre-
(c) For tliis i7, show tllat a non-zero element of F/i( V ) determines an orientatian
for V. Conclude tll at the bundle of odd scalar densities is not trivial if M is not
orientable.
(d) \Ve catl identify q k ( R n ),th
IRnk+' by taking
Recall that if f : V + V, we define T~( f ) : T~( V) + qk( V ) by
GL(Iz,R), we can consider it as a map A : Rn + Rn. Then
( I R ) + T~(IRn) determines an elemeilt qk(A) of GL(IZ~+',
R).
A
:
Let h : GL(>I,R)+ G L ( ~ ~ + ' , R
be) defined by
Ghen A
E
h(A) = (det A ) ~ ~ ~ ( Aw) an integer.
Tlie 1,undle F(TM) is called tlie bundle of (even) relative tensors of type (f )
and weight w. For k = 1 = O we obtain tlie bundle of (wen) relative scdars of
weight w [tlie (even) scdar densities are tl-ie (even) relative scalars of weigl-it 11.
1f (det A)w is replaced by 1 det A l W (w any real number), we obtain the bundle
of odd relative tensors of type (f) and weight w . Sliow that tlie ti-aiisformation
law for the components of sections of these bundles is
(or tlie same formula with det(axi/axti) replaced by 1 det(axi/axfj)l).
(e) Define
+I
gil
...in -
-1
O
if il , . . . ,i, is an even permutatjon of 1,. . .,n
if i l , .. . ,i, is an odd permutation of 1,. . . , T I
if i, = ib for some a # p.
Sliow tliat there is a covariant relative tensor of weight - 1 witli these compoiients in every coordinate system. Also sliow tliat E'' .-.'M = Ej,
are the components in every coordinate system of a certain con travariant relatjve tensor
of wcigh t 1. (See Problem 7-1 2 for a geometric in terpretation of these relative
tensors.)
CHAPTER 5
VECTOR FIELDS AND
DIFFERENTIAL EQUATIONS
w
e i-eíurn ío a more detailed study of the tangent bundle TM, and its
sections, i.e., vector fields. kt X be a vector field defined in a neighborhood of p E M. We would like to know if íhere is a curve p : (-E, E) + M
hrough p whose tangent vectors coincide with X, thaí is, a curve p wiíh
Since tliis a local quesíion, we wish ío introduce a coordinate system (x, U)
around p and íransfer the vector field X to x (U) c Rn. Recall that, in general,
a,X does not make sense for Cm functions a: M + N. However, if a is a
difTeomorpliism, then we define
It is noí liard to clieck (Problem 1) that a,X is Cm on a (M). In particular, we
have a vector field x, X on x (U) c Rn.There is a funcíion f : x(U) + Rn
with
i.e., (x*X), lias "componeiits" f * (y), . . . ,f" (y). Consider íhe curve c = x o p.
The condition
means that
hence
If ure use c' (í) to denote the ordinary derivative of the Rn-valued function c,
tl-ien this equation finally becomes simply
This is a simple example of a differential equation for a function c : R + Rn,
wliich may also be considered as a system of n differential equations for the
functions c*,
*
e* (i) = f (cl (i), . . . , cn ( 1 ) )
i = I, . . . , n.
We also want tl-ie "initial conditions"
Solving a differential equation used to be described as "integrating" h e
equation (tlie process is integration when h e equation has the special form
cf(i) = f (í) for f :R + R, a form to which our particular equations never
reduce); solutions we1.e consequently called "integrals" of the equation. Part of
tliis termiliology is still preserved. A curve p: (-E, E) + M with
is called an integral curve for X with initid condition p(0) = p. Similar termi~iolomis applied, of course, to tl-ie differential equations one obtains upon
introducirig a coordinate system. For quite sonle time, we will work entirely iil
Euclidean space, ar-id for a wliile x, y, etc., will denote points of Rn. If U c R*
is oper-i and f : U + Rn, tlien a curve c : (-E, E) + M with
is called an integral curve for f with initíd condition c(0) = x
E c t o ~Fields and Dguenlial Equalioíls
137
Before stating the main theorem about the existence and uniqueness of such
integral curves, we consider some special cases.
The equation for a curve c with range IR,
which would be written classically in terms of a function y : R + R as
is the special case f ( a ) = -a2. The standard method of solving this equation
is to write
Thus the curves
1
are supposed to be solutions. This can be checked directly if you don3tbelieve
the above manipulations. (Thcy really do make sense; the equation in question
asserts that y' = f o y , so
hence, if F' = i/f,then
( F o y)' = I
F ( y ( x ) )= x
+C
for some C.) To obtain tlie initial conditions c(0) = a, we must take
1
Tllis works in al1 cases except a = O. ln this case, the correct solution is
c(i)= O
for al1 i
(wliich we missed by dividing by y). In terms of vector fields, the curves c are
the integral curves of
Notice that no integral curve, except c(i) = 0, can be defined for al1 r, even
tllougll X is defined on al1 of R . It miglit be thought tliat this somehow reflects
the fact that X(0) = 0, but this has nothing to do with the case. For a > 0, the
curve c (1) = 1 /(r + 1 /a) is defined for al1 large i, and as r + cm it approaches,
but never reaches, 0. O n the other hand, as r + - 1 / a the curve escapes to
infinity because the vector field gets big ~ O Ofast. This will continue to be true
even if we modify tlie vec tor field near O so tbat it is never O.
Ano tlier phenomenon is illustrated by the equation
ct(r) = c(1)2/~,
written classicaily as
There are two dfirent solutions with the initial condition c(0) = O, namely
(1)
c(i) = 0
for al1 r,
1 3
for al1 r .
C(i) =
In this case, the function f, @ven by f (a) = a2i3,is no! differentiable. Unicjueness wiI1 always be insured when f : U + Rn is C ' , but it can also be obtained
with a rather Iess stringent condition. We say that tlie function f satisfies a
Lipschitz condition on U if there is some K such that
zi
for aiI S, y E U.
I f ( x ) - f1Y)I 5 Klx - Yl
Notice that /(a) = a2i3 is not Lipschitz; in fact, there is no K with
lf
for x near O, since
- f (011 5 Klx-l
A Lipschitz function is clearly continuous, but not necessarily dfirentiable (for
rsample, f (x) = 1
s1). O n the other hand, a C function is locaiiy Lipschitz,
~ h a is,
t it satisfies a Lipscliitz condition in a neighborhood of each point-this
Sollows from Lemma 2-5. A Lipschitz function is also clearly bounded on any
bounded set.
T h e basic existente and uniqueness theorem for differential equations depends on a simple Iemma about complete metric spaces.
'
l . THEOREM (THE CONTRACTION LEMMA). Let (M,p) be a nonempty complete metric space, and Jet f : M + M be a "contraction", that is,
suppose there is some C < 1 such that
P (f(x), f ( ~ ) 5) CP(x, Y
for aii x , y E M.
Then there is a unique x E M such that f (x) = x (the function f has a unique
" h e d point").
PROOF. Notice that f is clearly continuous. Let xo E M and define a sequence
{S,) inductively by
Xn+l = f ( x n ) ,
n times
Then an easy induction argument shows that
Thus
xEo
C n converges, so C"
. + c " ~+
- ' O as n + m.
Since C < I ,the sum
Thus the sequence {x,) is Cauchy, so there is some x with
+e
m
x = Iim x,.
n 3w
Continuity of f then shows that
f (x) = lim f (xn) = Iim xn+i = x.
n-+w
n-+m
We are going to apply the Contraction k m m a to certain spaces of functions.
Recall that if ( M , p) is a metric space and X is compact, then the set of al1
continuous functions f : X + M is a metric space if we define the metric o by
If M is bounded, then we do not even need X to be compact. Moreover, if M
is complete, then h e new metric space is also complete; this is basically jusí the
theorem that the uniform limit of continuous functions is continuous, plus the
fací that each lim fn ( S ) exisís since M is complete. In particular, if M is a
x-fm
compací subset of Rn, then the set of aii continuous functions f : X + M is
complete with the metric
Our basic strategy in solving difirential equations will be ío replace difirentiable functions and derivatives by continuous functions and integrals. If U c
Rn and f : U + Rn is continuous, then a continuous function a : ( 4b,) + U,
defined on some iníerval around O, clearly satisfies
if it satisfies the integral equation
where the integral of an Rn-valued function is defined by integraíing each component function separately. Conversely, if a satisfies (1), then a is differentiable,
hence continuous; thus a' f o a is continuous, so
-
For the proof of the basic tlieorem, we need only one simple estimate. If a
continuous function f : [a, b] + Rn satisfies 1 f 1 5 K, then
To prove this, we note tliaí it is true for consíaní functions, hence for step
functions, wliicl~are unjform limits on [a, b]
functions, and thus for co~~tinuous
of step functions.
2. THEOREM. Leí f : U + R" be any function, where U c R" is open.
Let xo E U and leí a > O be a number such that the closed ball 8 2 Q ( ~f
~~),
radius 2a and center xo, is contained in U. Suppose that
Choose b > O so that
Then for each x E 8,(xo) there is a unique a , : (-b, b) + U such that
axt(r>= f (a, ( r ) )
a, ( O ) = x.
PROOF. Choose x E & (xo),which will be fixed for the remainder of the proof.
Let
M = {continuous a : (-b, b ) + BZa(xo)}.
Then M is a complete metric space. For each a E M, define a curve Sa on
Sa ( r ) = x +
lof (a(4)
d1.4
(the integral exists since f is continuous on B20(x0)).The curve S a is clearly
continuous. Moreover, for any r E (4,
b) we have
Since ( x - ,yo( 5 a , it follows that ( S a ( i )- xo( < 2a, for al1 i E ( 4b),
, $0
Thus S : M + M .
Now suppose a , p E M . Then
sup la(u)-B(u)l
-bcucb
= bK ((a- fi (1.
by (2)
Since we chose bK < I (by (4.11, íhis shows that S : M + M is a coníracíion.
Hence S has a unique fixed point:
Tliere is a unique a : ( - b , b ) +
+l
(xo) wiíh
f
a(r) = x
/(a(u))du.
This, alas, is noí quite wliat íhe ílieorem síates. Having used íhe eleganí Contraction Lemma, we pay for ií by finishing off wiíh a finicky d e t d :
Tlie map a is the unique B : (4,
b ) + U saíisfying
rf
Reuson: We claim that any such B actudly lies in &(xO), in fact, in BZa(x.0).
Consider first numbers r > O. We have already seen (statement (*)) that for
each í with O 5 1 < b ,
(**)
p (1) =x +
lof ( B
( U ) ) d~
is in BzU(xo)
[the o
p ball]
provided that
so certainly if
B ( U ) E B2,(xO)
for dl u with O 5 ti 5 r
We can now use a simple Jeast upper bound argument. Let
Let a = sup A . Suppose a < b . We cIearIy have B ( u ) E B2, ( x o ) for O 5 u < a .
So B ( a ) E B2, (xO),by (**l. Tliis clearly implies that p ( a + S ) E B2, (xo) for
sufficiently smdl s > 0, wbicli contradicts tlie fact tllat a = sup A . So it must
be tllat sup A = b . A similar armgument works for -b c 1 5 0.
To sum up, the ui-iique fixed point a, of tlie map S is the unique curve with
tlle desired properties. +:+
~ I OFields
T
and Dfl~eniiQZ
i7quations
Notice íliat solutions of the dfirential equaíion
remain solutions under additive changes of parameter; that is, if
then
B ' ( 0 = ~ ' V O+ 1 ) = f ~ ( I +O 1 ) ) = f ( p ( ~ ) ) .
TJiis remark allows us to exíend the uniqueness parí of Theorem 2.
3. THEORFM. Suppose f : U + IRn is Jocally Lipschitz, that is, around each
point there is a bail on which f satisfies condition (2) of Theorem 2 for some K
(and Ilence also condition (1) for some L). Let x E U and Jet al ,a2 be two maps
on some open intervai I with al (1),a 2 ( I )c U and
ait(i)=
ai ( O ) = x
(í))
Then al = a2 on 1.
PROOF. Suppose al (lo)= a2(10) for some io E 1. If we define
then tlie functions pi satisfy the same difirentiai equation, p i r ( i ) = f ( p i ( i ) ) ,
and have the same initial condition pi (O) = al (lo) -- a2(iO)E U. Hence pi ( 1 ) =
p 2 ( i ) for sufficientlysmall i, by Theorem l. Thus the set
is open. It is clearly aiso closed and non-empty, so it equals 1. 43
We now revert to the situation in Theorem 2. \ e will write a, ( i )as a (1, x ) ,
so íhaí we have a map
satisfying
a (O, x ) = x
[i.e., Dla(r,A-)
d
-a(i,x) = f (a(1,x))
di
= f ( a ( i x, ) ) , but we will frequently use d / d í or d / d i in this
discussion]. This map a is called a local flow for f in (4,
b) x 3, (xo). To
picture this rnap a, the best we can d o is to draw the images of the integral
cuntes a,. If y = a,(i0), tllen the integral curve a, with the initial condition
a,(O) = x differs from t l ~ eintegral curve a, with initial condition ay(0) = y
only by a cl~angeof parameter, so the two images overlap. For each h e d x, the
rnap i I+ a(r,x) for -b < i < b lies alongpart of the curve through x. O n the
other hand, if we fix L,then the rnap
gives the result of pushing each A- along the integral curve tlirough it, for a time
intentd of r . To focus attention on tliis map, we denote it by #*:
This rnap #r is dways continuous. In fact, tlle whole flow a is continuous (as a
function of both i and x):
4. THEOREM. If f : U + Rn is Iocally Lipschitz, then tlie flow
given by Theorem 2 is continuous.
PROOF. Let us denote t11e rnap S defined in tlie proof of Theorem 2 by S,,
to indicate explicitly the role of x. Tlien
Recall that
- SP I 5 bKlla - B II.
lf S; denotes the n-fold iterate of S,, then
Recall also tliat in Theorem 1 the fixed point a, of S, is the limit of SFa for
any a. Hence a, = lim S~a,,so we obtain
t 1 4 00
Since Ila, -ayll = sup la([, A-)-a([, y)I, this certainly proves continuity of a. 43
1
If additional conditions are placed upon tlie map f,then further smoothness
conditions can be proved for a. In fact,
Unfortunately, this is a very hard theorem. A clean cxposition of the classical proof is given in Lang's introductiotl to L)@mntzable ManifoldF (2nd ed.), and
a recently discovered proof can be found in Lang, Real and Functional AnalysLs
(3rd ed.), pp. 371-379. In order to read this liigh-powered proof, you must first
Jearn the elements of Banach spaces, including the Halin-Banach theorem, and
then read about differential calculus in Banach spaces, including the inverse and
implicit function theorems (Real and Functional Annbsis, pp. 360-3651, but this is
probably easier than reading the classical proof (and, besides, when YOU'PZ finished you'll dso know &out Banach spaces, and differential caIculus in Banach
spaces).
We wilI just accept h i s fact. Notice that tlle maps #I are consequently Cm
if f is Cm.
Since tlie map
a : (-b,b) x B,(xo) + U
satisfies a (O, x ) = x, we llave
a : {O} x B.,2(xo) '&/2 ( ~ 0 c
) Ba ( ~ 0 ) -
Continuity of u and cornpactness of {O) x BUj2(xO)
irnply that there is some
E > O S U C ~that
[If x E Baj2(xa),then the integral curve with initial condition x stays in B, (xo)
for 1 1 c E.]
So if Jsl < E , and x E B n / 2 ( ~ Othen
) , h e point a(s, x) E BU(xo),so we can
also define
y ( ) = ( , ) )
IiIcE*
This satisfies
We have also noted that
satisfies
Consequendy, p (r) = a(i,a(S, x)) for 1 1 c E. In other words,
if
J s ~ , J t ~ , J< sE +
, t~
then
a(i,a(s,x))=a(s+t,x).
If we now Jet #t : Bal2(x0) + Hg" be #t (x) = a(i,x) for x E Ba/2(xO),we can
say:
Roughly speaking, #t+s = #r o = O # t . This shows, in particular, that for
Isl c E each #$ is a diffeomorphism, with invene 6,-' = #-s. Everything we
llave said, since it is local, can be resaid, without requiring any more proof, on
a manifold.
k t o r Fiel& and Dzferential Eqt~ations
147
5. THEOREM. Leí X be a C m vector field on M, and let p E M. Then there
is an open ser V containing p and an E > O, such íhat there is a unique colIection of difkomorphisms #t : V + #f( V ) c M Mor Ii 1 < E with íhe following
properties:
(1) # : ( - E , & )
V + M, defined by #(i, p ) = & ( p ) , is Cm.
X
(3) If y E V, ttlien Xg is íhe tangení vector at i = O of the curve i K- # f ( q ) .
The exarnples Fven previously sliow íhat we cannoí expecí #f to be defined
for al1 i, or on al1 of M. In one case however, this can be attained. The support
of a vector field X is just the closure of { p E M : X, # 0).
6. THEOREM. If X Iias compact support (in particular, if M is compact),
íhen íliere are dfiomorpI~isms#t : M + M for al1 i E R wiíh properíies (11,
(21, (3)*
PROOF. Cover support X by a finite number of open sets V , , . .. , Vn given by
TIieorem 5 wiíli comespoiiding E I ., . . , E , and dfiomorphisms #:. Let E =
m i n ( r l , .. . , E ~ ) .Notice thaí by uniqueness, # ( ( y ) = #{ ( y ) for y E Vi n 5. So
we can define
if y E Vi
if y # support X .
Clearly # : ( - E , & ) x M + M is Cm, and #fct=F
= #1 o # s if IiI,IsI, Ii + S I
and each #f is a dfiomorphism.
To define #, for Ii 1 >_ E, write
r = k ( ~ / 2+) I'
#f
=
{
#E/2
#-E/2
O
m
O
m
O #E/2
O
with k an integer, and 17'1 < ~ / 2 .
#r
- . - o #-&/2 O #r
[#E/2
iíeraíed k times]
iterated - k times]
It is easy to check t1iat this is the desired ( $ 1 ) . +
*:
for k 1 O
for k < O.
< E,
The unique collection { # 1 ) @ven by Theorem 6, or more precisely, the map
r I+ #t from R to tlie group of alI difkomorphisms of M, is called a 1-parameter
group of diffeomorphisms, and is said to be generated by X. In the JocaI case of
Theorem 5, we obtain a "local 1-parameter group of local difXeomorphisms".
The vector field X is sometimes called the "infinitesimal generator" of { # t )
(vector fields used to be called "infinitesimal transformations").
Condition (3) in Theorem 5 can be rephrased in terms of the action of X,
on a Cm function f : M + R. Recall that
Thus, to say that Xg is the tangent vector at i = O of the curve r H #I(q)
arnounts to saying tliat
( X f ) ( q )= X, f = lim f (#h ( Y ) ) - f ( Y )
h+O
h
Tliis equation will be used very frequently. The first use is to derive a corollary
of Theorem 5 whicli allows us to simplify many caicuIations involving vector
fields, and wliicli also Jias important tlieoretical uses.
7. THEOREM. Let X be a Cm vector field on M with X ( p ) # O. Then there
is a coordinate system ( x ,U ) around p such that
n
PROOF. It is easy to see that we can assume M = R" (with the standard coordiiiate system r', . . .,rfl, say), and p = O E R". Moreover, we can assume that
X(O) = a / a t l 1,. Tlie idea ofthe proofis that h a neigliborhood of O there is a
uiiique integral cunfe tlirougli eacli point (0,a2, . . . ,a"); if lies on the integral
curve through this point, we will use a 2 , . . .,a" as ihe last n - 1 coordinates of q
íuid the time intervd it takes the curve to get to q as the first coordinate. To
do ihis, Jet X generate #* and consider the map x defined on a neighborIiood
of O in Rn by
(0,a2,.. .,an).
x(a 1 ,. . . , a n )
W compute that for a = (a1,... ,a"),
+
1
= lim - [ ~ ( ~ ( a h,
' a2,. . . ,a")) - f (x(Q))]
k+O h
Moreover, for i > 1 we can at Jeast compute
1
= Jim -[f(x( O, ... , h ,..., O)) - f(0)]
1140 11
1
= lim - [f (O,. . .,h, . . . ,O) - $(O)]
h40 h
lo
Since X (O) = d/ai l by assumption, this shows tliat X,O = I is non-singular.
Hence x = X-' may be used as a coordiilate system in a neighborhood of O.
This is the d e s h d coordinate system, for it is easy to see tliat the equation
~ * ( a / d r') = X O x, which we have just proved, is equivalent to X = d/axl.
The second use of the equation
is more co~nprelieiisi\le. The fact tjiat Xf can be defined totally in terms of
the diffeon~orphisms#iI suggests that an aciioii of X on other objects can be
obtained in a similar way, To emphasize the fundamental similarity of these
notions, we first introduce the notation
Lxf
for
Xf.
-
\Ve cal1 Lx f tlle (Lie) derivative of f with respect to X ; it is another function,
wliose value at p is denoted variously by ( L xf ) ( p ) = L x f ( p ) ( X f ) ( p )=
Xp ( f ). Now if w is a Cm covariant vector field, we define a new covariant
vector field, tlie Lie derivative of w with respect to X , by
Tliis is the limit of certain members of M,*. Recall that if Xp E M,, then
A fairly easy direct argument (Problem 8) shows that this limit always exists,
and that tlie newly defined covariant vector field L x w is C m , but we will soon
compute this vector fieId explicitly in a coordinate system, and these facts will
then be obvious.
If Y is another vector field, we can define the Lie derivative of Y with respect
to X ,
1
Tlie \rector field #h+Y appearing here is a special case of the vector field a,Y
defined at the beginning of the chapter, for a : M + N a difkomorpliism and Y
a vector field on M . Tlius (#/,,Y), = #h* (Y$-,, (,) is obtained by evaiuating Y
at
( p ) = #-A ( p ) , and then moving it back to p by #/,*.
integral curve
r $1 ( P )
of X through p
Tlic definition of L x Y can be made to look more closely anaiogous to L x f
and Lx w in tlie follotving way. If a : M + N is a dfiomorphism and Y is a
Tr,tor l~icldsand Dzfenntial Equatians
151
vector field on the range N, thni a vector field a*Y on M can be defined by
Of course, a*( Y ) is just (&-'),Y. Now notice that
1
h+O /Z
Jim
Y )p
yp
- Yp]= h+O
lim
- (#h*Y)P
-h
1
[Yp - ( $ L ~ *Y ) , ]
k+O k
= lim
Nevertheless, we will stick to the original (equivalent) definition.
We now wish to compute Lx o and Lx Y in a coordinate system. The calculation is made a lot easier by first o b s e ~ n g
8. PROPOSITION. If Lx Y; and Lx wi exist for i = 1,2, then
If LxY and L x w exist, then
(3) L x f Y = X f . Y $ f . L x Y ,
(4) Lx f - o =X f . w + f . L x w .
FjnaIIy, if o ( Y ) deiiotes tlie function p I+ o ( p )(Yp)and L x o and LX Y exist,
then
PROOF. (1) and (2) are trivial. T h e remaining equations are al1 proved by the
a m e trick, t l ~ eoiie used iii finding ( f g ) ' ( x ) .We will do number (3)here.
+ /Jim
[f (p) - { ( m - h ( P ) ) #/r* y+-,, ( p ) .
140
1
The first limit is c'learly f ( p ) L x Y ( p ) . In ihe second limit, the term iii brackeis
approaches
lim f ( P ) - f (#k ( P ) ) = Xf ( P ) ,
k+O
-k
while an easy argument sliows that #h,Y+-,,(p) + Yp. *3
Mre are now ready to compute Lx in terms of a coordinate system ( x , U)
on M . Suppose X =
a i d / a s i . We first compute L x ( d x i ) . Recall (Problem 4-1) that if f : M + N and y is a coordinate system on N, then
zyEI
We can apply this to #h*, where y is s . Then
1
LX (dx')( p ) = lim - [(#h*)dxi( p) - dx' ( p ) ]
h+O h
Now the coeffrcient of d d ( p ) is
lim l 1 4 oh
["'d"'
-
$1
I
= lini Ir40 h
[ axj
{ibis step wiil be justified in a moment)
To justify (*) we note that the map A(h,q) = xi(#h(q)) is Coo from IR x M
to IR; thus a2~ / d h d x j= d 2 ~ / d x j d h wliich
,
is wliat the intercliange of limits
amounts to.
It now follows tllat
n
aai
LX dx' =
-d x j .
axj
We couId now use (2) and (4) of Proposi tion 8 to compute Lx w i n general, but
we are really iiiterested in computing L x Y. To compute
(d/axi) we could
imitate the calculations of L x dx'; but there would be a complication, because
#h* on \rector fields iilvolves one more composition thari #h* on covariant vector
fields. Tlie trick needed to deal with this complication has already been used to
prove (3), (41, and (5) of Proposition 8, and we can now use (5) to get the answa
immediately;
O = L. 6; = L x [dxi
(a)] (5)
+
S).
axj
= (Lx dx')
dx' (LX
thus,
Using (3) we obtain
Sumniing over j and tlien iiiterclianging i and j in tlie second double sum we
obtain
This somewhai complicated expression immediately leads to a much simpler
coordinate-free expression for Lx Y. If f : M + N is a C" function, then Yf
is a function, so XYf = X(Yf) makes sense. Clearly
Tlie secoiid partial derivatives which arise here cancel those in the expression
for Y ( Xf ), and we find tliat
LxY=XY-YX,
alsodenotedby [X,Y].
Often, [X, Y] (which is called the "bracket" of X and Y) is just defined as
XY - YX; note that this means
A straightfonvard verification shows that
so tliat [X, YIp is a derivation at p, and can therefore be considered as a member
of M,.
We are now in a very strange situation. Two vector fields LxY and [X, Y]
liave both beeii defined independendy of any coordinate system, but they have
Leen proved equal using a coordinate system. This sort of thing irks some
people to no end. FortunateIy, in this case tlie coordinate-free proof is short,
though hardly obvious.
In Chapter 3 we proved a Iemma wllich for the special case of W says that a
Cm fu~ictionf : (-E, E) + W with f (O) = O can be wntten
for a Cm fuiiction g : (-E, E) + W with g(0) = f'(O), namely
This lias a n immediate generalization.
9. LEMMA. lf f : (-E, E) x M + W is Cm and f (O, p) = O for al1 p E M,
tlien there is a Cm function g : (-E,&) x M + W with
PROOF. Define
1 0. THEORFM. If X and Y are Coovector fields, theii
PROOF. Let f :M + R be Cm. k t X generate #,, Ii 1 < E . By Lemma 9
tliere is a family of Cm functions gf on M such that
Then
Tl-ie equality Lx Y = [ X , Y ] = XY - YX reveals certain facts about LxY
which are by no means obvious from the definition. Clearly
[ X ,Y ] = -[Y, X ] , so
[ X ,X ] = o.
LxY = - L y X ,
LxX = 0.
Consequently,
so
+
Since we obviously Iiave Lx (aYl by2) = aLx Y, + bLx Y2,it follows immediately tliat L is also linear with respect to X :
Finally, a straiglitforward calculatioil proves tlie "Jacobi identity":
Tliis equation is capable of two interpretations in terms of Lie derivatives:
(a) L x [ Y ,21 = [ L x K ',l
+ [Y,Lx',],
(b) as opei-ato1.son Coafunctions, we have
L [ x , = Lx o L y - L Y o Lx (whicli miglit be written as [ L x ,L Y]).
Finally, note that Lx Y is linear over constants oilly, not over tlie Coofunctions .F.
In fact, Proposition 8, or a simple calculation using the definiíion of [X, Y],
shows that
[ f X , g Y 1 = fg[X, Y] + f f(Xg)Y- g(Yf )XThus, the bracket operation [ , ] is not a íensor-that is, [X, Y'Jp does not depend only on Xp and Yp (wliich is noí surprising-whaí can one do ío two
vectors in a lector space excepí íake linear combinations of íhem?), buí on íhe
vector fields X and Y. In particular, even if Xp = O, it does not necessarily
foIIow that [X, Y'Jp = O-in the formula
t11e first term Xp ( Yf ) is zero, but the second may not be, for Xf may have a
non-zero derivative iil tlle Yp direction even tliou~h( Xf ) (p) = 0.
T h e bracket [X, Y], although not a tensor, pops up in the definiíion ofpractically a11 otlier tensol-S,for reasons that \vil1 become more and more apparent.
Before procecdii~gto examine its geomeíric interpreta tion, we wil1 endeavor
to become more aí ease witIl the Lie derivative by taking time out ío prove
directly from tlle definiíioli of Lx Y two facís whic11 are obvious from íhe definition of [X, Y].
(1) LxX = o.
lf X generates #t, it certainly suffices to sl~owíhat (#n,X)p = Xp for a11 h.
Recall that (#h, X)p = #fd+ X4,/, (p). Now X4-, (p) is just h e tailgent vector at
time r = -h to t11e curve r I+
ío the curve
#I
(p), and tlius tlie tangent vector, at time I = 0,
Y P ) = 41-/i(P).
TIius @/l* X$-l,(P) is tlie íangení vecíor, aí time I = O, to the curve
But this tangení vecíor is jusí Xp .
k t o r I;ieZhand 1lz&i-entiaZ Equations
157
(2) If Xp and Y, are both O, then LxY (p) = 0.
Since Xp = O, tlie unique integral curve c wit1i c (O) = p and dcldi = X (c(1))
is simply c (i) = p (an integral curve Starting at p can never get away; conversely,
of course, an integral curve starting at some other point can never get to p).
Then Yp = O and
so LxY(p) = o.
To develop an iiiterpretation of [X, Y] we first prove two lemmas.
1 1. LEMMA. Let a : M + N l ~ ae dXeomorphism and X a vector field on M
wliicli generates {#[l. Then a,X generates {a o #t o a-'}.
PROOF. We have
12. COROLLARY. If a : M + M, tlien a, X = X if and only if $ t o a =
for al1 L.
13. LEMMA. Let X generate {#t) aiid Y generate {Srf}. Then [X, Y] = O if
and only if #l o
= $, o #t for al1 S, í .
= o #t for al1 S, tlien #,,Y = Y by CoroIlary 12. If this
PROOF. If #t o
is true for aII r , theii clearly Lx Y = 0.
Convei-sely, suppose that [X, Y] = O, so that
1
O=Iim-[Yq-(#h*Y)q]
h+o lr
forallq.
Giveii p E M, consider tlie cuive c : (-E, E) + Mp @ven by
For h e derivative, ct(r), of this map into the vecíor space M, we have
=O
)
using (*) with
= 4-1 (P)
= o.
Consequently c(í) = c(O), SO #t,Y = Y. By Corollary 12, #r o
alI s,r. 4 4
=
o
#[ for
We kave already shown tliat if X ( p ) # 0, then íhere is a coordinate system x
with X = a / d x l . If Y is another vector field, everywhere linearly independeiit
of X, then we mighí expect to find a coordinaíe sysíem with
However, a sllort caIcuIation immediately gives the result
so íliere is no Jlope offi~idinga coordinaíe system satisfying (*) unless [X, Y] = 0.
Tlie remarkable fact is that tlie condition [X, Y] = O is su#cient, as well as
necessary, for the existente of the desired coordinate system.
14. THEOREM. If X,, . . . ,Xk are IinearIy independent Coovector fields in
a neighborhood of p, and [X,, XB] = O for 1 < a,B 5 k, then tliere is a
coordinate system (s,
U) around p such that
PROOF. As in the pmof of Theorem 7, we can assume tllat M = Rn, thaí
p = 0, and, by a linear cl-iange of coordinates, that
X, (O) = -
a = 1, ..., k.
If X, generates (4," ), define x by
x ( a l , . . ,a ) = # , ( # * ( . . ( # ( , . . . , , ak4-1 , . . . , a n ) ) . . * ) ) .
As in the proof of Theorem 7, we can compute that
Thus x = x-' can be used as a coordinate system in a neighborhood of p = 0 .
Moreover, just as before we see that
Nothing said so fal. uses the l~ypothesis[ X , , XB] = O . To make use of it, we
appeal to Lemma 1 3; it shows that for each a between I and k , the map x can
also be wri tten
and our previous argument then shows that
We tlius see tliat the bracket [X, Y ] measures, in some sense, the extent to
which the integral curves of X and Y can be used to form the "coordinate
lines" of a coordinate system. There is a more complicated, more diEcult to
pi6ove,and less important result, which makes this assertion much more precise.
If X and Y are two vector fields in a neigl~borhoodof p, then for sufficiently
small h we can
(1) follow the integral curve of X
through p for time h ;
(2) starting from that point, follow the
integral curve of Y for time h;
(3) then follow the integral curve of X
backwards for time Ir;
(4) tlien follow the integral curve of Y
backwards for time Iz.
If iliere happens io be a coordinaie sysiem x with x ( p ) = O and
then these steps take us to points with coordinates
(1)
(2)
(31
(4)
(j,O,O,...,O)
( h , h , O , ... ,O)
(O,h,O,..., O )
(0,OY 0 , . - ,O),
so tliat this "parallelogram" is always closed. Even when X and Y are (linearly
iiidependent) vector fields with [ X , Y] # 0, the parallelogram is "closed up
to first order". Tlie meailiiig of tl-iis phrase [an exiension of the terminology
"c = y up to first order at O", which means tliat c'(0) = y f ( 0 ) j is the following.
Let c ( h ) be tlie poini whicl-i step (4) ends up at,
c(/z) = *-ff
($4
(*h (#h ( O ) ) )
Tlieii tlie cunJe c is tlie constant curve p up to first order, thai is,
15. PROPOSITION. ~ ' ( 0=) O.
PROOF If we define
al ( i ,/z)
= * r (#h ( P ) )
a2 ( i ,11) = 4-l (*h ( # f i ( P ) ) )
a3 ( i l h, = Sr-f (#-h (*k(#h
Moreover,
(P)))),
kctot. JieZd~and L)z&ien tial Equat ions
and for any Cm function f : M + R,
while
Consequently, repeated use of the chain rule gives
Wlienever we have a curve c: (-E,&) + M with ~ ( 0 =
) p and ct(0) = O E
Mp, we can define a new vector c"(0) or d2c/d121, by
A simple calculation sliows, wi?ghe arsun@ioti ~ ' ( 0 )= 0, that this operator ctt(0)
is a derivation, ct'(0) E Mp. (A more general construction is presented in ProbJem 17.) It turils out tllat for tlle curve c defined previously, the bracket [X, YIp
is r'elated to tliis "second arder" derivation. Until we get to Lie groups it wiI1
not be clear how anyone ever thought of the next tlieorem. The proof, which
eiids tlle cliapter, but can easily be skipped, is a n liorrendous, but clever, calculation. It is followed by an addendum containiilg some additional important
points about differeiltial equations which are used Jater, and a second addendum
coiicerning linearly independent vector fields in dimension 2.
16. THEOREM. c"(0) = 2[X, YIp.
PXOOF. Using the iiotation of the previous proof, since (f o c ) (1) = (foa3) (I,I )
we have
Now
(2) 2D2,l (f a3)(0?0)
= 2D1 (-Yf o a3)
= 2[D1 (Yf 0 a21(O? 0)
0 2 ( Y f o a2) (0, o)]
= 2XYf (p) - 202(Yf 0 a2)(0,0)
= 2XYf (p) - 2[Di (Yf o al)(O,O)
+
+ D2 (yf a1) (0, o)]
O
=2XYf(p)-2YYf(p)-2XYf(p)
Since (b) gives
by (e)
by (b) and the chain rule
by (4
by (a) and the cliain rule
by(c)and(f).
Trector fieldi and L)$uentiaZ Equariotls
Finally, from
Substituting (1)-(4) in (*) yields the theorem. +:+
163
ADDENDUM 1
DIFFERENTI AL EQUATIONS
Al thougli we have dways solved dxerential equatioiis
with the initial condition
a(0,x) = S ,
we could just as well kave required, for some io, that
To prove this, one can replace O by io everywliere in tlie proof of Theorem 2,
or else just replace a by i w a(í - lo,x).
Ano ther omissioli in our treatmen t oí' dflerential equations is more glaring:
tlle differentid equations ai(r) = f (a(t)) do not even include simple equations
of the form a' (i) = g(i), Jet alone equatioiis Iike ai(i) = ía (i). In general, we
would Iike to solve equations
where f : (-c, c ) x U + Rn. One way to do t his is to replace f (a(i, x)) by
f (1, a ( i , s))wl~ereverit occurs iii the proof. There is also a clever trick. Define
by
f ( s , s ) = (1, f (s,x)).
Then tliere is a flow (&',E2)= E: ( - b , b ) x W + R x R" with
For tlie f i i ~ coinponent
t
í'unction & ' this means that
a 1 (i, S,S ) = 1
-6
al
thus
&'(i,s,x) = s + i .
For the second component á2 we have
= f ( á l (i, S,s ) , á 2 ( t , s , s ) )
= f (S + 1, ii2(1, S, x)).
Then
~ ( I , x=
) a2(i,o,~)
is the desired Aow with
p (O, x ) = x.
Of course, we could also llave arranged for B (ro,x) = x (by first finding á
with á (io,s,X) = (S,x), not by considering the curve r I+
(i - io, S)).
Finally, consider the special case of a linear differential equation
wliere g is an n x 12 mati-ix-valued function on (a, b). In this case
If c is any n x n (constant) matrix, then
so c .a is also a solu tion of the same differeiitial equation. This remark allows us
to prove a n important property of linear differential equations, distinguishing
tliem from general differential equations a' (i) =: f (r, a (i)), which may have
solutions defined onIy o11 a small time interval, even if f : (a, b) x Rn + Rn
is Cm.
J 7. PROPOSITlON. If g is a continuous n x 11 matrix-valued function on
( u ,b), tlien tlie solutioi~sof the equation
can al1 be defined on (a, b).
PROOF. Notice tliat continuity of g implies that f (í,x ) = g(í) . x is locally
Lipschitz. So for any ío E (a, b) we can solve the equation, with any given initial
condition, in a iieighborhood of ío. Extend it as far as possible. If the extended
solution a is not defined for al1 r wit h ío 5 í 4 b, Iet r 1 be the Ieast upper bound
of tlie set of í ' s for which it is defined. Pick S with
TJien fl (r*) # O for r* < i l close enough to í l . Hence there is c with
coiricidcs witli a oii tlie iiiterval whe1.e they are defined.
By uniqucncss, c
I ltus a may be extended past í l as c - fl, a contradiction. Similarly, a must be
defined for al1 í with a c r 5 10. +3
ADDENDUM 2
PARAMETER CURVES I N TWO DIMENSIONS
If f : U + M is an immersion from an open set U c IWm into an 17-dimensional manifold M, the curve í H f (al,. . . ,aj-l,r, aj+l,. .. ,an) is caIIed a
parameter curve in the ith direction. Given 7t vecior fidds Xi ,. . . ,Xn defincd
in a neiglilorhood of p E M and Jinearly indepaident at p , we know that there
is usuaily no immersion f :U + M with p E f (U), whose parameter c u r e s
in the ith direction are the integral curves of the Xi-for we might not have
[Xi,
Xj] = O. However, we mighi hope to find an immersion f for which the
parameter culves in ihe ith direction lie along the integral curves of tlie Xj, but
have different parameierizations. A simple example (Problem 20) shows tliat
even this inodest kope cannot be fulfilled in dimension 3.
O n the oilier Iiand, in the special case of dimension 2, such an imbedding
can be found:
18. PROPOSITION. Let X1,X2 be Iiiiearly independent vector fields in a
neighboi.hood of a point p in a 2-dimensional manifold M. Then there is
aii imbedding f :U + M, where U c IR2 is open and p E f (U), whose ith
parameter Iines Iie aIong the integral curves of Xi.
PROOF. We can assume tliai p = O E IR2, and that Xi(0) = (ei)a. Ewry
poini y in a sufficiently small iieighborhood of O is on a unique integral curve
of Xl thiough a point (0,x2(y))-we proved precisely this fact in Theorem 7.
Similarly, y is oii a unique iiitegral curve of X2 through a point (xi (y),O).
The map y c ( x l(y),x2(y)) is Cm, wiih Jacobian equal to 1 at O (ihese facts
also foIIow from tlie proof of Theorem 7). Its iiiverse, in a sufficieiitly small
neighborhood of O, is tlie required dfiomorphism. ++:
\ e can always compose f with a map of the form ( x ,y) I+ ( a @ ) ,fl(y))
for dXeomorphisms a and fl of W , which @ves us considerable flexibility IIf,
for example, C c W 2 is the graph of a monotone function g , then the map
(.Y, y ) H ( x ,g ( y ) ) takes the diagonal { ( x ,A-))to C . Moreo\ler, for any particulai-
paiiimeterization c = ( e i , c 2 ) :W + W 2 of C , we can further arrange that c ( r )
maps to ( c ( r ) ,c ( r ) ) , by composing with ( x ,y) H c
x ) y ) . Coiisequently,
we can state
19. PROPOSITION. Let X I ,X2 be linearly ii~dependentvector fields in a
ileighboi-hood of a point p in a ?-dimensional manifold M, and let e be a
curve in M with c ( 0 ) = p and ~ ' ( 1 )never a multiple of Xi or X2. Then there
is an imhcdding f : U + M, where U c W 2 is open and p E / ( U ) , whose ith
parameter liiies lie along the i i tegral
~
curves of Xi, and for which f (r,1 ) = c ( r ) .
PROBLEMS
1. (a) If a : M + N is Cm, then a , : T M + T N is Cm.
(b) If a : M + N is a diffeomorphism, and X is a Cm vector field on M, tlien
a , X is a Cm vector field on N.
(c) If a : R + R is a(r) = r tlien there is a Cm wcior field X on R such that
a,X is not a Cm vector field.
',
2. Fii~da now11e1-eO vector field on R such tl~atal1 integral curves can be defined
only on some interval around O.
3. Fii~dail example of a complete metric space ( M ,p) and a function f : M +
M such that p ( f ( S ) ,f ( y ) )< p ( x , y ) foi- al1 x, y E M, but f has no fixed point.
4. Let f : ( - e , c) x U x V + R" be Cm, where U , V c R" are open, and let
(xo,yo) E U x V. Pro1.e tl~atthere is a neíghborhood W of (xo,yo) and a number b > O such that for each ( x ,y ) E W tl~ereis a unique a =
: ( 4b,) +
U wit11 a f ( t) E V for t E (4,
b) and
-
Moi-eover, if we write a(,,,,) ( t ) a ( t ,x, y ) , then a : ( 4b)
, x W + U is Cm.
Hi726:Consider tlie system of equations
5. \Ve soinetimes have to solve equations "depending on parameters",
wl-ierc f : (-c, c) x V x U + R", for opeil U c R" and V c Rm, and we are
solving for a(,,,,) : (4,
b) + U for each initial condition x and "parameter" y.
For example, the equation
is such a case.
(a) Define
j : (-C,C)
x
v x U + R" x W~
by
T u , Y, x) = (O,f(r, y , x)).
If (E', ¿i2) = ¿i : (4,
b) x W + R" x Rn is a flow for j in a neighborhood of
(YO, XO),SO that
show tliat we can write
for some a , and coiiclude that a satisfies (*).
(b) Sliow that equations of the form
(**)
a
a;a(r, X) = f (1, X., Mt, 4 )
can lze i-educed to equatioiis of tlie foi-m (*) (aiid tl-ius to equations
a
-41, x.) = f (a(!, x)),
at
ultimately). [Wlien one pi+ovesthat a C k function f : U + W" has a C k flow
a : (-b, b) x W + U, tlie hard part is to proi7e tliat if f is C i , tlien a is
differei~tiable~ t i t l respect
i
to tlie arguments in W, aiid tliat if tlie dei-ivativewi tli
1-csl~ectto tliese ai-gumeiits is denoted by D2a, tlien
(***)
DI D2a (r, s) = D2f (a(t,x)) - Dza(t, x)
(a i-esult wliicli follows directly fi-om tlie original equatioii
if f is c 2 , silice D ID? = D2D1). Since (***) is an equation for Dza of tlie
foi-m (**), it follows that D2a is diffei-eiitiable if Dr f is C 1 , i.e., if f is C2.
Differential~ilityof class ckis tlien proved similarly, by induction.]
14ct0r fields at2d Ihjiroztial Equnr ions
6. (a) Consider a linear dfierential equation
where g : R + R, so that we are solving for a real-valued function a. Show
that al1 solutions are multiples of
where Jg(r) dr denotes some function G with Gt(r) = g (one can obtain al1
podive multiples simply by changing G). The remainder of this problem investigates the extent to which similar results hold for a system of linear dserential
equations.
(b) Let A = ( u i j ) be an rz x 12 matrix, and let IA 1 denote the maximum of al1
lai, l. Show that
(c) Conclude tliat the infinite series of n x n matrices
converges absolutely [in the sense tliat ihe (i, j)& entry of the partial sums
converge absolu tely for each (i, j ) ] and uniformly in any bounded set.
(d) Show that
~X~(TAT=
- ' )T(exp A)T-'.
(e) If AB = BA, tl~en
exp(A
Hint: Write
C +
(A
p=o
P!
+ B) = (exp A)(exp B).
N
p=o
p=o
and show that l R ~ +
l O as N + m.
(f) (exp A)(exp -A) = 1, so exp A is always inverti ble.
(g) The map exp, considered as a map cxp : lRn2 + R"', is clearly differentiable
(it is even analytic). Show that
expt(0)(B)= B
(=exp(O). B).
(Notice thai for IA 1, tlie usual norm of A E lRnL,we have IA 1 5 IA l 5 n lA l. )
e)
(h) Use the limit established in part
to show that expf(A)(B)= exp(A) B
if AB = BA.
(i) Lec A : R + Rn2 be differentiable, and lec
m
If B3(r)denotes t l ~ ernatrix wliose enti-ies are the derivatives of the entries of B,
show that
B' (t) = Ar(r) - exp(A (r)),
plovided dznt A(r) A*(r)= At(r)A(r). (Tliis is clearly true if A(s)A (r) = A(r) A (S)
for al1 S , r .)
(j) Show that tlie linear differential equation
lias the solution
~witided[/nt g(s)g(r) = g(r)g(s) for al1 s,r. (This certainly happens when g(r)
is a constant matrix A, so every system of linear equations with constant coeficients can be solved explicitly-the exponential of i,g(s) ds = r A can be
found by put ting A in Jordan canonical form.)
7. Clieck tllat if tlie cooidinate system x is x = X-', for x : R" -t M? then
X = a/axi is equivaleni lo x, (alar l) = X 0 x.
8. (a) Let M and N be Cm manifolds. For a Cm function f : M x N + R
aiid y E N, let f ( - ,q ) denote the funciion from M to R defined by
If (x, U ) is a coorditiate systeni oii M, sliow that tlie function af/axi, defined
bv
is a Cm function on M x N.
(b) If # : (-E, E) x M + M i~ a 1-parameter group of diffeomorphisms, show
tliat for every Cm function f :M + R, the limit
exists, and defines a Cm function on M .
(c) If #, : ( - E , E ) x TM + TM is defined by
S ~ O M
tliat
~
#, is Cm, and conclude tliat for every Cm vector field X and covari-
ant vector field w on M , tlie limit
exicts and defines a Cm function on M .
(d) Treat L x Y similarly.
9. Give the argument to sliow tliat #h*Y4-,, ( P ) + YP in the proof of Proposition 8.
10. (a) Prove that
(b) How ~vouldProposition 8 have to be changed if we had defined ( L x l ' ) ( p )
as
11. (a) Show that
# * ( d f ) ( Y )= Y ( f o#).
(b) Usirig (a), sliow direc~lyfrom tlie definition of L x tliat for Y E M p ,
and conclude that
L x df = d ( L x f ) .
The formula for Lx ds' , dciived in tlie text, is jiist a special case derived in an
uiiiiecessarily clumsy way In the iiext part we get a much simpler proof that
L x Y = [ X , Y 1, usirlg the techxiique whicli appeared in the proof of Proposition 15.
(c) Let X and Y be \rector fields on M , and f : M + IR a Cm function. If X
generates (#,), define
/ t ) = yd-r(p)
( f o #h)-
174
Show tliat
Conclude that for c ( h ) = a(h,k) we llave
12. Check t1ie Jacobi identity.
13. O n R 3 let X, Y, Z be tlie vector fields
(a) Show that tlie map
is an isomorpliism (íiom a certain set of vector fields to IR3) and tliat [U,VI H
the cross-product of tlie images of U and V.
(11) Sllow tliai tlie flow of u X b y c Z is a rotation of R3 about come axis
througli 0.
+
+
14. If A is a tensor field of iype (): on N and # : M + N is a diffeomorpliism,
we define #*A on M as follows. If v i , . . . ,vk E Mp, axid A l , . . ., hr E Mp*, then
(a) Check tliat under tlie identificatioxi of a vector fie1d [or covariani vector
field] witli a tcnsor field of type
[or type
this agrees with our old #* Y.
@) If the vector field X on M generates ( # t ) , and A is a tensor field of type ( f )
on M, we define
(y)
(i)]
Sliow that
(so that
in particular).
(c) Show that
Lx,+x,A = Lx, A + Lx,A.
Hini: IVe already know tliat it is tme for A of type (:), (:), (:).
(d) Let
C : ( v )+ rJ k - I ( V )
he any contraction
= contraction of
Show tliat
Lx(CA) = C ( L x A ) .
(e) Noting that A(Xi,. . . ,Xk,wl , . . . ,w I )can be obtained by applying contractions repeatedly to A @ XI @ . @ Xk @ w~@ . @ wl, use (d) to show that
jl
...j, .
(f) If A has compoiients A,[ .,
n
in a coordinate system x and X =
o'a/axi,
i= l
show tliat the coordinates of Lx A are $ven by
I
n
E E ......4. .. ...
. -
+
l a - ] ila+]
a-1 i i l
ik
15. Let D be an operator taking tlie Cm functions F to F, and the C m vector
fields V to V, sucli that D : .F + F and D : V + V are linear over W and
(a) Show that D has a unique estension to an operator taking tensor fields of
type (f) to tliemselves such tliat
(1) D is linear over W
(2) D ( A @ B ) = DA @ 3 + A @ DB
(3) for any coatractioxi C, DC = CD.
If we lake Df = X.f and D Y = Lx Y1 tliea tliis unique exteasion is L x .
(b) Let A be a teiisoi. field of type (i), so tliat \vc cari consider A ( p ) E Elrd(Mp);
then A ( X ) is a vector ficld for each vector field X . Show that if we define
DAf = 0, DAX = A ( X ) , tlien DA has a uni-que extension satisfying (1), (2),
and (3).
(c) Show that
( D A # ) ( / ) )= - A ( P ) * ( # ( P ) ) (d) Show that
LfX = f LX - DX@df
Hillf: Clieck this for functions and vector fields first.
(r) If T is of type
slio~rtliat
(i),
a-l
Generalize to tensors of type
"-1
u-1
e).
16. (a) Let f : W + W satisfy f' ( O ) = O . Define g ( t ) = f ( A )for t 2 O. Sliow
tllat the right-hand derivative
~ ( 1 2 )- g ( 0 ) =: -.
f"(0)
fz+ O+
11
2
g'+ ( O ) = lim
(Use Taylor's Tlieorem.)
(b) Giw.11 c : W + M witli c'(0) = O E M,, define y ( t ) = c ( & ) for t 2 0.
SIIOWtllat tlic taiigeri t vectoi- cl'(0) defined by c"(0)( f ) = ( f o c)"(O) can also
be desci-ibed by c"(0) = 2 y t ( 0 ) .
17- (a) Let f : M + R have p as a critical poiat, so that f,, = O. Given
vectors X,, Yp E Mp, clioose vector fields X , Y witli Tp = Xp aild Y, = Y,.
Define
f**CXp, Y,) =
1.
rY
P
.
.
P
.
.
.",(O
Using the faci that EX, Y],( f ) = O, SIIOMJ that f,,(Xp, Y,) is symmetric, and
conclude that it is well-defined.
(b) Sliow that
(a2
(c) Tlie rank of
f / a s i a x j ( p ) )is independent of tlie coordinate system.
(d) Let f : M + N llave p as a critical point. For X,, Y, E M and g : N + R
define
f**(X, Y ) ( g )= X p ( Y ( g 0 f )).
P
.
.
P
.
.
Show that
$5.:
Mp x M, + N/(,)
is a well-defined hilinear map.
(e) If c : R + M lias O as a critical poixit, show that
takes ( l o , l o ) to the taiigent vector c"(0) defined by ctr(0)(
f ) = ( f o c)"(o).
18. Let c he tlie c u n ~of Theorems 15 and 16. If A- is a coordinate system
around p wiih x ( p ) = O, and
sliow that
x i ( c ( t ) )= uir
+ o(t2 ) ,
where o ( t 2 )denotes a function such that
lim o ( t 2 ) / t 2= O.
r+ O
19. (a) If M is conipact and O is a regular value of f : M + R, then there is
a neighborhood U of O E R such that f - ' ( U ) is dineomorpliic to f-'(O) x U,
by a diReomorphism #: f - ' ( O ) x U + f - ' ( U ) with f ( # ( p yt ) ) = t . Hini:
Use Theorem 7 and a partition of unity to construct a vector field X on a
neighborhood of f
(O) such that f* X = d / d t .
(11) More geiierally, if M is compact and q E N is a regular value of f : M +
N: then there is a neighborhood U of q and a diffeomorphism # : f
( q )x U +
f-' ( U ) with f ( # ( p , y')) = q'.
(c) Ii follo\r:s from (b) tliat if al1 points of N are regular values, tlien f - ' ( q l )
and f
( q 2 )are diffeomorphic for qi, q2 sufficiently close. If f is onto N , does
jt follo~rthai M is diffeomorphic to f ( q ) x N ?
-'
-'
-'
-'
let Y and Z be unit wctor fields always pointing along tlie y- and
r-axes, respeciively, and let X \vi11 be a vector field oiie of whose integral curves
is the s-axis, while certaixi other integral curves are parabolas in the planes
J' = constant, as shown in ihe first part of the figure below. Using the second
part of tlie figure, show ihai Propositioxi 18 does not hold in dimensjon 3.
20. In
CHAPTER 6
INTEGRAL MANIFOLDS
PROLOGUE
A matlieinatician's reputation rests oii
tlie i-iuinbei.of bad pi-oolq he has giveii.
[Pioxxeer woik is clumsy]
Beauty is the first test: there is no
permaneiit place in the world for
ugl); mat iiematics.
A. S. Besicovi tch,
quoied iii J. E. Li~lewood,
A A,faílze?nnlicin?z'sAdircellaniy
1
n ihe previous chapter, we have seen thai ihe integral curves of a vector field
on a inanifold M may be defixiable only for some small time interval, even
thougli the vector field is Cm on al1 of M. 14re will now vary our quesiion
a little, so that global results can be obtaiaed. Iastead of a vector field, suppose ihat for eacli p E M we have a 1-dimensional subspace Ap c M p . TIie
f~~nciioxi
A is cdlcd a 1-dimensional distribution (this kind of distrihution has
xiotliixig ~rhatsoeverto do with the disiribuiions of analysis, whicli iiiclude sucll
things as the "6-function"). Tlien A is spanned by a vector field ZocuZ~;that is,
wc can choosc (in niaay possilJe ways) a vecioi. field X sucli tllat O # Xq E Aq
for al1 q ixi some open set around p. 14fe cal1 A a Cm distrihution if such a
vectox. field X can be chosen to be Cm in a ncigliborliood of each point.
For a 1-dimexlsional distx-ibution tlle iiotioa of a n i~iiegralcurve niakes no
sense, I ~ u wc
i dcfine a (1-dimeiisioiial) subnianifold N of M to be a n integral
manifold of A if for every p E N we Iiave
i*(Np)
=: Ap
whei-e
i :N +M
is the ixiclusion map.
For a givcn p E M , we can always fi~ida11 iiitegral manifold N of a Cm
distribution A with p E N; we just choose a vector field X with O # Xq E Aq
for C] ir) a iieighboi-liood of p , find arr iiitegral curvc c of X with iiiitial condition
c ( 0 ) = p, a n d theri forget about the ~mrameterizationof c, by defining N to be
( c ( t ) j . Tliis al-qrinent actually slrows that for cvery p E M there is a coordinate
system (x, U) sucli tliat for each fixed sei of iiunibers u 2 , .. . ,un, the set
is an integra1 manifold of A on U: and that tliese are the only integral manifolds
in U.
TIris is still a local resuli: but because Mte are dcaling with submanifolds,
rathei- thar-i cuntes ~iritha particular parameter-ization, wfe can join over1appirig
iritcgral s~il~mariifolcls
together. The entire manifold M can he written as a
disioint uriion of con1lec ted integral submanifolds of A , which locally look like
(1-ather tlian like
or sonlething e1fer-i more coniplicated). For esample, there is a distribuiion
o11 thc IOI-LISIVIIOSC iritegral 1iinriifo1ds al1 look likc tlre densc 1-diii-ierisioriaI
submariifcitc1 pici~ircclin Cllaptcr. 2. O11 t1ie other llaiicl, tliere is a disiribution
o11 tlrc ioriis \,.liiclr 138s OIIC coiii~~act
coririectecl integral riianifold, arid a11 otlier
ir-iicgi-alniai-iifb1cIs riorr-compact. It liíippcris tIiat thc iritcgral mariifoIc1s of tI-iesc,
twfo clistril-i~iiions
are also tlie in tesral cunres for certain vector fidds, but on thc
h4d1ius strip thei'e is a distributiori \vliicli is sgarined by a \rector fie1d only
locally
IVe are leaving out tlie details invohred in fitting together these local integral
mariifo1ds because \ve ~rilleveritualty do tliis olrer again in tlie Iiigher dimensional case. For the niornent Mre \vil1 investigate higIier- dimensioiial cases onIy
locally.
A k-dimensional distribution oii M is a fiiiiction p H A,, ~tliereA, c M,
is a k-dimerisiorial subspace of M,. For any p E M tliere is a neighborhood U
arid li vcctor fields X1,. . . ,Xk such that XI ( q ) ,. . . ,Xk ( q ) are a basis for Aq,
for eacll q E U. IVe cal1 A a Cm distributioii if it is possible to choose Cm
vector fields XI, . . . ,Xq witli this property, iri a iicighborhood of eacli point p.
A (k-dimensional) submanifold N of M is calIed aii integral manifold of A if
for every p E N we have
i,(Np)= A,
where
i: N +
M
is theiriclusionmap.
Aliliough tlic dcfiiiitions givci-i so fal. al1 look the same as the 1-dimensional
case, the results \vi11 look verv different. In general, integral manifolds do ~ i o f
ex&!, even locally.
As tlie simplest cxarnple, consider the 2-din-iensional distributioii A in IR3 for
ivl-iich A, = A(,,b,,) is spanned by
l f we icler,tify TIR' witli IR"
IR3, tlien A, conrists of al1 (r, S, br-),. Thus A,
n1ay bc ~~icturccl
as tlie plaiie with the equation
The figure be1ow s h o ~ Ap
s for ~ o i n t ps =: (o, b, O). T h e plane
(u, 6, c.) is just parallel to the one through (o, b,0).
through
If you can pict~rrctl-iis disti-il~ution,you can probably see that it has no integral
manifolds; a proof can be @ven as fo1Iows. Suppose there were an integral
manifold N o i A witll O E Al. The intersection of N and ((0, y, r ) )would he a
cunfe y in the (y,:)-plane tlxrouglx O whose taiigent vectors wou1d Iiave to lie
ir1 the intersection of Ala,,,=) and tlie (y, z)-p1ane. The only such vectors have
tljird component O, so y 111ust 13e tl-ie 17-axis. Now corisider, for each fixed yo,
the iritersection N n ((A-,?lo,c ) ). This will be a curve in the p1ane ((A-, yo, z))
t11rou~l-i(O, yo, O), with a11 tangent vectors Iiavirig slope yo, so it must be the
linc ((x, yo, yox)). OUI.iiitcgi-al nianifold would have to look like the following
picture. R ~ i this
t
submanifold cloes not urork. For example, its tangent space at
( 1,0,0) contairis \fcctoi-swi th tl-iird componcn t non-zero.
To see in greater detaii what is happening liere, consider the somewhat more
general case where A(,,b,,) = Ap is
geometrically, Ap is the plane with the equation
Z-Crir
f ( f l , b ) (-~ f f ) + g ( f f , b ) ( y - b).
As in tlie first exarnple, the plaiie
through (o, b,c) wiil he paralle1 to
the one through (u, b, O ) , since f and g depend only ori a and b.
M'e riow ask wlleri tlic distribution A has an integral manifold N through each
poiiit. Since Ap is never perpendicular to the (-Y, y)-plane, the submanifold is
giveii locally as tlie grapli of a function:
No\%:tlie tangent space at p = (a, b, @(u,b)) is spanried by
Tlicse tangenr vectors are in Ap if and oiily if
So \ve need to fiild a funct~ona : R2 + R with
(4
a@
ax= f ,
- " g.
It is wcll-kr~ot~n
tliat this is not a1ways possible. Ry usirig the equality of mked
partial deri\?atives,\ve find a necessary condition on f and g :
In our previous exampIe,
so tliis iiecessary condition is not satisfied. It is also well-kriown that tlie neceshary co~~diiion
(a*) is st@riFltl for the existente of the function cr satisfying (a) in
a ~i~igllborliood
of any point.
O. PROPOSITI ON. If f , g : IR2 + IR satisfy
in a neig1iborhood of O, and zo E IR, tlien tliere is a f~inctioncr, defined in a
ncigliborliood of O E IR2, such tliat
cr (O, O) = -'o
PXOOE IVe fjrst define ~ ( xO), so that @(O, O) = o: and
riamely, wc define
X
Tlleii, for eacll x, we define a ( x , y ) so that
namely, we define
This cor~structioi~
does aot use (w),and always provide us with an a. satisfying (2), aa./ay = g. IVe claiin that if (**)holds, tlieii also aa/ax =: f. To prove
tliis, concider, for each fixed x, the function
This is O for = O by (1). To prove that it cquals O for al1 y, we just have to
S ~ I O Mthat
~
its derivative is O. Rut its derivative at y is
JI
IVc ar-e IIOW ready to look at essentially t11e rnost general case of a 2-dimensional distr-ibution in R3:
where f, g : R3 + R. Suppose tliat
N == ((x, y,z) : 6~ = a.(x,y))
is an integral manifold of A . The tangent space of N at p = (a, b, a(a,6 ) ) is
spanned, once again, by
These tangent vectors are in Ap if and on1y if
aa
f (a,b,d a , 6 ) ) = a~ ( a ,b),
aa
g(nl b, a ( o ,b ) ) = -(a,
ay
b).
In order to ohtain neccssary conditions for the existente of such a function a ,
we again use the equality of mixed partial derivatives. Thus (*) and the cllain
ru1e imply that
Tliis condition is not very usefu1, since it still involves the unknown function a ,
l ~ uwe
t can substitute from (*) to obtain
-(a, b,a(a,b ) )+ -(n,b,a(a,b ) ) g(a,b,@(a,b))
ay
az
Now wc are looking for conditions whicli will he satisfied by f and g when
there is an integral niailifoId of A Iltroqk euu-poinf, which means tliat for eacli
pair ( u ,b ) these equations must ho1d no matter what a ( a ,b ) is. Thus we ohtain
finally the necessary coiidi tion
In tliis more genera1 case, the necessary condition again turns out to be sufficient. In fact, there is no need to restrict ourselves to equations for a single
function defined on R2; we can treat a system of partial difGerentia1 equations
for n functioils o11Rm (i.e., a partial dflerential equation for a function from R"
to R"). In the following theorem, we wdl use I to denote points in Rm and x
for points in W"; so for a function f : Wm x R" + Rk we use
as. for
-
Dm+if.
axi
1. THEOREM. Let U x V c Rm x R" be open, where U is a neighborhood
of O E R"', ar-id let J i : U x V + Rn be Cm functions, for i = 1 , . . . , 1 1 1 . Then
for every A- E 1/, there is at most one function
defined in a neiglil~orhoodW of O in Rm, satisfying
(hllore precisely, ariy two sud1 furictions a1 arid a*, defined on WI and W2, agree
on tlie conil~oileritof Wi n W2 wliic1i contains O.) Moreover, such a function
exists (arid is autoriiatically Cm) in some lieighborhood W if and only if there
is a iieighborhood of (O, x) E U x V on which
I'HOOF. Uiliqueriess ~ l i l lbe obvious from tlie ~ r o o of
f existence. Necessity of
tlie conditioris (34)is lcft to the readei- as a siinple exercise, and we will concern
oursch~eswith provirig existerice if these coriditions do liold. Tlie proof will be
like tliat of Proposition O, 14th a different twist at the end.
IYe f i l ~ waiit
t
to define ~ ( rO,.
, . .,O) so that
a(0, o,. . .,O) = x
To do this, we corisider the ordinary differential equation
Tliis equation has a unique solution, defined for 111 < E I . Define
Then (1) holds for 111 -= EI .
Now for eacli fixed r l with 1 1 ~ 1 < E I ,consider thc equation
7'liis Iias a unique solution for sufficiently srnall 1. At tliis point the reader must
refer hack to Theor.ern 5-2, arid veri+ the following assertion: If wre choose ~1
sufficiently srnall, then for 11'1 < E I tlie solutiorls of tlie equations for P2 witli
tllc initial conditioiis p2(0) = Q ( I ~0,., . . ,O) will eacli be defined for 111 ~2 for
sor-ile ~2 > O. We then define
(2)
aot
-(fl,l,O
a12
,..., O) = f * ( f
1
1
,1,0,..., O,ot(1 ,1,0 ,..., O))
11'1 ( E l , 111 <E2.
l4'c claim that for eacli fixcd 1' witli
(3)
1 < E I wTealso Iiave, for al1 1 witli 111 < ~ 2 ,
o = g(1) = +fa@ l ,1, o,. . - ,O) - f l(11 ,1,0,. . O,ot(11 ,f,O,.. . ,O)).
a
,
81
Note first that
l e IIOW derive an equation for gf(1). In the following, a11 expressioris iilvoh/ing ot are to be n~aluatedat (I', 1, 0,. . . ,O) and all expressions involving fi are
to be evaluated at (rr,1,0,. . . , O , a ( l l , l , 0,. . . ,O)). l4re llave
and thus
a/z
" a/zaak as,
n
= - +axCal T ~ - ah~ - C
by (2)~
again~ ~
k
al1
k -1
k=i
by definition, (3)
Now equation (5) is a dxerential equatiori 1vit1i a uiiique solution for each
iiiitial conditioli. The soIution with initial coridition g(0) = O, given by (41, is
cleai-ly g(l) = O for al1 1. So (3) is true.
It is a simple exercise to continue rhe defiiiition of cr until it is eventually
defiiied on ( - E ~ ,E,) x
x (-E,, E,) and satisfies (*). 43
Tlieorem 1 csseiitial1y solves for us the problem of deciding which distributions
have iiitegi-a1mailifolds. Our investigation of the prob1em so far illustrates one
hasic fact about theox-ems in dxerential geometry:
T\Jaiiy of tlie fu iidamen tal theorems of differential geometry fdl in to
oiie of two classes. Tlie first kind of theorem says that if one has a
certain xiice situat ioii (e.g., a distrihution ~ 4 t hintegral submanifolds
thi-ough every poiiit) theii certaiil othei. conditions hold; these conditioils are obtaiiied by settixig mixed partials equal, and are called
"integrahility coilditions". The second kind of theorem justifies this
terminoIo0, by diowiiig that the "integx.ahility coxiditions" are sufficient for recovering the nice situation.
The reniaiiliiig parts of oui. investigation, iii which we will essentially begin
aiiew, illustratcs axi eicii inoiae impoi.tant fact about the theorems of dxerential
geometry:
Tliere are a1ways iiici-edibly concise axid elegarxt ways to state the integrahility conditions, aiid prove tlieir sufficieiicy, without ever even
meii tioiiiiig partid derivatives.
LOCAL THEORY
If f : M + N is a C m function, and X and Y are Cm vector fields oii M
and N, respectively, we say that X and Y are f-related if f*p(Xp) = Y'(p) for
each p E M. If g : N + R is a Cm function, then
so
IYg) 0 f = X I f og).
Conirersely, if this is true for al1 Cm functions g : N + R, then X and Y are
f-related.
Of course, a given vector field X may not be f-related to any vector field Y,
nor must a given vector field Y he f-related to any vector field on M. Iii one
case, the latter condition is fulfilled:
2. PROPOSITION. Let f : M + N be a Cm function such that f is an
immersion. If Y is a C m vector field on N with
fP*IMP)?
then there is a unique Cm vector field X on M which is f-related to Y.
Yf(p)
f
PROOF. Clearly we must define Xp to be the unique element of Mp with
Y/(,) = fp,Xp. To prove that X is C m , we use Theorem 2-lO(2): tliere are
coordiiiate systems (x, U) arouild p E M and (y, V) around f (p) E N sud1
that
y o f o r - ' ( o 1 ,...,un)= (o 1 ,..-,o n ,O,.-.,O).
This is easily seen to imply that
(a,)d,,,
a
fp*
Thus if
=
"
a
y=Cruiay'
i=l
where a' are Cm fuiictions, then
q
where a' o f = j?'. This implics that the functions j?' are C m (Problem 3). 4+
The moct iinportant property of f-relatedness for us is the following:
3. PROPOSITION. If Xj aiid Yj are j-1-elated, foi. i = 1,2, then 1x1,X2] aild
[Yl,Y,] are f-related.
PHOOI;. If g : N + iR is Cm, then
by (I), with g replaced by Y2g and Yig, respectively
= XI ( X 2 k o f))- X2(Xi(g o f)) by (1)
= Exi,X21(g ~ f )6
.
Now coiisider a k-dimensional distribution A . M7e wi1l say that a vector
field X belongs to A if Xp E Ap for al1 p. Suppose that N is an integral
inaiiifold of A , aiid i : N + M is t1ie indusion map. If X and Y are two vector
fields which belong to A , then for al1 p E N there are unique y,,
E N, such
that
Xp = i,Xp, Yp = i*Yp.
In otlier words, X and are i-related, and Y and I are i-related. Proposition 2
shows tliat F aiid y are Cm vector fields on N, and Proposition 3 then shows
that IR, 81and [X, Y] are i-related. Thus
- -
- -
i* Ex, Ylp = [X, Ylp.
Here EX, YIp E Np; tliis therefore shows tliat EX, YIp E Ap. Consequendy, if
tliere is ail integra1 nlaiiifold of A through every point p, then EX, Y] ah belongs
lo A.
For a moment look back at tlie distribution A in IR3 given by
The vector fields
x = -a +f-a
az
ax
bcloiig to A . Usiilg the foi.mula on page 156, we see that
This beloiigs to A only wlien tlle expressioil i11 pai-ei~tl~eses
is 0, which is precisely
the condition for A to have an integral nianifold through every point.
In general, A is called integrable if EX, Y ] beloilgs to A whenever X and Y
belong to A . This condition can be checked fairly casily:
4. PROPOSITION. If Xl,. . .,Xk span A in a iieighborhood U of p , theii A
is integrable on U if and only if each EXi, Xj] is a linear combination
k
[X;, Xj] =
E CGX.
rr=l
for C m functions CG.
PXOOI;. Such functions clearly exist if A is integrable, since EXi, Xj]q E As,
. liich is spaiiiled ly tlie X,(q). Coilversely, suppose such functions exist. If X
aiid Y belong to A we caii clearly write
To prove EX, Y] belongs to A , it obviously suffices to treat each [j;.Xi,gj Xj]
sepai+ately.Since we have
clearly [f X, g Y ]bcloiig~to A if X, Y and EX, Y ]do. *:*
We are now ready for tlie maiii t-lieorem. It is equivalent to Theorem 1; in
fact, Theoi-em 1 can be derived from it (Problem 7). Rut the proof is quite
diffei-ent.
5. THEOREh4 (THE FRORENIUS INTEGRARILITY THE0REh.I;
FIRST VERSION). Let A be a C m iiitegrahle k-dimensional distribution
oii M. For evcry p E M tliere is a coordinate system (A-,U) with
x(p) = 0
x(U) = (-E, E) x . - x (-E, E),
siicli tliat for each ak+', , .. , a n with al1 lail e E, tlie set
( q E U : x ~ + ' (=
~ nk+'
)
, . . . , xn(q) = n")
is an iiitegi-al manifold of A.
An): connected integral maiiifold of A i-estricted to U is contained in one of
tliese sets.
PX001;. We can clearly assume that we are in IR", with p = O. Moreover, we
can assume tliat Ao c Wno is spanned by
Let s : IR" + IRk be projection onto the first k factors. Then rr, : A. + Wko is
an isomorphisin. Ry contiiiuity, s. is one-one oii A, for q near O. So near O,
we can clioose unique
Xi(q), - .,xk(q) E Aq
so that
r*xi(q) = 2
i = I , ..., k.
a
Tlien the vector fields X j (oii a neigliborliood of O E W") and
rr -related. Ry Propositioii 3,
a/ar (on IRk) are
Rut, EXi, Xj]q E A, by assuinption, aiid rr, is one-one on A,. SO EXi, Xj] = 0.
Ry Tlieorem 5-14, tliere is a coordinate system x such that
Thc sets {q E U : xk+' ( q ) = n k+i ,. .. ,xH(r/) = 0") are clearly integral manifold~of A, siiice tlieii- tangeiit spaces are spaniied by the a/axi = Xi for
i = 1, ..., k.
If N is a coniiected in tegi-almanifold of A restrictcd to U, with iriclusion map
i : N + U, coilsider d(xn' o i) for k
I 5 rn 5 n. For any tangent vector X,
of N, we have
+
c/(smo i) (X,) = X, (xmo i) = i* X, (xm)
= o,
sirice i. Xq E A,, wliicli is spanned by the a/axjl, for j = 1, . .. ,k. Thus
d(x-"' o i) = 0, which implies tliat smo 1 is constaiit on tlie conriected manifold N.
GLOBAL THEORY
In order to express the global results succinctly, we introduce the following
terminoloa
If M is a Cm mailifold, a (usually disconnected) k-dimensional submanifold N of M is called a foliation of M if every point of M is in (some compoiient of) N , and if arouild every point p E M there is a cooi-dinate system
(A-, U ) , with
x ( U ) = (-E, E) X - X (-E, E),
such that the components of N n U are the sets of the form
{ q E U : xk++'( q ) = nk+l ,. .. , x n ( q )= n n )
10'1 < E.
Eacli compoiierit of N is called a folium oi- leal of the foliation N. Notice that
two distinc~componcnts of N n U might belong to tlle same leaf of the foliation.
6. THEOREh4. Let A be a Cm k-dimeilsional integrable distribution 011 M.
Then M is foliated by an integral manifold of A (each component is called a
maximal integral manifold of A).
PROOI;. Usirlg Tlieorem 1-2, we see tllat we caii cover M by a sequerice of
coordinate systems ( x ; ,U j ) satisfying the coiiditions of Theorem 5. For such a
coordinate systein ( x ,U ) , let us cal1 eac11 set
{ q E U : xk+l ( q ) = nk+I , . . , , x N ( q )= 0 " )
a s l i ~ of
t U.
It is possible for a single slice S of U; to intersect U, in more than one slice
of U j , as shown bclow. Rut S n U j has at most countably many components,
and each component is contained in a single slice of Uj by Theorem 5, so S nUj
is contained in at most countably many slices of Uj.
Given p E M, cl~oosea coordinate system (xo,Uo) with p E Uo, and let So
be the slice of U. contaii~ingp. A dice S of some Ui wil1 be ca11edjoined to p
if there is a sequence
. .
O = r o , ~ ~ , . . .=, ii ~
and corresponding dices
with
a =O,. . . , I - l.
Sia n S,+, # 0
Since ttiere are at most countably many such sequences of slices for each sequence io,. . . ,il, and only countably many such sequences, there are at most
countably many slices joiiled to p. Using Problem 3-1, we see that the union
of al1 such slices is a submanifold of M. For q # p, the corresponding union
is either equal to, or totaUy disjoint from, the first union. Consequently, M
is foliated by t1ie disjoint union of a11 such submai~ifolds;this disjoint union is
clearly an integral manifold of A . +:+
[If we are allowing non-metrizable manifolds, the proof is even easier, since
we do not have to find a countable number of coordinate systems for each leaf,
and can merely describe the topology of the foliation as the smallest one which
makes each slice an opei-i set. In this case, however, the discussion to follow
will not be valid-in fact, Appendix A describes a non-paracompact manifold
whic11 is foliated by a Iower-dimensional conneckd su bmani fold.]
Notice that if (x, U ) is a coordinate system of the sort considered in the proof
of tlie theorem, then ii~finitelymany slices of U may belong to the same folium.
However, a/ rizos/ coun1ab3,mang slices can belong to the same folium; othemise
tliis folium would contain an uncountable disjoint family of open sets. This
allows us to apply a proposition from C hapter 2.
7. THEOREh4. Let M be a Cm manifold, and Mi a folium of the foliati011 determiiied by some disti-ibution A. Let P be aiiother Cm mai~ifoldand
f :P + M a Cm function with J ' ( P ) c M i . Then f is Cm considered as a
map into M I .
PROOI;. According to Proposition 2-1 1, it suffices to sllow tliat f is continuous
as a nlap iilto M I . Given p E P, choose a coordiilate system ( x ,U ) around
f ( p ) such that tlie slices
( q E U : ,yk+' ( q ) = ak+],. . ., x n ( q ) = a " )
are integral maizifolds of A . Now f is coi~tinuousas a map into M, so f takes
some neighborhood W of p into U; we can choose W to be connected. For
k + 1 5 i 5 t i , if we had x i (f (P')) # ni for any p' E W, iheii xi o f would take
on al1 values between n' and xi( f ( p t ) ) , by continuity. This would mean tliai
f (W) contained points of uncountably many dices, contradicting the faci t l ~ a t
f ( W ) c M1
Consequei~tly,~ ' ( f ( ~ =
' ) ni
) for al1 p' E W. In other words, f ( W ) is
contained in t l ~ esingle dice of U whicl~contains p. This makes it clear that J'
is continuous as a map into M I . +3
PROBLEMS
1. (a) Let { = ír: E + 3 be an n-plane bundle, and {' = ír': E' 4 3 a
k-plane bundle such that E' c E. If i : E' + E is the inclusion map, and
Ig : 3 + 3 the identity map, we say that {' is a subbundle of { if (i, I B ) is a
bundle map. Show that a k-dimensional distribution on M is just a subbundle
of TM.
(b) For the case of Cm bundles { and {' over a Cm manifold M, define a Cm
subbundle, and sliow tl~ata k-dimensional distribution is Cm if and only if it is
a Cm subbundle.
2. (a) In the proof of Theorem 1, check the assertion about cl~oosingE I sufficiently small.
(b) Supply tlie proof of the uiiiqueness part of the theorem.
3. (a) In the proof of Proposition 2, show tliat
(13) Complete the proof of Proposition 2 by showing tl~atif
with oii o f = s i , then tlie fui~ctionsBi are Cm
4. In the piaof of Proposition 4, show that the functions CG actually are Cm.
.
5. Let A l , . . .,Al, be integrable distributions on M, of dimeiisions d i , . . ,di,.
Suppose tliat for eacli p E M,
Show that tllere is a coordinate system (x, U ) arouiid each point, sucli t l ~ a tAl
is spaiined by a / a x l , . . . , a/axdl, etc.
6. Prove Theorem 1 from Theorem 5, by considering the distribution A in
Rm x R" (with coordinates I, x), defined by
Notice that everl when tbe fi do not depend on x, so that the equations are of
the form
with the integrability conditions
we neirertheless work in Rm x R", rather than Rm. This is connected with the
classical technique of "introducing new independent variables".
7. This problem outlines another method of proving Theorem 1, by reducing
tbe partial dfirential equations to ordinary equations along lines through the
origin. A similar tec hnique will be very important in C hapter 11.7.
(a) If we want a ( u t ) = p(u, I ) for some function 8 : [O, E ) x W + Vy show
tliat p must satisfy the equation
We know that we can solve such equanons (we need Problem 5-5, since the
equatiori depends on the "parameter" I E Rm). One has to check that one E
can be picked wl-iich works for al1 I E W.
(b) Sllow that
p(u, V I >= ~ ( u u1).?
(Show tliat both functions satisfy tl-ie same differential equation as functions
of u, witli tl-ie same ii-iitial condítion.) Ry sl~rinkingW, we can consequently
assume that E = 1.
(c) Conclude that
(d) Use tl-ie integrability condition on f to show that
satisfy tlie same diffei.eiitia1 equation, as functions of v. Use (c) to conclude tliat
the two functions are equal.
(e) Define a(r) = P(1,l). Noting that a(vr) = P ( v , f ) , sliow that a satisfies the
desired equation.
8. This pi-oblem is for tliose who know something about complex analysis. Let
f : @ x @ + @ be complex analytic. If we denote the coordinate functions in
@ x C by -1, z2 = X] ylYx2, y2, then f = u
y
+ iv satisfies the Caucliy-Riemann
equations
Use Theorem 1 to p r o i ~tliat we can solve the equations
aa2
-=
ax
aal
v ( x , y , a l ( x ,y ) , a 2 ( x ,y)) = - ay
iii a neighborl-iood of O E @ (or of ariy poil-it zo E C), aiid coiiclude that the
differential equation
#'(d = f (z, #(E))
(iii ~ h i c h' denotes the coinplex derii~atiire)has a solution in a neigl~borhood
of ro, with any giireii iiiitial condition #(zo) = wo.
CHAPTER 7
DIFFERENTIAL FORMS
w
e turri our attention once more to teiisor fields, but we will be concerned
witli a special kind of tensor field, the discussion of which requires some
more algebraic preliminaries.
Let V be aii n-dimensional vector space over R. An element T E T k ( V ) is
called alternating if
If T is alternatiiig, then for aiiy vi,. . . , vk, we have
Tlierefore, T is skew-symmetric:
O f coui.se, if T is skew-symmetric, tlien T is also alteri~ating.[This is not true
iii the special case of a vector spacc ovei. a field ivhcre I I = O; i i ~this case,
skew-symmetry is tlie same as symmetry, and the condition of being alternating
is the strongei. one.]
W will denote hy R k ( V ) the set of al1 alternatiiig T E T k ( V ) . It is clear
tliat R k ( V ) c 7 " ~ )
is a subspace of T k ( V ) . Moreover, if f : V + W is a
': 7k(W) + 7 ' ( V ) preserves tliese subspaceslinear traiisfoi.niatioii, tlien 1
f * :(
W + R k ( . Notice tl~atR 1 ( V ) = T 1 ( V ) = V*, so R 1 ( V ) has
dimension j7. It is also convenient to set R O ( V )= 7 O ( V ) = R. At the moment
it is iiot cleai 12~liattlie dirnension of Rk (v) equals for k > 1, but one case is
1ve11-kiiowii. The most familiar example of aii alternating T is the determinant
fuiiction det E T"(Rn), c.onsidered as a function of the n rows of a matrixwc shall sooii see that tliis fui~ctionis, iri a certain seiise, tlie most general
al tei-iiatiiig furiction- hdost discu~sioiisof tlie determiiiant begin by s howing
tliat of any t~voalterriatiiig 17-linear fuiictions oii Rn, one is a multiple of the
+
other; i i i other ivords, dim a"(R" ) 5 1. Then one proves dim S2"(Rn) = I
bv actually constructing the non-zero function det (it follows, of course, that
dim 52" ( V ) = I if V is any 11-dimensional \rector space). The construction of
det is usua1ly bv a messx explicit formula, which is a special case of the definition
to follow.
Let Sk denote the set of all permutations of (1,. . . ,k); an element E Sk is
a function i H ~ ( i )If. ( U ] , .. . ,uk) is a k-tuple (of any objects) we set
This defiriition has a built-i11 confusion. On tlie riglit side, the first element,
for example, is t l ~ eo(I)" of tlie u's on t11e Ieft side; if these u's Iiave indices
1-unningin some order ofhm than 1,. . . ,k, then the first e1ement on the right is tior
iiecessarily tliat u ~rlioseiiidex is o ( ] ) .The simplest way to figure out somet11ing
like .(u3, u2. U ] ., . . ) is to renanie tliings: vs = W I , u2 = wl,U I = w3,. . . . Tlius
warried, we computc
by setting
so that
Thus
No\v foi- aiiy T E T'(v) we defiiic tlie "alternati011 of T"
wliei-e sgn
is + I
if a is an even permutation aiid -1 if o is odd.
1. PROPOSITION.
(1) If T E T k ( v ) ,then Alt(T) E Q k ( v ) .
(2) If o E Qk(v), then A l t o = o.
(3) If T E Tk(V), then Alt(Alt(T)) = Alt(T).
PK001;. Left to the reader (or see pp. 78-79 of Cafcuiuson Manifofds).+3
Mle riow define, f o r o E Qk(V) and fi E Q'(v), an element o ~ fE iQk+'(v),
the wedge product of w and fi, by
The funily coefficient is not esseiitial, but it makes some things work out more
nicely, as we sliall soon see. It is cIear that
(1) A is bilinear:
(01
+02)Afi=01
Afi+02Afi
(2) f*(w A fi) = $*oA f '7.
Moreover, it is easy to see that
(3) A is "anti-commutative": o A fi = (- I) k l fi A o .
111particular, if k is odd then
FinaIIy, associativity of A is proved in the following way
2. THEOREM.
(1) If S E T k ( v )and T E T'(v) and Alt(S) = O , tlien
Alt(S @ T )
-
N t ( T @ S ) = 0.
(2) ~ i t ( ~ l t@( o @ e )= A I ~ ( W@ fi @ e) = ~ l t@(~ ~l t@e)).
( ~
(3) If o E Q k ( v ) ,fi E Q2(V), 0 E Qm(V),then
E sgno - ( S 8 T ) . ( a - ( v ~ , . . . ~ v k + l ) )
= E sgn o S ( ~ U ( I. .)., v ~ ( k ) )T(v~(k+i),
. . .,vo(k+I)).
=
Y
ufiSk+r
NOMIlet G c Sk+lcoiisist of all o ivliicli leave k
+ 1 , . . . ,k + I fixed. Tlieri
E sgno - S(U,(I),. . . , ~ o ( k ) )T(vo(k+~), VU(~+I))
+
3
UEG
Suppose noMt tliat o0 $ G. Let ooG = (a0or: a' E G). Tlien
= sgn o0
.
sgriof . (S @ T ) ( o l . (oo. ( V I ,... ,vk+]))) .by (*).
UIEG
M7e baile just slio~lritliat tliis is O (since a0 ( v i , . . ., vk+l) is just some other.
(k + 1)-tuple of vectors). Notice tliat G n aoG = 0, for if o E G n ooG, then
o = oOalfor sorne o' E G, so a0 = o(of)-' E G, a contradiction. We can theii
coiitiiluc iil tliis way breaking Sk+]up into disjoint slibsets, tlie sum over eacli
l~eiiig0. Tlie relatioii Alt(T @ S) = O is proved similarly
(2) Clearly
Alt(Alt(l7 @ O ) - q 63 O) = Alt(q @ O) - Alt(77 @ O ) = 0,
tlie otlier equality is proved similarl.
++
+
++
(k I m)!
Alt((w A q) @ 0)
(k I)! m!
(k I m)! (k I)!
M t ( o @ q @ $1.
(k I)! n7! k!l!
Tlie othei- equality is proved similarly. S+
(#A
q ) A @=
+
+
+
+
Notice tliat (2) just states tliat A is associatii;e eiren if we had omitted tlie
factor (k I)!/ k! I! in t l ~ edefiilitioii. On tlie otlier liand, tlie fac.tor l/k! in tlie
definition of Alt is essential-without it, we would ilot have Alt(Alt T) = Alt T,
and the first equation in tlie proof of (2) would fail. [If we had defined
just
like Alt, but iwithout the factor l/k!, then A could be defined by
+
Tliis inakes sense, E I ~ Coum
~
njeId ofjnik clla~ackklic,because each term in tlic
sum At(o @ q)(vl,. . . ,
occurs k!I! times (siiice o and q are alternating),
and l/k! I! can be interpreted as meaning that tliese k!l! terms are replaced by
just one.] Tlie factor (k +l)!/k!I! Iias been iilserted into the definition of A
for the folloiving reason. If u,, . . . ,u,, is a basis of V, and #], . . .,#, is the dua1
basis, thcii
#] A * . . A#,
=
(1
+ . + I)!
1!.*-1!
sgno
=
a ( # ]
W # l
@
. @ #,)
@..-@#,Joo.
S,,
In particular,
($1
A
. A #n)(v1,
. . . ,un) = 1.
(So if V I , . . . ,v, is tlie standard basis for Rn, tlieil #] A . . . A #, = det .) A basis
for Slk(v) can iiow be described.
3. THEOREh4. Tlie set of al1
is a basis for R' ( v ) , wliich tlierefore Iias dimension
(In particular, Slk(V) = (0)for k > n.)
PKOOI;. 1f o E R'(v) c T'(v), we can write
A #jk for some
8 #),
is either O or = =t(l/k!)#jl A
Eacli At(#il 8
< j k span f i k ( v ) . If
jl c . c jk, SO tlie elenlents #jl A . . A #jk for j i <
t h e i ~applying b0t11 sides to (vil,. . . ,vik ) $ves uil ...jk = 0.
4. COROLLARY. If w,. . . ,wk E f i l ( V ) , tlieii 01,. . . ,wk are ljnearly independent if a i ~ donly if
O] A . * . A O k # O .
PHOOI;. If 01 ,. . . ,wk are liiiearly independent, there is a basis vi,. . . , VA,.. . ,v,
of V sucli tl~atthe dual basis vectoi-s $1 ,. . . ,#k, . . . , satisfy #j = W j foi'
I 5 i 5 k. Tlien 01 A . . . A wk is a basis elemciit of R ~ ( v )so
, it is not O.
O n t l ~ eother hand, if
then
To abbi-eviate foi-mulas, it is coi~vciiieiitto let 1 deiiote a typical "multi-index"
(ii, . . . ,ik), and let #I ddeote #jl A . A #i,. Tlien cveq elcmeni of f i k ( V ) is
Notice that Throi-cm 3 implies tliat nrci-y o E f i k ( ~ "is) a lineal- combiiiatioii
of tlie functions
Oile more siiiiple theoi-em is iii ordei; befoi-e we proceed to apply oui- coiistructiorl to manifolds.
5. THEOREM. Let vi,. . ., u, be a basis for V, let w E Qn(V), and let
j l i
Then
o(w], . .. , wn) = det(aij) w ( v ~.,. . , U,).
a
PR00E Define 17 E T"(Rn) by
Theii clearly 7 E S2"(Rn),so 17 = c det for some c E R, and
6. COROLLARY. If V is 11-dimensional and O # ¿o E Qn(V), tlzen there is a
unique orientation p for V such that
[ut,.
. . ,U,,] = ,u if and only if
VI, . .,un) > 0-
Witlz our izew algebraic coiistructioiz at lzaizd, we are ready to apply it to
\rector buildles. If
= ?r : E + B is a \lector buiidle, we obtain a new bundle
Qk(<)by replacing cach fibre rrwl(p)witli ~ ~ ( a - ' (A~section
) ) . w of ak(<)
is a function with w(p) E fik(?rwi( p ) )for eacli p E B. If 17 is a section of al(<),
tlien we caii defiiie a section w A 17 of QkU(<)by (w A q)(p) = w(p) A q(p) E
nk+'(?r-l( p ) ) .
In particulai; srctioiis of R ~ ( T M wliicli
),
are just alternating covariant tensor
fields of 01-der k, al-e called k -forms on M. A 1 -form is just a covariaizt vector
field. Siiice Qk(TM) can obviously be rnade into a Cm vector bundle, we can
speak of Cm forms; al1 forms will be uizdei-stood [o be Cm forms unless tlze
con trary is explicitly stated. Remember tlzat covariant tensors actually map
contravai-iaiztly: If J': M + N is Cm, aizd o is a k-form on N, tlzen f* w is a
k-forin oii M. iJe caii also define wi o 2 aiid o A v. The following properties
of k-forms a i r obvious fronz tlie coi-1-espoiidiizgproperties for n k ( V ) :
<
+
If (x, U) is a coordinate system, then the dxi(p) are a basis for M,*, so the
dxil( p ) A . A dx'a (p) (i] < . . < ik) are a basis for nk(p). Thus every
k-form w can be ~ ~ i t t uniquely
en
as
or, if we denote dxil A . . . A dx'" by dxx' for the multi-index I = (i],. . . ,ik),
The problem of finding the relationship between the wl and the functions o'!
when
w=
01dx' =
w ' ~dyJ
E
E
I
I
is left to tl-ie reader (Problem IG), but we will do one special case liere.
7. THEOREM. If f : M + N is a Cm fui~ctionbetween n-manifolds, (x, U)
is a coordinate system arouiid p E M, aild (y, V) a coordinate system around
q = f ( p ) E N , then
f * ( g c¡yl A - . A dy") = (g o f ) . det
PROOI;. It suffices to sliow that
Now, by Problem 4-1,
= dyl (q) A .-
("2")
dxl A
. A d,.'
8. COROLLARY. If ( x , U ) and (y, V) are two coordinate systems on M and
then
11 = g det
PKOOI;. Apply the tlieorem with f = identjty map. *t.
[This coro11ary S hows tliat 12-forms are the geometric objects corresponding to
the "even scalar densities" defined in Prob1em 4-10.]
If t = is : E + 3 is an 12-p1ane bundie, then a nowlrue zuo section o of 52" ( t )
has a special significance: For each p E 3,the non-zero w(p) E Rn(x-'(p))
determines an orientation pp of is-'(p) by Corollary 6. It is easy to see that
the collection of orientations (pp) satisfy the "compatability condition" set forth
in Chapter 3, so that p = ( p p ) is an orientation of t . In particular, if there is
a nowhere zero n-form o on an 11-manifold M, then M is orientable (Le., the
bundle TM is orientable). The converse also holds:
9. THEOREM. If a Cm manifold M is orientable, then there is an n-form w
on M which is nowhere O.
PKOOI;. By Theorem 2-13 and 2-1 5, we can choose a cover O of M by a collection of coordinate systems ((x, U)), and a partition of unity (#u) subordinate
to O. Let p be ail oriei~tationof M. For eacll (A-,U) cl-ioose an n-form wu
on U sucl-i that for v i , . . .,u, E Mp, p E U we have
wu(vi,. . . ,u,) > O
if and only if
[u],. . . ,u,,] = pp.
Now let
Then w is a Cm rz-form. Moreover, for every p, if v i , . . . , v, E Mp satisfy
[ V I , .. .,U,] = pp, then ea4
and strict iiiequality holds for at least one U. Thus o ( p ) # O.
+
Notice that the bundle Qn(TM) is I-dimensional. We have shown that if M
is orientable, then Qn(TM) has a nowhere O section, which implies that it is
trivial. Conversely, of course, if the bundle Qn(TM) is trivial, then it certainly
has a nowhere O section, so M is orientable. [Generally, if is a k-plane bundle,
tlien Qk(<)is trivial ifand only if is orientable, provided that the base space B
is L ' p a r a ~ ~ m p a(every
~ t ' ' open cover has a locally-finite refinemen t).]
Just as QO(V) has been introduced as another name for R, a O-Form on M
will just mean a function f on M (and f A o will just mean f o). For
eilery O-form f we have the I-form df (recall that df (X) = X(S)), which in a
coordinate system (x, U) is given by
<
If o is a k-form
I
then each d o r is a 1-forin, and we can define a (k
of o , by
+ 1 )-form d o , the diflerential
It iurns out tl~att11is definition does not depend on the coordinate system. Tllis
can be 131-ovedi i i several ways. The first way is to use a brute-force computation:
comparing the coefficients otl in the expression
1
witli the o].
Tlie second method is a lot sneakie~Mfe begin by finding some properties of
do (sti1l defined witll respect to this particular coordinate system).
1 0. PROPOSITION.
+
+
(1) d ( o l os) = d o , d o 2 .
(2) If wl is a k-form, then
PK001;. (1) is clear. To prove (2) we first note that because of (1) it suffices to
consider only
01
=fdx I
0 2 = g dx
Then wl A y = f g dx'
3
.
A d x 3 and
d ( o l A 02) = d ( f g ) A dx'
A dx3
=gdf~dx'~dx+
' f d g ~ d x ' ~ d x J
-
=dwl A w 2 + ( - 1 ) ~ f d x 2 ~ d g ~ d x 3
dwl A m2 +(-1)
k
m1 A dw2.
(3) It cleal-ly sufices to consider only k-forms of the form
w = f dx'.
Then
In this sum, the terms
cancel in pairs. +:*
iVe riext note tl~att11ese propei-ties characterize d on U.
1 1 . PROPOSITION. Suppose d' takes k-foi-ms on U to (k
for al1 k, a i ~ dsatisfies
+
+
(1) d'(w1 w2) = d'wl d1w2.
(2) d'(w1 A w 2 )= d'wl A w2 (- 1)'wl A dtw2.
(3) d'(dff ) = 0.
(4) d'f = (the old) d j :
Then d' = d on U.
+
+ 1)-forms on U,
PROOF. It is clearly enough to show that d'w = dw when w = f dx'. Now
by (21,
d'( f d x ' ) = d y A dx'
+ f A dr(dx')
= df ~ d x ' +f A d ' ( d x r ) by (4).
So it sufices to show that d r ( d x ' ) = 0, where
We will use induction on k . Assuming it for k - 1 we 11ave
d t ( d x ' ) = d'(d'wii A . . . A d t ~ - ' k )
= d'(d'xil) A d'Xi2 /\
.. . A d'x*
- d ' r i i A d'(d'xii A . . A d'xi*)
by (2)
by (3) and the inductive hypothesis. 6
= O - O,
12. COROLLARY. Tl-iere is a ui-iique operator d from the k-forms on M to
the ( k 1)-forms on M , for al1 k , satisfying
+
and agreeii~gwit1-i the old d on functions.
PROOI;. Foi- each coordinate system ( x , U ) we have a unique du defined.
Given t l ~ eform o , and p E M , pick any U with p E U and dcfii~e
Tlie thii-d way of proving tl~atthe defii~itionof d does not depend on the
coorclinate s w e m is to give an invariant definition.
For dw, as for any form, we have
It is clear from (*) that dw(i3/axffl,. . .,a/axffk+1) = O
Since the a's are increasing, this happens only if
in which case
which isjust the old definition. ++:
This is our first real example of a n invariant definition of a n important tensos,
and our first use of Theorem 4-2. We d o not find d w ( p ) ( v l , . .. ,vk+]) directly,
but first find d w ( X l , . . . ,Xk+]), where Xi are vector fie1ds extending vi, and
then evaluate this fuilction at p. By some sort of magic, this turns out to he
iiidependent of t1ie extensions Xl ,. . . ,Xktl. T11is may not seem to be much of
a n improvement over using a coordinate system and checking that the definition
is iildependent of the coordinate system. Rut we can hardly hope for anything
better. After al], although d w ( X l , . . . , Xk+])(p) does not depend o n the values
of Xi except at p , it dom depend oii the values of o a t points other tllan pthis must enter into our formula someliow. O n e otller feature of oui- definition
is common to most iiivariant definitions of teiisors-the presence of a term
involviiig brackets of various vector fields. This term is what makes tlie operator
I j ~ e a over
r
the Cm functions, but ii disappears iii computations in a coordinate
system.
In the particular case where w is a 1-form, Theorem 13 Gves the following
formula.
TIiis enables us to state a second version of Theorem 6-5 (The Frobenius Integrability Tlieoreni) in tei-nis of dxereiitial forms. Define the sing R ( M ) to be
the direct sum of the rings of 1-forms on M, for a11 I. If A is a k-dimensioi~al
distribution oii M, then $(A) c R (M) ~rilldenote the subring generated by
ihe set of al1 forms o wiili the property that (if o has degree i)
wlieiiever Xl, .. .,XI be1ong to A.
o(X,,. ..,XI) = O
It is clear that ol + o1 E $(A) if q,o1 E $(A), and that q A o E $(A) if
w E $ (A) [tiius, J(A) is aii ideal iii the riiig R (M)]. Locally, tlie ideal $(A)
is generated by n - k independent 1-forms wk+l, . . . ,wn. In fact, around any
point p E M we can choose a cooi-diiiate system (x, U ) so that
Then
d ~ ' ( A~ .) . A d ~ ~ is( non-zero
~ ) on AP.
By coniiiiuity, tlie same is true for q sufficiently close to p, which by Corolare linearly independent in A,. Therelary 4 implies that dxl (q), . . .,dxk
such that
fore, tliere are Cm functions
fa
k
dx"(q) =
E fa(q) dxB(q) reskkkd lo A,
a = k + l , ...,n.
,g= 1
We can therefore 1ei
k
14. PROPOSITION (THE FROBENIUS INTEGRABILITY TMEOREM;
SECOND VERSlON). A distributioi~A on M is iiitegrable if and on1y if
d($(A)) c $(A).
PXOOI;. Locally we can dioose 1-forms w', . . . ,w" which span Mq* for each q
sudi that ok" , . . . ,m" generate 4 (A). Let Xl ,. .. ,X, be the vector fields with
T h e n XI, . . . ,Xk span A. So A is integrable if and oiily if tliere are functions C:
with
k
Now
doa (Xi,Xj) = Xi (m' (Xj)) - Xj (w' (X;)) - wa ([Xf, XJ1).
For 1 5 i, j < k and a > k , the first two terms o n the right vanish. S o
dwff(Xi,Xj) =; O if and only if wff([Xi,XjJ) =: O . But each wlY([Xj,Xj]) = O
if and only if each [Xi, Xj] belongs to A (Le., if A is integrahle), whi1e each
dwa(Xi, X,) = O if and only if dw' E 4(A). $+
Notice that siiice tlie wi A W J (i
wri te
=
E 0;
j) spaii R 2 (Mq) for eacli q , we can always
A wJ
for certain forrns 0;.
j
If a > k , a n d io, jo 5 k are distinct, we have
so we can write the condition d(4(A)) c 4(A) as
dw' =
E 0;
A oB.
0 1 c e we have iiiti-oduced a coordinate sysiem (A+, U ) such that the slices
are integral sul~maiiifoldsof A, tlie forms dxk+', . . .,dx" are a hasis for 4(A),
SO wk+l , . . . ,wfl must lrie linear combinatioiis d tliem. i47e therefore llave the
fol1owing.
15. COROLLARY. If o'+',. . .,on are linearIy independent 1-forms in a
neighhorliood of p E M , then there are 1-forms 19; (a,/3 > k) with
if and oiily if t here are functions f g ,g B (a,B > k ) with
Nthough Theorem 13 warms the heart of many an invariant lovet, the cases
k > I will hardly ever he used (a very significant exception occurs in the last
chapter of Volume V). Prohlem 18 @ves another invariant definition of dw,
using induction on the degree of o, which is much simp1er. T h e reader may
refiect on the difficulties which would be involved in using the definition of
Theorem 13 to prove the following important property of d :
16. PROPOSITION. If f : M + N is Cm and o is a k-form on N , then
f * ( d o )= d ( f *o).
PH001;. For p E M , let ( x ,U ) he a coordinate system around f ( p ) . )Ve can
assume
= g d x " A . . . A dx'k.
We wil1 use induction on k. For k = O we Iiave, tracing through some definitions,
f * ( d g ) ( X ) = & ( f * X ) = [ f * X l ( g )= X ( g o f )
= d(g 0 f ) ( X )
(and, of course, f * g is to he i n t e r ~ r e t e das g o f ) . Assuming the formula for
k - 1, we have
d ( f ' o )= d (( f * g dxil A . - A dxh-l) A f *dxik)
= d ( f * ( g d x i l A ... A d x i h 1 ) ) A f*dxik + O
since d f *dxi' = d d ( x i k o f ) = O
= f * ( d ( gdxil A . . . A dx'k-1)) A f *dxlL
by t1ie inductive liyposthesis
= f ' ( d g A dxil A . . . A dxk-1) A f *dx"
f * ( d g A dxil A . . . A dxih-] A dxik)
= f * ( d o ) . *t*
One propert)' of d qualifies, by the criterion of the previous chapter, as a
hasic theorem of differential geometry The relation d 2 = O is just an elegant
way of stating that mixed partial derivatives are equa1. There is another set of
terminology for stating the same thing. A form o is called closed if d o = O
and exact if o = dq for some form q. (The terminology "exact" is c1assicaldjffei-ential forms used to he called simply "differentjals"; a dxerential was then
called "exact" if it actua11y was the differential ofsomething. The term "closed"
js hased on aii analogy with chains, which will he discussed in the next chapter.)
Since d 2 = 0, every exact form is closed. In other words, d o = O js a necessary
condition for soIving o = dq. If w is a 1-form
then the condition dw = O, i.e.,
is necessary for solving o = df, i.e.,
a fi - mi'
a xNow we know from TIxeorem 6-1 that these conditions are aIso sufficient. For
2-forms the situation is more complicated, however. 1f o is a 2-form on IR3,
then
o = d(Pdx+ Qdy+ Rdz)
if and only if
The necessary corzdition, d o = O, is
In general, we are dealing with a rather strange collection of partid dRerentia1
equations (carefully selected so that we can get integrahility conditions). It turns
out thai these necessary conditions are also sufficient: if o is closed, theii it is
exact. Like our results ahout solutions to differential equations, this result is
true only locally. T h e reasons for restricting ourselves to local results are iiow
somewhat different, however. Consider the case of a closed 1-form o on R2:
o = f dx
+ g dy,
with
af
ag
-= -
ay
ax
We know how to find a function or o n al/ of R2 with o = da, namely
O n the otller lland, the situation is ver\; different if o is defined only o n R2 -(O).
Recall that if L c R2 is [O,w) x (O), then
defined iil Chapter 2, is Cm; in fact,
is the inverse of the m a p
(u, b) H (u cos b , a sin b),
whose derivative at (u, b) has determinant equal to u # O. By deleting a dXerent
ray L1 we can define a different function B1. T h e n I9i = 8 in the region Al and
Oi = S + 2 x in tlie region A2. Consequently d19 and dol agree o n tlleir common
domain, so that togetlier they define a 1-form w on IR2 - (O). A c ~ m ~ u t a t i o n
(Problem 20) shows that
The 1-form w js usually denoted by do, but this is an abuse of notation, since
w = d8 only on IR2-L. In fact, w is not d f f o r a y C' function f : IR2-(O) + IR.
Indeed, if o = df, then
so d (f - 8 ) = O on IR2 - L, which implies that af / a r = aO/ax and 8f / a y =
;iO/aj) aiid hence f = 8+ constant on IR2 - L, wliicli is impossible. Nevertheless,
dw = O [the two relations
clearly imply that tliis is so]. So w is closed, but not exact. (It is still exact in a
neigliborliood of any point of IR2 - (O).)
Clearly w is also not exact jn any small region contajning O. This example
sliows that it is tlie shape of the region, rather tliaii its sizc, that determines
whether os not a closed form js necessarily exact.
A mariifold M is callcd (smoothly) contractible to a point po E M if tliere is
a Cm ftinction
H : M x [O, 11 + M
such tliat
For example, IR" is smoothly contractible to O E IR"; we can define
H : IR" X [O, 11 + IRn
bl1oi.e gerierally, U C IR" is coiitractil~leto po E U if U lias the property that
p E U implies po + I ( p - po) E U for O 5 I 5 I (such a region U is called
star-shaped with respect to po).
Of course, niaiiy other regions are also contractible to a point. If we tliink
of [O, 1 1 as represeriting time, tlieii foi- each time I we Iiave a map p I+ H ( p , I )
of M into itself; ai time 1 this is just t1ie identity map, and at time O it is the
Collstari t map.
l e will show ihat jf M is sinoothly contractible to a point, then every closed
forrn o11M is exact. (By tlie way, this result and our investigation of the form d0
pi-ove the iiituitively obvious fact that IR2 - {O) is l i o ~contractjble to a point; the
same result Iiolds for R" - {O), but we will not be in a position to prove this
uiitil tlie iiext cliapter.) The trick in proving our result is to analyze M x [O, 11
(for ariy manifold M), and pay liai-dly any attention at al1 to H.
For I E [O, 1 1 we define
il : M + M x [O, 11
147e clairn that jf w is a foi-m oii M x [O, I ] ~ 7 i t l do
i
= O, tlieii
ii*w -io*w
is exact;
we wil1 see later (and you may try to convince yourself right now) that the
t1ieorem follows trivially from this.
Consider first a 1-form w o n M x [O, I ] . We will begin by working in a coordinate system o n M x [O, I ] . There is an obvious function I o n M x [O, 11 (namely,
the projection ~ s on the second coordinate), a n d if (x, U )is a coordinate system
o n M , while ~ S Mis the projection o n M , then
is a coordinate system o n U x [O, 11. We will denote x i o B M by X z , for convenience. It is easy to check (os should he) tllat
where
mi( .,a ) denotes thc function
p H
(p,a ) .
wi d ~ +
' f dr we have
Now for w =
n
i=l
1
ami
-=al
I f w e define g : M + R by
n
- d.Zi A dr +
dw = [tesms not involving dr] S o d o = O imrrilies that
ami
ar
af
as'
--a
- d r z A dr.
i=I
azz
then
*(p)
axi
=/?f
$P
, I ) df.
o
Equations (1) and (2) show that
il* o - io*w = dg.
Now although we seem to be using a coordinate system, the function f,a n d
l~eilceg aiso, is really independent of the coordinate system. Notice that for
the tangent space of M x [O, I] we have
(4
( M x [O, Il)(p,r) = ker x, $ ker X M * .
ker rr,
-
If a vector space V js a direct sum V
VI $ V2 of two subspaces, then any
o E R1(V)can be written
o=w1+w2
where
Applyjng this to tlle decomposjtion (*), we write the I-form o o n M x [O, I] as
o]+ 0 2 ; there is then a unique f with o2 = f df.
In general, for a k-form o,it is easy to see (Problem 22) that we can write w
uniquely as
o = o1 (df A 7)
+
where o]( V I , .. . ,vk) = O jf some vi E ker x ~ , and
, 7 is a (k - 1)-form with the
analogous property. Define a (k - 1)-form Iw on M as follows:
r1
1% claim tliat d o = O implies tliat i i * o - io*o = d ( I o ) . Actually it is easier
to find a formula for i l * o - io*o that holds even when d o # O.
17. THEOREM. For any k-form o on M x [O, 11 we have
' * w - io*w = d ( I q ) + l ( d w ) .
l]
PROOE Since lo is already iiivariantly defiiied, we can just as we11 work in
a cooidi~iatesystem (3', . . ., T", 1 ) . The operator I is clearly linear, so we just
liavc to consjder two cases.
(1) o = f d2'1 A - . .A ~ T =
~ f
R d2'. Tlien
it js easy to see that
Sjilce lo = O, ihis proves the resuli in this case.
(2) U = j'dr A f/,fil A
Now
and
a
a
A dx*-1
f dr A dx'. Theii ji*o
= io* o = 0,
18. COROLLARY. If M is smoothly contractible to a point po E M , then
every closed form o on M is exact.
PKOOF. We are given H : M x [O, I] + M with
H ( P >1) = p
H(p,O) = Po
for al1 p E M.
Thus
H o il : M + M
Hoio: M + M
is the identity
is the constantmappo.
Bui
o - O = ilS(H*o)- io*(H*o)
= d(l(H*o))
by the Theorem. +:+
Corollary 18 is called the Poincaré Lemma by most geometers, while d 2 = O
is called the Poiiicaré Lemma by some (I
don't evexl kiiow whe tlier Poincaré had
aiiything to do with it.) In the case of a star-shaped open subset U of Rnlwhere
MT have an explicjt foi-mulafor- H, we can fiizd (Problem 23) an explicit formula
for I ( H * o ) ,for eveiy form o on U. Since the new form is given hy an integral,
we can solve tlie system of partial differeiiiial equations o = dq explicitly in
terms of integi-als. There are classical tlieorems about vector fields in R3 whicli
can l ~ derjved
e
fi-om thc Poiricaré Lemma and its converse (Prohlem 27), and
orjginally d was introduced in order to obtain a uniform generalization of al1
these results. Everi though tlie Poincaré Lemma aiid its converse fit very nice1y
into our pattern for basic thcorems about differential geometry, it has always
heen somethiiig of a mystery to me just why d turns out to he so important.
A11 aiiswer io tliis questiori is provided by a thcorem of Palais, Nutulril O@utiotis
011 Dijil-e?lfiaIForms, Trans. Amer. h4ath. Soc. 92 (1 959), 125-141. Suppose we
Iiavc any opei-ator D from k-forins to I-forms, such that tlie foUowing diagram
commutes for evcry Cm map f:M + N [it actually suffices to assume that
the diagram commutes only for diffeomorphisms f J.
k-forms on M
-
f* k-forms on N
PaIais' tlieorem says that, with few exceptions, D = O. Roughly, these exceptiona1 cases are tlie followjng. If k = f, then D can be a multiple of the identity
map, but ~iothingelse. If I = k 1, then D can only he some multiplc of d.
(As a corollary, d2 = O, since d 2 makes the abow diagram commute!) There is
only one other case ivhere a non-zero D exists-when k is the dimensjon of M
and f = 0. In ihis case, D can be a multiple of "integrationn, which we discuss
in the next chapter.
+
+
(d) Formula (c) can be used to give a definition of o]A o 2 by induction on k I
(which works for vector spaces over any field): If A is defined for forms of degree
adding up to < k + E , we define
Show that with this definition o1 A o 2 is skew-symmetric (it is only necessary to
check that interchanging vl and v2 changes the sign of the right side).
(e) Prove by induction that A is bihnear and that wl A 0 2 = (- 1)"w2 A ol.
( f ) If X is a vector field on M and o a k-form on M we define a (k - 1)-form
x ~ o b y
( X J o ) ( p )= X ( p )J W ( P ) .
Show that if wl is a k-form, then
X J ( W ]A w ~ =) ( X J W ] ) A W
+ ( - I ) ~ w A, ( X J o 2 ) .
5. Show that n funcrions 5,.
. .,S,: M + R form a coordinate system in a
iieighborhood of p E M if aiid only if dfi A - A dS, ( p ) # O.
6. An element o E R k ( v ) is called decomposable if o = #]A - - A#A for some
#i E V* = Q 1 ( v ) .
(a) 1f dim V 5 3, then every w E Q 2 ( v )is decomposable.
(b) If #;, i = 1 , . . . , 4 are independent, then w = (#l A $ 2 ) + (#3 A #4) is not
decomposable. Hinl: h o k at o A o.
7. For any o E Q k ( v ) ,we define the annihiiator of o to be
Ann(o) =: (# E V* : # A o =: 0).
(a) Show that
dim Ann(o) 5 k ,
aiid tliat equality 11olds if and only if o is decomposable.
(b) Every subspace of V* is Ann(o) for some decompasable o, which is unique
up to a multiplicative constant.
(c) If o, and w;! are decomposable, then An?i(wl) c Ann(w2) if and only if
0 2 = o]A q for some v.
are decomposable, then Ann(oi) í l Ann(w2) = ( O ) if and only if
(d) lf
wl A o 2 # O. In this case,
+
Ann(oi) Ann(w2)=: Ann(ol A w 2 ) .
(e) 1f V Iias dimensioii n , tlien any w E Qn-' ( V ) is decomposable.
(f) Sji~cevi E V can be regarded as elements of V**, we can consider vl A - . - A
ve E R k ( v * ) . Reformulate parts (a)-(d) in terms of tliis A product.
8. (a) Let o E f i 2 ( V ) . Show that there is a basis #l,. . . ,#, of V * such that
choose #] invol\?ing $ 1 ,
$3,.
. . , .Srn and $2 involving Ilrz,... ,$, so that
where o rdoes not involve $ 1 os
(b) Show that the r-fold wedge product o A .A o is non-zero and decomposable, and that the (Y 1)-fold wedge product is O. Thus r is well-determined;
it is called the rank of o.
(c) If o =
u i j q i A $ j , show that the iank of o is the rank of the ma~h
(uij).
+
xicj
9. If v i , . . . ,v, is a hasis for V and wi =
xj,,
n
a j i v j , show that
10. Let A = (ui,) he an ri xri matrix. Let 1 ( p 5 n be fixed, and let q = n - p .
For H = 11, <
< hp and K = k i <
< k q , let
(a) If V I , . . .,ün is a basis of V and
show that
(b) Let H' = (1, . . . , 1 1 1 - H (arranged in increasing order). Show that
VR
VK
K # H'
~ H , N vi
I A . . A vn K = H',
=
wliere e R , ~ isl the sign of the ~ermutation
(c) Prove "Laplace's expansion"
det A =
11. (Cartan's Lemma) Let # i , . . . ,#k
$ i , . . . , $ k E V * satisfy
Then
eH,n,B
HCH1
E V * be independent aild suppose tliat
k
12. In addition to forms, we can consider sections of hundles constructed fi-om
T M usirig S
I aild otlier operations. For example, if $, = n : E + 3 js a vector
the bundle whose fibre at p is f i k ( [ n - '( p ) ] ' ) .
bundle, we can consider f i k (t*),
Siiice we can regard
a
-
as an element of
(Mp)**,
axl
aiiy section of Q n ( T * M )can be writtei~IocalIy as
(a) Show tliat if
thcn
This shows that sections of R n ( T * M ) are the geometric objects corresponding
to the (even) relative scalars of weight - 1 in Prohlem 4-10.
@) Let Tktml
( v )denote the vector space of all multilinear functions
--
V x ... x V x V * x ... x V * + R m ( V ) .
I times
k rirncs
Show that sections of T ~ ~ " ' ( T M
corres~ond
)
to (even) relative tensors of type
(): and weight 1. (Notice that if vi, .. . , u. is a basis for V , then elements of
51" ( V ) can be represented by real numbers [times the element u* A . A u*,] .)
(c) If 7/tm1(v)
is defined similarly, except that R n ' ( V ) is replaced by R m ( V * ) ,
a
5jill
show tliat sections of
( T M ) correspond to (even)relative tensors of type (,)k
and weigh t - 1.
(d) Sliow tliat the covariant relative teiisor of type (:) and weight I defined in
Problem 4-10, with components E'' ...'?I, corresponds to the map
v* x
x v* + R " ( V )
n times
A #,.
Interpret the relative tensor with comgiven by ( $ 1 , . .. ,#,) +t #, A
ponen ts Eil ..,ir,similarly.
x V + R which are of
(e) Suppose R " ; w ( V )denotes a11 functions q : V x
the form
q ( u ~ , ~ . ~ ,=
u n[ )o ( ~ ~ , ~ - - , ~ nw) an
l ~ integer
for some o E Q n ( V ) . Let T , k [ " ' w lbe
( ~defiiied
)
like 7/kin1,except that R R ( V )is
,k[n;w]
replaced by R n ' W ( V ) .Sliow tliat sections of J ,
( T M ) correspond to (even)
relative tensors of type (f) and weight w. Similarly for
(f) For tliose who kiiow about tensor products V @ W and exterior algebras
hk( V ) , these resulta can al1 be restated. We can identif~qk( V ) with
7,il:wl.
k times
I times
Siiice R t " ( v ) = Am ( V * )= [Am( V ) ] * we
, can identify
( v ) with
V * @ @' V @ An'(V*).
Consider, more generally
Noting that An(V)@ - @ An(V) is always 1-dimensional, show that sections of
-k[n;wI
k
(TM) and T ~ ~ ; , ~ ( T Mcorrespond
)
to (even) relative tensors of type (/)
I
and weigl~tw and - w , respectively
13. (a) If V has dimension n and A : V $ V is a linear transformation, then
the map A * : Qn(V) + Qn(V) must be multiplication by some constant c.
Sliow that c = det A. (Tbis may be used as a definition of det A.)
(b) Conclude that det A 3 = (det A)(det 3 ) .
14. Recall that the characteristjc polynomial of A : V + V is
x ( I ) = det(h1 - A)
= In- (trace A)A"-'
= hn - cl hn-' +
+ - + ( - 1)" det A
+. + (-1)"~~.
(a) Show tliat ck = trace of A*: Q k ( v ) + Qk(v).
(b) Conclude that ck (AB) = ck (BA).
(c) Let s~'"'....lk be as defined in Pmhlem 4-5(xiii). If A: V + V has a matrix (u!) (with respect to some hasis), sliow that
(i),
Thus, if S is as deíined on page 130, and A is a tensor of type
then the
function p H ck(A(p)) can he defined as a (2k)-fold contraction of
k times
15. Let P(Xij) he a polynomial in n2 variables. For every n x n matrix A = (u,)
we then have a numher P(uij). Cal1 P invariant if P(A) = P(BAB-') for
a11 A and al1 invertihle B. This prohlem outlines a proof that any invariant P
is a polyi~omialin the polynomials ci ,. . .,c, defined in Prohlem 14. We will
need the algebraic result that any symmetric poIynomial Q(yi,. . . ,y,) in the n
variables yl,. . . ,y,, caii be written as a polynomial in 01,. . .,o,,, where oi is
the ih elementary symmetric polynomial of y,, . . . ,y,. Recall that the q can
he defined by the equation
Thus, they are the coefficients, up to sign, ofthe polynomial with roots yl,. . . ,
y,, Since the eigenvalues Al,. . . , h, of a matrix A are, by definition, the roots
of the polynomial ~ ( h )it, fol1ows that
We M ~ first
I consider matrices A over the complex numbers C (the coefficients
of P may also be complex).
(a) Define Q ( y l , .. . , y , ) to he P(A) where A is the diagonal matrix
Then there is a polynomial H such that
The polyiiomial 8 has real coeficients if P does.
(b) P(A) = H(ci (A), . .. ,c, (A)) for al1 diagonalizab1e A.
(c) The discriminant D(A) is defined as n i f j ( L i - A,)', where hi are the
eigenvalues of A , Show that D(A) can be written as a polynomial in the entries
of A.
(d) Sliow that P(A) = K(cl (A), . . . ,c, (A)) whenever D (A) # O. ConcIude, by
continuity, that the equation holds for a11 matrices A over C. (This last conclusion folIows even if C is replaced by some other field, since the set where D # O
is Zaris ki-dense; this is "the principal of irrelevance of algebraic inequalities",
compare pg. V 375.)
Now suppose that the coefficients of P are real and that P(A) = ~ ( 3 ~ 3 - ' )
for al1 real A and real invertible B.
(e) The same equation holds for complex A and complex invertible B. (Regard
the equation as n 2 polynomial equations in the uij and bij.)
16. (a) Let vi, . .. ,v, be a basis for Vyand let WI,. . . ,wk E V be given by
For o E a k ( v ) show that
where al is the determinant ofthe k x k submatrix of (a,) obtained by selecting
rows i],.. .,ik.
(b) Generalize Theorem 7 and CoroIIaqr 8 to k-forms.
(c) Check directly from (b) that the definition of d does not depend on the
coordinate system.
17. Sliow tliat d ( x i ,
%j
dxi A d x j ) = O if and only if
18. In PI-oblem 5-14 we defined Lx A for any tensor field A .
(a) Show that if o is a k-form, then so is Lx o.
(b) Show that
(c) Using 5-14(e), show that
(d) Deduce the foUowing two expressions:
(e) Show that
X J dw = L x w - ~ ( X J W ) ,
(This may be used to give an inductive definjtjon of d.)
(f) Using (e), show that d ( L x w ) = L x ( d w ) .
19. Let uij be n2 fuiictions on Rn with oij = uji. Show that in order for there
to b e fuilctions ul, . . . , u, in a neighborhood of any point in R" with
it is necessary and sufficient that
a2uij - a2uik
azul, - a2ulk
askaxl
axi axf
axkaxi axjaxi
for al1 i, j, k,l .
Hini: First set up partial dxerential equations for tlie finctions hk = a u J / a x k a u k / a ~ jand
, use Theorem 6-1.
20. C o m ~ u t ethat
(At niost places 6 = arctan y / x [
+ a constant] .)
21. (a) If o is a 1-form f dx on [O, I] with f ( 0 ) = f (1), show that there is a
unique number h such tliat w -h d x = dg for some fuiiction g with g ( 0 ) = g ( I ) .
Mni: Iiitegrate the equation w - h d x = dg on [O,11 to find h.
(b) Let i : S' + R2 - ( O } be the inclusion, and let o' = i'(d0). If C : [O, 11 + S'
is
c ( x )= (COS2x A-, sin S x x ) ,
show that
C* (u') c 2x dx-.
(c) If w is a closed 1-foi-m on Si sliow tliat there is a unique number h such
that w - 10' is exact.
22. (a) Show that every w E R ~ ( V$ ~V2) can be written as a surn of forms
ol A w;, where o1has degree oc and 0 2 has degree B = k - oc and
(b) If dim V2 = 1, and O # A E V2*, then o can be witten uniquely as o1
(w A A), where wl is a k-form and w;, is a (k - 1)-form such that
+
23. Let U c IR" be an open set star-shaped with respect to 0, and define H : U x
[O, 11 + U by H ( p , 1) = fp, If
on U, show that
24. (a) Let U c IR2 be a bouiided open set such that IR2 - U is connected. Show
that U is ditfeomorphic to IR2, and hence smoothly contractible to a point. (The
converse js proved in Problem 8-9.) Hinf: Obtain U as an increasing union of
sets, the kth set being a finite union of squares containing the set of points in U
whose distance from boundary U is 5 I/k.
(b) Find a bounded open set U c IR3 such that IR3 - U is connected, but U is
not contractible to a point.
25. Let U c R" be an open set star-shaped with respect to O. 1s U homeomorphic to R"? (It would certainly appear so, hut the "obvious" proof does
not work, since the length of rays from O to the boundary of the set could vary
discontinuously.)
26. Let ( , ) be the usual inner product on Rn,
(a) If v i , . . . ,U n - ] E Rn, show that there is a unique vector vi x
with
+
. . x v,-1 E Rn
@) Show that x . . x E R n - l ( R " ) , and express it in terms of the e * i , using the
expansion of a matriu by minors.
(c) For R3 show that
v x w = ( v 2 w 3 - v 3 w 2 , v 3 w 1 - v 1 w 3 , v 1 W 2 - u 2 W1 ) .
(First find al1 ei x ej .)
27. (a) If f :R" + R, define a vector field grad f , tlie gradient of f' on R"
by
n
a
a
gradf = E G .- = E D j f . -
" af
i- i
Introducjng the formal symhoIism
axi
i- 1
axi
we can write grad f = Vf. If (grad f ) (p) = w p , show that
wl~ereD,f (p) denotes tlle direciional del-ivaiive in the direction u at p (or
simply v,( f), if we regard vp E RP). Conclude that Vf (p) is the direction in
which f is changing fasiest at p.
(h) If X = EL1a'a/axi is a vector field on W", we define the divergente
of X as
n
div X =
aa!
axi
(Symbolically, we can wriie div X = (V, X).) We also define, for n = 3,
Define forms
Show iliat
c/f = Wgrad f
d(0x) = rlcurl x
c l ( v ) = (div X) dx A dy A dz.
(c) Conclude that
curl grad f = O
div curl X = 0.
(d) If X is a vector field on a star-shaped opeii set U c Rn and curl X = 0,
ilieii X = grad J' for- some fuiiciioii f : U + R. Similarly, if div X = O, iheii
X = curl Y for some vector field Y on U.
CHAPTER 8
INTEGRATION
T
he basic concept of this chapter generalizes Iine and surface integraIs,
~tllichfirst arose fiom very pIlysical considerations. Suppose, for example,
iliat c : [O, 11 +- R2 is a curve and o = f dx g dy is a 1-form on IR2 (WIICIF
f , g : R2 + R, and x and y denote tlle coordinate functions on R2). If we
choose a pai-tition O = to <
< r, = 1 of [O, 11, then we can divide the curve c
into pieces, tlie ith piece going fiom c(li-1) to
When the difirences
ti - ri-1 ai-e smaII, eacll sucll piece is approximateIy a straight segrnent, witll
+
>,
Iioiizoiital pi-ojection C' (li) - C' (li-1) and vertical projection c2(ti) - ~ ~ ( l ~ - ~ ) .
\Ve can clioosc poiiits ~ ( 6 on
~ )eacli piece by clioosing points Ci E Eh-1 ,li]. For
eacli parti tiori P aild each sucli cl-ioice 6 = ((1 , . . . ,Ct1),consider ihe sum
If tIlesc sums appi-oac11 a limit as tlie "mesli" 11 PIJ of P approaclles O, that is,
as tl-ie inaxin~umof ti - riWl appi.oacl-ies 0, then ihe limit is denoted by
(TIlis is a con111Iicated limit. To be precise, if JJ PJI = max{Ii - ri-l), then the
1
equaiion
P
lim S ( P , C ) =
llPlI+O
means: for al1 E > O, there is a 6 > O such that for al1 partitions P with 11 PJ1< S,
we have
for al1 choices 6 for P.)
The limii which we have just defined is called a "line integral"; it has a natural
plysical interpretation. If we consider a "force field" on R2, described by the
vector field
then S ( P ,6 ) is tI-ie "work" inwlved in moving a unii mass along the curve c in
tl-ie case wl-iere c is actually a straiglii Iine between ti-1 and ti and f and g are
coilsiani along tIiese straigIit line segrnenis; tlie limit is ihe naiuraI definition
of tl-ie work done in tl-ie general case. (In classical terminology, ihe differeniial
-f dh- + g c/-v would be desci-ilxd as ihe work done by tlle force field on an "ii-iliilitelv small" displacement wit11 components dx, cIJ); the integral is ihe "sum"
of iliese infinitely small displacements.)
Rcfore worrying about how to compute tllis limit, consider the special case
In tllis case, c 1 ( 1 ; ) - c 1 ( t i - , ) = l j - l j - 1 , while c2 ( t i ) - c2 ( l i - l ) = O, SO
Tl-iese sums approach
lc +
fdx
g dy =
1-
f(x,yo) dx.
O n the other hand, if
then C' ( l i ) - C' ( l i - 1 ) = ( b - a)(tj - l i - l ) , SO
These sums approach
In general, for any curve c , we have, by the mean value theorem,
A soinewhat messy argumei~t(ProbIem 1) shows that these sums approach what
it looks like they should approach, namely
PI-iysicistsYnotation (or abuse thereof) makes it easy to remember this resuli.
Tlie components c l , c 2 of c are denoted simply by x and y [;.e., x denotes
x o c and y denotes y o c ; tllis is indicated classically by saying "let x = x ( t ) ,
y = y ( t ) " ] . The above integral is then written
In preferente to this physical interpi-etation of "line integralsu, we can introduce a more geometrical interpretation. Recall that d c l d t ( 6 , ) denotes the
tangent vector of c at time ti. Then the sums
cIearly also approach
Consider the special case wl-iere c goes witli constan1 velocity on eacll ( t i , l , t i ) .
If we cIloose any ti E ( t i - l , t i ) , then
de
dl
Iength of - ( C i )
= tlie constant speed on ( I ~ - l~i ),
- Iength of the segment from c ( l i W lto
) c(li)
lj
dl
- ti-1
Y
( t i - t i - ] ) = Iength of segrnent from c ( / ; - i ) to c ( / ~ ) .
In this case,
is tIle leilgth of c, and tIle limit of sucli sums, for a general c , can be used as a
definitioii of the length of c . The Iine integral
o = limit of the sums (*)
can be tliouglit of as the "leilgth" of cy wlien our ruler is cllanging continuously iil a way specified by o: Noiice tl-iat the restriction of w ( c ( t ) ) to the
1-dimeiisional subspace of IR2,(,) spaniled by dcldt is a constant times "signed
le11gt1-i". The natural way to specify a contiiluously changing Iength along c
is to specify a Iength oii its taiigeiit vectors; this is tIie modern couilterpart of
tl-ie classical conception, whereby the curve c is divided into infinitely small
parts, the infinitely smaIl piece at c ( /), with compoiients d x , d y , having length
f ( ~ ( dx
0 )+ g ( c ( / ) )dl).
Refore pushiilg this geomeirical inierpretation ioo far, we sIlouId note that
tliere is no 1-form o on IR2 such that
LO = Iength of c
for al1 curves c
It is true ihat for a giveii oile-oiie curve c we can pi-oduce a form o which works
for c; we choose o ( c ( 1 ) )E 52i(R2c(,))
so that
(choosing ihe heme1 of o arbitrarily), and tlieii exteiid o to IR2. But if c is
not one-one this may be impossible; for example, in the situation shown beloy
tliere is no element of Ri(W2,(,)) which has the value 1 on al1 three vectors.
111general, giveii any o on JR2 wllich is everywliere iion-zero, t l ~ esubspaces
A, = ker o ( p ) form a 1-dimensional distribution on Ik2; any curve contained
i11 aii ii-itegral submanifold of A will have "length" O. Later we wilI see a way
of cii.cumventing this dificulty, if we are intei-ested in obtaining the ordinary
leilgth of a cunte. For ilie preseilt, we note that the sums (*), used to define this
gcileralized "length", make sense even if c is a curve in a manifold M (where
iliei-e is iio notioil of "leiigth"), aiid o is a 1-form on M, so we can define Jc o
as the limit of these sums.
One property of liiie iilregrals sliould be meiitioiied now, because it is obviouc wiih our 01-igiiial definition and mercly true for our new definitioii. If
p ; [O, 11 + [O, 11 is a one-oi-ie iiicreasiilg funciion from [O, 11 onto [O, 11, then the
cume c o p : [O, 11 + M is called a reparameterization of c-it has exactly the
same image as c, but ti-ansverses it at a dfiereiit rate. Every sum S ( P ,6 ) for c.
is clearly equal to a sum S(P', 6') for c 0 p , and con\tersely, so ii is clear froin
oui first definition that foi- a cuive c: [O, i ] + JR2 we I-iave
("tlie integral of o ovcr c is iildepcndent of tlic parameterization"). Tliis is ilo
loiigci- so cleai. wlieii we consider the sums (u) for a curve c : [O, 11 + M, iioi. is
it cleai- eveii for a cunre c : [O, 11 + W2, but iii this case we can proceed right to
tlie iiliegral tl-iese sums approacli, narnely
The 1-esult tlien folluws from a calculation: tlle subsiiiution I = p(u) @ves
=
1-
[ f (e o p(u))(c o p)If(u) + g(c 0 p(u))(c o p12'(u)] du.
For a curve in Rn, and a 1-form o = E:=, mi dxi, there is a similar calculaiion; for a general maiiifold M, we can iiitroduce a coordinate system for our
calculations if c([O, 11) lies in one coordinate sysiem, or break c up into severa1
pieces otherwise. Mle are being a bit sloppy about al1 this because we are about
to iilti-oduce yet a t1iii.d defiiiition, which will e\tentually become our formal
choice. Consider oiice again the case of a 1-foi-m on IR2, wliere
Notice tIiat if 1 is the standard cooi-dinate system on R, tlien for tlie map
c : [O, 11 + R2 we have
c*(f d x
+ g dy) = (f o c)c*(dx) + (g o c ) c * ( d ~ )
= (f o C)d(x o C)
+ (g o c) d ( y o c )
+ (g o c)c2' dr,
so iliat formally wc just integrate c*(f d x + g dy); to be precise, we write
= (f
O
c)cl' dr
+
c.*(fdx g dj)) = h dr (in the unique possilJe way), and take the integral
of h on [O, 11.
Everyiliing we Iiave said foi- cuwes c : [O, 11 + IR" could be generalized to
fuiictions c: [O, 112 +- R". If x and y al-e tlle coordinate functions on IR2, let
For a paii. of pariitioi-is so < - . - < S,, aiid lo < . - - < E, of [O, 11, if we clioose
1. PROPOSJTION. Let c : [O,11" + IRn be a one-one singular 11-cube witll
det c' > O on [O,11". Let w be the 11-form
w =fdx' A...A~x".
PROOF. By definition,
/
( f o c ) Jdet c'J d x ' A
[o,-l]l1
=
=
/
c(I0,' P')
f
A dx"
by assumption
by the change of variable formula. 4 3
2. COROLLARY. Let p : [O,11' + [O,11' be one-one onto with det p' 2 0,
let c be a singular k-cube in M and Iet UI be a k-form on M. Then
I- =Lpo.
PROOF. We llave
J o= J
COP
'
[O, lk
=
/
(C 0 p ) * o =
J P*(c*o)
'
[O, jk
c * ( w ) by the Pmposition, since p is onto
The map c o p : [O, l l k + M is called a repararneterization of c if p : [O, 1 l k +
[O, 11' is a C m one-one onto map with det p* # O everywhere (so that p-' is
also Cm); it is called orienta tion preserving or orientation reversing depending
on whetller det p' > O or det p' < O everywhere. The corollary thus shows
independence of parameterization, provided it is orientation preserving; an orientarion reversiiig repai-ameterization clearly changes the s i p of the integral.
Notice that there ~ f o u lbe
d no such result ifwe tried to define the integral over c
of a C m fuilcrioil f : M + IR by the formula
For example, if c : [O, 11 + M then
1
f ( ~ ( 1 ) ) dl
is generally
#
o
f ( c ( p ( 1 ) ) 0) 1
o
From a formal point of view, differential forms are tlie things we integrate because they transform coi.rectly (i.e., in accordance witll Theorem 7-7, so that
tlle change of variable formula will pop up); funcrions on a manifold cannot be
integrated (we can integrate a function f on the manifold R' only because it
gives us a form f d x l A
A dxk).
Our definition of the integral of a k-form UI over a singular k-cube c can
iinmediately IIC generalized. A k-chain is simply a formal (finite) sum of singular
k-cubes multipIied by integers, e.g..
Tlle k-cllain 1c1 = 1 cl will also be deiioted siinply by c l . We add k-cllains,
and multipIy them by integers, purely formally, e.g.,
Morcovcr, wc define the integral of
way :
over a k-chain c =
xiaici in t11e obvious
The 1-easoil Toi. iilti.oduciilg k-chaiiis is that to evei-y k-chain c (wl~iclimay bc
just a singulai A--cube) we wisll to associate a (k - 1 )-cliain ac, ~ } h i c his called
tl-ie boundary of c , aild which is sul~posedto be t11c sum of the various singular
(k - 1)-cubes around the boundary of each singular k-cube in c. In practice, it
is convenient to modify this idea. The boundary of í2,
for exarnpk, 4 1 not be
the sum of tlie four singuIar 1-cubes indicated below on the left, but the sum,
witIl the indicated coefficients, of tlie four singular 1-cubes shown on the right.
(I\'otice tliat tliis will not cliange the integral of a 1-form over al2.) For each i
witli 1 5 i 5 n we first define two singular (n - 1)-cubes ICgo)
and II;,,) (the
(i, O)-face and (i, 1)-face of í")as foIlows: If x E [O, 11"-' , then
I:,~)(X) = ~ " ( x ' , .. . ,xi-i ,O,Xi y . . . y ~ n - l )
= (x 1 ,...y xi-l
y
o, 2,.. . xn-l),
y
I;.,,)(x) = í " ( x l , . . . , xi - 1 ,1, Si
= (S 1 ,..., x i-l , l , xi ,
, . a
. ,xn-l)
A-"-1)
..m,
The (i,a)-face of a singular n-cube c is defined by
Now we define
n
FinaIly: the boundary of al1 n-cllain Ciaici is defined by
These definitions al1 make sense only for n >_ 1. For the case of a O-cube
e : [O, 1' + M, wliicli we will usually simply identify with the point P = c(O),
we define ac to be tlie number 1 E IR,and for a O-chain
aicj we define
xi
Notice that for a 1-cube c : [O, 11 + M we have
Mre also Iiave, for a singular 2-cube e : [O, 112 + M'
From a picture it can be cl-iecked that this also Iiappens for a singular 3-cube,
a good exercise because this involves figuring out just what tfle boundary of a
3-cube looks like. In general, we have:
3. PROPOSITION. If c is any n-chain in M, then a(&) = O. Briefly, a2 = 0.
PROOE Let i 5 j 5 i 7 - 1, and consider (I{,a))(j,B). For x E [O, 11"-~, we
have, from the definition
a
,;('l
)(jln<") = a,:l'
(x))
n
= '(;,.)(x
1
.
Y
xj-I, ByxJ , , , r 2 )
y .
= ~ ~ ( x l ~xi-l
. . .>CY,X
, i
? . .
. Y
byx j y . . x ' - ~ )
XJ-~,
y
Similarly,
(';/+i
,a>)(i,a>
=
'c+i .m(';:)(~))
.. . x i-1 ,ayx ,.. . ,x " - ~ )
= rn(xIy... x i-l CY,
..
p y ~ j y. ,x. ." - ~ ) .
= r;+,
1
(A- ,
I
y
I
y
y
x * ? .
y xJ-ly
Thus (':lct))( jla> = +
;('I
lg))(i,ct) for i 5 j 5 n - 1 . Jt follows easily for any
~ ( )~ + ~ , a >
for) (i ~5, j~ )2 n - 1. NOMI
singular 17-cube c ihat ( C ( ~ , ~ ) ) (=~(, @
ri
In this sum,
n-l
, B ) ) ( ~occur
, ~ ) witli opposite signs. Therefore
al1 rei-nis cailcel in pairs, arld a(&) = O. Since tlle tlleorem is true for singular
( j , ~and
) ('(
71-cubes, it is clearly also irue for singular n-chains. 4%
Norice tliai for some 11-cl-iainsc we I-iave not only a(ac) = O, but even ac = 0.
For example, this is the case if c = c1 - c2, wliere c, and c2 are two 1-cubes
witli cl(0) = ~ ~ ( and
0 ) cj (1) = ~ ~ ( 1Jf) .c is just a singular 1-cube itself, then
ac = O precisely when c(0) = c(l), Le., when c is a "closed" curve. In general,
any k -chaii-i c is called closed if ac = 0.
RecaII tl-iat a differei-itial form o with do = O is also caIIed "closed"; tliis
terminology has been purposely chosen to paralle1 the terminology for chains
(on tlle otl-ier haild, a chairi of t I ~ eform ac is iiot desa-ibed, recipi.ocally, I q ,
tlie classicaI term of "exact", but is simply called "a boundary"). This paralle1
terminology \vas not cllosei~mcrely because of tlie formal similarities between d
and 8, expressed by tlie relatioiis d 2 = O and a2 = O. The connection betweeii
fo1.n-i~and cliains goes muc11 deepcr tl-ian that. For example, we have seen that
on IR2 - {O)tlier-e is a 1-form "de" wl-iich is closed but rlot exact. There is also a
1 -chaiii c ~ ~ I i i cishclosed but iiot a boundary, namely, a closed curve encircling
t I ~ epoint O once. AIthough ii is irituitiveIy clear tIiat c is not the boundary of a
2-chain in IR2 - {O), the simplest proof uses the theorem which establishes the
corinection between forms, chains, d , and a.
4. THEOREh4 (STOKES' THEOREM). If o is a (k - 1)-form on M and c is
a k-chain in M, then
PROOF R4ost of tlie proof involves the special case where o is a (k - 1)-form
oii IRk and c = rk. In tllis case, o is a sum of (k - 1)-forms of the type
and it suffices to prove the tlieorem for each of these. Mle now compute. First,
a little ríotaiion translation shows that
*(f dx' A . - . A ~ X ; ' A + ~ . kA) ~ X
1,.
/'dxl
=
-
A * . .n d x l A - . . ~
k
(-])]+u
j = l a=0,1
J
[O, l]k-1
d
x
~
I~,"
* ()f d x l A . . . n d x i A . . . A ~ A - ~ )
On the other hand,
d(fdx-' n ..A
m
2 n . . A c/xk)
m
By Fubiili's theorem and the fundamental theorem of caIculus we have
-
-f ( x l ,.. .,O,.. . ,x k ) ] d x l .. . d x t . . . clxk
Thus
For an a r l ~ i t r a qsing~lark-cube, chasiiig througl~the defiilitions sliows tIiat
Therefore
The iheorein cIearly follows for k-cIiains also. $4
Notice that Stokes' Tlieorem not onIy uses tlie fundamental tIieorem of calculus, but actually becomes that theorem when c = 1' and o = f.
As an application of Stokes' Theorem, Mte show that the curve c : [O, 11 +
IR2 - (0) defined by
althougI1 closed, is not ac2 for any 2-chain c2. Ifwe did have c = ac2, then we
But a siraightforward computation (wliicl-iwill be good for the soul) shows that
[There is also a ilon-computational arguinenr, using the fact tbat "d6'" really
is d6' for 6': IR2 - ([O, cm)x (O)) + IR: We havr
and 8(1 -E) - @(E)+ 21r as E + O.]
Altliougl~we used iliis calculation to s h o ~tIlat
t c is not a boundary, we couId
just as well llave used it to show that o = "cf8" is not exact. For, if we had
o = df foi- some Cm fuilction f : IR2 - (O) + IR, then we would Iiaw
147e wei-e III-eviouslyable to give a simpler ai.gument to show that "de" is not
exact, but Stokes' Theoi-em is the tooI \vIlicI1 ~tillenable us to deal with forms
o11 R" - (O). Foi- example, we will eventually obtain a 2-form w on IR3 - (O),
whicli is closed but not exact. For the moment we are keeping the origin of w a
secret, but a sti-aightfonvard calculation shows that d o = O. To prove tliat o is
not exact we \vil1 want to integrate it over a 2-chain which "fills up" the 2-sphere
S2c IR3 - {O). Tliere are lois of ways of doing this, but they al1 turn out to give
the same result. In fact, we first want to describe a way of integrating n-forms
over ii-maiiifoIds. Tliis is possible orily wlien M is orientable; tlie reason will
be clear from the riext result, whicli is basic for our definition.
5. THEOREM. Let M be aii 11-maiiifoldwitli an orientation p, aiid let ci, c2 ;
[O, 11" + M be two singular 17-cubes which can be extended to be difiomorphisms iii a iieiglil~orlioodof [O, 11". Assume tliat cl and c2 are botli on'enlalion
preservilzg (with respect to the oiientation p on M, and tlie usual orientation
on IR"). If w is an 11-form oii M sucli tliat
support o c ci ([O, 11") n c2([O, l]"),
tIieri
PROOF 14%waiit lo use Corollary 2, and write
TIic only problcm is tliat c2-' o cl is i ~ o defined
t
on al1 of [O, 11'' (it does satisfy
det(cz""' o cl )' > O, since c.1 aiid ~2 are 13otli orientatioii preservirig). However, a
glaiice at tlie proof of Corollary 2 will show tliat tlie result still follows, because
of tlie fact tliat support w c cl ([O, 11") n c2([O, 11"). *:
Tlie romiilon iiumber
[w, for singular ii-cubes c : [O, 11" + M with sup-
Jc
poi-t w c c([O, 11") aiid c 01-ieiitatioii 11i-eserving,wiI1 be derioted by
lf o Is aii al-l~itrary11-form oii M, tlieii there is a cover 0 of M by open sets U.
eacli coiitaiiied iii some c([O, l]"), wliei-e c is a sirigular ii-cube of tliis sort; if @
is a partitioii of uiiity subordinate to this cover, tlien
is defiiied for each 4 E @. We wish to define
14%wilI adopt tIiis definition only when o has compact support, in wliich case
the sum is actually finite, since support w can intersect only finitely many of t11e
sets { p : $ ( p ) # O), which form a locally finite collection. If we haw anothei.
partition of unity Q (subordinate to a cover O'), tIlen
iliese sums are al1 finite, and the last sum can clearly also be written as
so tl-iat our definiiioii does not depeiid on the pai-tition. (1We really sl~oulddenote
tliis sum by
for tlle oi-ientation - p of M we clearly liave
Howevei; we usuaIly omit explicit mention of p . )
Witli ininor modifications we can define iM
o evcn if M is an i?-manifoldwitli-boundary. If M c IR" is aii 11-dimensional maiiifold-~4th-bouiidaryand
f : M + IR has compacr support, tlien
wllerc tl-ie 1-iglithaild side dellotes the oi.di1-iai-yii~iegi-al.TIiis is a simpIe consequerlce of Proposition 1 . Likewise, if f : M" + N" is a diffeomorphism onto:
aiid o is aii 17-foi-mwitli compact support or-i N, tlieii
rlo
-
..-
if f is orientation presewing
ii f is orientation reversing.
AItIlough n-forms can be integrated only over orientable manifolds, there is
a way of discussing integratioil on non-orientable mai-iifolds. Suppose that o is
a function on M such that for each p E M we have
4
~ = )Iqpl
for some
qp E Qn(Mp),
i.e., for any n vectors u],. . ., u, E Mp we have
Sud1 a furlction w is called a volume element-on each vector space it determilles a way of measuring n-dimeiisional volume (not signed vol ume). If (x, U)
is a cooi.dii1ate system, then on U we can write
= fldx'
~ - . . ~ d x " l for f 2 O;
\ve cal1 o a Cm volun-ie element if f is Cm. One way of obtaining a vo1ume
element is to begin witll an rr-form q and tllen define w(p) = Jq(p)i. Howeve~
not every volume elemei-it arises in this way-the form qp may not vary contii~uouslywith p. For exainple, consider the Mobius strip M, imbedded in W3.
Siiice M, can be considered as a subspace of R3p, we can define
w(p)(up, wp) = area of parallelogram spanned by u and w.
lt is not Ilard to see thai w is a volume element; Iocally, w is of the form w = J q J
for an 11-form q. But tliis canilot Ile true on al1 of M, siilce there is no 11-form q
on M which is everywhere non-zero.
T1-ieoi.em 7-7 Iias ail obviohs modification for volume elemeiits:
7-7'. THEOREh4. If J'; M + iV is a Cm function I~etweenrl-manifolds:
(x, U) is a coordiiiate sysrem arourld p E M, aiid (y, V) a coordinate system
arouiid q = f (p) E N , tlien for iion-llegative g : V + W we have
PROOF. Go tl-irouglitlie proof of Tlieorem 7-7, putting in absolute value signs
in rhe right place. +f,
7-8'. COROLLARY. If (x, U) and ( y , V ) are two coordinate systems on M
and
Idy' ~ . . . l i d y " =
J bldxi A
~
~
~
A
g~ , hX2 0"
~
tlien
[Tliis coi-ollary sliows tliat volume elements are tlie geometric objects corresponding to tl-ie "odd scalar densities" defined in Problem 4-10.]
It is now an easy matter to iiitegrate a volume element o over any manifold.
First we define
Tlien foi- an n-chain e : [O, 11" + M we define
'
Tlieoirm 7-7' sliows tliat Proposition 1 holds for a vol ume element o = f ldx A
A dx"] even if det e' is not 3 O. Thus Coi.ollary 2 liolds for volume elements
eveii if det p' is iior 3 O. From this we conclude that Theorem 5 holds for
volume elements o on aiiy manifold M, without assuming el, c2 orientatioii
presen~ing(or even diat M is orientable). Coi~sequentlywe can define JM o
for ai-iy voluiiie elcment o wit1-i compact suppor~.
Of course, when M is orientable these consider-ations are unnecessary. For,
there is a nowliere zero 11-foi-mv on M, and consequently any volume element o
can be written
= f Ivl, f 2 0.
1f we clloose an orientation p for M such that o ( v l , . . .,U,)) > O for vi,. . . ,U,
l~ositivelyoriented, tlien we can define
I~olumeclements will be important later, but for tlie remainder of this chaptei.
we are coiiccrned oilly ~ 4 t hintegrating forms over oriented manifolds. In fact,
our main resuli about iiltegi-alsof forms over manifolds, an analogue of Stokes'
Theor-em about the integral of forms over chains, does not work for volume
elements.
Recall from Problem 3-16 that if M is a manifold-with-boundary, and p E
aM, then certain vectors v E Mp can be distinguished by the fact that for any
coordinate system x : U + W" around p, the vector x+(v) E Wnf(pl points
"out~~ards".
14%cal1 such vectors v E Mp "outward pointing". If M has an
01-ientationp , we define iIíe induced orientation ap for aM by the condition that
[ u I , . .. ,
E ( 8 ~ if) and
~ only if [w, vi,. . .,~ ~ - 1 E1 pp for every outward
poiiiting w E Mp. If p is the usual orientation of H", then for p = (a, O) E W n
we have
Since (-en)p is aii outward poiiiting vector, tliis shows that the induced orieiitation on IRR-' x {O) = a W R is (- l ) n times t1le usual one, The reason for this
choice is the follotiliilg. Let c be an 01-ieiitationpreserving singuIar n-cube in
( M , p) sucli thai aM n c([o, 11") = C ( ~ , ~ ) ( [ O11"-').
,
Then c(.,o): [O, 11"-' +
(aM, ap) is 01-ieilrationpi-esenring for even n , and orientation reversing for
odd n. If w is ai-i (n - 1)-foi-mon M whose support is contained in tlie interior
of tl-ie image of c (this interior contains points in the image of
ihat
it follows
But c(,,0) appears witll coefficient (- 1)" in ac. So
If it were not for this choice of ap we would have some unpleasant minus s i p s
in the followiiig theorem.
6. THEOREM (STOKFS' THEOREM), If M is ail oriented n-dimensional
manifold-with-boundary, aild aM is giveii the induced orieniation, and o is an
(il - 1)-form on M with compact support, then
PROOF. Suppose first that there is an orientation preserving singular n-cube c
in M - aM suc11 rhat supporr o c interior of image c. TIien
=O
since suppori o C interior of image c,
Suppose nexi that tlierc; is an orientation presenring singular n-cube c in M
sucli tliat 8~ n c ([O, 11")= C(,,~)([O, 11"-'), and suppori o c interior of image c.
Then once again
Jn geiieral: rIiere is an open cover O of M and a partition of unity @ subordinaie io 13 such ihai for each $ E @ the form # . o is one of the two sorts
already considered. \We have
Sii-ice o lias compacr suppori, tIiis is reaIIy a finite sum, and we conclude tliat
Therefore
One of ilie simplesr applicaiioris ofSrokes7Theorein occurs wlien ilie orjeiiied
iz-n~niiifold( M , p ) is coinpaci (so tliat every foim has compact support) and
a& = 0. 111iliis case, if is aiiy (u - 1)-form, ihen
Tlierefore we caii fii-id aii 11-fo1.m o on M wllich is t~ofexact (eveii thougli ir
iiiusr be closed, l~rcauseal1 (12 + f )-firms on M are O), simply by findiiig aii o
wi t11
Such a fui.i~~
o iil~vaysexisis. Indecd we havc seen ihai il-iere is a fm-iii o sucl~
ihai foi- vi,. . .,u,, E M, we liave
If c. : [O, I]" + ( M , 11) is oi-ieiitatioii pi.esei\*iiig,trheri t l ~ efoiam c*w o11 [O, 11" is
c.leaiiy
g dxi A - . A dx"
for some g > O oil [O, I]",
1,
so w > 0. Ii follows that JM w > O. Tliere is: moreover, no need to choose
a foi-m w with (*) holding eveiywhere-we can aIIow the > sign to be replaced
by >_ . Thus Mte can even ob tain a non-exact 11-form on M which Iias support
contained in a coordinate neighborhood.
This seemiiigly minor result already pi.oves a theorem: a compact oriented
maiiifoId is not smoothly contractible to a point. As we have already emphasized, it is tlie "shape" of M, rather than i ts "size", which de termines whether
or not every closed form on M is exact. RoughIy speaking, we can obtaiii
more informatioii aboui tlie shape of M by analyzing more cIoseIy the extent
to ivhicli closed foorms are not necessarily exaci. In particular, we wouId now
Iike to ask jusr Iiow many non-exact 17-foi-msthere are on a compact oriented
11-manifoId M. Naturally, if w is not exact, then tlie same is true for w + d q for
any ( n - 1)-form q , so we realIy want to consider w and w d q as equivalent.
Tliei-e is, of coui.se, a standard way of doiiig tliis, by coiisidering quotient spaces.
14%will apply tliis coiistruction not onIy to 11-forms,but to forms of any degree.
For eacli k, ihe collectioii z'(M) of a11 closed k-rorms on M is a vector
space. Tlie space B'(M) OSal1 exact k-forms is a subspace (since d2 = O), so
we can form tlie quotient vector space
+
this vecior space H k (M) is cvlled iIie k-dimensional de Rham cohomoIogy vector
space of M. [de Rlzam % T/zeore~nstates tliat tliis vecior space is isomorphic to
a ccrtain vectoi. space defiiied pureIy in ierms of the iopology of M (for any
space M), called tile "k-dimensional coliomology gi-oup of M with real coefficients"; tlic noiatioii z', 3' is chosen to cori-espond to the notation used iii
algebraic ropolog): wliere ihese groups are defined.]
An element of H k ( M ) is an equivalente class [m] of a closed k-form o, two
cIosed k-foi.ms w1 aiid w2 beiiig equivaleiit if and onIy if their dserence is exact.
Iii terms of these vector spaces, tlie Poiiicart Lemma says that H' (W") = O (the
vecior space coiitaiiiing only O) if k > O, or more generally, H' ( M ) = O if M
is contractible aiid k > 0.
To computc H0(A4) we iioie fiist iIiat BO(M) = O (there are no non-zero
exacr O-hrms, since tliere are no non-zero (-1)-forms for them to be the difSerential 00. So H'(M) is iIie same as iIie vector space of al1 Cm functions
.f : M + W ~ 4 t hdf = O. If M is coiiiiected, the condition d f = O implies
tliat / is constant, so H0(M ) % R . (Iii general, tlie dimension of H'(M) is the
iiuml~erof components of M.)
Aside from these trivial i-emarks, we 111-esentlyknow only one other fact about
H k (M) -ir M is compact and oriented, tIien H n( M ) Iias dimension >_ l . The
furtlier study of H k ( M ) i.equires a careful look at spheres and Euclidean space.
O n S"-' C R" - {O) tliere is a naturaI choice of a n (n - 1)-rorm a' with
E S"-lp, we define
Js,l-l 0
' > O: for VI)^, . . . ,
ClearIy iliis is > O if
.. . , ( v , - ~ ) is~ a positiveIy oriented basis. In fact,
we defiiied ihe 01-ientationof S"-' iii precisely iliis way- tliis orientation is just
tlie induced oiientation ~ t l i e nS"-' is considered as the boundary of the unit
ball { p E IW" : Ipl 5 1 ) ~ i i l tlie
i usual oi-ieiiiaiion. Using the expansioii oT a
rlctermiiiaiit by minors along tlie top row we see tIiat o' is tlie restriction to
S"-1 of ilie form a on R" defiiied by
Tbe form a' on S"-' will now be used to find an (n - 1)-form on R" - {O)
wliich is closed but not exact (thus sl~owiiigthat H"-' (W"- (0)) # 0). Colisideithe inap 1 . : IW" - {O) + S"-' defined by
ClearIy i0(p)= p if p E S"-'; otheiwise said, if i : S"-' + R" - {O) is tlie
inclusion, tllen
r o i = identity of S"-'.
A c X aild 1.: X + A satisfies r ( a ) = a for a E A, tllen r ir;
called a retraction of X onto A.)
Clearly, r*a' is closed:
(Ii1 seneraI, if
Ho~t-vcr,i t is not exact, for ir r"a'
11ui íve ki~owthat a' is not exact.
-
dq, then
It is a worthwhile exercise to compute by bru te force tliat
xdy-ydx
xdy-ydx
= d6'
x2 y2
v2
forn=:2,
r *o1 =
for n = 3,
r *oI = x d y ~ d z - y d x ~ d z + z d x ~ d y
(,Z + }'2 + $)3/2
+
1
=-[xdy~dz-ydx~dz+zdx~dy].
u3
Since we ~ 7 i I lactually need to know r*olin general, we evaluate it in anothei.
way :
7. LEA4h4A. If o is the form on IR" defined by
o=
E(-
1)'-lx'
dxl A -
-.
A dxJ A
A dx",
aiid o' is the restrictioli i*o of o to S"-', then
IWP
PROOF. Ai any poiii t p E IWn - (O), tlie tangent space
is spanned by pp
aiid tlie vectors vp ili tlie tangent space oT tlie spliere S"-' (1pl) of radius 1 pl.
So ir sufices to check that botli sides of (*) give the same result when applied
io ir - 1 vectors eacll of wliich is one of tl-iese two sorts. Now pp is the tangent
vector of a curve y lyiiig aloiig ilie straight line tliiough O and-p; this curve is
takcn to tlle single point r ( p ) by r, so r+(pP)= O. O n the other hand,
So i t suffices to apply both sides of (*) to vectors in the tangent space of
S"-' (1~1).Tlius (Pi-ol~lem15), i t suffices to s l i w that for such vectors vp we
liave
1
But tliis is almost obílious! since the [rector vp is t l ~ etailgent vector of a circle y
lyins
.... in S"-' (lpl), aiid the c u n a r. o y líes in S"-'
and goes I/lpl as far in the
same time. *:*
8. COROLLARY (INTEGRATION I N "POLAR COORDINATES"). Ler
f : 3 + IR,where
= { p E Ik" : 1pl 5 J ] ,
and define g ; S"-' + W b!.
Tlieii
/
~ f = ~ f d x ' ~ - - . ~ dSi,x t "g o=r m
PROOF. Coiisider S"-' x [O, 11 and the two projectioiis
ni : S"-'
: S"--'
x [O, I ] + S"-'
X
[O, 11 + [O,11.
Lei us use tlie abbreviation
S"-'
a' A dt = ni*orA n2*di.
a
IT (y, U) is a coordiiiate system on S"-', \vi tli a corresponding coordinate system (J',t ) = ( y o T I ,n2)on S"-' x [O, J ] , and a r = cr dy' A - - A dyn-' , theii
clearly
o r h d r = & o n i d j l~ . . . n d y " - ' ~ d t .
From rliis i t is easy to see tliat if we defiiie h : S"-' x [O, 11 + IR by
h ( p , u ) = un-' f ( u p),
theii
/
SI!- 1
= (- )"-l
/
/zar A dt
SI)- t x[o,IJ
NOM'í~recaii define a dfleonioi-pl?ism @ : 3 - { O ) + S"-' x ( O , J ] l y
Then
-
(- 1)"-'
""+1
n
E ( X 'd x)l~
A
m
A ds"
i=l
Hence
= ""-1 f .
f dxl A .
(- l)"-'
Y"-1 d x l n - . A dx"
. . A dxn = (- I)'-'
/
J
$*(ha1A d l )
3-(01
= (- l)"--I
ha' h dt
S"-' x ( 0 , l j
Stl- t
(This lasi siep requires some justificaiion, w11icIi sliould be supplied by the
reader, since tlie rorms involved do not bave compact support on the maniIoIds B - {O) aiid S"-' x (O, 1 ] where they are defined.) +:+
We are aboui i-eady io coinpuie H' ( M ) in a kw more cases. We are going to
reduce our calcuIations to calculaiions within coordinate neighborhoods, which
are sul~nianifoldsor M , bui not compact. It is tllererore necessary to introduce
ailotIier co11eciion of vector spaces, which are interesting in their own righi.
The de Rham cohomoIogy vector spaces with compact supports 13: ( M ) are
wliere Z: ( M ) is the vector space of closed k-forms with compact support, and
B:(M) is tlie vecior space of al1 k-rorms dq where q is a (k - 1)-Torm with
compact support. Of course, if M is compact, then H: ( M ) = Hk(M). Notice
tliat ~ , ( kM ) is nol ihe same as tlie set of al1 exact k-forms with compact support.
For example, on Rn, if f 2 O is a function with compact support, and f > O
ai some point, then
o = f dx' A - . .~ d x "
is exact (ever): closed form on Rn is) and 11as compact suppor t, but o is not dq
for any form q with compact support. Indeed, if o = dq where q has compact
support, tlien by Stokes' Theorem
Tliis example shows tliat HC(Rn) # 0, and a similar argument shows that
if M is any orieniable maiiifold, ilien H;(M) # O. M7e are now going to sl~ow
tl~atfor any connected orientable manifold M we actually have
Tliis means ilia t if we clioose a fixed o wi tli JM o # O, then for any n-form o'
witli compaci support there is a real number a sucli that w' -ao is exact. The
number a can be described easily: if
then
ilie problem, of course, is sliowing ihat q exists. Notice that the assertion thai
R is equivalent to the assertion that
H:(M)
II
is an isomorpliism of H!(M) with R, i.e., to the assertion that a closed form o
with compaci suppori is the dflerential of another form with compact support
if JM o = O.
9. THEIOREM. If M is a coiinecied oi-ieniable ti-ma~iifold,then H," ( M ) i-- R.
PROOF. \Ve wi11 establish tlie theorem in three steps:
(1) The theorem is true for M =: W
(2) If the t11eorem is a u e for (n - 1)-manifolds, in particular for S"-', then
i t is true for W n .
(3) If t11e tlieorem is true ior IR", then it is true Tor any connected oriented
11-manifold.
S~efi1. Le t w be a 1-form on W wi th compact suppor t such that JR w = O. There
is some fui~ctionf (not iiecessarily with compact support) such that w = df.
Since suppori o is conipact, df = O outside some interna1 [-N, N], so f is a
constan1 cl on (-m, -N) and a constant c2 on (N, m ) . Moreover,
TIierefore ci = c2
-
c aiid we Iiave
wIiere f - c Iias compaci suppori.
Skp 2. Let w = .f dxlA - A dx-" be an 11-hm witl~compact support on W"
such tliat IR,, w = O. For simplicity assume that support w c ( p E W" : Ipl < 1).
Mre ki~owr11ai tliere is an (rt - 1)-form q on W n suc11 tliat w = dq. In fact, from
Prol~lem7-23, we tiave an expIicit formula for q,
Using tlie subsriru tion u = 1 pl l this becomes
Define g: S"-' + W by
0 i i tlie ser A
= { p E Wn : 1 pl > 1) we I1a17e f = O, so on A we Iiave
h4oi~eover,11y CoroIIary 8 hreIiave Ioi ilie (11 - 1)-Iorm ga' on S"-',
=
J..
o = o.
Tlius, Ily rlie I~yporliesisfor S ~ c j i2,
go' = d l
Ior some (il - 2)-form 1 o11 S"-'.
I x t lr : R" + [O, I ] l ~ caiiv CbO Iunctioli witli h = 1 011 A aiid h = O in a
iirigliboi-Iioodof O. Tlieii jii-*A is a Cm form on W" and
the foi'm q - d(hlm*h)Iias compact support, since on A we have
Sley 3. Choose an n-form o such that JM o # O and o has compact support
coiitaiiied iil an open set U c M, with U diffeomorphic to Rn. If o' is any
otlier 12-Sorm with compact support, we want to show that there is a number c
aiid a form q witli compact suppor t sucli tliat
Using a partiiion of unity, we can write
wliei-e eacli $;o' lias coinpact support contained iii some open set Ui C M
with Ui diffeomorpliic to R". Jt obviously suffices to find ci and with $irn/ =
t i a + dqil Sor eacli i . In otlier words, we can assume m' Iias support contained
in some open V c M wliicll is difíeomorphic to Wn.
Usiiig tlie coiiiiectedness of M, ii is easy io see iliat there is a sequence of
open sets
u = vi, ..., V, = if
diffcomoi-pliic to R", with I/i n
n
# 0. Clioose Sorms w j witli support
c
and Jv. oi # O. Siiice we are assuiniiig the tlieorem for W" we haw
I
W ~ I C I P al1 q j liave coiiipact
desired resuli. *3
support (C K). From tliis we clearly obtain the
The method used in tl-ie last step can be used to derive another resu1t.
1 O. THEOREh4. IT M is any connected non-orientable n-manifold, then
H ; ( M ) = O.
PROOF. Clioose a n 11-Torm o ~ 4 t compaci
h
suppori contajned in an open set U
difiomorphic to IR", such that Ju w # O (this integral makes sense, since U is
orieiitable). It obviously suffices to sIlow that o = dq for some form q with
compact support. Consider a sequence
u = v , , . . . , vr = v
of coordinate systems (Vi, x i ) where cacIi xi o xi+l -1 is orientation preserving.
Choose tIle forms mi in S1el3 so that, using the orientation of Vi which makes
xi : 6 + IR" orieniaiion pi-esen~iiig,we llave
iYii > O ; then also JVirl mi > O.
Consequently, the numbers
iK
c =
w
/
It ~ O ~ I O W S that
oi =: Cw
o
are positive.
+ dq wliere c > 0.
Now if M is unorientable, tllere is such a sequei-icewhere V, = VI but A-, 0x1-'
is orieiitaiion rezielrnizg. Taking o' = -o, we llave
-w = cw + dq
Tor c > O
*
(-c
- ] ) o= dq
Tor - c - 1 # O. +3
We can also compute H " ( M ) Toi- non-compact M
11. THEOREh4. If M is a connected non-compact n-manifold (orientable or
not), ihen H n ( M ) = 0.
PROOF. Consider firsi an 11-form o witli support contained in a coordinate
i~eigliboi-lioodU wIiic1-i is diffeomorphic io IR". Sii-ice M is not compact, t11ere
is an infiniie sequence
u =: U,,Uz,U3,U4,...
of sucli coordinate neighborhoods such that Ui n Ui+1 # 0, and sucli that the
sequence is eventually in the complement of any compact sei.
Now clioose ir-forms mi witll compact support contained in Uj n Ui+i, such
tliat Jui q # O. There are constants cf and forms
with compact support
c Ui sddi tliai
Since aiiy poiiit p E M is eventually in the complement of the UiYs,we have
where the riglit side makes sense since the Ui are eventually outside of any
compact set.
Now it can be sliown (ProbIem 20) that there is actually such a sequence
U1,U2, U3,. . . whose union is al1 of M (repetitions are allowed, and Uj may
intersect several Uj for j < i, but the sequence is still eventually outside of any
compact set). Tlle cover 13 = ( U ) is tllen locally finite. Let ($u) be a partition
of unity suboi.di~iareto O. If o is an 19-form oii M, then for each Ui we have
seen that
$vio =: dqi
Hence
wl~ereV i has support contained in Ui U Ui+1 U Ui+2 U - . - .
SUMMARY OF RESULTS
(2) If M is a connected n-manifold, then
H;(M) =
R
O
ir M is orientable
ir M is non-orientable
e if M is compact
Hn(M'-{~
if~isnotcompact.
\/Ve also know that Hn-' ( R n- ( 0 ) )# 0, but we have not listed tllis result, since
we will eventually improve it. In order to proceed furiher with our computations
we nced to examine the behavior of the de Rham cohomology vector spaces
under Cm maps f ; M + N . If o is a closed k-form on N , then f * w is also
closed ( d f *o= f * d o = O), so f * takes Z k ( N ) to Z k ( M ) . O n the other hand,
f * also takes B k ( N ) to B k ( M ) , since f * ( d q ) = d( f *q). This shows that f *
induces a map
( N ) / B ~ ( N+
) z~(M)/B~(M),
zk
also deno ted by f * :
f*:H ~ ( N+
) H~(M).
For example, consider the case k = 0. Ii N is connected, then H O ( N )is just
the coIIection of constant functions c : N + R. Then f " ( c ) = c o f is also a
constani funciion. If M is connected, then f * : H O ( N )+ H ' ( M ) is just the
identity map under tlie natural identificatioii of H O ( N )and H O ( M )with R .
If M is disconnected, with components M,, a E A , then H O ( M )is isomorphic
to the direct sum
R
where each W , % R ;
U EA
ilie map f * takes c E W into the element of @ R . with a" component equal
to c . V N is aIso disconilected, with components Ng, E B , then
takcs the elemeiit {cg) of @gEB Rg to
(41,
where c; = cg when / ( M u ) c Ng.
A inore interesting case, and the only one we are pi-esently in a position to
Iook at, is the map
f * : H n ( N ) + Hn(M)
when M and N are both compact connected oriented n-manifoIds. There is
no natural way io make H n ( M ) isomorpl-iic to IR,so we reaIly want to compare
for o an 11-form on N. Clioose oiie wo wi th JN o 0 # O. Then there is some
number a sucll tliai
r
F
Siiice o H lM
o is an isomorphism of H n ( M ) and R (and similarly for N) it
folIows tliai for evay form o we Iiave
TIie ilumber a = deg f, w1iicIi depends onIy on f, is called tIle degree of J:
If M and N are not compact, bu t f is propei- (the inverse image of aily compact
set is compact), tlien we have a map
f * : H,"(N) + H,"(M)
and a number deg f, sucli that
h r al1 forms o on N with compact support. Until one sees the proof of the
nexi iheorem, it is almost unbelievabIe that tlis number is always an inlegm.
12. THEOREM. Let f : M + N be a proper map between two connected
oriented n-maiiifolds ( M , P ) and (N, u). Let q E N be a regular value of f.
For each p E f -' (q), let
signp f =
(
1
if f*p : Mp + N4 is orientation preserving
(using the orientations pP for Mp and v4 for N4)
-1
if
is orientation reversing.
TIien
degf=
1 sipp/
(=0iff-'(~)=0).
~€f-'(4>
PROOF. Notice first that regular values exist, by Sard's Theorem. Moreover,
f -'(q) is finite, since it is compact and consists of isolated points, so the sum
above is a finite sum.
Let f -' (q) = ( p l , . . . , pk}. Choose coordiiiate systems (Ui, xi)' around pi
such tliat all points in U j are regular values of S, and the Ui are disjoint. \Ve
want to clioose a coordinate system ( V, y ) around q such that f -' ( V) = U1 U
. - U Uk. To do this, first choose a compact neighborliood W or q, and let
W' c M be the compact ser
w'= f - ' ( w ) - (u1 u . ' . u U k ) .
TIien f ( W') is a dosed set which does not contain q. We can therefore choose
V C W - f ( W'). T h i s eiisures tliat f -' (V) C U1U . -UUk. Finally, redefine Ui
to be Ui n f -'(VI.
Now clioose w on N to be w = g dyi A
A (13'" wliere g 2 O has compact
support contained in V. TIieil support f *w c U1 u . - u Uk. SO
Sinc,e f is a diffeomorpliism li-om each Ui to V we have
S, f*w=S,
ir f is orieiitation preserving
=-S,
if f is orientation reversiiig.
w
w
Siiice f is orientatioii pi-esenriiig [or reversing] precisely wlien s i p , f = 1 lor
-11 tllis proves the theorem. *t,
As an immediate application of tIie iheorem, we compute the degree of tlie
"antipodal niap" A : S" + S" defined by A ( p ) = - p . ' We have already
seeii that A is orientation presenring or reversing at aU points, depending on
wliether rl is odd or even. Siiice A-' ( p ) consists orjust one point, we condude
tha t
deg A = ( - ] ) " - l .
We can draw an interesting concIusion from this result, but we need to intsoduce ailotller imporiaiii concept first. Two fuiictions f,g : M + N between
two Cm manifoIds are called (smoothly)homotopic ir there is a smooth function
tlle map H is calIed a (smooth)homotopy between f and g. Notice that M is
smootlily contraciible to a poiiit po E M ifand only if the identity map of M is
liomotopic to tlie coiistaiit map po. Recall tliat Tor every k-form o on M x [O, 11
we defiiied a (k - 1)-form ío on M such thai
ir*# - io*o -- d ( 1 o ) + 1 ( i l o ) .
M7e used tliis fact to sllow tliat al1 cIosed forms oii a smoot1~1ycontraciible maiíifold are exact. We can iiow prove a more general result.
1 3. THEOREM. If J; g : M + N are smoothly homotopic, then the maps
f*:H ' ( N ) + H ' ( M )
g*: H ' ( N ) + H ' ( M )
are equal, f* = g*.
PROOF. By assumprioii, ihere is a smootIi map H : M x [O,11 + N with
f =:HoiO
g =H oii.
Aiiy element of H' ( N ) is tlie equivaleiice class [o]
o i some closed k-form o
on N. Tlien
g*w - f * o = ( H o i i ) * o- ( H O i O ) * o
=: i l * ( H * w )- io*(H*w)
d ( 1 H *o)+ í ( d H * o )
= d ( í H * w )+ 0.
=:
But tliis means that g * ( [ o ] )= f * ( [ o ] ) .+a:
14. COROLLARY. Ir M and N are compact oriented n-manirolds and t11e
maps f,g : M + N are homotopic, then deg f = deg g.
15. COROLLARY. T i 11 is even, then there does not exist a nowhere zero
vector field on S".
PROOF \/Ve have aIready seen that the degree or the antipodal map A : S" +
S" is ( - ] ) " - l . Since tlie identity map has degree 1 , A is iiot homotopic to
the identity ror I I even. But if there is a nowhere zero vector field on S", then
we can construct a I-iomotopy between A and the identity map as roIIows. For
each p, there is a unique great semi-circle yp irom p to A ( p ) = - p whose
iangeni vector at p is a muItipIe of X ( p ) . Define
For n odd we can expIicitly construct a nowhere zero vecior fieId on S". For
p = (xi ,. . . ,xn+i ) E S" we define
this is perpendicular to p = ( x i,x2, . . . ,x , + ~ ) and
,
therefoore in S",. (On S'
this gives the standard picture.) The vector field on S" can then be used to give
a Iiomotopy between A and the identity map.
For another applicaiion or Theorem 13, consider the retraction
r : R" - ( O ) + S"-'
Y(P)
=P/IPI.
Ir i : S"-' + R" - ( O ) is tl-ie inclusion, tlien
r o i : S"-' + S"-' is the identity 1 or S"-'.
Tlle map
is, of course, not the identity, but i t is I-iomotopic to the identity; we can define
ihe llomotopy H by
A 1-etraction with tllis property is called a deformation retraction. Whenever 1.
is a deformation i-etractioil, the inaps (r oi)* and (i o).)* are the identity. Thus,
for tlle case of S"-' c R" - ( O ) , we have
and
r.* o i* = ( i o r)* = identity of H k ( R " - {O))
i* o r.* = ( I . o i)* = identity of H k ( s " - ' ) .
So i* aiid i." ai-e inverses of eacll other. Thus
H k ( S " - ' ) ';. H k ( R " - ( O ) ) for al1 k.
Iii particular, we Iiave H"-'(R"
tlle closed form r*aJ.
-{O))
R . A generator of H"-' (R" - { O ) ) is
We are now going to compute H k (R" - ( O ) ) for al1 k . We need one further
observa tion. The manifold
M x ( O ) c M x R'
is clearly a deformation retraction of M x R'. So H k ( M )
for al1 I .
H k ( M x IR')
Now
~ ' IA 'IB) - 0
onAnB
and
H' (A n 3 ) = H'([R3 - (O)] x R)
H'(R3 - (O)) = O.
So ~ J A- ~ J B= d h for some O-form h on A n B. Unlike the previous case, we
cannot simply consider VA - d h , since this is not defined on A . To circumveni
this difficulty, note that there is a partition of unity ($A,$B) for the cover (A, 3 )
of R3 - (0):
$A+$B
+
=1
d $ ~d $ =
~O
support $A c A
s u p p o r t $ ~C B.
Now, if
denotes
$ ~ honAnB
O
onA-(AnB),
and similarly for $AA, then
$ ~ h is a Cm form on A
$AA i s a C W f o r m o n 3 .
VA - d ( $ ~ h )=: VA - $B dh
- d $ A~
+ ($A - 1) d h + d$A
= 'IA - dh + d($Ah)
= 'IA
= 'IB
A
+ d($Ah).
So we can define a Cm form on R" - (O) = A U 3 by letting it be VA - d ( $ ~ h )
on A, and q~ + d ( $ ~ h )on 3. Clearly,
so w is exact.
Tlie general inductive step is similar. Q
\/Ve end this chapter with one more calculation, which we wiU need in Chapier 11.
17. THEOREM. For O < k < n we have H~<(w")= O.
PROOE Tlie proof that H:(R") = O is left to the reader
Let o be a k-form on Rn with compact support, O < k < n. We know that
o =: d q for some (k - 1)-form q on R". Let 3 be a closed ball containing
support u.Then on A = R" - B we have d q = O. Since A is dfleomorpliic to
R" - ( O ) alid k - I < n - 1 we Iiave from Theorem 16 thai
q =: d h
for some (k - 2)-form h on A .
Lei f : R" + [O, 11 be a CbDfunciion wiili f = O in a neighborllood of 3 and
f =: 1 on Rn - 23, where 2B denotes the ball of twice the radius of B. Then
d (fA) makes sense on al1 of R" and
ihe form q - d ( fA) clearly Iias compaci suppor t contained in 2 3 .
PROBLEMS
1. The Riemann iniegral umstrs iheDmboux iniegrab Let f : [a,b] + W be bounded.
For a partition P = ( l o <
< 1,) of [ a ,b ] , let n7i = 1 7 7 ~ f( ) be the inf of f
on [li-I, li] and define Mi = Mi( f ) similarly. A choice for P is an n-tuple
6 = (ti,.. . ,t,,)with ti E
li]. Me' define the "lower sum", "upper sum",
and "Riemann sum" for a partition P and choice 6 by
CIear-ly L(f,P ) 5 S (J; P , 6 ) 5 U ( f , P ) . 14%cal1 f Darboux integrable if the
sup of al1 L(f , P ) equals the inf of all U ( f , P ) ; this sup or inf is called the
Darboux integral of f on [ a ,b ] . 1We cal1 f RXemann integrable if
Iim S ( f , P , t ) exists;
llPIl+O
i11e limii is called tlie Riemann integral of f on [a,b].
(a) We can ddine S ( f , P , 6 ) even if f is noi bouiided. Sliow however, thai
lim S ( f , P , 6 ) cannot exist if f is unbounded.
IIPl+O
(b) If f is coniinuous on [a,b ] , then f is Riemann aiid Darboux integrable on
[a,b ] ,and the two iategrals are equal. (Use uniform continuity of f on [a,b].)
(c) If f is Riernaiin integrable on [a,b ] , then f is Darboux integrable on [a,b]
and tlie two integrals are equal.
<
< S,) and Q = ( l o <
(d) Let n7 5 $ 5 M on [ a J b ] Let
. P = (SO <
I,) Ile two pariitions of [a,b ] . For e a ~ h
i = 1,. . . ,n, let
ei = lengtli of
, li]
- sum of Iengtlis of al1 [S,,
S,
3 wliich are con tained in
,- --
-
,S,] 'S coi~iainedin [li-l, li]
slladed lei~gihs add up to e!
[S,-
ti].
Sliow ihat, if Mi denotes ihe sup of f on [rl-1, li], then
T l ~ e r eis a similar result for Iower sums.
(e) Show that
ei + O as 11 Pll + 0, and deduce Darboux's Theorem:
E;=l
lim U( f, P ) = inf(U(f, Q ) : Q a partition of [a, b])
IP11+0
lim L ( f , P ) = sup(L(f, Q) : Q a pariition of [a,b]).
llp 11-
0
(f) If f is Darboux iniegrable on [a, b], then f is Riemann integrable on [a, b].
(g) (OsgoodYsTheorem). Let .f and g be iniegrable on [a, b]. Show that foi
choices 6 , t ' for P,
Hial: If Igl < M on [a, b ] , then If (tJi)g(Fi)- f (ti)g(tJi)l < Mlf (t'i) - f (tt)l.
(h) Sliow thai
1, f d x + g d j ~ defined
,
as a limit of sums, equals
2. Compute Ic d0 = Jfo,llC* do, where c(r) = (cos 2nr, sin2xr) on [O, 11.
3. For 11 a n inieger, and R > O, jet C R , ~: [O, l ] + R2 - {O) be defined by
~ ~ , n (=t (R
) cos 212711, R sin 2~711).
(a) S h w thai there is a singular 2-cube c : [O, 1l2 + R2 - {O) such that CR, ,n CR2,n = ac.
@) If c : [O, 11 + R2 - {O) is a i ~ ycuive witli ~ ( 0 =
) c(J), show ihat there is
some n sucli that c - cl,, is a boundary in JR2 - (0).
(c) Show tliat n is unique. Ji is called the windingnumber of c around O.
4. Let f : @ + @ be a polynomial, f(í) = ~ " + a ~ í "+ .- - ~- + a n , where n 2 l .
Define c ~ , :f [O,l ] + G by c ~ , =
f f o cR,1.
-
(a) Show that if R is large enough, ihen c ~ , -f CR,n is the boundary of a chain
in @ - (O).Hinl:Note that C ~ , , , ~ ( [[cR,1
) ( I ) ] " ,and wriie
(b) Sliow tl~aif (z) = O for some z E G ("Fundamental TI~eoremof AIgebra").
Hinl:If f ( 2 ) # O for al1 z with 121 5 R, ihen ~ R , J- co,f is a boundary,
5. Some approaches to iniegraiion use singular simplexes insiead of singular
cubes. Alihough Stokes' Tlleorem becomes more complicaied, tl~ereare some
advantages in using singular simplexes, as indicaied in the next Problem.
Lei An C R" be ihe sei of al1 x E Rn such ihat
A singular rl-simplex in M is a Cm function c : A , + M, and an 12-chain
is a formal sum of singular 17-simplexes. As before, let I n : An + RQe the
inclusion map. Define ai :
+ A , by
n-1
1
a0(x-) = ( [ 1 - zi=,
x Z ] ,x , . . . , X n - l )
a i ( x ) = (X 1 , . . . ,A+i - t , O i , . . ,x n l )
O < i < n,
and for singular rz-simplexes c , d e h e sic -- c o a i . Then we define
(a) Describe geometrically the images a i ( A n - 1 ) in A , .
(b) Show that a2 = 0.
-
(c) Sllow thai if o = f d x l A - - A dxi A .
then
A dx" is an (n - 1)-form on W n ,
(Xmiiate the proof for cubes.)
(d) Define Jc o for any k-chain c in M and k-form o on M , and prove thai
for any (k - 1)-form w .
6. Every x E Ak+1 can be writien as tx', for O 5 r 5 1, and x'
E ao(Ak).
h4orcover, x' is unique excepi wlien r = O. For any singular k-simplex c : A k -t
define C : Ak+1 + W" by
We then define F for chains c in the obvious way.
(a) Show tliat 8c = O implies that c = 8F.
(b) Let c : [O, 11 + JR2 be a closed curve. Show tliat c is ltol the boundary of
any sum a of singular 2-cubes. H i n l : If 80 =
a i c i , what can be said about
Ciai ?
(c) Sliow ihai we do llave c = 80 + c' wliere c' is deSe?zm~le,tliat is, c' ([O, 11) is
a poin t.
(d) If c~ (O) = ~ ~ ( and
0 ) cI(l) = c2(1), show that ci - c2 is a boundary, using
either simplexes or cubes.
7. Let o be a I-form on a manifold M . Suppose that Jc o = O for every closed
If we do have o = dS, tllen for any
curve c in M . Show that o is exact. Hin~:
curve c we llave
E
8. A manifold M is called simply-connected if M is connected and if every
smooth map f : S' + M is smoothly contractible to a point. [Actually, any
space M (not necessarily a manifold) is called simply-connec ted if i t is connected
and any continuous f : S' + M is (con tinuously) contractible io a poin t. It is
not hard to show that for a manifold we may insert "smooth" at both places.]
(a) If M is smoothly conti-actible to a point, then M is simply-connected.
@) S is not simply-connected.
(c) S" is simply-connected for n > l . Hinl:Show that a smooth f : S1+ S"
is not onto.
(d) If M is simply-connected and p E M , then any smooth map f : S1+ M
is smoothly contractible to p.
(e) If M = U U Ir where U and V are simply-connected open subsets with
U n V conilected, then M is simply-connec ted. pliis gives ano ther proof that
S" is simply-connected for n > 1.) Hint:Given f :S' + M , partition S' into
a fmite number of intervals each of which is taken into either U or V.
(f) If M is simply-connected, ihen H 1( M ) = O. (See Problem 7.)
9. (a) Let U c R2 be a bounded open set such that W' - U is not connected. SIlow ihat U is not smoothly contractible to a point. (Converse of
'
Prooblem 7-24.) Hinl:If p is in a bounded component of R2 - U , show that
tliere is a curve in U which "surrounds" p.
@) A bounded connected open set U c IR2 is smoothly contractible to a point
if and only if i t is simply-connected.
(c) This is false for open subsets of R 3 .
1 O. Let o be an n-form on an oriented manifold M". Let @ and q be two
partitions of uiiity by functions with compac t support, and suppose thai
JM 9
(a) This implies tliai
(b) Show that
w converges absolutely.
xJMmw=
E t:JMqmw.
#e@
#E@
*e*
and show the same result with o replaced by lo 1. (Note that for each 9, there
are only finitely many xdiicl-i are non-zero on support 9.)
(c) Sliow iliat Ceo*JM @ lo1 < m, and that
\Ve define tliis common sum to be j, u.
(d) Let A , c (n,iz + 1) be closed sets. Let f : R + R be a CbQfunction witIi
iAII
f = (-])"/ti
and support f c U, A,. Find two partitions of unity @
and q sucli tliat &E,
9 f d x and ,E&'
i,il. f dx converge absolutely
to different values.
IR
IR
1 1. Followiiig Problem 7-1 2, define geometric objects corresponding to odd
1-elab'veteiisors of type (f) and weight w (w any real number).
12. (a) Let M be ((x, E R2 : I(x, y) 1 < I ), together witli a proper portion
JJ)
of its boundary, aiid let w = A- dy. Show that
elen ihough bot1i sides inake sense, using PI-obleni 10. (No computatioiis
iieeded-iioie tliai equaIity would hold if we had the entire boundary.)
(11) Similarly, fiiid a countei-example to Stokes' Theorem when M = (0, 1)
aiid w is a O-form whose support is not compact.
(c) Exaniiiie a partition of uiiiiy for (O,?)by fuiiciions with compact support io
see just wliy ilie proof of Stokes' Tlieorem breaks down iii this case.
13. Suppose M is a compact orientable 77-manifold (with no boundary), and 19
is an (n - 1)-form on M. Show that de is O at some point.
14. Let M I , M2 C R" be compact n-dimensional manifolds-with-boundary
with M2 c Mi - aMl. Sl-iow that for any closed (n - 1)-form w on Mi,
15. Account foi- the factor I/lp 1" in Lemma 7 (we have r,(vp) = ( I / ~ p l ) ~ , ( ~ ) ,
but this only accounts for a factor of l/lpln-', since there are n - 1 vectors
vi , ,vn-1).
16. Use the formula for r*dxi (Problem 4-1) io compute **ot. (Note that
tl-ie map i o r : R" - (0)+ R" - (O) is just r, considered as a map into R" - {O}.)
1 7. (a) Let M" and N m be oriented manifolds, and let w and q be an n-form
and an m-form with compact support, on M and N, respectively We will
orient M x N by agreeing that vi, . . . ,u,, wr, . . ., w, is positively oriented in
(M x
Mp $ Ng if V I ,. .. ,Un and wi , . . . , Wm are positively oriented
in M, and N,, respectively. If ni: M x N + M or N is projeciion on the ith
factor, show that
where
g ( p ) = JN
-)V.
~ I ( P-,1 = q
h ( p Y01.
(c) Every (m + n)-form on M x N is h n1* m A n2*TJ for some w and q .
18. (a) Let p ER"-(O}. Let wi,...,wn-2 E IR? andletu E IR> be (hp), for
some h E R. Show that
(b) Let M C IR" - (O) be a compact (n - 1)-manifold-with-boundary which is
ilie uiiion of segrnents of rays through O. Show that jM r*o' = 0.
(c) Let M c IRn-(O] be a compact (n- 1)-manifold-with-boundary whicl-i in tersects every ray througli O at mosi once, and let C(M) = (hp : p E M, h 2 O}.
Sliow tl-iai
/M
r*o' =
/
r*ol.
C(M)flSz
Tl-ie latiel. integral is il-ic nieasure of tlie solicl arigle subtended by M. For ibis
reason we often denote r*ol by d e n .
19. Fol. all {-Y,y,-) E lR3 except tliose wiih A* = O, y = O, r E ( - 0 0 , O], we
deline $(x, y,z ) to be ilie angle be tween il-ie posiiive z-axis and the ray from O
through (x, y, r ) .
+
(a) # ( x , y , z ) = arctan(Jx2 y 2 / z ) (with appropriate conventions).
(b) If v ( p ) = IpI, and 8 is considered as a function on W 3 , @ ( Sy, , z ) =
arctan y / x , then (u, 8 , #) is a coordinate system on the set of all points ( x ,y , z )
in IR3 except those with y = O, x E [O, cm) or with x O, y = 0, z E ( - c m , O ] .
(c) If u is a longitudinal unit tangent vector on the sphere S 2 ( r ) of radius r,
then d#(u) = l . If w points along a meridian through p = ( x ,y , z ) E SZ( r ) ,
-
t hen
(d) If 8 and 4 are taken to mean the restrictions of 8 and # to [certain portions
ofl S', then
o' = h d8 A d 4 ,
where h : S2+ IR is
h ( x , y , z )=
-m
(tiie minus si91 comes fmrn the orientation).
(e) Conclude that
a' = d ( - cos # d e ) .
(f) Let r2: R2- (O) + S' be the retraction, so that d9 = rz*i*o,for the form o
on It2. Sliow that
r2*d@= de.
If n : R3 + R2 is the projection, then the form d9 on [par t of] IR3 is just n*d@,
foi- tlie form d9 on [part ofJ R2. Use this to show that
(g) Also prove tliis directly by using the result in part (c), and the fact thai
r,(vp) = v,<,>/lpl for v tangent to S2(1pl).
(h) Conclude that
(i) Siniilarly, express d 0 , on R" - (O) in terms of do,,,
on R"-'
- (O).
20. Prove iliai a connected manifold is the union UI U U2 U U3 U . , where
ille Ui are coordinate neighborhoods, with Ui í l Uj # 0, and the sequence is
eventually outside of any compact set.
2 1. Let f ; M" + N" be a proper map beiween oriented n-manifolds such
that f, : Mp + NI(,) is orientation preseniing whenever p is a regular point.
Sllow that if N is connected, then either f is onto N, or else al1 points are
critica1 points of f.
22. (a) Sliow tliat a polynomial map f : C + C, giveii by f(z) = z" +ai 2"-' +
- . +a,, is proper (n 2 1).
(b) Let fl(z) = 71ín-I
(n - I ) a ~ z " - ~ .+an-1. Show that we have f' ( E ) =
lim [f ( c + w) - f (i>]/w, wliere w varies over complex numbers.
+
+
w+o
(c) M7riteJ'(x+iy) = u(x, y )+iv(x, y ) for real-valued func tions u and v. Sllow
that
av
-(x,
ay
au
y) - i-(x,
ay
Hinl: Choose w to be a real h , and then to be ih.
(d) Conclude that
If '(x i y ) l2 = det Df (x, y),
+
y).
where f ' is defined in part (b), whde Df is the linear transformation defined
for any dflerentiable f : R2 + R2.
{e) Using Problem 2 1, give ano ther proof of the Fundamental Theorem of Alge bra.
(f) There is a still simpler argument, not using Problem 2 1 (which relies on many
theorems of this cliapter). Show directly that if f : M + N is proper, then the
number of points in f (a) is a locally constant function on the set of regular
values of f. Sliow thai tliis set is connected for a polynomid f:C + C, and
conclude that f takes on all vdues.
-'
23. Let M'-' C R" be a compact oriented manifold. For p E Rn - M, choosi
an ( u - 1)-sphere E aiaund p such that aU points inside C are in R" - M. Let
rp : R" - ( p ) + E be the obvious retraction. Define the winding number w (p)
of M around p to be the degree of rp1 M.
(a) Sllow tllat this defii~itionagrees with that in Problem 3.
@) Show tllat tllis definition does not depend on the choice of C.
(c) Sllow that w is constant in a neighborhood of p. Conclude that w is constant on each component of IRn - M.
(d) Suppose M coiitains a portion A of an (ti - 1)-plane. Let p and q be points
close to tl-iis plane, but on opposite sides. Show that w(q) = w(p) z t l . (Show
tliat rq 1 M is homotopic to a map which equals rp1 M on M - A and which does
not take any point of A onto the point x in the figure.)
(e) Show tliat, in general, if M is orientable, then W n - M has at least 2 components. Tlie next few Problems show Iiow to provc tlie same result even if M
is not orientable. More precise conclusions are drawn in Chapter 11.
24. Let M and N be compact n-manifolds, and let f, g : M + N be smoothly
honiotopic, by a smootli liomotopy H : M x [O, 11 + N.
(a) Let q E N be a regular value of H. Let #/-'{y)
of points in f (q). Show tliat
-'
#f
denote the (fiiiite) numbei.
-'(q) = #g-i (q) (mod 2)
HUZI:H-' (y) is a compac t l -manifold-\vi tli-bouiidary The iiumber of poin h
iii its boundary is clearly even. (Tliis is one place where we use tlie strongeiform of Sard's Theorem.)
(b) Show, more geiicrally, tliat tliis result holds so long as y is a regular value of
I~otlif and g.
25. For two maps f, g : M + N we will write f
smoothly Iiomotopic to g.
(a) If f
-
g to iiidicate that f is
g , theii thei-e is a smooth homotopy H': M x [O, 11 + N sucli tliat
' { p , ) = f . ) for I in a neighborliood of O,
H' { j ~ 1,) = g ( p ) for iri a iieigliborhood of l .
01) -. is an equivalence i-elatioli.
26. If f is smootlily Iioniotopic to g by a smooth homotopy H sucl-i ihat p w
H { P , I ) is a diffeomorpIiism for eacli I, we say that f is smoothly isotopic to g.
(a) Reing smootlily isotopic is an equivalence relation.
(b) Let 4 : IW" + IW he a Cm function wliich is positive on tlle interior of the
uiiit hall, aiid O elscw1iei.e. I'or p E S"-', let H: W x Rn + W" satisfy
(Lach solution is defined for al1 r, by Tlieor-em 5-6.) Show that each s w H{t, s)
is a diffcoinoi-pliism, wliich is smootlily isotopic to the identity, and leaves al1
points outside the unit hall fixed.
171iegm[ion
2 95
(c) Show tl-iat by choosing suitable p and t we can make H(t, O) be any point
in the interior of the unit bdl.
(d) If M is connected and p , q E M, then there is a diffeomor~hismf : M +
M such that f (p) = q and f is smoothly isotopic to the identity.
f Skp 3 of Theorem 9.
(e) Use part (d) to give an alternate ~ r o o of
(f) If M and N are compact n-manifolds, and f : M + N, then for regular
values ql, qz E N we have
(where #fml(q) is defined in Problem 24). This number is cdled the mod 2
degree of f.
(g) By replacing "degree" with "mod 2 degree" iii Problem 23, show that if
M c IR" is a compac t {n ])-manifold, then R" - M has at least 2 components.
-
27. Let ( X ' ) be a CbDfamily of CbDvector fields on a compact manifold M.
(To be moi-e precise, suppose X is a Cm vector field on M x [O, 11; then X f ( p )
will denote T ~ * X ( ~ , From
~ ) . ) the addendum to Chapter 5, and the argument
wl-iich was used in the ~ r o o of
f Theorem 5-6, it follows that there is a CbDfamily
(4, ) of diffeomorphisms of M [not necessarily a l-parameter group] ,with & =
identity, which is generated by (X'), i.e., for any CbDfunction f : M + IR we
{X'f )(p) = lim
f( 4 i + h ( ~ )-) f (4' (P))
ii
h+O
For a family o, of k-forms on M we define the k-form
¿I,-- lim
Ut+h - a'
h+O
11
(a) Show that for q{t) = 4,*ofwe haw
(b) Let o o and o1 be riowhere zero 11-forms o11 a compact oriented n-manifold M, and define
o' = (1 - t)wo + /o1.
Sliow that the family 4, of di&omorpliisn~s generated by (X') satisfies
~ I * W= wo
if and only if
for al1 t
(c) Using Problem 7-18, show that this holds if and only if
(d) Suppose that JM 00 = JM 0 1 , SO that wo - o1 = d l for some l . Show that
there is a dfleomorphjsm J : M t M such that wo = fi*ol.
28. Let f : M' + W n aiid g : N' + R" be Cm maps, where M and N are
compact oriented manifolds, n = k + I + 1, and f ( M ) n g ( N ) = 0. Define
a l r g :M x N + S"-' C W" - (O)
We define the linkingnumber of f and g to be
wliere M x N is oriented as in Problem 18.
(a) [ ( f ,g ) = (-l)k'+ie(g, f ).
@) Let H : M x [O, J ] t W" and K : N x [O, 11 + W" be smootli homotopies
with
such that
( H ( P , I ): p E M } n { K ( q , r ): q E N ) = 0 for e v e r y ~ .
Show that
[ ( f , g ) = [tf 2).
(c) For f , g : S' + W 3 show that
where
A ( u , u ) = det
( f l )'(u)
( f 2>'(u>
( f3 ) ' ( ~ >
( g l)'(u)
(g2)'(v)
(g3)'(v>
s l ( v > - f l ( ~ )g 2 ( v ) - f 2 @ ) g 3 ( v ) - f 3 ( u )
(tlie factor 1/ 4 x comes from the fact that ls,o' = 4 n [Problem 9-14]).
(d) Show that L( f , g ) = O if f and g both lie in the same plane (first do it
for (x,y)-plane). Tlie next problem shows how to determine L ( f ? g ) without
calculating.
29. (a) For ( a , b , c ) E W 3 define
For a compact oriented 2-manifold-with-boundary M c
le t
and ( a ,b , c )
M,
E
a ( a ,b , C ) = JM dQ(a,b,c).
Let ( a ,b , c) aiid (a', b', c') be points cIose to p E M , on opposite sides of M.
Suppose ( a ,b,c) is on tlie same side as a vector wp E W 3 p - M, for which the
triple w,, (vi ),, ( ~ 2 is) positively
~
oriented in R~~ when (vi,,) (u2), is positivel y
oriented in Mp. S how that
fi ( a , b, c ) - S2 (a', b', c') = -471
lim
ta,b,c)+ p
(a',b7,c')+
Hint: First sliow that if M = a N , then L?( a ,b , c ) = -471 for ( a ,b , c ) E N - M
and S2 ( a ,b , c ) = O for ( a ,b , c ) N .
(b) Let f :S' + W' be an imbedding such that f ( S 1 )= aM for some compact oriented 2-manifold-~4th-boundary M . (An M with this property always
exists. See Fort, Zpologv qf 3-A..lnn23[dr, pg. 138.) Let g : S' + lT3' and suppose
The figure on the left shows a non-orientable
surface whose boundary is the "trefoil" knot,
l ~ i i tthe surface on the right-including
the
hemisphere behind the plane of the paperi r orientable.
that wben g(t ) = p E M we llave dg/dr $ M,. Let u+ be tlie number of intersecrions wl-iere dg/dt poiiits iil the same direction as tlie vector wp of part (a),
and
the number of other intersections. Show that
(c) Show that
S1iow tliar 17 =
for tlle pairs shown below.
30. (a) Let p , q E R" be distinct. Choose open sets A, 3 c 8" - ( p , y) so
tl-iat A and 3 are diffeomorphic to Rn - (O), and A fi 3 is diffeomorphic to IR".
Using an argument similar to that in the proof of Theorem 16, show that
H'{R" - ( p , q)) = O for O < k < 11 - 1, and that H"-' {R" - (p, q)) has
dimension 2.
@) Find the de Rham col-iomology vector spaces of R" - F where F C Rn ís
a finite set.
31. We define the cupproduct u : H k ( M ) x H'{M) + Hk+'(M) by
(a) Show t11at u is well-defined, i.e., w A q is exact if w is exact and r~ is closed.
(b) Sllow that u is bilinear.
(c) If ci E H ~ { M )a n d p E H'{M), then c i u p = (-1) k l p u a .
(d) If f : M + N, and ci E H k { N ) , E H'{N), then
(e) Tlie cross-product x : H' { M ) x H' { N ) + Hk+'{ M x N ) is defined by
Sliow tllat x is weI1-defined, aild tllat
(f) If A : M + M x M is the "diagonal map", given by A ( p ) = ( p , p ) , show
that
L Y V=~At(a x p).
32, O n the n-dimensional torus
n times
let d e i denote ni*d9, where n i : T" + S' is projection on the ithfactor.
A de'" represent different elements of H k ( T " ) ,by
(a) Sliow that al1 dei' A
finding submanifolds of T" over which they have different integrals. Hence
dim H k ( T " )2
Equality is proved in the Problems for Chapter 1 l .
(b) Show that every map f : S" + T" has degree O. Hint: Use Problem 25.
6).
CHAPTER 9
+
RIEMANNIAN METRICS
1
n previous chapters we have exploited nearly every construction associated
with vector spaces, and thus with bundles, but tliere has been one notable
exception-\ve have never mentioned inner products. The time has now come
to make use of this neglected tooI.
An inner product on a vector space V over a field F is a bilinear function
from V x V to F, denoted by (v, w) i-, (v, w), which is symmetric,
arid non-degenerate: if v # 0, theil there is some w # O such that
For us, the field F will always be R.
For each r with O 5 r 5 17, we can define an inner product ( , ), on lT3" by
this is non-degenerate because if a # 0, then
n
1
((a ,. . .,a n ) ,(a 1 , . . . , a r , -a r+ 1,
. ,-an)Ir - C ( a i 1 2 > O.
i=l
In particular, foi- i. = II we obtain the "usud inner product", ( , ) on R",
(a, b) =
E aibi.
For this inrler product we have (a, a ) > O for aily a # O. In generd, a symmetric
bilinear function ( , ) is called positive definite if
A positive definite bilinear function ( , ) is clearly i~on-degenerate,and consequently an iniier product.
30 1
Notice that an inner product ( , ) on V is an element of T ~ ( v ) ,so if
f : W + V is a lii~eartransformation, then f * ( , ) is a symmetric bilinear
function on W. TIiis symmetric bilinear function may be degenerate even if f
is one-one: e.g., if ( , ) is defined on R2 by
(a, b) = a ' b1 - a 2 b.2 :
aiid f: IR + IR2 is
f (a>= (a, a > However, f * ( , ) is clearlg iiondegeiierate if f is an isomoi.pl~ismozzlo V. Also,
if ( , ) is positive definite, then f * ( , ) is positive definite if and only if f is
or~e-one.
For aily basis V I , . . , v, of V, witli corresponding dual basis U*], . . . ,U*,,,we
can write
In this expressioii.
so symmetry of ( , ) implies tl-iat tl-ie matrix {gij) is symmetric,
Tlie matrix {gij) lias anorhcr important iilierpretation. Since an inner producr
( , ) is lii-iearin rlie seco~ldargument, wc can define a liiiear fuilcrional$, E V*,
for eacli v E V, by
$v{w) = (u, w).
Sirlce ( , ) is linear in tl-ie fii-st argument, tlie inap v I-+
4, is a liiiear transfor~nationfi-om V io V*. No~-i-degeiiei.acyof ( , ) implies that $, # O if U # 0.
Tlius: if V is fii-iiiedimensional, an i1111er111-oduct ( , ) gives irs al] isomor-phism
CY : V + V*, witl-i
(v,w) = Q(v)(w).
Clearly, rl-ie matrix (gij) is jusr tbe matrix of CY : V + V* witl-i 1-espect to tl-ie
I~ases{vi} [or 1' ai-id { v *} ~for V *. Thus, 11o1-i-degeileracyof { , ) is cquiualei-ii
to the cor-idition thar
Positive d ~ f i r - i i t ~ ~of
- i ~(s s, ) correspo~~ds
to rhe more complicated coi-idition
thar the n-iatrix (gij) be "positive definitc", meaning that
II
''>O
gija a
i= 1
for al1 a l , . . . , a n witl-i at least one a' # O.
Given aiiy posiliue deJilzib iiiner product ( , ) oii 1/ we define the associated
norm ll Il by
(the positive square root is to be taken).
llvll = JT;;;)
In IR" we denote the norm corresponding to ( , ) simply by
Tlie principal pi-operties of 11 11 are tlie followiilg.
l . THEOREh4. For al1 u, w E V we have
(1) ll0vll = 101 + llull.
(2) 1 ( u ,w)l 5 11 v 11 11 w 11, with equality if aild only if v and w are linearly
dependent (Schwarz inequality).
(3) Ilv wll 5 llvll 11 w 11 (Triangle iiiequality).
+
+
PRQQE (1) is trivial.
(2) If v aiid w are linearly dependent, equality clearly holds. If ilot, then O #
hv - w for al1 h E R, so
O < l l h ~ - w 1 1 ~ =( A v - w , h v - w )
= h211v112- 2 h ( v , w )
+ llw112.
So the right side is a quadratic equation in h with no real solution, and its
discrimiiiaiit must be iiegative. Tlius
4 ( v , w ) -~411vl1211wl12 o.
Tlie fuiic.tioi.i 1 11 has certaii-i uiipleasai-it properqies-for example, the functioii 1 1 017 IR" is ilot dflereiitiable at O E IRn-wliich do riot arise for the function
11 112. This lattei fui-ictioii is a "quadratic function" on V-in ter-ms of a basis
( v i ) for V it can be wi-itten as a "homogeiieous polynomial of degree 2" in the
More succii-ictly,
n
An invariai-it definition of a quadratic furlction can be obtained (Problem 1) from
the following obsentatioi-i.
2. THEOREhil (POLARIZATION IDENTITY). If 11 11 is the norm associated to an inner product ( , ) on V, then
PXOOF Compute.
Theorem 2 shows that t\vo iilnei- products which induce the same norm ar-c
theiiiselves equal. Similarly, if f : V += V is 1101-m131-esen~ing,
that is, 11 f ( v ) 11 =
llvll fol. al1 v E 1/, then f is also iniler product presenfing, that is, (f (u), f (w)) =
(v, w) for al1 v, w E V.
\Ve wil1 now see that, "up to isomorphism", diere is only one positive definite
ii-iner product-
3. THEOREhil. If ( , ) is a positive defiiiite iiiiier product on ai-i n-diinei-isional vector space V, then there is a basis vi,. . . ,v, for V such that (vi, vi) =
6,. (Such a basis is called orthonormal with respect to ( , ).) Consequentl!;
there is an ison~oi-pliismf : Rti + V such that
In other words,
f*( > = { >.
7
9
PRO0.F Let Wr,. . . , wtl 13e ai-iy basis for V. \Ve obtain the desired basis by
applviiig tlie "Granl-Scl-imidt orthonormalization proc.ess" to this basis:
Since wi # O, we can define
aiid clcady Ilui 11 = l . Suppose that we have constructed u], . . .,u k so that
(vi, v j > = 6,
I si,jsk
and
span V I , . . . ,vk = span W I ,. . ., wk.
T h e n wk+] is linearly independent of V I , .. . ,vk. Let
It is easy to see that
So we can define
and continue inductively. *3
A positive definite inner pi-oduct ( , ) on V is sometimes called a Euclidean
71zeh-i~011 V. Tilis is because we obtain a metric p on V by defining
T h e "triangle iilequality" (Theorem l(3)) shows that this is indeed a metric. 147e
also caH llvll the length of v.
\die have onfy one more algebraic trick to play. Recalf that an inner product
( , ) on V prcwides an isomorphism a : V 4 V* with
Using the natural isomorphism i : V + V4*,defined by
1dre can ilow use /3 to define a bilinear functioii ( , )* o n V* by
(A,p}*= /3(A)(p) = i a - l ( A ) ( ~ )= P ( ~ - - ~ ( A ) ) .
Now, tlie symmetry of ( , ) can be expressed by tbe equation
Letting
#(u) = A,
u(w) = p,
this can be written
1
( A )= Au-l (A)),
which shows that ( , )* is also symmetric,
Consequently ( , )* is aii inner product on the dual space V* (in fact, the one
which produces j3).
To see what this al1 means, choose a basis (ui) for V, jet (u*i) be the dual
basis for V*, and 1et
n
Then
(gij) is the matrix of
u : V + V*
with respect to (vi) and (u*¿)
V* + V
witli respect to (u*i) and ( q )
(gjj)-' is thc matrix of u-':
so
(gG)-' is the matrix of
B : V* + V** witli respect to
and ( ~ * * i ) .
Tlius, if we let g i j be the entries of the inverse matrix, (g") = (gjj)-l, so tliat
theii
=
g " ~ i @ u j , ifwe considerui E V**.
Onc can clieck directly (Pi-oblem 9), without the invariant definition, tliat this
equation defines ( , )* indepelidently of the choice of basis.
Notice that if ( , ) is positive definite, so that
then, letting a(v) = h, we have
so ( , )* is also positive definite. This can also be checked directly from the
defiiiition in terms of a basis. In the positive defiilite case, the simplest way to
describe ( , )* is as follows: TIle basis u*], . . . , v*, of V* is orthonormal with
respect to ( , )* if aiid only if v i , . . . ,u, is orthoilormal ~ 4 t hrespect to ( , ).
Similar tricks can be used (Problem 4) to produce an inner product on al1
tlie vector spaces T k ( v ) , T,(V) = T k ( v * ) , and a k ( v ) . Howwer, we are
intcrested in oiily oile case, which we will i ~ odesci-ibe
t
in a completely invariant
way. The vector space Qn(V) is 1 -dimensional, so to produce an inner product
on it, we need only describe whidl two elements, o and -o, will have length 1.
Let V I , . . ,v, aild W r , . . . ,w, be two bases of V which are orthonormal with
respect to ( , ). If we write
So the transpose matrix A' of A = (ujj) satisfies A - A' = 1, which implies that
det A = fl . It follows fi-om Theoi-ein 7-5 that for aily w E S2"(V) we have
It clearly follows that
We have thus distinguished two elemen ts of R" (V); they are both of the form
v * A~ . - 'AV*, for (vi) an ortho11ormalbasis of V. \IZ1e will cal1 these two elements
the elements of norm 1 in Rn(V). If we also have an orientation p, then we can
furtlier distinguish the one which is posiiive when applied to any (vi,. . ., U,)
Hfith[u1,. . .,Un] = p; \ve will cal1 it the positive element of norm 1 in n n ( V ) .
To express tlie elements of norm 1 in terms of an arbitrary basis wi,. . . ,w.,
we cl-ioose an ortbonormal basis vl, .. . , U, and write
Problem 7-9 implies that
then
det(gjj) = det(At A) = (det Al2.
In particular, det(gij) U o l w ~posiiiu~.
) ~ ~ Consequently, the elements of norm 1
in R" (V) arc
c
\Ve noiv apply our new tool to \~ectorbuiidles. If = a : E + B is a vector
l~uiidle,we define a Riemannian metric on [ to be a function ( ) whicli assigns
to eacli p E 3 a positive definite iiiner product { , ), on a-'(p), and which
is coiltinuous iil tbe cense that for aily two continuous sections si ,$2 : 3 + E,
the functioil
( ~ 1 3 ~ 2=) p
(SI(].'),S~(P))~
is also coiltinurius. If 6 is a Cm vector builclle over a Cm manifold we can also
speak of Cm Riemannian metrics.
[Another approach to the definition can be given. Let Euc(V) be the set of al1
positive definite inner products on V. Ifwe replace each n-'(p) by EUG(X-]
(p)),
and Jet
EucO) =
~uc(rr-'(p)),
u
PEB
then a Riemannian metric on can be defined to be a section of E M ( { ) .Tbe
only problem is that Euc(V) is not a vector space; the new object E~ic(c)that wc
obtaiii is not a vector bulidle at all, but an iiistance of a more general structure,
a fibre buiidle.]
4. THEOREM. Let { = n : E + M be a [Cm] k-plane bundle over a Cm
maiiifold M. Tlieil tliere is a [Coo] Riemailnian me tric on
c.
PROOE Tliere is aii opeii locally fiiiite cover O of M by sets U for which there
exists [Cm] trivializations
ru: =-'(U) + U x IR^.
On U x IRk, tliere is aii olwious Riemannian metric,
((p,4,(p,b))p = ( ~ , b ) .
For u, w E n-'(p), define
(u, w)pO = ( I U ( ~~u
) ?(w))p.
Then ( , ) u is a [Cm] Riemannian metric for CIU. Let ( # u ) be a partition of
uiiity subordinate to 0.We define ( , ) by
(u,
=
E #u(p) (u, w)p
u, w E x - ~ ( ~ ) .
UEO
Then ( , ) is coiitiiiuous [Cm] and eacli ( , )p is a symmetric bilinear function
oii n-' ( p ) . To sliow that it is positive definite, iiote that
eacli #u (p)(u, u):
2 O, aiid fol. some U strict inequality liolds.
'+
:
[The same argument shows that aily vector bundle over a paracompact space
lias a Riemannian metric.]
Notice that the argunient in the filial step would not work if we had merely
picked iioii-degeiierate iiiiier piaducts ( , )
111 fact (Pi.oblem i),there is no
( , ) on TS' wliicli gives a symmetric I~ilinearfuiiction on each sZP
which is
iiot positive defini te 01. iiegative defiiiite bu t is still non-degenerate.
As an application of Theorem 4, we settle some questions which have ti11 now
remained unanswered.
3 . COROLLARY.
If 6 = x : E + M is a k-plane bundle, then {
PROOF. Let ( , ) be a Riemannian metric for
c*.
c. Then for each p E M, we
have an isomorphism
U p : x-i ( p ) + [x-1 (p)]*
defined by
ol,(v)(w) = ( u , W p
v, w E x-l (p).
Continuity of ( , ) iinplies that the union of al1 ap is a homeomorphism from E
lo E' = U,,,[x-'
(p)]*. O
6. COROLLARY. If { = x : E + M is a 1-plane bundle, then { is trivial if
aiid only if is orieiitable.
PROOF The "only if" part is trivial. If t has an 01-ientation p aiid ( , ) is a
Riemaiinian metric o11 M tlien there is a unique
with
p
= 1,
[s(p)I = Pp
Clearly s is a section; \ve then define an equivalente f : E + M x R by
ALTEMATIVE PROOF \Ve know (see the discussion after Theorem 7-9) that
if is orientable, theil there is a nowhere O sectioil of
c
so that t* is trivial. But {
t*.+3
All these coi~sidei.atioiistake on special significai-ice wl-ien our bundle is the
taiigeiit bui~dleT M of a C* manifold M . Iii this case, a C* Riemanniaii
inetric ( , ) for T M , which gives a positive definite inner product ( , ), oi-i
each Mp, is called a Riemannian metric on M. If (x, U) is a coordinate system
on M, then on U we can write our Riemaniliail metric ( , ) as
wherc the Cm fuiictioils gij satisfy gij = gji, since ( , ) is symmetric, and
det(gli) > O since ( , ) is positive definite. A Riemannian metric ( , ) on M
is, of coui.se, a covariai-ittensor of order 2. So for every Coo map f :N + M
there is a covai-iant tensor f*( , ) on N, which is c1early symmetric; it is a
Riernai-iniai-imetric on N if and only if f is an immersion
is one-one for
a11 11 E N).
The Riemaililian metric ( , )*, wl~ich( , ) induces on the dual bundle T4M,
is a coi1travariant teiisor of order 2, and we can write it as
Our- discussion of innei- products irlduced on V* shows that for each p, the
matrix (g"(p)) is the inverse of the matrix (gjj(p)); thus
Similarly, foi- each p E M the Riemanniail metric ( , ) on M determines
two c1cments of S2"(Mp), the elemei-its of norm l . \Ve have seen that they caii
11e written
I Jdet(glj(p)) d x l ( p ) A . - . A dxn(p).
If M has ai1 or.ieiitation p , then pp aUows us to pick out the positive element of
r1oi.m 1, and we ol~tainan i?-fosm on M; if x : U + R" is orzenfaiio~zpres~ving,
tlien o11 U tliis form caii be written
Even if M is iiot oi.icntabIe, we obtaii-i a "voIun1e e1ement" on M, as defined
in Chaptei- 8; in a coor-diiiate system (x, U) it can be written as
Tliis volume elen~eiitis deiioted by d V, eveil thougll it is usual1y not d of
ai-iytliii-ig(even wl~enM is orientable and it can be considered to be an n-form),
and is calied tlie volume element determined by ( , ). We can then define the
volume of M as
F
This certainly makes sense if M is compact, and in tlie non-compact case (see
Problem 8-10) it eitl~erconverges to a defiilite iiumber, or becomes arbitrarily
large over compact subsets of M , in which case we say that M has "infinite
volume".
If M is ail 11-dirneii.sioiialmanifold (-witli-boundary) in R", with tlie "usual
Riemannian metric"
n
then gij = d i j , SO
d V = Idxl A
- - A dxnI,
and "volume" becomes ordinary volume.
Tliere is an eveii more importaiit construction associated with a Riemannian
metric on M , ~rhicliwill occupy us for the I-est of tlie chapter. For every Cm
cunte y : [a,b] + M,we have tangent vectors
and can t1ierefoi.e use ( , ) to define tlieir length
14Jccaii then define the length of y from a to b,
If y is mesely pulc~wrsesttzooilr, n~eaiiiiigtha t there is a parti tion a = ro < . .
t,, = b of [u, b] sucli tliat y is smootli on eacli [ ~ i - i ,li] (with possibly different
left- and right-hand derivatives at t i , . . . ,tn-]), we can define the length of y by
IYhenever tliere is no possibility of misuiiderstaiiding we will denote L,b simply
by L. A little ai-pmeni shows (Problem 15) tliat for piecewise smooth curves in
IRn, with the usual Riemannian metric
dx' @ d x i 7
tliis defiiiition agrees ivith tlie definition of length as the least upper bound of
tlle lengths of inscribed polygonal curves.
We can also define a function S : [a,b] + IR, the "arclength function of y"
by
Co~~sequeiitly
d y / d r has constant length 1 precisely when S ( t ) = t
thus precisely when s(1) = E - a. Then
+ constant,
We can reparameterize y to be a curve on [O, b - u] by defiiiing
p(r) = y(t - a ) .
For the new curve y we have
iiew s ( l ) = L;(?) =
= old s(i f u ) - old s ( u )
= t.
If y satisfies S ( [ ) = r we say that y is parameterized by arcfength (and then
often use S iiistead of r to denote the argument in the domain of y).
Classically, the norm 11 11 on M was denoted by d s . (This makes some sort
of sense aren in modern notation; equation (s)says that for each cuwe y and
corresponding s : [ a ,b] + IR we have
on [ a ,b].) Consequently, in classical books one usually sees the equation
Nowadays, this is sometimes interpreted as being the equivalent of the modern
' d x j , but what it always actually meant was
equation ( , ) = x y , j = l g i j d ~ @
Tlie symbol dx'dx j appearing here is n o l a classical subai tute for dx' O d x j the value ( d ~ ' d s j ) (of~ d) x ' d x j at p should not be interpreted as a bilinear
function at ali, but as the quadratic function
H
d x i ( p ) ( u ) d x j ( P ) (u)
*
v E Mp ,
and we would use the same symbof today. The classical way of indicating
d x i @ d x j was very strange: one wrote
E gij dx'dxj
where d x and Sx are independent infinitesimals.
i,j=l
(Classically, the Riemannian metric was not a function on tangent vectors, bu1
tlie inner product of two "infinitely smaU dispiacements" d x and Sx.)
Consider now a Riemannian metric ( , ) on a cotlneckd manifold M. If
p, q E M are any two points, then there is at least one piecewise smootli cuwe
y : [ a ,b] + M from p to q (tliere is even a smooth curve from p to q). Define
d ( p ,q ) = inf { L ( ~: y) a piecewise smooth curve from p to q ) .
It is clear that d ( p , q ) > O and d ( p , p ) = 0 . Moreover, if r E M is a third
point, then for any E > O, we can choose piecewise smooth curves
y, : [u,b] +- M from p to q with L ( y i ) - d ( p , q ) < E
y2: [ b , ~+
] M from q to r with
L ( y 2 )- d ( q , r ) < E .
If we define y : [a,c] + M to be y, on [a,b] and y2 on [b,c],then y is a
piecewise smooth cuwe fmm p to r and
+
L ( Y ) = L(Y,) L(y2) < d ( p , q ) + 4 % 4 + 2 ~ .
Since tliis is true for al1 E > O, it follows that
d ( p ,r ) 5 d ( p , 9 ) + d ( q ,4.
[ I f w did not allow piecewise smooth cuntes, there would be difficulties in fitting
together y, and y2, but d ~rouldstill turn out to be the same (Problern 1 T).] The
function d : M x M + R has aíl properties for a metric, except that it is not so
clear that d ( p ,q ) > O foi- p # q. Tliis is made clear in the following
7. THEOREM. The function d : M x M + R is a metric on M , and if
p : M x M + IR is the original metric on M (wliich makes M a manifold), then
( M ,d ) is homeomorphic to ( M ,p).
PROOF. Both parts of the theorem are obviously consequences of the folíowing
7'. LEMMA. Let U be aii opeii neighborhood of the closed ball 3 = { p E R" :
jpl 2 11, let ( , ), be the "Euclidean" or usual Riemannian metric on U,
n
aiid let ( , ) be any otlier Riemannian metric. Let 1 1 = 11 11, and 11 Il be the
corresponding norms. Then there are numbers m, M > O such that
aiid consequelitly for any cunte y : [a,b ] + 3 we have
??lLe(y)5 L ( Y )
PROOE Define G : B x S"-<
R by
< ML,(Y).
G ( p , a ) = llap llp.
Then G is contiiiuous and posiiive. Since 3 x S"-' is compact there are numbers m, M > O such that
on B x S"-'.
il7 < G < M
Now if p E B and O # bp E Rnp, let a E S"-' be a = b/lbi. Then
1?7lbl < lblG(p,a) < Mlbl;
siiice
l b j G ( p , ~=
) lb1 . l l ~ ~ =l lIiObIlOpIIp
~
= IIbIIp,
tliis gives the desired inequality (wliich clearly also holds for b = 0). O
Notice that tlie distance d ( p , q ) defined by our metric need liot be L ( y ) for
any piecewise smooth curve from p to q. For example, the manifold M might
be IR2 - {O), and y might be - p . Of course, if d ( p , q ) = L ( y ) for come y,
then y is clearly a shortest piecewise smooth curve from p to q (there miglit be
more than one shortest curve, e.g., the two semi-cirdes between the points p
and - p on S ' ) .
ln order to investigate the question of shortest curves more tl~oroughly,we
liave LO employ tecliiiiques fi-om the "calculus of variatioils". As an introduction
to such teclii~iques,we consider first a simple problem of this sort. Suppose we
are giveli a (suitably differentiable) function
Wc seek, amoiig al1 functions J.: [ a ,b ] + IR with f( a ) = a' and f ( b )= b' oiie
whicli will maximize (or minimize) the quantity
For examplc, if-
then we are looking for a function f on [u,b] which makes the curve t H
( E ,f ( 1 ) ) between (a, u') and (b, b') of shortest length
As a second example, if
then we are trying to minimize the area of tlie surface obtaiiied by revolviiig
tlie graph of f around the A--axis,which is given (Problem 12) by
To appl-oach this sort of problem we recall first tlie metliods used for solving t1.ie n~uchsimpler problem of detei-ininiiig t l ~ einaximum or niinimum of
a function f : R + R. To solve tliis problem, \ve examine tlie critical points
of f, i.e., those points x for wliicli fl(x) = O. A critica1 poilit is not iiecessarily
a maximum or minimum, or even a local maximum or minimum, but critical
points arc the only caiididates for maxima or mii.iin.ia if f is everywliere difTereiitiable. Similarly, for a functioii f : IR2 + IR we consider poiiits (x, y) E R2
for which
This is tlie same as saying that the cuntes
have derivative O at O. \Ve might try to get more information by considering
the condition
o = (f c)'(O)
0
for every cuiw c : (-E,&) + R2 with ~ ( 0 =
) (x,y), but it turns out that these
conditions follow from (*), because of the chain rule.
To find maxima and minima for
we wish to proceed in an analogous way, by considering curves in Ihe sel ufall
funclions f : [u,b] + R. This can be done by considering a "variation" of f,
that is, a function
a : (-E,E) x [u,b] + R
such that
@(O, 1) = f U).
The functions 1 H a(u,I) are then a family of functions on ( - E , E) which
pass through f for u = O. We will denote this function by &(u). Thus & is a
function from (-E,E )to the set of functions f : [u,b] + R. If each ¿?(u) satisfies
&(U) (u)= u', & (u)(b)= b', in other words if
for al1 u E (-E,&),then we cal1 a a variation of f keeping endpoints fixed
For a variation a we now compute
As is usual in classical notation, the arguments of functions are either put in
indiscriminately or left out indiscriminately-in this case, not only are the arguments t and (1, f ( r ) , f f ( t ) ) omitted (resulting in the disappearance of tht
fuilction f for which we are solving), but the dependeiice of SJ on a is not
indicated (which can make diings pretty confusing).
If f is to maximize or minimize J , then S J ( a ) must be O for every variation cx
of f keeping eiidpoints fixed. As in the case of 1-dimensional calculus, there is
no reason to expect that the condition S J ( a ) = O for al1 tr will imply that f is
eveii a local maximum or miilimuin fos J , and h7eempliasize this by introduciiig
a definition. \Ve caB f a critica1 point of J (01-an extremal for J) if S J ( a ) = O
for al1 variations a of f keeping endpoints fixed. The particular form (**)into
which we have put SJ iiow allo~tsus to deduce an important condition.
8. THEOREM (EULER'S EQUATION). The C2 function f is a critica1
point of J if and only if f satisfies
JF
- 7
ax
.
f'
(
dr
r
ay
,
, ( r ) ) =o.
)
PROOI;: Clearly f must niake t l ~ eintegral in (**) vanisli for e u q l
which vaiiishes at a and b. So the theorem is a consequence of the followiiig
simple
8'. LERIMA. If a continuous function g : [u,b] + R satisfies
for every Cm fuiictioi~q on [a,b]with ?](a)= q(b) = 0, tben g = 0.
PROOE Choose q to be #g where # is positive 011( a ,b ) and # ( a ) = # ( b ) = 0. +3
As an example, consider tlie case where F ( r , s , g ) =
cquatioil is
m.
Tlie Eules
hence
0 = ( 1 + f r 2 ) f " - ftj-'/
= (1 - f t + f12) f",
wliich implies that f" = O, so f is linear.
Notice that we ~ ' o u l dhave obtained the same result if we had considered t l ~ e
case F(t,x, y ) = 1 Y2,for then the Euler equatioii is simply
+
This is analogous to the situation in 1-dimensional cafculus, where the critica]
points of f i are the same as tliose of f, since
For tlie case of tlie surface of revolution, wliere F(l, x , Y ) = x J 1
Euler equation is
+ Y', [he
this Ieads to the equatioii
whic.11 we will a1so write in the cIassica1 form
To solve this, we use oiie of the Ko standard tricks (leaving justification of the
details to t1ie reader). We let
Then
so our equation becomes
1
2
- log(l + p 2 ) = log y + constant
y = constant
J;+pi
and thus (see Problem 20 for tlie definition and properties of tlie "liyperbolic
cosine" function cosh and its inverse)
Replacing c by 1 / c , we write this as
The graph of
cosli x =
ex
+ eYX
2
is sliown below; it is syn-imetric about the y-axis, decreasing for A- 5 0, and
iricreasing for x 2 0.
So our surface must look like the one drawn below. It is, by the way, not
trivial to decide whether there a7-e constan ts k and c which will make the graph
of (*) pass through (u, a') and (b, b'). Problem 2 l investigates the special case
where a' = b'.
It is easy to generalize these considerations to the case where f : [a, b] t Pn
and
In this case \ve consider a : ( - E , & ) x [a,b] t Pn with G(0) = f, and compute
that
Tlius, any critica1 point f of J must satisfy the n equations
We are ~iowgoing to apply these results to the problem of finding shortest
paths in a manifold M. If y : [u,b] t M is a piecewise smooth cunte, with
y ( a ) = p and y ( b ) = q , we define a variation of y to be a function
for come E > O, such that
(1) ~ ( 0 7 =
~ ~) ( 0 ,
(2) there is a partition a = ro < E] <
. < EN = b of [a,b] so that a is Coo
on each strip (-E, E) X [ ~ i - lli].
,
\IZ1e cal1 a a vai-iatioii of y keeping endpoints fixed if
a(u,a) = p
@(u,b) = q
for al1 u E (-E, E).
As before, we let G ( u ) be the path t H &(u,E). We would like to fiild which
paths y satisfy
for a11 variations a keeping endpoints fixed. However, we will take a liint from
oui. first exan-iple ai-id first fii-id the critica1 points for the "energy"
which has a much ilicer iiltegrand; aftenvards we will consider tlie relati011
between tlie two integrals.
\Ve can assumc that cach y 1 [ti-], Ei] 1ies i i i some coordiiiate system (x, U)
(otheiurise we just refine the partition). If (u, E )is the standard coordinate system
in (-E,&) x [a, b] we write
Then aa/aE(u,E) is the tangent vector at time E to tlie curve &(u). If we adopt
the abbreviations
then
If we usc tlie coor-diiiate system x to ideiitify U witli Rn,aiid consider t he g ,
as fuiictio~iso11 Rn,then we are considering
wliere
1
F ( x , y )= -
Il
1g G ( x ) .
2 . .-
ilj-1
arid
$ j j .
In arder to obtain a symmetrical looking result, we note that a little index
juggling gives
From (***} we now obtain
d E (á.(u)l [di- I di])
1
Remember that y is oniy piecewise Coo. Let
¡/Y +
-(li
) = rigilt hand tangeilt vector of y at li
clt
¡/Y
-(ijr)
= left lland tangent vector of y at ii.
¡/l
Y (ti-])
Notice that tlie fiilal sum iil the above formula is simplv
To abbi*eviatethc iii tegral somewhat we introduce the symbols
Tliese depend oii the coordinate system, but the integral
dZyr
r=l
n
-1
dyi d y j
x [ i j y l I ( ~ ( F )dl
1 di
i,j 2 1
di,
which appears in our result, clearly cannot. Consequently, we will use the exact
same expressiorl for each [ii-], ii], even though dXerent coordinate systems may
actually be involved (and hence dXerent gij and y').
Now we just have to add up these results. Let
Then we obtain the following formula (where there is a convention being used
in the integral).
9. THEOREM (FIRST VA RIATION FORh4ULA). For any variation a ,
we have
dzy'
Il
I=i
(111 tlie case of a variation a leaving endpoints fixed, tlie sum can be written
from 1 to N - 1.)
Tliis result is not very pretty, but there it is. It should be rloted that [ij, /] are
noi the c.omponents of a tensor. Nevertheless, later on we will have an invariant
iiiterpretation of the first variation forniula. For the time being we present,
with apologies, this coordinate dependent appr-oach. From the first variation
foi.inii1a it is, of course, simple to obtain conditions for critica1 points of E.
10. COROLLARY. If y : [ a , b ] + M is a Cm path, then y is a critical p i n t
of E; if and only if for every coordinate system ( x , U ) we have
PR00E Suppose y is a critical point. Given t witli Y ( E )E U , choose a partition
of [ a ,b ] with E E ( E j - , ,l i ) for some i, and such that y 1 [ E i - ] , E j ] is in U. If a
is a variation of y keeping endpoints fixed, then in the first variation formula
\ve can assume that the part of the integral from E j - 1 to E j is written in terms
of ( x , U ) . The final term in tlie formula vanishes since y is Cm. Now apply
tlie me tliod of proof in Lemma 8', clioosing al1 a a l / a u (0, 1 ) to be O, except one,
wliich is O outside of ( E i - ] , t i ) , but a positive function times the term in brackets
on (Ei-],1 1 ) . *3
In order to put the equations of Corollary 10 in a standard form we introduce
anothe r set of symbols
Our equations can now be written
We know from tlie standard theorem about systems of second order dairential
equations (Problem 5-4), that for each p E M and each v E M,, there is a
unique y : ( - E , E ) + M , for some E > O , such that y satisfies
Moreover, this y is Cm on (-E,&). This Iast fact shows that if y, : [O,&)+ M
and y2 : (-E, O] + M are Cm functions satisfying this equation, and if moreover
then y, and y2 togetfier give a Cm function on (-.$,E). Naturalfy, we coufd
replace O by any other t . We now have the more precise result,
1 1. COROLLARJ7. A piecewise Coopath y : [a,b] + M is a critica1 point
for E: if and only if y is actually Cm oii [o,b] and for every coordinate system
(x, U ) satisfies
.
..
for y ( 1 ) E U.
PROOE Let y be a critical point. Choosing the same u' as before (al1u' are O
outside of (ti-l, ti)), \ve see that y [ti , ] , ti] satisfies the equation, because tlie
final tcrm in tlic first variation formula still vanishes, Now choose u so that
We already kiiow that the integra1 in tlie first variation formula vanishes. So
we obtain
%
which irnplies that al1 AIi are 0. By our previous remarks, this means that y
is actually Cm on al1 of [a,b]. 44
As t11e simplest possible case, consider tlie Eucljdean metric on Rn,
Here gij = 6,, so al1 agil/axk = 0, and r$ = O. Tlie critical poinis y for the
energy function satisfy
dZyk
= o.
d12
Tlius y lies along a straiglit line, so y is a critica1 point for the length function
as well. The situation is iiow quite dXerent from the first variational problem
we considered, when we considered only curves of the form E H ( E , f ( E ) ) .
Any reparameterization of y is also a critical point for length, since length is
independent ofparameterization (Problem 16). This shows that there are critical
points for length which definitely aren't critical points for energy, since we have
just seen tliat for y to be a critical point for energy, the component functions
of y must be liiiear, and lience y niust be parameterized proportioiially to mlength. This situation always prevails.
12. THEOREM. If y : [a,b] + M is a critical point for E, then y is parameterized proportionally to arclength.
PROOE Obsen~efirst, from tile definitions, tliai
Now we have
Replacing ag,/axl by tlie ~ ~ a l ugiven
e above, tliis can be written as
Since y is a critical 11oiiic for E, I~otliterms in paientlieses are O (Coi-ollaiy 10).
Tlius rlie leiigth lIdy/dilj is coiistant. <+
The formula
occurring in this proof will be used on several occasions later on. It will also be
useful to know a formula for agij/axk. To derive onne, we first dXerentiate
to obtain
Thus we have
We can find the equations for critica1 points of the length function L in exactly the samc way as we treated the energy function. For the moment we
consider only paths y : [u, b] + M with d y / d ~# O everywhere. For the portioii y 1 [li-] ,li] of y contained in a coordinate system (x, U), we have
(
Y-
dyf d y J
1
dl.
i, j=I
Coiisideriiig our coordinate system as Rn?we are now dealing with the case
NTeintroduce the arclength function
S([) =
G(Y).
Then
After a little more calculation we finally obtain the equations for a critical point
of L:
It is clear from this that critical points of E are also critical points of L (since
they saiisfy d 2 s / d t 2 = O). Conversely, given a critical point y for L with
d y / d t # O everywl-iere, the function
is a difGeomorphism, and w e can consider tl-ie reparameterized cuive
y O S-' : [O, ~ : ( y )+
] M.
This reparameterized curve is automatically also a critical point for L, so it
inust satisfy tlie same differential equation. Since it is now parameterized by
arclength, the lhird cerm vanishes, so y o S-' is a critical point for E.
Tl-iere is only one detail which remains unsettled. Conceivably a critical
point for L might have a kink, but be Coobecause it has a zero tangent vector
there, as in tlie figure below. In this case it ~touldnot be possible to reparame-
tcrize y by arcleng-th. Problem 3'7 shows that this situation cannot arise.
Henceforth we will cal1 a critical point of E a geodesic on M (for the Riemannian metric ( , )). This name comes from the science of geodesy, which
is concerned witli the measurement of the earth's surface, including surveying
and the measurement of degrees of latitude and longitude. A geodesic on the
earth's surface is a segrnent of a great circle, which is the shortest path between
two points. Before we can say whether this is true for geodesics in general,
wliich are so far merely known to be critical points for leng-th, we must initiate
a local study of geodesics.
Tl-ie most elementary properties of geodesics depend only on facts about
dxercntial cquations. Observe tliat the equations for a geodesic,
llave an important homogeneity property: if y is a geodesic, then r I+ y ( c l ) is
also clearly a geodesic. This feature of the cquatioil allows us to irnprove the
result given by tlie basic existence aiid uniqueness theorems.
13. THEOREM. Let p E M. Then there is a neighborhood U of p and a
number E > O such that for every q E U and every tangent vector v E Mq with
lIvII < E there is a uniquc geodesic
satisfying
Y,(O) = q,
d ~ u
-j$OI
= v.
PR00.F. The fundamental existencc and uniqueness theorem says that there
is a neighborhood U of p and & ] , E * > O SO that for q E U and v E Mq with
1 1 ~ 1 1< EI there is a unique geodesic
~ t i t hthe required initial conditions.
Clioose E < E] E*: Tlien if IvI < E and 11 [ < 2 we have
So we can define y, (E) to be
(~21).+f,
If v E M, is a vector for wl-iic1.i there is a geodesic
then \ve define tl-ie exponential of v to be
(Tl-ie reason for tliis termiiiology will be explaiiled in tl-ie next chapter.) The
geodesic y can thus be described as
Since M, is an 11-diineiisioiial \lector space, iliere is a natural way to give ii
a C* sii-ucture. If 13 c M, is tlle set of al1 \~eciorsv E M, for whicli exp,(v)
is defined, then the map
exp, : 13 + M
is Cm, since il-ie solutions of ilie differential equations for geodesics have a Cm
flo~!1dentif~ii-i~
the tangent space (M,), at v E M, with M, itself, \ve have ail
ii-iduced map
(expq)v* : M, + MeXPq(U).
Iii. pariicular, we claim tliat the map
111fact, io ol~raina cuiw c ii-i rl-ie manifold M, with dc/d~(O)= v E M, =
(Mq)o,we can let c(r) = t u . Theii exp, o c ( ~ =
) expq(tu), the geodesic witli
iangent vector v at time O, so
(espq)o*(v)= -
exp, (c(t)) = v.
PROOK Theorem 13 says that the vector O E Mp has a neighborhood V in the
manifold TM such that exp is defined on V. Define the Cm function F: V +
M x M by
F(u) = ( W , exp(u)).
Let (x, U) be a coordinate system around p. We will use the coordinate
system
(f',. . ., i n , x l , - . .,Y),
described above, for lr-'(U).
factor, then
If r j : M x M -> M is projection on the ith
is a coordinate system on U x U. Now, using the fact that
is the identity, it is not hard to see that at O E Mp we have
Consequently, F* is one-one at O E Mp, SO F maps some neighborhood V'
of O diffeomorpliically onto some neighborhood of (p, p) E M x M. We may
assume tllat V' consists of al1 vectors v E Mq with q in some neighborhood U'
of p and [IvII < E. Cboose W to be a smaHer neighborhood of p for which
F(V1) ii W x W. 43
Given a W as in [he theorem, and q E W, consider the geodesics through q
of the form t I+ expq( t v) for l[u 11 < E. These fill out Uq. The close analysis of
geodesics depends on the following.
15. LEMMA (GAUSS' LEMMA). In U,,the geodesics through q are perpendicular to the hypersurfaces
{exp, (u) : 11 u11 = constant < E ) .
FIRST PR00E Le t u : IR + M, be a smooth curve with 11 u (E)[I = a consta111
k < E for al1 E,and define
a(u, E) = exp, (u u([))
-l<u<l.
We are claiming that for every such a we have
A calculation precisely like tllat in tlle p-oof of Theorem 12 proves the foilowing
equation, in which tlle arguments (u, E)and a(u, E)are omitted, for conveniente:
The first term on tlle right is O since each curve u H #(u, E) is a geodesic.
Similarly, we obtain
wl~ichis just hice the second term on the right of (1). But aa/au(u, E)is just the
tangent vector at time u to tlle geodesic u H exp,(u u ( ~ ) )where
,
Ilu(~)Il= k;
so ilaa/aull = k. Tlius the second term on the right of (2) is also O. So
(
,
isindependentofu.
But a(0, E )= expq(O) = q , SO aa/ar (O, r ) = O. It follows that
SECOA'D PROOF. Let u : R + Mq be any smooth cuive with 11 u([) 11 = a constant
k < E for aii E,and define
B(u,E)= expq(l . v(u))
(note carefull~the roles played by r and u).
Then fl is a variation of the geodesic y([) = expq(r u(O)), defined on [O, 11. By
tl-ie firsi variation formula, we have
tlie iniegral vanishing since y is a geodesic. But each curve j(u) has energy
16. COROLLARY. Lei c: [u,b]+ Uq - { q ) be a piecewise smooth cuive,
for O < u(!) < E and /lu(E)lI = 1. Then
with equality if and only if u is monotonic and u is constant, so that c is a radial
geodesic joiiiing IWO conceritric sphericai shells amund q.
PROOF If @(u,E) = exp, (u . v(E)),then C(E)= a (u(E),E)and
Since
with equality if and only if a ~ l / a i= O, and hence ~ ' ( r =
) O. Thus
~ ~ i tequality
li
if and only if u is monotonic and v is constant. +:+
17. COROLLARY. Let W and E be as in Theorem 15, let y : [O, 11 + M be
the geodcsic of leilgth < E joiiling q, q' E W, and let c : [O, 11 + M be any
piecewise Coopath from q to q'. Then
with equality holding if and only if c is a reparameterization of y.
PROOE We can assume that q' = exp4(rv) E U, - {q} (otherwise break c up
into smaller pieces). For S > 0, the path c must contain a s e p e n t which joins
the splierical sliell of radius S to the spherical sliell of radius r, and lies between
them. By Corollary 16, the Iength of this segment lias length 2 r - S. So the
length of c is 2 r, and clearly c must be a reparameterization of y for equality
to hold. +:+
\Ve thus see that suficimi~smallpieces of geodesics are minimal paths for arclength. We can use Corollary 17 to determine the geodesics on a few simple
surfaces, without any computations, if we first introduce a notion which will
play a crucial role later. If ( M ,( , )) and ( M ' , ( , 1') are Cm manifolds with
Riemannian metrics, then a one-one Cm function f : M + M' is called an
isometry of M into M' if f*( , )' = ( , ) . For example, ~flectionthrough a
plane E* c Rn+' is an isometry I : S" + S". It is clear that if c : [O, 11 + M
is a Cm cunte, then the length of c with respect to ( , ) is the length of f 0 c
with respect to ( , )'; and if c is a geodesic, then f o c is likewise a geodesic.
For the isometry I : S" + S" mentjoned above, the fixed point set is the great
circle C = S" n E*. Let p, q E C be two points with a unique geodesic C'
of minimal length between them. Then I ( C f )is a geodesic of the same length
as C f between I ( p ) = p and I ( q ) = q. So C' = I ( C f ) ,which implies thai
C' c C , so that C ¡Sa geodesic. Siiice there is a great circle through any point
of S" in any given direction, these are al1 the geodesics.
Notice tliat a portion of a great circle ~ l i i c his larger than a semi-circle is
defiliitely not of minimal length, evaz among tiearb pa0i.r. Antipodal points on
tlie sphere have a continuum of geodesics of minimal length between them. Al1
o~lierpairs of points have a unique geodesic of minimal length between tliem
b i i ~a11infii~itefamily of non-minimal geodesics, depending on how many times
tl-ie geodesic goes around tlie spliei-e and in which direction it starts.
The geodesics on a right circular cylinder Z are the generating lines, the
ciirles cut by planes perpendicular to the generating lines, and the helices
on 2 . In fact, if L is a generating line of Z, tl-ien we can set up an isometry
I : Z - L -t R2 by rolling Z onto R2. Tl-ie geodesics on Z are just the images
ui-ider 1-' of tl-ie straight lines in R2. Two points on Z have infinitely many
geodesics be tween them.
We are now in a position to wind up our discussion of Riemannian mei1-ics on M by establisl-iiilg an important coililection between the Riemannian
metric ( , ) and the metric d : M x M + R it determines,
d ( p , q ) = inf{L(y) : y a piecewise smooth cunte from p to q}
Notice that on both tlie spl-iereand tl-ieit-ifinitecylii-iderevery geodesic y defined
on an iiiterval [u, b] can be extended to a geodesic defined on al1 of R . This is
false on a cylinder of bounded l-ieight, a bounded portion of Rn,or Rn - {O].
l n general, a manifold M wit1-i a Riemai-inianmetric ( , ) is called geodesically
complete if eveiy geodesic y : [u, b] + M can be extended to a geodesic from R
to M.
18. THEOREM (HOPF-RINOW-DE RHAM). If ( , ) is a Riemannian
metric on M, then M is geodesically complete if and only if M is complete
in the metric d determined by ( , ). Moreover, any two points in a geodesically complete manifold can be joined by a geodesic of minimal length.
PROOF. Suppose M is geodesically complete. Given p, q E M with d(p, q) =
r > O, choose Up as in Theorem 14. Let S c U, be the spherical shell of radius
S < E. There is a point
on S such that d(po, q) 5 d(s, q) for al1 S E S . We claim that
this will show that the geodesic y(l) = expp(rv) is a geodesic of minimal length
between p aild q. To prove this result, we will prove that
First of all, since every cunte from p to q must intersect S, we clearly have
So d(po,q) = i - - S. This proves that (**) holds for E = S.
Now let 10 E [S, 1.1 be tl-ie least upper bound of al1 1 for which (**) holds. Then
(**) llolds for 10 also, by continuity. Suppose 10 < r. Let Sr be a spherical shell
of radius S' around y (10) and le t
E S' be a point closest to q. Tlleil
+
d(y(lo),q) = m i l l ( d ( ~ ( ~ o ) , s )d(s,q)) = 6' +d(po', cl),
SES'
Hence
d ( p , po') > d ( p q9 ) - d ( p ~ ' 99) = lo + 6'.
Bui tlie path c obtaiiied by following y from p to ~ ( 1 0 and
)
then the minimal
geodesic from y (10) to por l i s length precisely 10 6'. So c is a path of minimal
Icngtli, and must therefore be a geodesic, ~ f h i c means
h
that it coincides with y.
Hence
y(I0 6') = po'.
+
+
Hence (***) gives
d(~(10+6'),q=
) r - (~g+6'),
+
sliowing that (**) holds for 10 S'. Tliis contradicts t he choice of 10, so it must
be tliat 10 = i-. In other words, (**) Iiolds for 1 = r, which proves (*).
From tliis result, it follows easily that M is complete witli the metric d . In
fact, if A c M has diameter D , and p E A , then the map expp: Mp + M
maps the closed disc of radius D in Mp onto a compact set containing A. 111
otlier words, bounded subsets of M have compact closure. From this it is clear
tliat Cauchy sequences converge.
Conversely, suppose M is complete as a metric space. Given any geodesic
y : (u, b ) + M, choose E, + b. Clcarly ~ ( 1 , ) is a Caucliy sequence in M, so ii
coiivesges io some point p E M. Using Theorem 14, it is not difficult to show
that y can be cxtcndcd past b. Consequently, by a least upper bound argument,
any geodesic can be extended to R. *:*
As a particular consequclice of Thcorem 18, note that tliere is always a minimal geodesic joiniiig any two points of a compact manifold.
ADDENDUM
TUBULAR NEJGHBORHOODS
Lei M" c N"+k be a submanifold of N, with i: M + N the inclusion map,
so that for every p E M we have i*( M,) C N, . If ( , ) is a Riemannian metric
for N, tlien we can define M', C N, as
Lci
E=
U M,'
and
m: E + M
take~,'top.
Ir. is not hard to see that v = -¿ir : E + M is a k-plane bundle over M, the
normal bundle of M in N.
For example, the normal bundle v of S"-' E R" is the trivial 1-plane bundle,
for v Iias a sectioii consistiiig of unit ouiwasd iiormal vectors. O n the oiher haiid
if M is tlle Mobius strip and S' c M is a circle around tlie center, then it ir
noi hard io see ihai tl-ie iiormal bundle v will be isomorpbic to the (non-tiivial)
I~uridleM + S'. If we consider S' C M c P', then the normal bundle
of S' iii IP2 is exactb' tlie same as tlie iiormal 11uiidle of S' in M, so ii too ir
non-trivial.
Our aim is to prove that for compact M the normal bundle of M in N is
always equivalent to a bundle a : U + M for which U is an open neighborhood
of M in N, and for which the O-section S : M + U is just the inclusion of M
into U. In the case where N is the total space of a bundle over M, this open
neighborhood can be taken to be the whole total space. But in general tlie
neighborhood canilot be al1 of N. For example, as an appropriate neighborhood
of S' c IR2 we can choose IR2 - {O}.
A bundle rr : U + M with U an open neigl-iborhood of M in N, for wliich the
O-section s : M + U is the inclusion of M in U, is called a tubular neighborhood
of M in N, Before proving the existente of tubular neighborhoods, we add some
remarks and a Lemma.
If a : U + M is a tubular neighborhood, then c1early
rr o s = identity of M,
soa
is smoothly homotopic to the identity of U,
so a is a deformation retraction, and H k ( U ) * H k ( ~ )thus
; M has the same
de Rllam col~omologyas an open neighborhood. Moreover, if we choose a
Riemailnian metric ( , ) for a : U +- M and define D = {e E U : (@,e)5 1):
tllen D is a submanifold-with-boundary of U, and the map 7110: D + M is
also a deformation retraction. So M also has the same de Rharn col~omolog-)~
as a dosed neighborhood.
1 9. LEMMA. Let X be a compact metric space and Xo c X a closed subset.
Let f : X + Y be a local homeomorphism such that f
is one-one. Then
1x0
there is a neigl-iborhood U of Xo such that f IU is oae-one.
( ( x , y ) E X x X : x # y and f(x) = $(y)).
Then C is closed, for if (x,, y,) is a sequence in C with x, + x and y, + y,
then f (x) = lim J'(x,) = lim f(y,) = f (y), alid also x # y since f is locally
one-one.
If g : C + R is g(x,y) = d ( s , Xo) d ( y , Xo), then g > O on C. Since C
is compact, there is E > O such that g > 2~ on C. Then f is one-one on the
E-neigl-iborhoodof Xo. 43
+
20. THEOREM. Let M c N be a compact submanifold of N . Then M has
a tubular neigliborhood T : U + M in N: which is equivalent to the normal
bundle of M in N.
PROOF, Choose a Riemanniaii men-ic ( , ) for N , with tlie corresponding
norm 11 11, and metric d : N x N + R . Lei
It follows easily from Theorem 13, and compactness of M, tl-iat exp is defined
o11 E, for sufficieiitly small E > O. We claim that for sufliciently sm al1 E, the map
exp is a diíTeomorpliism from E, onto U,. This will clearly prove the theorem.
Let V c E be the set of a non-critica1 points for exp. Then V 3i M (considered as a subset of E via thc O-section), and VI = V n El is compact; since exp
is one-one on M c I/i, it follows from Lemma 19 that for sufficiendy small E
tlie map exp is a difTeomorphism on E,.
It is clear also that exp(E,) c U,. To prove tliat exp is onto U,, choose any
q E U,, and a poirit p E M closest to q . If y : [O, 11 + N is the geodesic of
1engtl-i < E witli y(0) = p and y(1) = q , it is easy to see that y is perpendicular
to M at p (compare the second proof of Gauss' Lemma). This means that
q = exppdy/dt(0) where dy/d1(0) E E,. 43
O11c of tlie ii-iiei-estingfeatures of Theorem 20 is that al1 the paraphernalia
of Riemanriian metrics and geodesics are used in its proof, while they do not
even appear in tlle statement. Theorem 20 will be rieeded only in Cl-iapter 11,
wl-iere we will also need the following modification.
Rienzalzn ian Adel?.ics
347
2 1. THEOREh4. Let N be a manifold-with-boundary, with compact boundary aN. Theii aN has (arbitral-illlsmall) opeii [and dosed] neighboiqlioodsfor
which there are deformation re tractjons onto a N .
PRO0.F Exactly the same as the proof ofTlieorem 20, using only inward poiiiting normal vectors. +3
PROBLEMS
1. Let V be a \lector space over a field F of characteristic # 2, and let h : V x
V + F be srmmetric and bilinear.
(a) Define q : V + F by q(v) = h(v, u). Show that if e l , . . . ,#n is a basis
for V*, then
n
for SOme U i j .
(b) Show that
(e) Suppose q : 1/ + F satisfies q(-u) = u, and that h(u, u) = q(u+ v) - q(u) q(v) is bilinear. Show that
Conclude that q(0) = 0, and q(2u)
-
4q(u). Tl-ien show that q(v) = h(u, u).
2. Let ( , ) be a Euclidean metric for V*. Suppose #i, Sri E V* satisfy #i A
- A #k = Sri A - A # O, and 1et W4 and W* be the subspaces of V*
m
m
spanned by the
ei and Sri.
(a) Show tl-iat w E W4 if and only if w A #i A . - . A #k = O. Conclude that
w4= w*.
(b) Let u], .. . ,uk be aii or~lionoi-malbasis of WQ = W y. If #i = & U j i U j ,
s11ow that the signed k-dimensioiia1 volume of the parallelepiped spanned by
,. . . ,#k is det(aij). (Tlie sign is if #i,. . . , Iias the same orientation as
01, . . .,Ok, and - otherwise.)
(c) Usirlg Problein 7-9, sliow that tliis volume is the same for Srl ,.. . ,@k.
(d) Conversely, if W4= W*,and the si\gned volumes of the parallelepipeds are
A . - .A Srk.
the same, sliow that #] A - A #k =
el
+
If we ideiitify V witll V**,so tliat we llave a wedge product vi A - - - A vk of vectors
vi E V, tlieil we liave a geometi-ic coiiditioii for equality with wl A
. A wk.
111L p n s sur la GWniArie des Q l i c e s de Rieinliiiii, É. Cartan uses this condition to
d@n~nk( Y * ) as formal sums of equivaleiice classes of k vectors; he deduces
geornen-ically ihe coi~espondiiigconditions on the coordinates of vi, wi .
3. Let V be an it-dimensioi-ial vector space, and ( , ) an inner product on V
wliich is iiot riecessarily positive defiiiite. A basis v i , .. . , u , for V is called
orthonormal if (vi, vj) = f6,.
(a) If V # {O}, then there is a vector u E V with (u, u) # 0.
(b) For W C V, let W' = (u E V : (v, W ) = O for al1 w E W}. Prove that
dim W' 2 n - dim W. Hini: If {wi} is a basis for W, consider the linear
functionals hí : V + R defined by h i ( ~ =
) (U,wi).
(e) If ( , ) is non-degenerate on W, then V = W @ w', and ( , ) is also
non-degenerate on w'.
(d) V has an orthonormal basis. Thus, there is an isomorphism f : R" +
V with f * ( , ) = ( , ), for some r (the inner product ( , ), is defined on
page 301).
(e) Tbe index of ( , ) is the largest dirnension of a subspace W c V such that
( , ) 1 W is negative definite. Show that the index is n - r, thus showing that r
is unique ("Sylvester's Law of Inertia9').
4. Let ( , ) be a (possibly non-positive definite) inner product on V, and let
vi,. . . , u, be an orthonormal basis (see Problem 3). Define an inner product
( , l k on nk(v) by requiring that
be an orthonormal basis, wit1-i
(a) Sliow tliat ( , l k is independent of the basis vi,. . .,uk. (Use Problem 7-16,)
(b) Show thai
(c) If ( , ) has index i , theii
( u * ~A .
. A u*,, u*] A
"
. A U*,)"
= (-lli.
(d) For tliose wlio k n w about @ and iIk. Usiiig the isomorphisms
( g k v ) *and i I k ( v * ) 4 ( i I k v ) * ,define inner products on
and iIkv by
using tlie isomorphism V +- V* given by the ii-iner product on V. Show that
tl-iese ii-ii-ierproducts agree witl-i the oi-ies defined above.
5. Recall tlie definition of u1 x
x u,-1
in Problem 7-26.
(a) Sliow tliat (VIx - - - x v,-1, vi) =: O.
(b) S ~ O M
tli¿~t
' Iu] X * - X U,,-1 1 = l/deto,wliere gij = (ui, uj). H&L: Apply
tlie result oi-i page 308 to a certain (11 - 1)-dimensional subspace of R".
c
B be a vector bundle. An indefinite metric on 6 is
a continuous choice of a non-positive definite inner product ( , ) p on each
71-'(p). Show that the index of ( , ) p is constant on each component of B.
6. Let
=: 71 : E +=
7. This problem requires a li ttle knowledge of simple-connec tedness and covering spaces.
(a) There is no way of continuously c hoosing a 1-dimensional subspace of S%,
for each p E s2.(Consider the space consisting of the two unit vectors in each
subsp ace .)
@) There is no Riemannian metric of index 1 on s 2 .
<
8. Let ( , ) and ( , )' be two Riemannian metrics on a vector bundle =
71 : E + B. Let S be the set of e E E with (e,e) = 1, and define S' simiIar1y.
Show that S is homeomorphic to S'. If { is a smooth bundle over a manifold M,
show that S is dHeomorphic to S'.
9. Show by a computation tliat if tlie functions gij and gtij are related by
~ i t l det(gjj)
l
# 0, and the fiinctions gij, grij are defined by
then
Tliis, of course, is the classical way of defining the tensor [Iiaving the components] gV.
10. (a) Let ( , ) be a Riemannian metric on M, and A a teiisor of type
by
tliai A@): Mp += Mp. Define a tensor B of type
(0)
If the expression for A in a coordinate system is
(I),so
sliow that B =
Bik dxi @ d x k , wliere
@) Similarly, define a tensor C of type
(2) by
Show that if C has components ck/,
then
The teiisors B and C are said to be obtained from A by "raising and lowering
indices".
11. (a) Let Xl, . . .,X, be linearly independent vector fields on a manifold M
with a Riemannian metric ( , ). Show that the Grarn-Schmidt process can be
applied to the vector fields al1 at once, so that we obtain n everywhere orthonormal vec tor fields Y], .. . ,Yn.
(b) For the case of a non-positive definite metric, fii-id Y], . . . ,Y" with (Yi,
Yj) =
k 8 i j in a neighborhood of any point.
12. (a) If f : [u, b] + R is positive, show that tl-ie area of the surface obtained
by revolving the graph of f around the x-axis is
lb
2 a f J G i .
(b) Compute tlie area of s2.
13. Let M c R" be an (11 - ])-dimensional submanifold with orientation p.
The outward unit normal v(p) at p E M is defined to be that vector in Rnp
of leilgth 1 such that v(p), ( u ] ) ~ ,... ,
is positively oriented in W
I P when
)p is positively oriented in Mp.
(U] lp, . . . ,
(a) If M = ¿lN for a11 n-dimeilsional manifold-witl-i-boundary N C R", then
v(p) is outward pointing in the sense of Chapter 8.
(b) Let d V,.,-] Ix the volume element of M determined by the Riemannian
metric it acquires as a submanifold of IR". S how that if we consider v (p) as an
element of R", then
dVn-* (p) ((u1 Ip,.
j.
,(%-1 lp) = det
un-1
Conclude tllat dVn-1 ( p ) is the restriction to Mp of
i=l
(c) Note tliat VI x - x v,-1 = olv(p) for some <Y E W (by Problem 5). Sliow
that for w E IR" we have
(w, v(p)) - (VI X - - - X un-1,v(p)) = (w,v1 x - - - x un-1).
Conclude that
vi(p) . d Vn-l (p) = resti-iction to Mp of
-
( - ~ ) ' - ~ d x ~A( ...
~ ) ~ d x > ( pA ). . . i \ d x n ( p ) .
(d) Let M c IRn Ile a compact ri-dimensional maiiifold-with-boundary with v
tl-ie outward unit normal on aM. Denote the volume element of M by dV,,
and that of aM by
Let X =
uia/ax-' be a vector field on M. Prove
th e Diuergence T/ieurem:
xi
/M
div X dVn =
JdM
(x, u ) dVn-l
(the function div X is dcfined in Problem 7-27) Hint: Consider the form w
on M defined by
(e)
. . Let M c lR3 be a compact 2-dimerisional manifold-with-boundary, with
oi-ientation u, and outui7ard-unitnormal v. Let T be the vector field on aM
coilsisti~lgof positiidy orieir ted unit vectors. Dci.iote tlie volume element of M
by dA, arid tliat of aM Ily ds. Let X be a vector field on M . Prove (the original)
Stokes ' TIzeurem:
(V x X, u ) dA =
(X, T ) ds
/M
JdM
(V x X is defined in Prol~lem7-27).
14. (a) Let
I~etl~
i~olume
e
of tlie uiiit ball in IRn. Show tliat
ll 1
(b) If 1. =
(1
x2)("-')/' dx, show that
(c) Using Vi = 2, V2 = x, show that
n even
~ ( n + r ) /(n-1)/2
2~
1-3-5---ri
(In terms of the r function, this can be written
n odd.
n"/2
r(1+n/2)
(d) Let A.-I be the 01 - 1)-volume of S"-'. Using the method of proof in
Corollary 8-8, but reversing the order of integration, show that
(e) Obtain tliis same result by applying the Divergence Tlieorem (Problem 13),
with X ( p ) = pp.
15. (a) Let c : [O, 11 + IRn be a differentiable curve, where IRn has the usual
Riemannian metric ( , ) =
xi dxi @ dx'. Sliow that
(b) For tlie special case c : [O, 11 + IR2 $ven Ily c ( t ) = ( t , f ( t )), show that this
length ,
i s tbe least upper bound of the lengths of inscribed polygonal curves.
Hint:If the inscribed polygonal cunre is determined by the points ( t i , ~ ( t i ) for
)
a partition O = to <
- < E, = 1 of [O, 11, then we have
for some ti E [ti-1, ti].
(c) Prove tlle same result in the genera1 case. Hi7zi: Use the results of ProbIem &l, and uniform continuity of J- on a compact set.
It is iiatural lo suppose tliat the area of a surface is, similarly, the least upper
bound of the areas of inscribed polygonal surfaces, but as H. Schwarz first
obsened, tliis lcast upper bound is infinite for a bounded portion of a cylinder!
To illustrate Schwarz's example 1 have plagiarized t l ~ efollowing picture from a
book called Maazi~~u?nuuccitu.Ü
A~mw
ita M~ozoohj1a3urix,wntten by someone
called M. C I I M I ~ ~ K .
Top view
To ir-icrrasc t l ~ enumber of triangles, we maintain the hexagonal arrangement,
t ~ i i inove
i
tl~eplanes of tlie l-iexago~is
closer together, so that the triai~glesare more
r~earlyir1 a plaiie parallel to the bases of the cylinder. 311 tliis way, we can increase
the ilum ber of triaiigles iridefinitely,wl-iile t1-ie area of each approaches h1/2.
Tl-ie topic of suriace area for non-differen~iablesur-faces is a complex one, wliicb
we will not go into here.
16. Let c : [O, 11 + M be a curve in a manifold M with a Riemannian metric
( , ). lf p : [O, 11 + [O, 11 is a diffeomorphism, show that
L(c) = L(c 0 p).
17. Show that the metric d on M may be defined using Cm, instead of piecewise Cm curves. (Show how to round offcorners of a piecewise Cm path so that
tlie length increases by less than any @ven E > O; remember that the formula
for length involves only first derivatives.)
18. (a) Let B c M be liomeomorphic to the ball ( p E Rn : Ipi 5 1 ) and let
S c M be the subset corresponding to ( p E IRn : j p l = 1 ) . Show that M - S
is disconnected, by showing that M - B and B - S are disjoint open subsets of
M - S.
(b) If p E B - S and q E M - B, show that d(p, q) 2 min d (p, q'). Use
g1eS
tliis fact and Lemma 7' to complete the proof of Theorem T . (In the theory of
infinite dimensional manifolds, tl-iesedetails become quite important, for M - S
does no, liave to be disconnected, and Theorem T is false.)
19. (a) By applying integration by parts to the equation on pages 318-319,
show that
-
S'
a
-(t.
ax
f (t), f '(1)) dr] d i ;
tbis result makes cense even if f is only C' .
(b) Du Bois R~mondi hiima. If a con tinuous furlction g on [a,b] satisfies
for al1 Cm functions q on [a,b] with q(a) = q(b) = O, then g is a constant.
Hi711:The constant c must be
We clearly llave
/ob
"(t)[g(t) - -rl di = O,
so we need to find a suitable q with q3(t) = g(t) - c.
(c) Coiiclude tliat if the C' func~ionf is a critica1 point of J, then f still saósfies
the Euler equations (wliicli are not oprioioA meaningful if f is not c2).
20. T h e hyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions
sinh, cosh, and tanh are defined by
ex
sinhx =
- ,-x
2
3
cosh x =
ex
+ eeX
-3
'
sinh x
t a n h x = -.
cosh x
(a) Graph sinh, cosh, and tanh.
(b) Show that
cosh2 - sinh 2 = 1
tanh2+l/cosh2 = 1
+ y) = sinh x cosh y + coshx sinh y
cosh (x + y) = cosh x cosh y + sinh x sinh y
sinh(x
sinh' = cosh
cosh' = sinh.
(c) For those who know about complex power series:
sin ix
sinhx = -,
i
cosh x = cos ix.
(d) T h e iiiverse fuiictions of sinh and tanh are denoted by sinh-' and tanh-',
respectively, while cosh-' denotes the inverse of cosh 1 [O, m). Show that
21. Consider tl-ie problem of findii-ig a surface of revolution joining two circles
of radius 1, situated, for conveniei-ice, at a and -a. We are looking for a function
Rie~nannian Mehics
of the form
x
f (x) = ccosh C
where c is supposed to satis+
u
~ c o s h- = 1
(e > O).
C
(a) There is a unique yo > O wit11 tanh yo = 1 /yo. Examine the sign of 1/ y tanh y for y > 0.
(b) Examine the sign of coshy - Ysinhy for y > 0.
(c) Let
u
A , (c) = c cosh C
c > O.
Show that A , has a minimum at alyo, find the value of A , there, and sketch
the graph.
(d) There exists c with c coshu/c = 1 if and only if u 5 yo/ cosh yo. If a =
yo/ cosh yo, tlien tl~ereis a unique such c, namely c = u/yo = 1/ cosh yo. If
u < yo/ coshyo, then there are two such c, with c, < a/yo < C?. It turns out
that tl-ie surface for cz has smaller area.
-yo/ cosh yo
-a
(e) Using Problem 20(d), show that
-m.
Yo cosh yo
[Tliese plieriomena can be pictured more easily if we use the notion of an
envelope-c.f. Volume 111, pp. 176K The envelope of the l-parameter famill.
of cunes
1f,(x) = c cosh C
is determiried Ily solving the equations
O=-
M'e obtain
af,(x) = cosh -X - -x sinh -.x
X
x
y = ccosh- = ccosh yo,
C
C
- = f3'0,
so tlie envclope conskts of the straight lines
y=+
cosh yo
X.
Yo
Tlie uriique member. of tlie family througli (yo/ cosh yo, 1 ) is tangent to tlie
erivelope at tliat ~ o i r i t , I;br a < yo/ cosh 170, tlie grapli of f,,is tangent to
tire envelope at poirits P, Q E (-a,a), but the grapli of h2is tangeiit to tlic
cnvelope at points outside [-u, a]. Tl-ie poirit Q is called conjugate to P along
tlie exti-emal f,,, and it is slrown in thc calculus of variations tliat tlie existericc
of this conjugate point implies tl~atthe portion of f,, from P to (o, 1) does no1
@ve a local minimum for
\ f m.
[Compare with the discussion of
J
conjugate points of a geodesic in Volume IV, Chapter 8, and note the remark
on pg. V396.)]
22. All of our illustratioiis of calculus of variations problems involved an F
wliich does not involve t , so tliat the Euler equations are actually
(a) Show that for any f and 1;: R2 + IR we have
d
dt ( F - y':)
= f f aF dt
[-ax- L Eay] ,
and conclude tl-iat the extremals for our problem satisfy
(b) Apply tliis to F(A-,y) = x
m to obtain directly the equation d y / d x =
JCJ*- I which we evelitually obtained in our solution to the problem.
23. (a) Let x and x' be two coordinate systems, witli corresponding gij and gfil
for the expression of a Riemannian metric. Show that
(11) For the c.or.responding [ij,k] and [@, y]', show that
so tliat [ij,k] are not tlie components of a tensor.
(c) Also show tl-iat
24. Sliow that any Cm so-uctur-eon R is difTeomorphic to the usual Cm structure. (Consider tlie arcleligth function on a geodesíc for some Riemannian
metric on R.)
25. Let ( , ) = x i d x i @ dxi be the usual Riemannian metric on R", and
Eitj
let
gildui @ d u j be another metric, where u', . . ., u" again denotes the
standard coordinate system on Rn. Suppose we are told that there is a difKeomorphism f : Rn -t R" such that E , gij du' @ d u l = f * ( , ). How can we
go about finding f ?
(a) Let af/aui = ei : Rn -t R.. If we consider ei as a \rector field on Rn, show
that f,(a/auf) = e i .
(b) Show that gij = (ei, e j ) .
(c) To solve for f it is, iii theory at least, sufficient to solve for the ei, and to
solve for tliese we want to find dflerential equations
satisfied by t l ~ eri's Show that we must have
Il
def
Bi),k =
Egkr~;,
r=l
-1=1
(d) Show thai
(e) By cyclically permuting i, j, k, deduce tbat
,
so that A; = r&.
I n Lqons sur In Géo7nétrú des Espaces de Riemnmr, E. Cartan uses
tliis approacli to motivate the introduction of the r$.
(f) Deduce the result A;] = fh directly from our equations for a geodesic.
(Note tliat tlie cuntes obtained by setting al1 but one f' constant are geodesia,
siiice they correspond to lines parallel to the xi-axis.)
26. If (Vf,( , )') and (V", ( , )") are two vector spaces with inner products,
wedefine ( , ) on V = V f @V" by
(a) Show that ( , ) is an inner product.
(b) Given Riemannian metrics on M and N, it follows that there is a natural
way to put a Riemannian m e t i c on M x N. Describe the geodesics on M x N
for this metric.
27. (a) Let y : [a,b] + M be a geodesic, and let p ; [a,B] + [a,b] be a
diffeomorphism. Sl-iow that c = y o p satisfies
(b) Conversely, if c satisfies this equation, then y is a geodesic.
(c) If c satjsfies
d2ck
dt2 +
Il
E r$( ~ dci( dcJt )= )xdckp~( t~)
for p : IR+ IR,
i, j==1
then c is a reparameterization of a geodesic. (Tlie equation p"(t) = p'(t)p(t)
can be solved explicitly: p(t) = J' e M ( S ) ds, where M1(s) = p(s) .)
28. Let c be a curve in M with d t l d t # O eveqwhere, and consider the hypersurf aces
{exp,(,)u : Il u ll = constant, where u E M,(,) with (u, dc/dt) = 0).
Show tliat for u E M,(r) with (u, dcldt) = O, the geodesic u H exp,(*)u . u
is perpendicular to these liypersurfaces. (Gauss' Lemma is the "special case"
where c is constant.)
29. Let y: [u, b] + M be a geodesic with ~ ( u =
) p, and suppose that expp is
a difTeomorpliism on a neighborhood 13 c Mp of (tyt(0) : O 5 t 5 1). Show
that y is a cunre of minimal length between p and q = y(b), among al1 curves
in exp(0). (Gauss' Lemma still works on exp(O).)
30. If ( , ) is a Riemannian metric on M and d : M x M + IR is tlie corresponding metric, then a curve y : [u, b] + M with d(y (a), y (b)) = L (y) is a
geodesic.
31. Scflwal-z'sulequalilif for continuous functions states that
witli equality if and only if J' and g are linearly dependent (over IR).
(a) Prwe Schwarz's inequality by imitating the proof of Theorem l(2).
(b) For any curve y show tliat
with equality if and orlly if y is parameterized proportionally to arclength.
(c) Let y : [u,b] + M be a geodesic with L : ( ~ ) = d(y(o), y(b)). If c(u) =
y(u) and c(b) = y(b), show that
Conclude that E(y) < E (e) unless c is also a geodesic with
111particular, sufficiently small pieces of a geodesic miiiimize energy.
32. Let p be a poilit of a niaiiifold M with a Rieniannian metric ( , ) . Clioosc
a basis vi,. . .,v, of Mp, SO tliat we have a "rectangular" coordinate system
o11 Mp given by
aivi I+ (ui, . . . ,un); let x be tlie coordinate system ~o exp-l.
defined in a neighborhood U of p.
xi
(a) Sliow tliat in tliis coordi~iatesystem we llave r$(p) = O. Hini: Recall
the equations for a geodesic, aiid note tliat a geodesic y ilirough p is just exp
composed with a straight line through O in M,, so that each y k is linear.
(b) Let r : U + W be r(q) = d@,q), so that r o y = C k ( y k ) 2 . Show that
(c) Note that
Using part (a), conclude that if Il y'(0) 11 is sufficiently small, then
so that d (r o y)2/dr is strictly increasing in a neighborhood of O.
(d) Let 3, = { U E M p : llull 5 E ) and SE= { U E Mp : llull = E). Show that
tlie followirig is true for al1 sufficiently small E > O: if y is a geodesic such that
y(0) E exp(SE)and such that yt(0) is tangent to exp(S,), then there is 6 > O
(dependiiig on y) such that y(t) # exp(BE)for O # 1 E (-S,S). Hznt: If y'(0) is
tar~gentto exp(S,), then d(r o y)/dr = 0.
(e) Let q and q' be two points witl-i r (q), r(q') < E and let y be the unique
geodesic of Iength < 2~ joining them. Show that for sufficiently small E the
maximum of r o y occurs at either q or q'.
(f) A set U c M is geodesically convex if every pair q, q' E U has a unique
geodesic of n~inimumlength between them, and this geodesic lies completely
in U. Show that exp ( ( u E Mp : 11 u 11 < E)) is geodesically convex for sufficiently
small E > 0.
(g) Let J.: U + W" be a diffeomor-pl-iism of a neighborhood U of O E R"
into R". Sl-iow that for sufficiently small E, the inlage of the open E-ball is
convex.
33. (a) There is an everywhere daerentiable curve c ( t ) = ( t , f ( 1 ) ) in IR2 such
that
leng-th of c 1 [O, h]
lim
# 1h-O
lc ( h ) - ~ ( 01 )
Hint: Make c look something like the following picture.
1
length fmm
(f ,O) to (170)
length fmm
.. .
,
(b) Consider the situation in Corollary 15, except that c ( t ) = q if and only if
t = U , and suppose u r ( t ) > O for r near O. If c is c',then v ( t ) approaches a
limit as t + O (even though v(0) is undefined). Show that if c is
then there
is some K > O such that for al1 r near O we have
c',
Hnt: In Mq we clearly have
Since expq is locally a difTeomorp11ismthere are O < K I < K2 such that
for al1 tangent vectors v at points near q.
(c) Conclude that
L ( c I [O, I11)
lirn
= lim
h-O d ( p , ~ ( h ) ) h-0
lh
Ju'(t)2
+ I$(U(t), 1)
u(h)
1
2
di
- 1.
(d) If e is c', show that L(c) is the least upper bound of inscribed ~iecewise
geodesic curves.
34. (a) Using the methods of Problem 33, show that if c is the straight line
joining u, w E Mp, then
L (exp o e)
lim
= 1.
v,w+o
L,(c)
(b) Siinilarly, if yviWis the unique geodesic jojnjng exp(v) and exp(w), and
Yv. w = exp o cu,,, then
lim L ( Y ~w), = 1.
v,w+o L(cuiW)
(c) Conclude that
d (exp u, exp w )
ljm
= 1.
v,w+O
flu-wlf
35. Let f : M + N be an jsometry. Show that f is an isometry of the metric space structures determined on M and N by their respective Riemannian
36. Let M be a manifold with Riemannian metric ( , ) and corresponding
metric d. Let f : M + M be a map of M onto itself which preserves the
metric d.
(a) If y is a geodesic, then f o y is a geodesic.
(b) Define f t : Mp + MAp) as follows: For y a geodesic with y(0) = p, let
Show that ff f'(X) ff = ff X 11, and that f'(c X) = cf '(X).
(c) Given X, Y E Mp, use Problem 34 to sliow that
- IIxI12+ IfU12 - lim [ d ( e x p t X , e x p r ~ ) I 2
flxll - IlYlI
r-0
IlrX lf . 111 y 11
Conclude tliat (X, Y), = (f '(X), f '(Y))J(,), and then that f'(X + Y) =
f '(X) + f'(Y1.
(d) Par t (c) sho~vstfiat f ' : M, + Ml(,) is a diffeomorphism. Use this to show
that f is itself a diíTeomorphism, and hence an isometiy.
37. (a) For u, w E Rn ulth w # O, show that
lim IIv+lwII-IIvlI
1-0
1
--( v , ~ )
llvll
The same result tlien Iiolds in aiiy vector space with a Euclidean metric ( , ).
Hin~:If u : Rn + R is the norm, tlien tlie limit iS Du(v)(w). Alternately, one
can use the equatiori (u, u) = ijuli . iivii - cose wliere 8 is the arigle between u
and v .
(b) Coiiclude tliat if w js liriearly independent of u, then
lim
iIv+fwll - llvll - lltwll
&
(e) Let y : [O, 11 + M be a piecewise CI critical poirit for lerigth, aiid suppose
that yl(to+) # yl(lo-) for some 10 E (O, 1). Clioose r l < 10 and consider the
varjatioii cr for wliicli 6 ( 2 1 ) is obtairied by following y up to 11 , then tlie unique
geodesic fi-om y(fI) to y(lo + u), and firially the rest of y. S110~7that if 11 is
1
close eiiough to 10 then d L (iu (a))/du I,=o # O, a contradic tiori. Tlius, critica1
paths for leng-th carinot h a ~ khks.
e
38. Consider a cylinder Z c IR3 of radius r . Find tlie metric d induced by tlie
Riemannian metric it acquires as a subset of p.
39. Consider a cone C (without the vertex), and let L be a generating lbe.
Unfoldiiig C - L onto lR2 produces a map f : C - L + lR2 whicli is a local
isometi-); but which is usuaUy not oiie-orie. Iinrestigate tlie geodesics on a cone
(tlie number of geodesics between two points depends 011 the angle of the cone,
arid soine geodesics may come back to their initial point).
40. Let g : S* + Pn be tlie map g ( p ) = [ p ]= (p, - p ] .
(a) Sliow tliat there is a uiiique Riemaririian metric (( , )) on P" such tliat
g*(( , )) is tlie usual Riemanriian metric on S" (tlic one that makes the iriclusion
of S" into lRnil an isometry).
(11) Sliow tl-iat every geodesic y : IR + Pn is closed (tliat is, tliere is a riumber a
sucb that y(t +a) = y(t) for al1 E), and tliat every two geodesics isltersect exactly
once.
(c) Sllow tbat tliere are isornetries of P"onto itself taking arly tangent vector
at orie point to any tangent vector at any other point.
Tliese recults sbow that P
' provides a model for "elliptical" non-Euclideaii
georiieti-)! Tlie sum of the angles in aiiy triangle is > n.
41. Tlie Poincark upper halFp1ane 3f2 is the maiiifold ( ( x ,13 E lR2 : y > 0)
m:itli the Riemannian metric
(a) Compute that
k = 0.
al1 otl-ier rij
(b) Let C be a semi-circle in X 2 with center at (O, e) and radius R. Considering
it as a curve t t-+ (1, y(t)), show that
(c) Using Problem 27, show that all the geodesics in X 2 are the (suitably parameterized) semi-circles with center on the x-axis, together with the straight
lines parallel to the y-axis. ,
(d) Show that these geodesics have infinite length in either direction, so that the
upper haif-plane is complete.
(e) Show that if y is a geodesic and p 4 y, then there are infinitely many
geodesics through p which do not intersect y.
(f) For those who know a little about conformai mapping (compare with Problem IV7-6). Consider the upper haif-plane as a subset of the complex numbers C. Show that the maps
are isometries, and that we can take any tangent vector at one point to any
taiigent vector at any other point by some $,. Conclude that if length AB =
length ~ ' 3and
' length AC = length A'C' and the angle between the tangent
vectors of p and y at A equais the angle between the tangent vectors of p'
and y' at A', then length BC = length B'C1 and the angles at B and 3' and
at C and C' are equal ("side-angle-side"). These results show that the Poincaré
upper lialf-plane is a model for Lobachevskian non-Euclidean geometry. The
sum of the angles in any triangle is < x.
42. Let M be a Riemannian manifold sucli tliat every two points of M can be
joined by a unique geodesic of minimai length. Does it necessarily follow that
the Riemannian manifold M is complete?
43. Let M be a manifold with a Riemanliian metric ( , ), and choose a fixed
point p r M . Suppose that every geodesic y ; [a,b ] + M with initial vaiue
y (a) = p can be extended to ail of R. Show that the Riemalinian manifold M
is geodesically complete.
44, Let p be a point in a complete non-comlacl Riemanilian manifold M. Prove
that there is a geodesic y : [O, m) + M with the initiai value y(0) = p , having
the property that y is a minimai geodesic between al-iy two of its points.
45. LRt M alid N be geodesically complete Riemannian manifolds, and give
M x N the Riemannian metric described in Problem 26. Show that the Riemannian manifold M x N is aiso complete.
46. This problem presupposes knowledge of covering spaces. Let g : M + N
be a covering space, where N is a Cm manifold. Then there is a unique Cm
structure o1-i M wliicli makes g an immersion. If ( , ) is a Riemannian metric
oli N, tlien g*( , ) is a Riemannian metric on M , and ( M , g *( , )) is complete
if and only if (N, ( , )) is complete.
47. (a) If M" C
is a submanifold of N, show that the normal bundle v
is indeed a k-plane bundle.
(b) Usiiig tlie i-iotioii of Whitney sum @ introduced iil Problem 3-52, show that
48. (a) Show that tlie normal bundles v i , vz of M" c
defined for two
different Riemannian metrics are equivaient.
(b) If t = rr : E + M is a smooth k-plalie bundle over M*, show that the
normal bundle of M c E is equivaient to 5 .
49. (a) Given an exact sequence of bundle maps
as in Problem 3-28, wllere tlie bundles are over a smooth manifold M [or, more
generally over a paracompact space], show that E2 2 E l $ E3.
(b) If t = r r : E + M is a smooth bundle, conclude that T E R. rr*(t) $
x*(TM).
50. (a) Let M be a non-orientable manifold. According to Problem 3-22 there
is S' c M so that ( T M ) I S ' is not oricntable (the Problem deals with the case
where (TM)IS' is aiways trivial, but the same conclusions will hold if each
(TM)IS' is orientable; in fact, it is 11ot hard to show that a bundle over S' is
trivial if and only if it is orientable). Using Problem 47, show that the normal
bundle v of S? M is not orientable,
(b) Use Prol~lem3-29 to condude that there is a neighborhood of some S' c M
which is not orientable. (Thus, any non-orientable manifold contains a "fairly
smail" non-oi.ientable open submanifold.)
A+
CHAPTER 10
T
LIE GROUPS
liis cliapter uses, and illuminates, many of the results and concepts of tlle
pixeding chapter-s. It will also play an important role in later Volumes,
where we are concerned ~ 4 t geometric
h
problems, because in the study of these
pi-oblems the groups of automorphisms of various structures play a central role,
aiid tliese groups can be studied by the methods now at our disposal.
A topological group is a space G which also lias a group structure (the product
of u , b E G beiiig denoted by a b ) such that the maps
from G x G to G
(a,b ) M a b
from G to G
u Ha
al-e coiitinuous. It cleai-lysuffices to assume instead that the single map ( a ,b ) w
nb-' is continuous. 147~will mainly be iiiterested in a very special kind of
topological group. A Lie group is a group G which is also a manifold with a
Cm structure such that
are Cm functioiis. It clearly suffices to assume tliat the map (x, y) w xy-'
is Cm. As a mattei. of fact (Problem 1), it eveli suflices to assume that tlie map
(x, 1)) M xy is Cm.
Tlle simples1 example of a Lie group is R", with the operation . The
circle S' is also a Lie group. Oiie way to put a group structure on S' is to
consider it as tlie quotient group R/Z, wliere Z C R denotes the subgroup
of iiitegers. The f~iiictions M cos2nx and x M sin 2nx are Cm functions
on R/Z, and at each poilit al least one of them is a coordinate system, Thus
the map
( A - , y ) W .Y - }' W x)'-'
m
+
m
m
R x R + R + S'=R/Z.
wkich can be expressed in coordiliates as one of thc two maps
+
(x, 1 1 ) M cos 2x(x - 11) = cos 2nx cos 2n y sin 27rx sin 2ny
(x,y) M sin 2n(x - y) = sin 27rx cos 27r y - cos 27rx sin 2xy,
is Cm: consequently the rnap (A-,y) w xy-' from S' x S' to S' is also Cm.
If G and H are Lie groups, then G x H, with the product Cm structure:
and the direct product group structure, is easily seen to be a Lie group. In
particular. the torus S' x S' is a Lie group. The torus may also be desuibed
as the quotient group
the pairs (a. b) and (a', b') represent the same element of S' x S' if and only
if a' - a E Zalid b' - b E Z.
Many important Lie groups are mati-is groups. The general linear group
GLOI,R) is the group of al1 non-singular real 17 x 17 matrices, considered as a
subset of R"'. Since the function det : R"' + R is continuous (it is a polynomial
map), the set GL(i1, R) = deth'(R - (O)) is open, and hence can be given tlie
Cm struciuir wliich makes it an open submanifold of R"'. Multiplication of
mati-ices is Cm, siiice tlie entries of A 3 are polynomials in the entries of A
and B. Sinoothness of tlie inverse map follows similarly fi-om Cramer's Rule:
(~-')~
=idet Aij/det A ,
where A'J is the matrix obtained from A by deleting row i and column j.
Oiie of the most importaiit esamples of a Lie gi-oup is the orthogonal group
O(II),consistiiig of al1 A E GL(r1, R) witll A A' = 1, wliere A' is tlie transpose
of A. Tlis condirion is equivaleilt to tlie coiidition tliat the rows [alid columns]
of A are 01-tlioiiormal, which is equivalent ro tlle coiidition that, with respect
ro the usual basis of R", the inatrix A i-epresentsa linear ti-ansformation wliich
is aii "isomerry", Le., is norm presenling, and thus inner product presenling.
Problem 2-33 presen ts a proof tliat O (17) is a closed submanifold of GL(i1, R),
of dimension i l I r ? - 1)/2. To sliow tliar O(I?)is a Lie group we must sliow rliat
the map (x, y) i-, xy-' which is Cm on GL(i1, R), is also Cm as a map from
O(]?)x O(i7) ro O(II).By Propositioii 2-1 1, i t suffices ro sliow rllat ir is coiitinuous;
but this is true because the inclusion of O(i7) + GL(n, R) is a homeomorphism
(sii~ceO ( I ~is) a sii11niaiiifold of GL(n , R)). Later in the cliapter we will havc
anotlier way of proving that O(i?) is a Lie group, aiid in particular, a maiiifold.
Tlie argument in the previous paragraph shows, generally, tliat if H c G is a
subgroup of G aild also a submanifold of G, rheii H is a Lie group. (Tliis gives
aiiotlier prooftliat S' is a Lie groiip, for S' c R2 caii be coiisidered as tlle gioup
motions, Le., isometries of R". A little arpmenr shows (Problem 5) tliat even.
element of E(]?)can be writren uniquely as A T where A c O ( n ) , and r is a
translation,
T ( X ) = Ta(X) = X
+U .
]Ve can give E(]?)the Cd¢st~.iicturewhich makes it diffeomorphic to O ( n ) x R*.
Now E(n) is iiot tlie direcr pi-oduct O(i7) x IR" as a gi-oup, since translations aiid
ortl~ogoiialtrallsformations do not generally commute. In facr,
SO
Ara A-' = r ~ ( a ) ,
Ar, = %(o} A .
Consequen tly:
wliich shows that E(i7) is a Lic gmup. Clcarly tlie compoiient of e r E(r7) is
the subgroup of Al Ar with A E SO(i?).
For any Lie group G, if a E G we define [he left and right translations.
L,: G + Gaild R , : G + G , b v
L,(b) = ub
Ru ( b ) = bu.
Notice tliat L, and R, are bot1.i diffeomol-pliisms,witll iiiverses La-, and R,-1.
resliect ively. Corisequently, tlle maps
are isomorphisms. A vector field X on G is called left invariant if
La,X = X
for al1 a E G.
Recall tliis means thai
It is easy to see tliat tbis is rrue if we merely liavc
La, X, = Xa
for al1 a E G.
Co~lscquently,giveli X, E G,. tl1e1-e is a uliique left invarianr \rector field X
on G wllich lias rlle value Xp at e .
2. PROPOSITION. E1ler-y left invariant vector field X on a L e group G
is Cm.
PROOE It suffices to prove tbat X is Cm iii a neigliborhood of e , since the
diffeomorphism L, theti takes X to the Cx \lector field L., X around a (Problem 5-1). Ler (x, U ) be a coordinate system around e . Choose a neighborI~oodV of e so rliat a!b E V implies ab-' E U.Tlien for a E V we have
Since the map (a,b) H ab is Cm on V x V we can write
lbr some Cm fuiictioti f' on x(V) x x(V).Tllerl
xxi(a)= xe(A-' o L.)
n
=
Ccj ax
j= 1
n
a(xi o L,)
where X, =
j
e
ci
a
j =1
wliich shows tliat x x i is Cm. This implies that X is Cm. O
3. COROLLARJ'. A Lie group G aiways has a trivial tangent bundle (and is
consequently orietitable).
PROOE Cl~oosea basis Xle,
. . . , X,,for G,.Lct Xl,.
. . ,Xn be the left invarialit vector fields wirll these values ar e . Tlien XI ,. . . , Xn are clearly everywliere
litiearly il~dependetit,so we can define an equivalente
A left invarianr vector field X is just one that is La-related to itself for all a.
Consequently, Proposition 6-3 shows that [X, Y] is left invariant if X and Y are.
Henceforrh we will use X, Y, etc.: to denote elements of G,, and X, Y, erc., to
denote the left invariant vector fields with R(e) = X, y(e) = Y, etc. We can
then define an operation [ , ] on Ge by
The \lector space G,, together with this [ , ] operation, is called rhe Lie algebra
of G, and will be denoted by k ( G ) . (Sometimes the Lie algebra of G is defined
insread [o be the set of left invariant \rector fields.) We wiIl also use the more
customary notation Q (a German Fraktur g) for J ( G ) . This notation requires
some conventions for particular groups; we write
QI(II,IR)
o(n)
fortheLiealgebraofGL(n,R)
for the Lie algebra of O(n).
In general, a Lie algebra is a finite dimensional vecror space V, with a bilinear
operation [ , ] satisfying
[X, X] = o
[[X,Y], Z ]
+ [[Y,Z], X] + [[Z,X], Y] = O
'yacobi identity"
for al1 X, Y, Z E V.
Since the [ , ] operation is assumed alternaring, it is also skew-syrnmetric,
[X, Y] = -[Y, X]. Consequently, we cal1 a Lie algebra abelian or commutative
if [X, Y] = O for all X, Y.
Tl-ie Lie algebra of IR" is isomorphic as a vector space to IR". Clearly L(IRn)
is abelian, since t11e vector fields a/axi are left invariant and [a/dxi, d/axj] = 0.
The Lie algebra 6 ( S 1 ) of S' is 1-dimensional, and consequently must be
abelian. If
are Lie algebras with bracket operations [ , 1; for i = 1,2, then
we can define an operation [ , ] on the direct sum V = 6 @ V2 (= VI x V2 as
a set) by
[(Xi, X2), (Y19 y211 = ([Xl, Ylli, [X2, y212).
It is easy to check that this makes V into a Lie algebra, and that L ( G x H)
is isomorpl~icto J ( G ) x L ( H ) with this bracket o~eration.Consequently, the
Lie algebra L(S1x - - x S' ) is also abelian.
The structure of gi(n,IR) is more complicated. Since GL(n,R) is an opcil
sulnnanifold of IR"', the tangent space of GL(n,R) at the identity I can br
identified with Wn2.If \*e use the standard cooi-dinatesxij on Wn', then an i7 x n
(possibly singular) matrix M = (Mij) can be identified with
Let fi be tbe left invariant \rector field on GL(n, W) corres~ondingto M. 1%
compute the function M x k i on GL(n,B) as follows. For every A E GL(ri,W),
NOMIthe function xk' o LA: GL(li, W) + GL(r1,R) is the linear function
witli (constant) partial derivatives
So if N is another n x 17 matrix, we have
Eom this we see that
al
[M, ElI = ~ ( M -NN M ) ~ 'axki
k,'
;
thus, if we identify 91( n , W ) with Wn2,the bracket operation is just
[ M , N=
]M N - N M .
Notice that in any ring, if we define [a,b ] = a b - b a , then [ , ] satisfies the
Jacobi iden ti h.
Since O(ii)is a submanifold of G L ( n ,W ) we can consider O ( n ) l as a subspace
of G L ( n , R ) ] , and tlius identify o ( n ) with a certain subspace of Wn2. This
subspace may be determined as follows. If A : ( - E , E ) + O ( n ) is a curve with
A ( 0 ) = 1, and we denote ( A ( t ) ) , by A j j ( i ) , then
which shows that
A i j f ( 0 )= - A ~ ~ ' ( o ) .
Thus O ( i l ) C
~ w"' can contain only matrices M wliich are skew-symmetric,
Tliis subspace has dimeiisioii ~ ( 1 1 - 1)/2, wliich is cxacrly the dimensioii of O ( n ) ,
so O ( U )must
~ coilsist exactly of skew-symmetiic matrices. If we did not hiow
[he dimeilsioil of O(iz), we could use tl-ie foltowing line of reasonii-ig. For each
i , j with i < j, wc can define a cunre A : W + O(n) by
~ 4 t hsin t and - sin t at (i, j ) and ( j , i ) , 1 'S on tlie diagonal except at (i,i) and
( j ,j),and O's elsewhere. Then the set of all ~ ' ( 0 span
)
the skew-symmetric matrices. Hence O ( I I )must
~ consist exactly of skew-symmetric matrices, and O(n)
must have dimension n(n - 1)/2.
\Ve do not need any new calculations to determine the bracket operation
in o(11). In fact, considel. a Lie subgroup H of any Lie group G, and let i : H +
G be the inclusion. Since i, : H, + G, is an isomorp hism into, we can identify
f& witli a subspace of G,. Any X E H, can be extended to a left invariant
vector field X" on H and a leR invariant vector field on G. For each a E H C
G, we have left translations
L,: G + G
La : H + H ,
and
L a o i = i o La.
rY
LI
i, X (a) = i, La, X = L,,(i* X) = X (u).
rY
Iti, other woids, X" and f are i-related. Co~isequentl~
if Y E &, then [X, Y]
aiid [X, Y]are i-related, which means that
LI
LI
Tlius, H, c Ge = Q is a subalgebra of Q,thar is, H, is a subspace of Q wliich is
closed uiider the [ , ] operation; moreover, He with this induced [ , ] operation
is jusr 1) = L ( H ) .
This coi-respoiideticebetween Lie subgt-oups of G and subalgebras of 9 turns
out to work in the othei- direction also.
4. THEOREh4. Let G be a Lie group, and 11 a subalgebra of 9. Then there
is a utiique connected Lie subgroup H of G whose Lie algebra is $.
PROOF. For u E G, let A, be tlie subspace of Gu consisting of al1 ?(o) for
X E 11. The fact that 11 is a subalgebra of g implies that A is an integrable
distribution. Let H be rhe maximal integral manifold of A containing e. If
b E G, then clearly Lb*(Aa) = Aburso Lb* leaves the distribution A invariant.
It follows immediately that Lb permutes the various maximal integral manifolds
of A among rliemselves. Iti particular, if b E H, tlien Lb-[ takes H to the
tnaximal iiitegrai manifold containitig Lb-, (6) = e , SO Lb-] ( H ) = H. This
iinplies rhar H is a subgi-oup of G. To prove tliat it is a Lie subgroup we just
iieed to show tliat (u, b) I-+ ub-' is Cm. Now tliis map is clearly Cm as a map
into G. Using Theorem 6-7, ir follows that it is Cooas a map into H.
The pt-oof of uniquetiess is left to the reader. *3
There is a very dificult theorem of Ado which states that every Lie algebra
is isomorphic to a subaigebra of GL(N7R) for some N. It then follows from
Theorem 4 that e v q Lie aigebra ti tiomo@ic to /he he aigebra ofsome Liegroup. Later
on we will be able to obrain a "local" version of tliis result. We will soon see to
what extent the Lie aigebra of G determines G.
1% continue the study of Lie groups along the same iSouteused in the stud!.
of groups. Having considered subgroups of Lie groups (and subaigebras of tlieir
Lie algebras), we next consider, more generaily, homomorphisms between Lie
groups. If # : G + H is a Cm homomorphism, then #,,: Ge + He. For anv
a E G we clearly have
-
so if X E Ge, and 5 = #.,X
value #,, X at e, t hen
is the left invariant wctor field on H with
Thus X" and 5 are #-i-elated. Consequently, tlie map #,, : Q + 1) is a Lie
algebra homomorphism, tliat is,
Usually, we will denote #,, simply by #, : g + I).
For example, suppose tliat G = H = R. There are an enormous iiumbeiof homomorphisms # : R + R, because R is a wctor space of uncountable
dimciision wer Q, and ewry linear transformation is a group homomorphism.
But if # is Cm, then the conditioi-i
implies thar
which ineans that #(t) = ct for some c (= #'(O)). It is iiot hard to see tliat eveii
a coi-itii-iuous# must be of rhis form (one first shows that # is of this form on tlie
rational numbers). ]Ve can identify L(R) with R. Clearly the map #, : IR + R
is just rnultiplication by c.
Now suppose that G = R, but H = S' = R/Z. A neighborhood of the
identity e E S' can be identified with a neighborhood of 0 E R, giving rise to an
identification of l ( S 1 )with R. The continuous homomorphisms # : R + S'
are clearly of the form
once again, #, : R + R is multiplication by c.
Notice that the oiily continuous homomorphism # : S' + R is the O map
(since (O) is the only compact subgroup of R). Consequently, a Lie aigebra homomorp hism Q + 11 may not come from any Cw homomorphism # : G + H.
However, we do have a local result.
5. THEOREM. Let G and H be Lie groups, ai-id a: Q + b a L e aigebra
homomorphism. Then there is a neighborhood U of a E G and a Cm map
#: U + H such that
#(ab) = # ( n ) # ( b )
when a, b,ab E U,
and sucli that for every X E 9 we have
Moreover, if tliere are two Cw liomomorphisms #, S r : G + H with #,
$*e = a, and G is connected, then # = @ .
=
PROOE Let f (German fiaktur k) be the subset f c Q x b of al1 (X, @(X)),for
X E Q. Since is a homomorphism, f is a subalgebra of Q x b = L(G x H). By
Tlieorem 4, there is a unique connected Lie subgroup R of G x H whose Lie
algebra is f. If xl : G x H + G is projection on tlie first factor, and w = x11 K ,
rhen w : K + G is a Cm homomorphism. For X E Q we have
so w, : K(e,el + G, is aii isomorphsm. Consequeiitly, there is an opcn neighl~orhoodV of ( e , e ) E K such that w takes V diffeomorphicaily onto an open
neigliborliood U of e E G. If x2: G x H + H is ~rojectionon the second
factor, we can define
# = x2 O u-' on U.
Tlle first condition on # is obvious. As for the second, if X E Q, then
Given #,S, : G + H , define tlie one-one map 8 ; G + G x H by
The image Gr of 8 is a L e subgroup of G x H and for X E g we clearly have
so L ( G r ) = f. Thus G' = K,which iinplies that $ ( u ) = #(a) for al1 a E G. Q
6. COROLLARY. If two L e groups G and H have isomorphic L e algebras.
then they are locally isomorpliic.
PXOOE Given an isomorpliism 0 : g + I), let # be the map $ven by Tlieorem 5. Since #, = is aii isoinoi-phism, # is a dfleomorphism in a neighborhood of r E G. e3
Xenlark: For tllose wl-io know about siniply-coilnected spaces it is fairly easv
(Problem 8) to conclude that two simply-coriiiected Lie groups with isomoi-phic
L e algebras are actually isomorpliic, aild that al1 connected Lie groups with a
given Lie algebra are covered by [he same simply-connected Lie gi-oup.
7. COROLLARY. A connected Lie group G with an abelian Lie algebra is
itself abeliaii.
PROOE By Corollary 6, G is locally isomorpbic to IR", so a b = bu for a , b
in a neighborhood of e. It follows that G is abeliaii, sii-ice (Problem 4) a i i ~
neighborhood of e geiierates G. +$
8. COROLLARY. For every X E G,, thei-e is a uiiique Cm I-iomomorphism
# : R + G such tl-iat
FIRSTPROOE Define 0 : R + 6 ( G ) by
@(a) = (xx.
Clearly 0 is a Lie algebra homomorphism. By Tlieorem 5, on some neigliboiliood (-E,&) of O E R there is a map # : ( - E , & ) + G with
#(S
+ 0=# ( ~ ) # ( l )
lsl, l t l , 1s
+11
E
and
To extend # to R we write every t with It 1
+
1 =k ( ~ / 2 ) Y
E uniquely as
k an integer, Ir 1 < & / 2
and define
.
# ( ~ / 2 ) # ( ~ / 2 ) #(r)
m
#(i) =
#(-~/2)
# ( - ~ / 2 ) - #(Y)
[ # ( ~ / 2 )appears k times]
[#(-&/S)
appears -k times]
k 3 0
k < 0.
Uiiiqueiiess also follows from Theorem 5.
SECOJVD (DIRECV PROOE If .f : G + IR is C m , aiid # : R + G is a C m
homomorpl-iism, then
= lim S(#(t)#(ld) - f( $ 0 1)
h+O
h
Thus # iriust be an integral cunre of X", whicb proves uniqueness. Con~rersely~
if # : R + G is aii integral curve of
tben
X,
is aii iiitegi-al curve of X wliich passes through # ( S ) at time i = O. The same is
clearly true for
i E- #(S
+t):
so # is a hoinomoi.pliisrn. MTekno\rr that integral cunres of X" exist locally; they
can lx extended to al1 of R usiiig [he method of the first proof. *3
A homomorphism # : IR + G is called a l-parameter subgroup of G. We
thus cee that there is a unique 1-parameter subgroup # of G with given tangent
wrtnr d#/di (0) E G,. l e have already examined the 1-parameter subgroups
ui R. More interesting things liappen when we take G to be IR - {O), with
multiplicatioil as tlie group operation. Then all Cm homomorphisms # : IR +
IR - {O), with
#(S + t ) = #(s)#(I),
must satisfy
The solutions of tliis equation are
Notice that IR - (O) is just GL(1, IR). A l Cm l~omomorphisms# : IR + GL(n, IR)
must satisfy tlie analogous differential equation
where iiow denotes matrix multiplication. The solutions of these equations
can be written formally in the sarne way
where exponeiiriation of matrices is defined by
This follows from t he facts in PI-oblem5-6, some of which will be briefly recapitulated here.
If A = (ujj) and IAl = max lau 1, then clearly
)~+*
+ O asN+m.
+ + ( N1A+I K)!
(11
* . m
so tlie series for exp(A) converges (the (i,j)'hentry of the partial sums converge),
and convergence is absolute and uniform in any bounded set. Moreover (see
Problem 5-6), if AB = BA, then
exp(A
+ B) = (exp A)(exp B).
Hence, if # (1) is defined by (m),then
exp(t#'(O) + 17#'(0)) - exp(t#'(O) 1
h+O
h
#'(I) = lim
= lim
h+O
= lim
h-to
[exp(l7#'(0)>- 11
exp(t#'(O)
h
1!
2!
h
exp (1#'(O) 1
so # does satisfy (*).
For any L e gi-oup G, we now define the "exponential map"
exp: g + G
as follows. Given X E 9, let # : R + G be the unique Cm homomorphism
witfi d#/dt(O) = X. Then
exp(X1 = #(]l.
We clearly have
9. PROPOSITION. The map exp: Ge + G is Cm (note that G, R" has a
iiatural Cm structure), and O is a regular poirit, so that exp takes a neighborhood
of O E Ge diffcoinorpliically oiito a rieigfiborhood of e E G. If S,: G + H is
any Cm homomorphism, tlien
exp o S,* = S, O exp.
expl
G-H
lexp
S,
PROOE The tangent space (G, x G)tx,,) of the Cm manifold G, x G at the
point (X,u) can be identified with G, @ G,. We define a vector field Y on
G, x G by
4
\
Y,,.,
=
o @ ?(a).
(S')Lx S i
-
Tlien Y ]las a flow cx : R x (G, x G) + G, x G, which we know is Cm. Since
exp X = projection on G of a(],
O @ X),
it follows tliat exp is Cm.
If \ve ideiitifv a \lector u E (G,)o witli G,, tlien tlie curve ~ ( i =
) t v in G, has
tangent \lector v at O. So
So exp+ois tIle identity, aild lle~iceone-oiie. Tlierefore exp is a diffeomorphism
i i i a neigl~borhoodof O.
Given $: G + H, and X E G,, let #: R + G be a Iiomomorpliism with
Con~equeiitly~
exp($*X> = S, O # ( i ) = $(exp X). e3
10. COROLLARY. Every o~ie-oneCm Iiomomorpliisrn #: G + H is ail
immersion (so #(G) is a L e subgroup of H).
PROOE If
13ut rlien
= O for some iioii-zeio X E Q, then also #*,(X) = 0.
= exp #+e(lX) = #(exp(lX)),
contradictiiig tlie fact tllat # is oiie-one. *=r
11. COROLLARY. Every coiltinuous I-iomomorphism #: R + G is Cm.
PROOF, Let U be a star-sllaped open neighborhood of O E G , on which exp
is one-one. For any a E e x p ( ~ U ) if, a = exp(X/2) for X E U, then
a = exp(X/2) = [exp(x/4)12,
exp X/4 E exp(; U).
So a has a square rooi iii exp(; U). Morewer, if a = b2 for b E exp(4u), then
b = exp(Y/2) for Y E U, so
exp(X/2) = a = b2 = [exp(y/2)12 = exp Y.
Since X/2, Y E U it follows that X/2 = Y, so X/4 = Y/2. This shows that
every a E exp(; U ) has a unique square root in the set exp(; U).
Now dioose E > O SO that #(í) E exp(; U) for 11 1
E. Let #(E) = exp X,
X E exp(f U). Since
[#(&/2)12= #(E) = [exp X/212,
it follows from ihe abwe that # ( ~ / 2 ) = exp(X/2). By induction we have
#(&/Y) = e ~ p ( X / 2 ~ ) .
Helice
#(m/2" E) = #(~/2")" = [exp(X/2")lm = exp(n7/2" X).
12. COROLLARXi. Every continuous Iiomomorphism # : G + H is Cm
PROOE Clioose a basis Xl, .. . ,X, for G,. The map t i-, #(exp tXi) is a
coiitiiluous l~on~omorphism
of R to H, so tl-iere is Yi E He such thai
#(exp iXi) = expiYj.
Thus,
# ((exp11Xi ) - . - (exp i, x,~)) = (exp ti YI ) - . (exp hYn).
(*>
Now tlle map S,: Rn + G given by
is Cm and clearly
so is a diKeomorpliism of a tieighboi-liood U of O E R" onto a neighborllood V of e E G. Then on V,
# = (# O S,) O S,?
aiid (*) sliows tllat # o S, is Cm. So # is Cm at e, and tllus everywhere. 9
1 3. COROLLARXr. If G and G' are Lie groups wllich are isomorphic as topo-
logical groups, then they are isomorphic as Lie groups, that is, there is a d f i o morphism between them wllicll is also a group isomorphism.
PROW Apply Corollary 11 to the continuous isomorpliism and its inverse.
+:+
Tlle propert jes of tlie particular exponential map
exp : V' (= 1
,R ) + GL(n, R)
may 11ow be used to show tllat O(n) is a Lie group. It is easy to see thal
exp(~'=
) (exp M)'.
h/loreo\~er,sirlce exp( M + N) = (exp M) (exp N) when M N = NM, we have
(exp M)(exp -M) = I .
(exp M)(exp M)' = 1;
Le., exp M E O(n). Co~lversely,any A E O(!]) sufficiently close to I can be
written A = exp M for come M. Let A' = expN. Then I = A A' =
(cxp M)(exp N), so es11 N = (exp M)""' = exp(- M). For sufficiently small M
and N tliis implies tliat N = -M. So exp M' = A' = exp(- M); lieilcc
M' = -M. It follows tliar a ncighborhood of I in O(n) is an n(n 1)/2
dimensional su bmailifold of GL(i1, R). Since O(iz) is a subgroup, O (11) is itself
a submanifold of GL(ir ,R).
Just as in GL(n, R), tbe equation exp(X Y ) = exp X exp Y liolds wlie~ievci
[X, Y] = O (PI-ol~lein13). 111 general, [X, Y] measures, up to first order, the
cstent to which tliis equatioii fails to hold. In tbe following Theorem, aiid i11
its piaof, t~ iiidicate diat a functioii c. : R + G , has tlie property that c ( t ) / t is
l)ouiided for small t, we wiD denote it 11y O(t3). Thus O(f3)will denote differeni
fi~~ictions
at differeiit times.
-
+
14. THEOREM. If G is a Lie group and X, Y E G,, theii
+ Y ) + if '[ ~ , YI + 0 ( t 3 ) J
(1) exp t~ exp / Y = e x p { f ( x
(2) exp(-tX) exp(-t Y) exp tX rxp t Y = exp{f2 [ ~Y], + O(t 3)i
(3) exp tX exp t Y exp(-tX) = exp(ty
+ r2[x,Y] + ~ ( t ~ ) } .
(i)
Ff(a) = Fa( f ) = L. x(/) = X(J o L ~ =)
l o
f a exp UX).
Similarly,
m
Irf(a) = -
f ( a exp uY).
m
For fixed S, let
#(i) = f(expsX expiY).
Then
(iii)
f=.l,
d
#'(i)=-f(expsXexpiY)=
di
f(exp sx exp I Y exp u Y)
Applying (iii) io yf insiead of / gives
m
(iv>
m
#"(/) = [Y (Yf )](exp sX exp /Y).
Now Taylor's Theorem says thar
Suppose that f (e) = O. Then we have
Similarly, for any F.
-
Substituiingin (v) for F = f , F = y~ and F = Y ( Y f ) gives
(vi) f'(expsXexpiY)=s(yf)(e)+i(Yf)(e)
Now for small i we can wriie
exp iX exp t Y = exp Z ( i )
for some Coo function Z witll values i1.i G,. Applying Taylor's formula to Z
gives
Z ( r ) = ízI i2z2 0 ( t 3 ) ,
+
+
for some Zi ,Z 2 E G,. If f (') = O , then clearly f ( A ( t )
O({'), SO by (vi) we have
+ 0 ( t ')) = f ( A( 1 ) ) +
Siiice we can tnke tfle .f?s to be cooi-dinate fuilctioi~s,comparison of (vii) and
(viii) gives
w11i cli givw
1
thus proving (1).
Equatioi~(2) follows immeciiately from (1).
To prwe (3), again cl-ioose f with f ( r )= O. T l ~ e nsimilar caiculations give
(ix) f (exp t X exp t Y exp(-t X))
then we also have
Comparing (ix) and (x) gives the desired result. e3
Notice tliat for-mula (2) is a special case of Tl~eorem5-16 (compare aiso with
Problems 5-1 6 and 5-13).
T l ~ ework involved in prmii-ig Tl-ieorern 14 is justified by its role ii-i the following beautiful theorem.
15. THEOREh4. If G is a Lie group and H c G is a closed subset which is
also a subgroup (algebraically),the1-i H is a Lie subgroup of G. More precisely
tl~ereis a Cm sti.ucturc on H , triith die relntiue topoiogy, tllat rnakes it a Lie subgroup
of G.
PROOE We attempt to recoiistruct the Lie a1gebi.a of H as follows. Let B c Ge
be the set of al1 X E G, such tliat expiX E H for al1 t .
Assulioll I . Let Xi r Ge with Xi + X and let t j + O witl~each ti # O. Suppose
exp tiXi E H for a11 i. Then X E 11.
Pro!$
le[
We can assurne ti > 0, since exp(-tiXi) = (exp li Xi)-'
1
ki (1) = largest integer 5 -.
ti
E H. For t > 0:
ki(r)
exp(ki (t)tiXj) = [exp(ii~ j ) ]
E H,
ki(t)tiXi + t X .
Thus exp tX r H , since H is closed and exp is continuous. We clearly also haw
expiX E H for i c O, so X E 1). QE.D.
]Ve now clairn that 1) c G, is a vector subspace. Clearly X E 1) irnplie~
sX r 1) for al1 s E R . If X , Y E 11, we can write by (1) of Theorern 14
exp tX exp t Y = exp{t(X + Y ) + iZ(i)i
+
wliere Z(i) + O as i + O. C1loose positive tj + O aiid let Xi = X Y + Z(ii).
Then Assertion 1 irnplies that X + Y r b. Alternatively, we can write, for fixed i.
+
taking 1irnits as n + ca gives exp t ( X Y ) E H.
[Sirnilarly, using (2) of Tlieorern 14 we see tl~at[X, Y] E b, so that 1) is a
subalgebra, but we will ilot even use this fact.]
Yow let U be an open neigliborllood of O r Ge on whicla exp is a d8'Teornoi.pl~isrn.Then exp(b n U ) is a submanifold of G. It c.lear1y suffices to sllow thai
if U is srnall enough, tlleil
H n exp(U) = exp(1)n U).
Clloose a subspace 1)' C G, cornplernei~taryto 11, so tliat G, = 1) $ 11'.
Assutioil 2. Tl-ie rnap # : G, + G defined by
#(X + X') = exp X exp Xf
X E Ij, X ' E 1)'
is a dXeornorpllisin i i i sorne neigl-iborliood of O.
Rooj: Cl~oosea basis X1,. . . ,Xk, . .. ,X, of G, witli XI ,. . ., Xk a basjs for [l.
Tlieii # is given by
Siiice tlie rnap .x:=, a i X i H ( a l,. , .,on)is a diffeomorphisrn of G , onto W".
it suftices to show thai
@(al,.. . ,a,) = exp
is a diffeomorphism in a neigliborhood of O E IR". This is clear, since
Assc~~olr
3. Tlicre is a i~eighborlloodV' of O in 11' such that exp X' $ H if
0 # X' E V ' .
Plo$ Clioose aii iiiner product on lj' aiid let K c 11' be the compact set of al1
X' E I)' with 1 5 IX'I 5 2. If the assertion were false, there would be Xi' E b'
~vitliXj' + O and esp Xi' E H. Clioose integers 17i wcitli
Choosirig a suheqiience if necessary, we can assirrne Xi' -+ X' E K. Since
it follows fi-orn A . T . T ~ .1~ that
z o ~ ~X' E 11, a contradiction. Q1E.D.
\Ve cari now cornpletc tlie proof of the tlieorem. Choose a neighborhood
U = W x W' of G , or-i which exp is a d~eornorphisrn,with
W a neighborhood of O E 11
W' a neighborhood of O E 11'
siicl~that W' is contained in V ' of Assn-tion 3, and # of Asscriioli 2 is a daeornorpliism on W x W'. Clear1)-
To pr-we the reverse iriclusion, let a E H n exp(U). Then
n = exp X esp X'
X r W, X' r W'.
Since n , exp X E H we obtain exp X' E H, so O = X', and a E exp(b flU). +$
Up to iiow?we liave concentrated on tlie left invariant vector fields, but rnan?
properties of Lie groups are better expressed in terrns of forrns. A forrn w is
called Iefi invariant if L,*w = o for al1 a E G. This rneans that
Clearly? a left invariant k-forrn o is deterrnined by its value o ( r ) E SZ'
(G,).
Heiice: if o',. . . , o" are left iiivariant 1-forrns such that w 1(E), . . .,w n ( e )span
G,*. theri e\-eI?I left invariaut k-forrn is
foi- certain conslanh al. 1f o' (e), . . . , o" ( r )is t1ie dual basis to Xi , . . ., Xn E G,.
t11er-i any Coovector field X can be written
X =
for Cm functions j.'.
f'Tj
Tl~ei~
w i ( x )= f ' ,
so o' is Cm. It folloius that aiiy left invariant forrn is Cm.
If o is leíi invariant, tllen for a E G we have
L,*dw = d ( LO*#)= d o ,
so clw is also left in~~ariailt.
The for.rnula on page 215 irnplies that for a lefi
invarian t 1-forrn w and left in\-ariant \rector fields y and Y we llave
--
d w ( X ,Y ) = X " ( w ( p ) )- r(o(2))
- o ( [ X ,Y ] )
N
-
= -o([X,Y]).
tlie bracket being the operation in Q.
Thc ir~ter-plg.betweeii left ii-i\.ai.iant arid 1-ight in\7ariarit \lector fields is tlie
sul~jijectof 1'1-oblern 1 l. Her-e \ve coiisider the case of for-nis.
16. PROPOSITION. Let *: G + G be @ ( a )= a - ' .
(1) A for-rn w is left invariant if and only if $*w is rjglit invariant.
E Q ~ ( G ~then
) , $*oe= (-])'oe.
(2) 1f
(3) If w is left and right invariant, then d o = 0.
(4) If G is abelian, then 0 is abelian (converse of Corollary 7).
If o is left invariant, tlien
so @*o
is riglit invariant. T h e converse is sirnilar:
(2) It cleasly sufices to prove this for k = 1. So it is enough to show that
SI,,(X) = -X for X E Ge. Now X is the tangent vector at i = O of the curve
1 i-, exprX. So **e X is tlle tangent vector at r = O of 1 i-, (expix)-' =
exp(-f X); tl-iis tangent \lector is just -X.
(3) If w is a left and right invariant k-forrn, then
Sir-ice @*o
ar-id o are 110th left invariant, we have
T l ~ eforni do is also lefi a i d nght invariant, so
But
$-'(do) = d(**w) = d((- 1 l k o ) = (- 1)k dw.
(4) If G is akliail, then al1 left invaiialt 1-fol-rns w are also rigl-it invariant. So
d o = O for al1 left invariant 1-foi*rns. It follows fiorn (x) tllat [X, Y] = O for al1
x , Y E n.
A ! ~ e ~ ? ? a k p r o d(4).
o f B y Tlieorern 14, if G is abelian, then for X, Y E G, \ve
h ave
H ence
2 [X, Y]
+ o(t3)p2
= ;[Y,x ] + 0 ( 1 ' ) / 1 ~ .
Letting r + O, we obtain [X, Y] = [Y, X]. $+
Since do is left invariant for any left invariant o , it follows tliat for a basis o ' ,
. . . ,o" of iiivariaiit 1-forms we can express eacli dwk in terms of the o' A d .
First clioose Xi , . . ., X, E G, dual to o' ( e ) , . . ., & ( e ) . There are constants C$
such that
ti
clearly we dso liaw
n
Tlie iiumbers C$ are called tlie constants ofstrucnire of G (witli respect to h e
basis Xl ,. . . , X, of g). fi-om skew-sylniiietry of [ , ] aad tIicJacohi idcntity we
obtain
Frorn (*) on page 394 we obtain
Ii iiiriis out that (2) is esactly wliat we obtaiii korn the relation d 2 0 k = O. Coiidition (2) is tlius an ir-itegrability coiidition. lo fact, we can prwe (Probleni 30)
that if C$ ale coiistaiits satisfyiiig (1) aiid (2), tlien we can find everywlicrr
liiiea1.ly indcpciideiit 1-forms o ' ,. . . ,o" iii a neigliborhood of O E W" sucli tliat
Morewer, the existente of such o' jrnplies (Prohlern 29) that we can define a
rnultiplication (a,b) I-+ab in a neighborf~oodof O which is a group as far as
it can be and which has the o' as left invariant 1-forrns. Frorn this latter fact
and (a suitable local version of) Theorern 5 we could irnrnediately deduce the
following Theorern, for which we supply an independent proof.
17. THEOREM. Let G be a Lie group with a basis of left invariant 1-forrns
o', . . .,o" aiid constants of structure c;. Let M" be a differentiable manifold
aiid let O', . .. ,O" be everywhere linearly independent 1-forrns on M satisfying
Tlien for every p E M there is a neigllborhood U and a diffeornorphisrn
f : U + G such that
0' = f*J.
PROOF. k t T C ~: M x G + M and xz : M x G + G be the projections. Let
Then
d(ek - ók)= -
EC; ([ei oj] - [ói
A
A ój])
icj
By Proposition 7-14., M x G is foliated by 11-dimensional rnanifolds whose
tangent spaces at each point are anniliilated by al1 8* - o*. Choose a E G
and let r he tlie foliurn through ( p , a). Now
.. . ó', .. . ,ónare linearly
independent everywliere; so on r(p,alrwhich is tlie set ofvectors in ( M x G)(p,a)
where
- ók = O, tlie sets o ' , . . . ,8" and o',... ,ón are each linearly indepeiident. Hence T C ~: r + M and x2: r + G are eacli diffeornorphis~ns
i i i sorne neigliborhood of (p,a). Tl~isrneans that r contains the graph of
a diffeornorpliisrn f fi-orn a neigliborliood U of p to a iieighborhood of a.
e', ,en,
ek
Let f:U + M x G be the rnap
It is also possible to sap by how rnucll any two such rilaps differ.:
18. THEOREM. Let M be a coi~nectedmanifold, let G be a Lie group, and
let .j;, .j i : M + G be two Cm niaps sucll tliat
for al1 lefi in\.ai-iant 1-for-rnsw . Tl~eiiJ j and J" differ by a left translation, tlial
is, ther-e is a (uriique) a E G sucl~tllat
PEDISTliLclAr PROOF. C m 1. M = IR and tlze 1x10ntaps y,, y2: IR + G soiisfi
y1(0) = y, (0). ]Ve rnust show tliat y, = y2. For every left iiivariant 1-forrn w
we llavc.
1r follows tliar
If wc r-egai-d y, as givei~?and wi-ite this equation out iri a coordinate system.
tl-ii~iit bccoiiies an or-diiiaiy difl'cr-critialeqiiation for y2 (of t l ~ etype considered
in the Addendurn to Cl~apter5), so it lias a unique solution with tlie iiiitial
coridition y2(0) = y, (O). But this solution js clearly y2 = y,.
Cme 2. M = IR, but 11le nzajis y,, y2 are a r b i l r a ~ .Choose a E G so that
If o is a left invariant 1-forrn, then
(L. 0 Y,)*(#) = y,*(La*w) = y,*(o) = y,'(o).
Since La o y, (O) = y2(0),it follows frorn Cme 1 that La o yl = y2.
C a s ~3. Ge~ia.01cme. Let po E M. Choose a E G so tliat
For aiiy p E M there is a Cm curve c : IR + M i v i t l ~c(0) = po and c(1) = p.
Let yj = fi o C. Then
By C a e 2, we llave
y2(t ) = a . y, ( t )
i i i particular for t = 1, so $2(]))
=a -
for al1 f .
(P).
E L E G A A T P R O O F : Lec : G x G + G be projection on the ithfactor. Clioose
a l~asiso ' , . . . ,o'' for tlle left iinrariant 1-forrns. For (a, b) E G x G, let
Tlieil A is aii integr-able distribution on G x G. In fact, if A(G) c G x G is tlie
diagoiial subgi-oup {(a,a ) : a E G), tlieii tlie rnaxirnal integral rnanifolds of A
are t l ~ elefi cosets of A(G). Now define h : M + G x G b?.
By assurnptioii,
Since M is connected, it follows that Iz(M) is contained in sorne left coser
of A(G). 1 n otliei- words, tl1ei-e are a , b E G witll
19. COROLLARY. If G is a connected Lie group and f : G + G is a CE
map preserving left invariant forrns, then f = L, for a unique a E G .
\Yliile left ini-ai-iant 1-forrns play a fundamental role in thc study of G, t h c
left iilvariant 12-for-rns arvz also very irnportant. Clearly, al1 left invariant 11-form:
ai-c a constant rnultiple of anv non-zero one. If o" is a left invariant n-forrn.
then o" deter.rnii1es al1 orieiitation on G, aiid if f : G + R is a Cm fui-iction
ivitll coimpac t support, we can define
Siiice a'' is usually kept fixed in any discussioi-i, this is often abbreviated to
T h e latter iiotation has adiyantages iii cer-tain cases. For exarnple, left invariai~cc
of o" irnnlies that
f ( n ) da =
/G
jG j'(ba) da.
m
in other words,
S, S,
fa" =
ga',
wherr glo) = .f(b«);
1 note that Lb is an orientation presening difkornoi-phisrn, so
~cllichproves the rorrnula]. \Ve can, of c.ourse, also consider right iiivariai~t
n-f'oi.rns. Tliese gcncr-ally turn out to be quite different frorn t l ~ eleft invariant
12-ioi-ms (see the csarnple in Pr-oblern 25). But in one case they coincide.
20. PROPOSITION. If G is conipact niid connected and w is a left irivariai~t
12-hl-in,then o is also rigllt invariant.
1'1~00.E Suppose o # O. For each a E G, t l ~ elorrn Ra*o is left invariant. so
tliere is a unique real nuinber f ( a )with
Siiice R,* o RbA= (Rab)*,we have
S o f ( ~ C) IR is acoinpact connected subgroup of R-(O). Hence f ( G )= { 1). +$
]/Ve can abo consider Riernannian rnetrics on G. In the case of a cornpact
group G there is ahvays a Riernannian rnetric on G which is both left and
right invariant. In fact, if ( , ) is any Riernannian rnetric we can choose a
bi-invariant 17-forrn an and define a bi-invariant (( , )) on G
We are finally ready to account for sorne terrninology frorn Chapter 9.
2 1. PROPOSITION. Let G be a Lie group with a bi-invariant rnetric.
(1) For any a E G, the rnap I, : G + G giwn by Ia(b) = ab-'a is an isornetq.
wllich reverses geodesics through a , i.e., if y is a geodesic and y (O) = a , then
Iu (Y (t)) = Y (-0.
(2) T h e geodesics y with y(0) = e are precisely the l-pararneter subgroups
of G, i.e., the rnaps f J-+exp(tX) for sorne X E g.
PROOE (1) Since
I,(b) = b-',
the rnap l e , : G, + Ge is just rnultiplication by - I (see tlle proof of Proposition l6(2)), so it is an isornetry on G,. Since
for any a E G, the rnap J,, : G, + Gu-1 is also arl isornetry. Clearly Ie reverses
geodesics through e .
Since
la =
R,-l
:
it is clear that lais an ison-ietry reversing geodesic through a .
(2) Let y : IR + G be a geodesic with y(0) = r . For fixed t, let
T l i e i ~y is a geodesic and Y (O) = ~ ( 1 ) So
.
It follows by induction that
Y ( n l ) = Y (0"
for any integer n.
If 1' == 12'1 a n d 1'' == nfft for integers n f and n", then
so y is a hornornorphisrn on 0. By continuity, y is a 1-pararneter subgroup.
T l ~ e s eare the only geodesics, since there are 1-pararneter subgroups with
any tangent vector at t = 0, and geodesics tbrough e are deterrnined by theirtarigent vectors at t = 0. *:*
]Ve conclude this chapter by iiitroducing sorne neat forrnalisrn which aHows
us to write the expression for dok in an invariant way that does not use tlic
constants of structure of G. If V is a d-dimensional vector space, we define a
V-valued k-form on M to be a furiction o such that each o ( p ) is an alternating
maP
k times
If u], . . .,ud is a basis for V, tlieri tliere are ordinary k-forrns o ' , . . ., od sucli
tliat for- A'], . . . ,Xk E Mp we have
we will write sirnply
For- anv V-valued k-forrn w we define a V-valued ( k
+ 1)-forrn d o by
a simple c.alculation slio~vsthat tl~isdefinition does not depend o n tl-ie choice of
basis u', . . . vd for V.
Líe G'7.aups
403
Sirnjlarly, suppose p : U x V + W is a bilinear rnap, where U and V have
bases u l , . . . , u , and u ] , . .. ,ud, respectively. If w is a U-valued k-forrn
and q is a V-valued 1-forrn
+
is a MI-ualued (k 1)-forrn; a calculation shows that this does not depend on the
choice of bases u], . . . ,u, or v i , . . .,ud. M'e will denote this W-valued (k + ¡)forrn by p(w A q).
Tliese concepts have a natural place in the study of a Lie group G. Althougli
there is iio natural way to choose a basis of left invariant 1-forrns on G, there U
a natur-al q-valued 1-forrn on G, narnely the forrn o defined by
U$ng tlie bilinear rnap [ , 1: CJ x + g, \ve have, for any R-valued k-form q
a11d any n-valued ¡-forrn h on G, a new g-valued (k + 1)-forrn [q A A] on G.
Now suppose tbat X i , . . . ,X, E Ge = g is a basis, and that o', . . . ,wn is a
dual basis of leíi invai-iant 1 -forrns. T h e forrn o defined by (*) can clearly be
writteii
n
Tlien
O n the other hand,
n
Cornparing (1) and (2): we obtain the equations of structure of G:
T h e equations of structure of a Lie group ~ 4 1 1play an irnportant i-ole in
Voiume 111. For the pi-esent we rnerely wish to point out that the terrns d o and
[oA o]appearing in this equation can also be defined in an invariant way. Foitlle terrn d o wejust rnodify tlle formula in Theorern 7-13: If U is a vector field
on G and f is a g-va1ued fuiiction on G, then (Problern 20) we can define a
g-valued function U ( f ) on G. O n the other hand, # ( U ) is a g-valued function
on G. For vector fields U and V we can then define
Recall that the value at a E G of the right side depends only on the values U,
and Va of U and V a t a . I f w e clioose U = R, V = Y for sorne X 7 Y E G r ,
then
Fa)= O - O - o-( a-) ( [ X Y, ] , )
rY
rY
do(a)(yQ7
= - w ( e ) ( [ X ,Y ] , )
-o(e)([X, Y ] )
= -[X, Y ]
E
= - [ o ( a ) ( y a ) ,o ( a ) ( y a ) ]
--
since [ X , Y ] is left invaimiani
by definition of [ , ] in G ,
by definition of o.
It foHows that foi. any lector fields U and V we have
Pi-oMem 20 gives an iinrariant definition of & A V ) and shows tliat tliis equation
is equivalent to tlie equations of structure.
MIARNING: In sorne books the equation which we have just deduced appears
) ,( ~ ) ]
T h.e appearance of the factor f Iiere Iias
as d w ( U , V) = - L2 [ ~ ( U o
noi1rin.g to d o with the f in the other forrn of the structure equations. It comes
about because sorne books d o not use tIie factor ( k l ) ! / k !l ! in the definition
of A. This rnakes their h A q equal to f of ours for 1-forrns h and v. T h e n the
defiiiition of d ( x dx') as
d o i A d x i rnakes their d o equal to f of ours
for 1-forrns o.
+
PROBLEMS
1 . Let G be a group which is also a Cm manifold, and suppose that ( x ,y) J-+ ñ-1.
is CE.
(a) Find 1 - ' w h e n f : G x G + G x G i s J ( x , y ) = ( x , x ) l ) .
(b) Show that ( e ,e ) is a regular point of f.
(c) Conclude that G is a Lie group.
2. Ler G be a topological group, and H c G a subgroup. Show that tlir
of H is also a subgroup.
closure
3. Let G be a topological group and H c G a subgroup.
(a) If H is open, then so is every coset gH.
(b) If H is open, then H is closed.
4. Let G be a connected topological group, and U a neighborhood of e E G.
Let U" denote al1 products al a,,for ai E U.
(a) Show that Un+' is a neighborl~oodof Un.
(b) Conclude that U, Un = G. (Use Problem 3.)
(c) If G is locaily compact and connected, then G is a-compact.
5. Let f : Rn + Rn be distance presenring, with f(O) = 0.
(a) Show that J takes straight lirles to straight lines.
(b) Sliow that f takes planes to planes.
(c) Show that f is a linear transforrnation, and hence an elernent of O ( n ) .
(d) Show that any elernent of E ( I z )can be written A z for A E O ( n ) and z a
translation.
6. Show tliat tbe tangent bundle T G o f a Lie group G can always be made into
a Lie group.
7. \Ve have cornputed that for M E gl(n, IR) we have
(a) Show that this means that
(It is ñctually clear n
that M defined in this way is left invariant, for LA* =
LA since LA is linear.)
(b) Fi~indthe i.ight invariant lector field with value M at J.
8. Let G and H be topological groups and #; U + H a rnap on a corinected
open neighbrhood U of e E G such that #(ab) = #(a)#(b) when n, b,nb E U.
(a) For each c E G, consider pairs (1/, S,), where V c G is an open neigliboi-hood of c with V V-' c U: and where S,; V + H satisfies $(o) S,(b)-] =
#(ab-' ) for n, b E V. Define (Vi, S,]) y (V2,S,2) if S,] = S,2 on sorne srnaller
neigliborhood of c. Show that the set ofall 7 equi\lalence classes, for al1 c E G ,
caii be rnade into a covering space of G.
(b) Conclude that if G is sirnply-connected, tl~en# cari be extended uniquel\to a liornornorphism of G into H.
9. In Theorem 5, show that # and S, are equal elten if t h q are defined only
o11 a neigliborliood U of e E G, provided that U is connected.
10. Sliow that Corollary 7 is false if G is not assurned connected.
11. If G is a gi.oup, \ve define the opposite group G 0 to be tlie sarne set 134th
tlie rnultiplication + defined by a +b = b .a. If g is a Lie algebra, ~ 4 t hoperation
[ , 1, we define th e opposite Lie aIgebra f1° to be tlie sarne set with t l ~ eopei-atiori
[X, YI0 = -[X, Y].
(a) G0 is a group, and if S,: G + G is a I-+ a-': then S, is an isornoi.phism
from G to GO.
(b) no is a Lie algebra, and X I-+ -X is an isornorphisrn of g onto gO.
(c) 6 ( G 0 ) is isornorphic to [6(G)1° = gO.
(d) Let [ , ] be the operation on G, obtained by using right invariant \lectorfields instead of left invariant ones. Then (n, [ , ]) is isornorphic to 6 ( G 0 ) , and
hence to gO.
(e) Use this EO give aanother proof that g is al~elianwllen G is abelian.
12. (a) Show that
e X p (-a
O
:)=( -coso
sin
cosa
sino
a
(11) Use the matrices A and B below to show tllat exp(A + B) is not generally
equal to (exp A)(exp B).
13. Let X, Y E G, with [X, Y] = 0.
(a) Use Lemrna 5-13 to show that (exp sX)(expt Y) = (exp t Y)(expsX).
(b) More generally, use Theorem 5 to show that exp is a hornornorpllism
on the subspace of G, spanned by X and Y. In articular, exp(X + Y ) =
(exp X)(exp Y).
+
14. Problem 13 implies that exp i ( X Y) = (exp iX)(expt Y ) if [X, Y ]= O. A
more general result liolds. Let X and Y be vector fields on a Cm manifold M
with corresponding local 1-parameter families of local diReomorphisms {#r):
Suppose that [X, Y ]= 0, and let vi = #[ o
= $[ o #[.
(b) Using Corollary 5-1 2, show that
In other words, (vr) is generated by X
+ Y.
15. (a) If M is a diagonal matrix ~ ? i tcomplex
h
entries, show that
det exp M = etrace M
(b) Show that the same equatioii holds for al1 diagonalizable M with complex
entries.
(c.) Conclude that it holcis for al1 M witli complex eritries. (The diagonalizable
matrices are deiise; compare Problem 7-1 5.)
(d) Using Pi-opositioil 9, sliow tliat for tlie homomorphism det: GL(n,R) +
R - {O), the map ciet,: ~l(12,
R) + J ( R - (0)) = R is just M J-+trace M.
(e) Usc tliis fact to give a fancy proof tliat trace M N = trace NM. (Look at
trace(MN - NM) = trace[M, N].)
(0Prove the result iii part (d) ciii-ectly: ivitl~outusii~g(c). (Sincc det, and tracc
are hornomorphisrns, it suffices to look at matrices with only one non-zero entry.)
(g) Now usc tliis i-esult and Pr-oposition 9 to give a fancy proof of (c).
16. (a) Let U be a iieighborliood of the ideiitity (1, O) of S' (considered as a
subsct of R2). Show that no matler liow small U is, there are elements a E U
which have square roots outside U in acidition to their square root in U.
(b) Show that foi. each n 2 1, there is a neighborhood U of e E G such thal
eitery elemeiii in U has a unique nth root in U.
(c) For G = S ' . show tl~attliere is no neighborl~oodU which has this propcrt);
for al1 n.
17. (a) Let (x, V ) be a cooi-diiiatesystein arouiid e E G with x i ( e ) = O. Let
for Cm functions J'. Show that
(b) If a,p : (-E, E) + G are difherentiable, show that
(c) A s o deduce this result fiorn Theorern l4(I). (Not even the full strength of (1)
is needed; it suffices to kiio\v that exp tX exp t Y = exp{t ( X Y) O(t)). Tlie
argurnent of p a n (a) is essentially equivalent to the initial parc of the deduction
of (1)s)
+ +
18. Let G be a Lie group, and let H C G be a subgr.oup of G (algebraically),
such that every a E H can be joined to r by a Cm path lying in H. Let 6 c G,
be tlie set of tangent vectors to al1 Cm paths Jying in H.
(a) Show tliat 11 is a subalgebra of G,. (Use Theorern 14.)
(b) Let K C G be tlie connected Lie subgroup of G with L e algebra 6. Show
that H c K. Hint: Join any a E H to r by a Cm curve c, and show that the
tangent \rectors of c lie in tlie distribution constructed in the proofofT1ieorern 4.
(c) Let t i , . . . ,ck be curves in H with {cit(0)) a basis for b. By considering
tlie rnap / ( t i , ..., t k ) = c ~ ( i 1 ) . . . c k ( i k show
),
that K c H. Thus, H is a Lie
subgroup of G. It is even true that H c G is a Lie subgroup if H is path
connected (by not necessady Cm paths); see Yarnabe, On an arcwire cotrnected
sllbg~oupofa L e group, Osaka Math. J. 2 (1950), 13-14.
(d) If H c G is a subgroup and an irnrnersed subrnanifold, then H is a Lie
subgio up.
19. For a E G, consider the rnap b M aba-l = L,R,-'(b).
The rnap
is denoted by Ad(a); usually Ad(a)(X) is denoted sirnply by Ad(a)X.
(a) Ad(nb) = Ad(n)oAd(b). Tlius we llave a 1iornoinorphisrn Ad : G + Aut(g),
where Au/(n), the autornorphisrn group of g, is the set of al1 non-singular linear
transforrnatioi~sof the vector space g onto itself (tllus, isornorphic to GL(n, R)
i f n has dirnension n). Tlie map Ad is called the adjoint representation.
(b) Show tl~ai
exp(Ad(a)X) = n(exp ~)a-' .
Hini: T1iis follows irnrnediatcly frorn one of our propositions.
(c) For A E GL(n, R) and M E gl(n, R) show that
(11suffices to show this for M in a neighborhood of O.)
(d) Show that
Ad(expíX)Y = Y + í [X, Y] + 0 ( i 2 ) .
(e) Since Ad: G + g, we have the map
Ad*e : (= Ge)
+
tangent space of A u t ( ~ )at the
identity map i of Q to itself.
Tl~istangent space is isornorphic to E n d ( ~ )where
,
End(9) is the vector space of
al1 linear transformations of g into itself: If c is a curve in A u t ( ~with
)
c(O) = l,,
tllcn to regard c'(0) as an elcment of A u t ( ~ )we
, let it operate on Y E Q by
(Compare with the case 9 = Rn, A u t ( ~ )= GL(n, IR), End(g) = 12 x n matrices.)
Use (d) to show that
Ad*e(X)(Y) = [X, Y].
N
N
(A proof may also be given using the fact that [X, Y] = L F ~ . )The map
Y J-+[X, Y] is denoted by ad X E End(g).
(f) Conclude that
Ad(exp X ) = exp(ad X) == l o
+ ad X + (ad2!x ) ~
+
-
e
.
.
(g) Let G be a connected Lie group and H c G a Lie subgroup. Show that H
is a iiormal subgroup of G if and only if I) = L ( H ) is an ideal of Q = L ( G ) ,
tllat is, if and only jf [X, Y] E lj for al1 X E n, Y E I).
20. (a) LRt f : M + V, where V is a finite dimensional vector space, wjth basis
v i , . . . ,ud. For Xp E M p , define Xp(f) E V by
xd
where f = I=I j' vi for f i : M + R. Show tliat this definition is indepeiident of the cl~oiceof basis vi, . . . ,vd for V.
(b) If o is a V-valued k-forrn, sliow that d o rnay be defined invariantly by the
formula in Theorern 7-13 (using the definition in part (a)).
(c) For p: U x V + W. show that p(w A q) rnay be defined invariantly by
Conclude, in particular, that
(d) Deduce tlle structure equations frorn (b) and (c).
21. (a) If o is a U -valued k-forrn and q is a V-valued I-forrn, and p: U x V +
W, then
d ( p ( o A $1) = p ( d o A 9) (-lIk p ( o A dq).
+
(b) For a pvalued k-foi-rn o and 1-forrn q we have
[ o A q] = (-l)kl+'[qA o].
(c) Moreover, if A is a 9-valued n7-forrn, then
22. Let G c GL(II,IR) be a Lje subgroup. The inclusion rnap G + GL(n, IR) +
IRn2 will be denoted l y P (for "point"). Then d P is an IRn2-valued 1-forrn (it
corresponds to the ideiltity rnap of the tangent space of G jnto itself). We can
also consider d P as a rnatrix of 1 -forrns; it is just tlie rnatrix (dxij), where each
dxij is restricted to the tangent bundle of G. \Ve also have the ~ " ~ - v a l u e d
1 -forrn (or rnatrix of 1 -forrns) P-' dP, wliere - denotes rnatrix rnultiplication,
and P-' denotes the rnap A I-+A-' on G.
(a) P-' - d P = p(P-] A dP), where p: 8"' x 8" + 8"' is rnatrix rnultiplication.
(b) LA*dP= A dP. (Use f *d = df *.)
(c) P-' d P is left invariant; and (dP) P-' is right invariant.
(d) P-] d P is the natural Q-valued 1-forrn o on G. (It suffices to check that
P-' d P = w at 1.)
(e) Using d P = P o , show that O = d P - o P - d o , where the rnatrix of
2-foi-rns P . d o is cornputed by forrnally rnultiplying h e matrices of 1-forrns d P
and o. Deduce tllar
do+o.o=O.
m
+
If w is the matrix of 1-forrns o = (oU),
this says that
Check that tliese equations are equjvalent to the equations of structure (use the
forrn d o ( X , Y) = -[o(X),o(Y)].)
23. Let G c GL(2, R) consjst of al1 matrices
(Ol)w i h # O. ~ o conwr
n
iiieiice, denote the coordinates xll and x12 on GL(2, IR) by x and 1..
(a) Show that for the natural g-valued forrn o on G we have
so tliat dx/x and d y / s are IeA iilvarjant 1-forrns on G, and a left invariant
2-form is (dx A dy)/x2.
(b) Find tlie structure constants for these forms.
(c) Show that
and find tlle right iiivariant 2-forrns.
24. (a) Show that tlie natural gI(n, R)-valued 1-forrn o on GL(n, R) is giverl by
..
olJ=
(11)
1
det (xaB) k=i
ddx"'
Sliow that botli tlie leR and nght invariant ,12-forrns are rnultiples of
1
(dxJ1A . . . / \ d x n ' ) A . . . A (dxl" A . . . ~ d ~ " " ) .
(det(xa8))"
25. The special linear group SL(,z, R) c GL(n, R) is the set of all matrices of
deterrninant 1.
(a) Using Problern 15: sliow that its Lie a1gebi.a .r;l(~z,
R) consists of al1 matrices
with trace = 0.
(b) For the case of SL(2,R), sbow that
P
~
P
- y dv
( -=uvddxx+- x yd du u - uvdy
dy+xdv
where we use x, y, u, v for x", x12, x2', x ~ Check
~ . tliat the trace is O by
djfferentiating the equation xv - yu = 1.
(c) Show that a left invariant 3-forrn is
26. For M , N E o ( n ) = L(O(n)) = { M : M = -M'), define
(N, M ) = - trace M N t .
(a) ( , ) is a positive definite inner p~-oducton u(n).
(b) If A E O ( n ) ,tliei~
(Ad(A) is defined in Problern 19.)
(c) Tlie left inva-iant rnetric on O(!?) witli value ( , ) at
invarian t.
is also right
27. (a) If G is a cornpact Lie group, then exp: g + G is onto. Hint: Use
Proposition 21.
(b) Let A E SL(2,R). Recall that A satisfies its cl-iaracteristic polyilornjal, so
1 -2.
A 2 - (trace A)A + I = O. Conclude that trace
(c) Show that the following elernent of SL(2, R) is not A 2 for aiiy A . Conclude
that it is ilot iil the irnage of exp.
(d) SL(2, R) does not llave a bi-invariant rnetric.
28. Let x be a cooi-diilate systern around r in a Lie group G, let ~ r: G
i xG+G
be the projectioiis, a i ~ dlet ( y , z) be tlie coordiiiate systern arouiid (e, r ) given
by y ' = x ' o i r i , z i = x i o i r i . Define#': G x G + R by
and let Xi be tlie left iilvariant vector field on G with
(a) Show that
(b) Using &Lb = Lab, s l i ~ wthat
Deduce that
xi (b)(xi La) = @: (ab).
0
and then tliat
-
.
Letting $ = (3;) be tlic inverse rnatnx of @ = (y?:),
we can write
This equation (or arly of nurnerous tl-ijng equivalent to it) is known as Lde'sfirsi
fundame~talUieorem. Tlie associativity of G is jrnplicitly contained in it, since we
used the fact that L, Lb = Lab.
(c) Prwe tlie conuene uf G e S jrsi funda~nenhl Ilreurem, which states the followiiig.
Let # = (#', . . .,#") be a differentiable fuilction in a neighborhood of O E IR2"
[with standard coordinate system
. . , v",zl,. . . , s n ] such that
#(a,o) = a
for a E IR".
Suppose thei-e are differciitiable functions @; in a iieighborhood of O E IRn [with
standard coordinate system rl,.. .,xn] sucli tliat
$j(O) = sj
n
(*)
a#'
m("7b)=C@:(4(a,b)).$;(b)
i= l
Toi. (a, b) in a neighborhood
of O E Ik2".
Tlien ( a ,b ) M # ( a , b ) is a local Lie group structure on a neighborhood ofO E IRn
(it is associative arid has inverses for points close enough to O, which serves as
the ider-itity);the corresponding left invariant \rector fields are
[To prove associativity, note that
arid tlien sliow tliat # ( a ,# ( b , z)) satisfies the sarne equation.]
29. Lid secolrdfundonzenid tlteo~e~n
states tl-iat tlie left invariant \rector fields Xi of
a Lie group G satisfy
n
for certain constoak ~ 6 - i n otlier words, tlie bracket of two left invariant vector
fields is left invariant. The airn of this problern is to prove tl-ie ~o?¿uer.eof&3
secondfulzdo)nento2 tlteo~em,w1iicl-i states tl-icfollowing: A Lie algebra of vector fields
on a rieigliborhood of O E IRn, which is of dirnension iz over IR and cor-itains a
basis for RUo,is tlie set of left invariant \lector fields for sorne local Lie group
structure on a rieighborhood of O E IRn.
(a) Choose X I ,. . . ,X, in the Lie algebra so that Xi (O) = d/dxl l o arid set
tlien tlie oi are tlie dual rorrns, and coiisequently
(b) Let q : IR' x IRn + IR^ be tlie projections. Then
'
Consequeri tly? tlie ideal 9enerated by tlie forrns d (r ox2)(11.: ox2) xl*oi
is rhe same as the ideal 9 eeiierated by the forms rn*oj-xI'oj. Using the faci
iliat tlie cjk are coirstants: SIIOMIthat d(9) c 9. Hence W" x W" is foliated b\11-dimensional manifolds oii which tlie forms d ( x i o x2) (@/ O x2) xl*oi
a H j1ar-iish .
(c) ConcIude, as in tlie prooiof Theorem 17, that for fixed a,there is a iunct~on
@, : W" + W" satisfying @,(O) = a and
n
a( b ) = n @f ( @ o ( b ) )- i q b ) .
~A-J
E
i= l
Now ser # ( a , b) = @,(b): arid use tlie converse of Lie's first fuiidamental tlieorem .
30. Lie's tliirdjiudnn?cr~i~l
11zeoren~states that tlie C;k satisfy equations (1) aiid (2)
on page 396, ;.e., tliat tlie left invariant vector fields form a Lie algebra undes
[ , 1. The aim of tliis problem is to prove tlie cu:o,zverseo f h ' s tízirdfundamental
:Acooi.ern. \vIricli states that any 17-dimerisional Lie algebra is the Lie a1gebz.a Porsorne l'ocal Lie group in a ~ieigliborhoodof O E IRX.
Let C; be constants satisfyiiig equations (1) aiid (2) on page 396. We would
like to find vector fields XI, . . . , X , on a neigl~borlioodof O E IR" such rhat
[ X i . X,] = E;=, C[ X k . Equivalently, we want to fiiid forms o' witli
Tlierl tlie 1-esultwill follow from tlie conver-se of Lie's secorid fundamental tlieoreni.
(a) Lec h4 be functions on W x IR" such that
h,k (O, X) = 0.
Tliese ai-e equatioris "deperiding on the paranieters x" (see Problern 5-5 (b)).
Notr tliat l$(r, O) = 6:r' so tliai h f ( i,O) = 6J. Let ok be tlie 1-form on W x WN
defiried b!.
and wrile
d o k = hk + (di A a k ) .
where hk and a k do not involve d t . Show that
(b) Show that
(c) Let
Show that
+ terrns not involving d f.
Using
and equation (2) on page 396, show that
+ terrns not involving di
FinaUy deduce that
dek = dt A -
E C; xjg1 + terrns not involving d t .
j,i
(d) MIe can wnie
wliere gk.f J (O. .Y) = O (Mrli!;?). Using (c), show ihai
Cor-iclude that 8' = 0.
(e) M;e 11ow ha\:(-
CHAPTER 11
EXCURSION IN THE REALM
OF ALGEBRAIC TOPOLOGY
T
liis cliapter exl~loresfurther properties of tlie de Rharn cohornolog-y \lector.
spaces of a riianifold. Our rnain results will be restaternents, in terrns of thr.
de Rharn colrornology of fundamental properties of the ordinary cohornolog?
whicli is studied 111 algebraic topologv. Because we deal only with rnanifolds.
rnar-iv of the proof3 I~ecornesignificantlv easier. On tlie other hand, we will be
usir-ig sorne of tlie riiairi tools of algebraic topology rl-ius retaining rnuch of the
flavor of rlia t subject. Along tl-ie wav \ye \vil1 deduce al1 sorts of in teresting consequences? irrcludirig a tl~eor-ernabou t tlie possil3ility of irnbeddirig ir-rnanifolds
in ~ ? l + l
Lct M be a nianiiold with M = U U 1/ ior open sets U . V C M. Before
exarniiiing tlie coliornology of M we will sirnply look at tlie vec.tor s p c e ck( M )
of k-forrns orr M. Ler
l ~ etlie inclusions. T l ~ c nwe have two linear rnaps cu and /?.
defiried b!
Her-e iw*(o)
is just the restr-iction of o to U'. etc. Clearly /? o a = & ln other
w ~ ~ 'iriiag~'
d ~ I a C ker /?. h4oreover; tlic converse liolds: ker /?C iniage a. Foq
if P ( h r , h 2 )= 0: [he11i r= A 2 on U n V. so we can define o on M to be h I
on U and h2 on 1/. arld tllen a(w)= (AI,h2). The equation irnage ar = ker /?is
expressed by sayirig tliat tlie above dia~i-arnis exact at tlie rniddle vector space.
\Ve can esrend thic diasrarn by putting tlie vector-space coritaining orily O at thc
e~ids;rl~eai.r.o\trs at eitlier crid of the following sequerlce are tlie only possilil~linear rnaps.
1. LEh/lh/lA. The sequericc
is exacr at al1 places.
I'ROOF. lt is clear tliat a is one-one. Tliis is equi\~aleritto exactness at ck( M ) .
sirice tlie irnage of the firsi rnap is (O) c ck( M ) . Sirnilarly, exactness at C k (U
11) is equi\,alerit to #l
be ir^^ onro. To projle tliar /?is onto, let (#u,
#VI be a
partiiiori ol'uriity suliordiiiate to ( U , V ) . Tlien o E c k ( U n V ) is
wl~er-e# v o denotes tlie Iorrn equal to #vo ori U n 1/, arld equal to O o11
U - ( U n V I . +>
By p u t t i r ~ir1~rlie rnaps (1. ~ - c cari
expand our diagram as follows:
so rliat rlie I-OM'S arme all exacr. lt is easy to check rhat rliis diagram cornrnuies.
tliat is, any rwo cornpositions Irorn orie vector space to anotlier are equal:
Our first rnain theor~rndeperids onlv on tlie siinple algebraic structure inherent in this d i a ~ ~ - a r nTo
. isolate this purelv al9ebr-aic structure, we rnakr*
ilie following definitions. A cornplex C is a sequence of wctor spaces C"
k = 0,1,2,. . . , togerller with a sequence of linear rnaps
satisfyilig dk+' o d A = 0: or biiefly, d 2 = O. A rnap a: CI + C2 briweeii
cornplexes is a sequerice of linear rnaps
sucli that tlie followirig diagrarn cornrnu tes for al1 k .
Tlie rnost irnportaiit exarnples of cornplexes air obtained by clioosin~C"
C' ( M ) foi. soine nianifold M, with d k tlie opei-atoi- d oii k-forins. Anotlier
exarnple, irnplicir in our discussioii, is rlic direct sum C = Cl C2 of tmio
coinplexes? defiiied 11).
l'or any cornplex C IW caii define tlie cohomology vector spaces of C by
k
(')
ker clX
dk-1
= image
Katuially, if C = (c'(M):, tlien H'(c) is just H ~ ( M ) .lf a : Ci + C2 is a
rnap betwren cornplexes. tl~eriwe have a rnap. also denored by a.
To deíiiie a wc iiotr diat e17ery clerneiit of HQC,) is deterrniiied by sornr
.Y E C? i'4tlt d d i k ( , ~
=) O. Cornrniitativity of [he above diagi-arn s1io1z.s tliat
k k
di"(s) = O, SO a k ( x ) detei-rniiies aii elernent of H ~ ( c ~ ) ) ,
d2 (a (x)) = a
~1hicl-i~ $ define.
~ e ro b r @([heclass deterinined bv A*). This rnap is ~vell-defined.
for if \ve cllange x to x + dik-' (y) for sorne y E Clk-', then olk ((x) is clianped
lo
ak(x dlk-i(B))= a k ( x ) ak(dik-'(y))
+
+
k
= a (x)
+ d2k - I ( f f k - l ( Y ) ) .
wliich deiel-iniiies tlle sanie elerneiit of H k (4).\Ylien Clk = C k ( ~ )~2~
, =
@((N). and a : C k ( M ) + C ~ N is
) f * for f :N + M, then this map is jusi
I*: H ~ ( M +
) H~(N).
h o w suppose that \\le havc an eiact seqlrence of cornplexes
ivl~iclii.eally ineaiis a vasi coinmutative diagrarn in ~+lliichall rows are exacr.
W i a i does iliis iniply aboui tlic rnaps a : HB(C1) + H P( c ~
alid
) B : H k (c?)3
Hk(C3)? Tlir iiiccsi tliiny iliat could liappeii would be for ihe followiiig diagrarn lo bc exact:
a
B
O + H k ( C 1 )+ Hk(C'2) + HL(C'3) + O .
l i s S o e . l:or csanildc, if U and V are oiei-lsppiiig poiíions of s2 IOI
wliich ilier-e is a deforrnariori i.erractiori of U flV inio S', then we llave aii exacf
sequen cc
but nol an exact sequencc
h-e\lertheless, sornetliing very nice is true:
ff
B
2. THEOREh4. lf O + CI + C2 + C3 + O is a short exact scqucncc o i
cornplexes, tlieii tliere are linear maps
so that the follo\ving infinitely long sequence is cxact (eveqwhere):
PROOE Throughout tlie proof, diagrarn (*) should be kept at hand. Let x E
C3'< with d3'<(x)= O . By esactness of tlie rniddle ro\v of (*), there is y E c:"
~ 4 t hp k ( y ) = X . Tlieii
So d2k(y) E ker pk++'= iiiiageolk+'; tlius d2k(~7)
= ak+'( i )for sorne (uniquej
r E CIk + ' . h4oreo~~ei:
Since <uk++' is one-oiie! tliis irnplies that dik+'(s) = O, so z determines an elenieilr of H k + ' (Cl ): this eleiaent is defined to be 6k of the elernent of H k ( ~ 3 )
dcrcrrnined by s.
ln order to prove tliai 6k is well-defined, ure iiiust check tliat the result does
iiot dcpend on thc choice of x E c~~
represeiiting the elernent of Hk(C3). So
\ve Iiave lo show that we obtain O E Hk+'(C1) if7we start 14th an elernent of
tlie í'orrn d3kd1(x') lor x' E G ~ - ' . 1n this case, let x' = pkdi(y'). Then
-Y = d3k-l (,U')
= dgkdl/?k-l(J~i)
= /lkd2k-1(y1),
so we clioose d2k-1(Jl') as J.. This rneans that dik(?) = 0, and hence r = 0.
I t is also rrecessanl to check tliat our defiiiitiori is independent of tlie choice
of y with /3k((i;) = x; tliá k left to the reader
Tlie proof tliat tlie sequerice ic exact consists of 6 sirnilar diagrarn chases. 14;~
will suppl:~the proof tliat ker ol c imase 6. Let x E C'ik satiSfy di k ( ~=) 0, and
suppose that ol"(x) E c~~
represclits O E kJk(c2). T h k meaiis that
=
dzkdiO') lor sorne 1)E c ~ ~
NOM-~.
So ~ " ~ ( j t ) represeiiis air elernent of kJkdi(C3).h4oreover. tlie definition of S
irnrnediately sliows that the irnay of tliis elernent under 6 is precisely tlie class
represented by A-. $4
Ir is a wortliwliile exercise to clicck that the rnain step in tlie proof of Tlieorein 8-16 is preciselv tlie proof rliat ker ol c irnage 6, togetlier with tlie first pai-t
of thc proof tl-iat 6 i s dl-defiiied. Al1 of Theoi.ern 8-16 caii he derived dir.ectl\.
fi-on-i the folloi.i.iii~corollanr of Lernrna 1 and Theorern 2.
3. THEOREh4 (THE MAX'ER-VJETORIS SEQUENCE). lf M = U U 1/:
~rliereU ar-id V are operi. tlien \ve Iiave an exact sequence (eventually endiny
in 0's):
As several of tlie Problerns sliow. tlie colioniology of ncarly everythiiig can
I J ~computed b?. a suital~leapplicaliori of thc Mayer-Vietoris sequence. As a
simple cxamplc. 1 . i ~corisider tlie torus T = S' x S r , aiid tlie opeii sets U
arid V illustrated 11elot.v. Since iliere is a deforination retractioii of U and 1'
orito cirrlcs. arid a defor-mation 1-etr-actioriof U n 1/ onto 2 circles. tlie h4averVietoris sequeiicc i:
Tlle niap H ' ( U n V) + H 2 ( T ) is iiot O (it is onto H2(T)), so its kernel is
1-dimensional. Thus the image of the rnap H 1 ( U ) H ' ( v ) + H 1 ( U f l V)
is I -dimensional. So the kernel of 11tG rnap is 1-dimensional. and consequentiv
tlie rnap H ( T ) + H' (U) $ H' ( V ) lias a l -dimensional irnage. Similar reasoning sIio~vstliat this rnap also has a l-dimensional kernel. 1t follows thar
dirn H' ( T ) = 2. The reasoning used liere can fortunarely be systematized.
'
4. PROPOSlTl ON. lf tlie sequeilcc
is exact, tlieii
PROOF. BY inductiori on k. For k = 1 we llave the sequencc-
Exactness nieans rliar (O) C VI is the keriiel of tlie rnap Vi + O, which implier
that VI = 0.
Assume tlie tlieorern for k - 1. Siiice tlle rnap V2 + V3 has kernel @(VI),it
iiiduccs a rnap V2/ac(V1) + V3. h/loreover, tl~isrnap is one-one. So we have an
exact sequcnce of k - J \lector spacei
O = d h V2/~(V1)
-dirn V3 + . . .
= - dirn VI + din? 5J2 - dim V3 + - . . .
whicli proves tlie [heoren? for k . +$.
Rarher than cor-i-ipure tI-ie col~omologvof otl-ier rnanifolds: we \vil1 use tliv
Mawr-\ktoris sequence to relate the dimensions of fik(M) to an entirel?.
dirrerent set of iiumbers. arising fiom a "tiíanplation" of M, a new structurc
which we will now define.
The standard 11-simplex An is defined as the ser
A, = x E IR""
{
: O 5 x' 5 1 and
x;f: x 1I = 1 I
'
(111 Pi.oblem 8-5. A,
is dcfined to be a diflei-er-it.altbough horneomorpliic, ser.)
Tlir siibset of A, obraiiied by settiiig 17 - k of the coordinates x' equal to O
is Iion~comorphicro A k . aiid is called a k-face of A,. lf A C M is a difleornorphic in-iage OS sorne A,, tlien tlie iii-iage of a k-face of A, is called a
k-face of A . Bow 11v a triangulation of a c-oinpact ir-manifold M we mean a
fii-iitecollection
of diffeornorpbic i m a ~ e of
s A, wliich cover M arid whicl~
satisfv the fo1lowii-i~coiiditioi-i:
lf a"! ri a';. # fl. tlieii for some k the iiltersection uniri anjis a k-lacc
of both aniar-id a", .
1t is a dficult ~heorerrirlrat evenl CK rnailifold has a triai-iplarion; for a
proof see Muiikres. E/crmnncn/o~~~
~ i j i r m ; i a %/~o/ogy.
/
or IYhitney, Gonrelnk lniqrolion
T l l e o ~ .Assurning: that our manifold M has a tr-iangulation { u n i )we will cal1
each uni an 77-simplex of the trianplation; any k-face of any un;will be caIIed
a k-simplex of the triangulation, MJe Iet ark be tbe nurnber of these k-sirnplexes.
Kow let U be the disjoin union of open balls, one within each 71-sirnplex uni:
and let Vndl be the cornplernent oftlle set consisting of the centers of these balls,
so that V,-1 is a nei~hborlioodof the union of al1 (n - 1)-sirnplexes of M. Tlien
M = U U l/,dl where U f~V,-l lias the sarne cobornology as a disjoint union
of an copies of S"-]. Consider first [he case where 11 > 2. The Mayer-jiietoi-is
sequence breaks into pieces:
(2)
For 1 < k i17 - 1.
Applyins Proposition 4 to these pieces vieldr;
O ~ k ~ r z - 2
dim Hk(Vn-1) = dim hJk (M)
dim ~ " - ' ( l / , - ~ )= dim H"-'(M) - dim H n ( M ) + a..
For the case n = 2 \ve easily obtain rl-ie same result ~ i t h o u splitting
t
up tlie
sequence. \4;e now ii~troducethe Euler characteristic x ( M ) of M, defiiíed b!l
X ( M )= dim H'(M) - dim H ' ( M )
+ d i m ~ ~ ( -M .). . + (-1)" dirn H"(M).
This makes sense for ano manifold in ~lliichal1 hJk(M) are finite dimensional;
we aiiticipate Iiere a later result that H k ( M ) is finite dimensional wlieiiever M
is compact. Tlíe above equarions tl~enimply thar
~ ( " ~ - 1 ) = x(-l)k
dim H ~ ( V , ~ - , )
k=o
dim h J k ( ~ )
=
+ (-1)"-'[dim Hn-'(M) - dim H n ( M )+ u,]
5. THEOREM. For any triangularion of a compact manifold M we llavc
I'ROOF. Iii tlle mailifold V,-
we dehile a ilew opeil set U wllicll coiisists of
a disjoint unioií of sets diffeornosphic ro IR"?oi-ie for ead-i ( n - 1 )-face, joining
the balls of the old U.
componeiits of
iiew U (n 1= 3)
-
componenrs of
new U (17
3)
M'c \vil1 let 1',,-2 be t11e cornplerne~~r
oral-cs: in t11e ilew U , joining ihe ceniers
of the balls in the old U .
Vn-2 js ~ l i r
complemenr of
1;,-2
js the
complement of
An argurneilt precisely like thar wIlic11 proves the equation
Similady \ve iilrroduce Vt1-3,.. ., VO;the last of r11ese is a disjoinr uilion of a0
sers eacll of ~~hic11
is srnoothly conrractil~leto a point. Hence x ( V o ) = cwo, w11iIe
in al1 orl~ercases we have
Cornbining these equations, we havc
6. COROLLAR17 (DESCARTES-EULER). If a convex polyll edron has V
vei-rices, E tdges, and F Paces, r l ~ e i ~
lf we turn frorn HQO H: we encounter a very dfiereni situation. If U c M
is open, a iorrn o wirli cornpacr supporr C M rnay 1101 restrict to a forrn wirli
compact supporr c U : rhe inclusion rnap of U inro M is nor proper. O n r 1 1 ~
other hand, if o is a form wirh compacr suppori c U: tlien o can be exteiided
ro M by lerting ir be O ourside U; we will deiiore rhis extended íorrn 131-
If c!(M)
dciioies tlie vcctor space of k-fornls wiili compact suppori on M, ivc
can define a iie~vsequciicc..
is exact.
PROOF Ii is clear iliar jw'@ -jv' is one-one; iii faci, each rnap ju' and jr~'
is oile-one.
To pr-ove tliat i v ' + i y r is onio, ler w l ~ ac k-fhrrn wirh cornpact supgort on M,
aiid let (#u,#v: hc a partirion of unity foi. rhe cover (U, V I . Tlie~í
is clearly ilie i n i a p of ( # o o , # v ~ )E C;(U) @ Cck ( v ) .
Ir is clear that iiiiage (jw'
@ -jv')c ker(io' + iV8).To prove rlie coiiversc,
suppose rliat
(Al, A l ) E C: ( U ) @ c:( V )
satisfies i U ' ( l i )
+ iv'(A2) = 0.
This ilicaiis tliat ii= - k 2 . Siilcc supporr A l c U aiid suppori A;! C U. tliic
slinws tl~arsiipposi iL1c U n V aíid siipporr h2 C U n V. So (A1 ,Al) is tl1r
iiiia~eof A l E C/(U n V ) +3
8. THEOREhI (MAYER-VIETORIS FOR COMPACT SUPPORTS). If ihe
manifold M = U U V for U , V open in M I rlieii there is a long exact sequence
PROOE Apply Tlieorern 2 io tlie sllort exaci sequence of cornplexes given by
t l ~ eLernrna. +3
Tllis sequence is rnuch h arder to work ~YltllI han the M ayer-Vietoris sequence.
For exarnple, suppose vire waiit to fiiid H,k for W" -(O]? wliich is dfleornorphic to
x B. Ifwr write S" = UUV in the usual way. so iliat U n V is diffeornorpliic
ro S"-' x IR: rllen S" x W = (U x W) U (If x W), whei8e (U x B) n ( V x B)
is diffeornorphic to SN-Ix B2. The oiily i*ay to use iiiduction is io find H/
for al1 S" x R", stariiiig xzitli S' x IRm. The details will be left to the reader;
\ve ~411rnerely recosd one furrlier i-esult! for latei- use. aiid tlieil proceed ro yet
aiior 11es application of Tl~eorern2.
9. COROLLARXt, If M = U U V for U , V opeii i i i M ! tlien rkei-e is a dual
loiig exact sequeiicc
PROOE M7ejusr have to show t11at if tlie sequeiice of linear rnapc
is exaci at W2.theii SO is tlie sequeilce of dual rnaps aiid space5
For aiiy h E W3* we llave
SOff* o p * = 0.
h'ow suppose A E W2* satisfies a * ( h ) = O. Tlien h o ar = O. We clairn tl~at
tliere is 1: MI? + IR ~ i i l hi = #j*(h),i.e., A = #j o h. Giveii a w E W3 wliich is
of tlie forrn / ? ( w l )we
: define
i ( w ) = A(/?').
Tliis rnakes sense, for if / ? ( w l )= #I(w"), then w - w" = a(,-)
for some r ? so
A{w) - A{wM)= Acv(s) = O. Tliis defines h oii #?(W2)c W3. Now clioos<
W c lV3 wiili W3 = /?(Wi)@ W. and define h 10 be O on U[.
+
M;e I ~ O M ?co~isidera i-arlier differenr situation. Let N C M be a comnpaci subrnaiiifold of M. Then M - N is also a rnaiiifold. Mre r 11 erefore llave the sequencc
wliere c is "exteiision". Tliis scquence is tioi exact ai C$ (M): the kernel of i'
coiitaiiis al1 o E c , ~ ( M ivhicli
)
are O on N , wliile rlie iiriage of e contains al1
o E C: ( M ) ~sliicliare O in a iieigliborhood of hr.
To circumirent rhis clifficulty, ive will llave to use a technical devicc. M+
appeal first ro a resulr fi-om rhe Addeiidum to Chaprer 9. There is a compacr
iiei~hboi~hood
V of N and a map x : V + N such rhat V is a manifold-wit11boundar).; aíid if j : N + V is tlie inclusion, then rr o j is the idenrity of n'.
~vliilej o rr is srnoor hly homotopic ro tlie idenriry of V. ! e now coiistruct
scquence o i sucli neighborhoods V = V I 5i V2 5i V3 5i
with
Vi = A'-
ni
KOWronsider ~ W Oforms ~i E ck(Vi),
W j E ck( V j ) . \Ve will cal1 wi atid w j
agirinalemii if rliere is I > i , j sucli tliar
It is clear that we can make rlie set of al1 equivalence classes into a vector sparc.
$"N)? the "gerrns of k-forins in a iieighborliood of M". Moreoirer, it is easy
[o define d : P ~ ( N +
) g k + ' ( N ) , so tliat we obtain a complex $. Finally. wi
define a map of cornplexes
i i i tlle o l ~ i ~ i ulajr:
o ~ s w I+
tlle equivalence class o í aily o1 V i .
1 0. LEAtMA. The sequence
e
i*
O + C;(M - N ) -t C ; ( M ) -t $ l k ( N ) + O
IS exact.
PROOF. Clearly e is one-one.
If o E c:(M - N ) . then o = O in some neighborliood U of N . Since N is
compact and
Vi = N , there is some i sucli tliat Vi c U, arid consequeritl!.
o = O on l/i. Tliis iiieans tliat i * e ( o ) = O. Co~i~fersely,
suppose h E c ) ( M )
satisfies i*( A ) = O. By defiiiition of $lk ( N ) ,this means tliat h 1 Vi = O for some i .
Hence hl M - N lias conipact support c M - N , and h = e(hlM - N ) .
Fiiially, any eleme~itof $lk ( N ) is represented by a form q on sorne V,. Let
.f : M + [O, 11 be a Cm function wliicl~is 1 on I/i+l, having support f c
interior l/i. Then f q E c!(M ) , and f q represents tlie same element of $ l k ( N )
as 17; conseque~itlytliis element is i* ( f q ) . 4 3
nl
11. LEMMA. Tlie coliomology vector spaces ~ ' ( 8of) the complex { $ k ( N ) )
are isomor~hicto H k ( N ) for al1 k .
PROOF, Tliis follows easily from the fact tliai j* : H' (vi)+- I f k ( N ) is aii
isomorpliism for eacli lfi. Details are left to tlie reader. 43
12. THEOREM (THE EXACT SEQUENCE 01:A PAIR). If N c M is a
coinpact submaiiifold of M , then there is an exact sequence
. .. ,
H:(M
- N ) + H;(M)
-
6
H~(N)
+
H,k + 1 ( ~- N ) ,
.-
PROOF Apply Theor-em 2 to tlie exact sequence of complexcs given by Lemma
10, and theii use Lemnia 11. *t.
In tlie proof of tliis tlieorem, the de Rliam coliomology of the manifold-witlil ~ o u r i d a r1/1
~ entered oiily as an intermediar? (aiid we could have re~laced
tlie Vi by tlieir iiitei-iors). But in the next tlieorem, whjch we will need later, it
is the object of primary interest.
13. THEOREM. Let M be a manifold-witli-bouiidaiy, with compact boundary aM.Tlien there is aii exact sequence
PROOF, Just like the proof of Theorem 12, using tubular nejghborhoods I/i of
aM in M. *t*
As a simple application of Tlieorem 13, we can rederive H;(Wn) from a
kiio\vledge of Hk(Sn-'), by clioosiiig M to be the closed ball 3 in R", witli
H: ( 3 )= H A( B ) = O for k # O. The reader may use Tlieorem 12 to computi
H:(s"
x IRm), by considering tlie pair (Snx Rm,{ p ) x IRm). Then Tlieorem 13 may be used to compute tlie cohomology of S" x Sm-' = a(Snx
closed ball iii Rm). For oui- iiext applicatioii we will seek bigger game.
Let M c R"+' be a rompart 11-dimensional submanifold of IRn+' (a compact "hypers~irface"of IRn+'). Using Tlieoi-em 8-17, the sequence of the pair
(IRn+',M ) @ves
It follo\vs tliai
($1
iiumber of compoiients of IRn+'
- M = dim H n ( M )+ 1.
But we also know (Problem 8-25) tliai
(**>
iiumber of compoiielits of IR"+'- M 2 2.
14. THEOREM. If M c Rn+' is a compact hypersurface, then M is orientable, aiid Rn+' - M has esactly 2 coinpoi-ients. h/loreover, M is the bounday
of each componeni.
PROOF. From ($1 and (**) we obtain
+ 1 1.2.
dim H n ( M )
Sii-ice dim U n (M ) is eitlier O or 1, we conclude tliat dim U n ( M ) = 1, so M is
01-ieiitable;then (*) sliows tliai IR"+' - M has exactly two compoiienrs. Tlie
proof in Problem 8-25 shows tliat every point of M is arbitrarily close to points
iii diflerent componeiits of IRn+' - M, so every point of M is in the boundary
of eacb of the m components. 4.
15. COROLLARJ7(GENERALIZED [Cm]JORDAN CURVE THEOREM).
If M c Rn+' is a submaiiifold liomeomorpliic to S", then Rn+' - M has two
coinpoilents, and M is tlle boundary of eacli.
16. COROLLARY. Ncitlier the projective plaiie nor tlle KJein bottle can be
jmbedded in R3.
Our iiext inain i-esult will combine some of tlie tlieorems we ali-eady have.
However, tl-iere are a nuiill~erof teclinjcalities iiwolved, wliicli we will llave ro
dispose of first.
Coiisidei- a boui-ided opei-i set U c R" wllicli is star-sllaped with respect to O.
Tlien U can be described a s
U = ( t s : A- E S"-' and O 5 t < p(x)j
for- a ceriain fuiiction p : S"-' + R. \Ve will cal1 p the radial function of U.
If p is C m , tlien we can prove that U is diffcomoi-pliic to the opeii ball B of
radius 1 in R". The basic idea of the p o o f is to take tx- E B to p(x)t A- E U.
Tliis produces difficulties at O, so a modification is necessary.
17. LEMMA. If the radial function p of a star-shaped open set U C Rn is Cm?
tlieii U is difTeomorpliic to tl-ie open ball B of radius 1 in R".
PROOF. \Ve can aassume: ~ i t l i o u tloss of generality, tliat p > 1 on S"-'. k t
f : [O, 11 + [O, 11 be a Cm function with
f = O i1-i a neighborhood of O
f' >
-o
fI1) = l .
Define h : B + U b\:
Clearly h is a one-oiie n-iap of B oiito U. It is tl-ie identity in a neigl~borhood
of O, so it is Cm. \vitli a non-zero Jacobian, at O. At any other point the same
coiiclusioil follo\vs from tlie fact tliat t H t + Ip(x-)- 1 ) f ( t ) is a Cm functio~i
witll strictly positive derivative. 4 3
111general, rlie fiiiiction p i-ieed i-iot be Cm; it miglit not even be coiitinuous.
However: tlle discontinuities of p can be of a certain form oi~l!:
18. LEhdhlA. At each point s E S"-'. tlie radial f~inctioiip of a stai--shaped
opeii set U c R" is "lower semi-contiiluous": for every E > O tliere is a neighborl~oodW of s i r i S"-' sucli tl-iat p ( y ) > p I x ) - E for al1 E .'M
PROOF. Choose rx E U witl-i p ( x ) - t < E . Since U is open, tliere is an open
l~allB with s E B c U. There is clearlv a neighborl~oodW of s witli tlie
p ~ ' o ~ e r tllat
t y foi- J . E W tlle point t j q is in B. ancl Iicnce iri U,Tllis means thar
for. J. E W \ve have p o . ) 2 t > p ( x ) - E . *e:
Even when p is discontinuous, it looks as if U should be difTeomorphic to R".
Proving this turns out to be quite a feat, and we will be content with proving
the following.
19. LEMAJA. If U is an open star-shaped set in Rn, then U k ( u ) =Z Hk(R")
and U/ (U) U:(Rn) for al1 k.
PROOF. Tlie proof for U k is clear, since U is smoothly contractible to a point.
\Ve also know t11at U," [ U ) -" R =Z H:[JRn). By Theorem 8-17, we just have to
show that H ~ ( u )= O for O 5 k < ir.
Let w be a closed k-form with compact support K c U. \Ve claim that there
is a Cm function b : S"-' + JR such that < p and
K c V = {rx : x E Sn-' and O _< t < ~ I x ) ] .
This will prove the Len-irna, for then V is dfleoinorpl-iic to Rn,and consequentl!.
w = dq \vlier.e q has compact support contained in V, and hence in U.
For each x E S"-', choose lx< p(x) sucl~tliat al1 points in K of the form u x
for O 5 u 5 p(x) actually llave u < 1,. Since K is closed and p is lower semi-
continuous: tliei-e is a neighborhood W, of x iii S"-' sucli that t, may also
lie used as 1,. for al1 y E W. Let M~,,,.. . ,W, cover S"-', let d i , .. .,#r be a
pai.tition of unity subordiriate to tliis cover, and define
Any poini A. E S"-' is in a certain siibcollection of t l ~ eWxi, say W,, ,. . . ,W,,
for com~eilience. Tl-ien p/+r [x), . . .,pr [A-) are 0. Each rxl ,. . . ,fx, is < p(x).
Siiice #](A-) . $1 (x) = 1. it follows tl-iat b(x) < p [ x ) . Similarly, K C V. 43
+ +
\Ve can applv this last Lemma in the following way. Let M be a compact
manifold: and clioose a Riemannian metric for M. According to Problem 9-32,
every point has a neighborhood U which is geodesically convex; we can also
choose U so tliat for any p E U tlie map exp, takes an open subset of Mp
diffe~mor~hically
onto U. k t {U', . . . , U,) be a finite cover by such open sets.
If any V = Uil n. n Uil is non-empty, then V is clearly geodesically convex. If
p E V, then expp establishes a diffeorn~r~hism
of V with an open star-shaped
set in Mp. 11 follows from k m m a 19 tliat V has the same HP and H: as Rn.
In general, a manifold M will be called of finite type if there is a finite cover
{UI,.. . , U,) such that each non-empty intersection has the same Hk and H/
as Rn;such a caver will be called nice.
It is fairly elear that if we consider N = {1,2,3,.. . ) as a subset of R2, then
M = R2 - N is not of finite type. To prove this rigorously, \*.e fin<use thr
Mayer-Vietoris sequence for W2 = M U V, wliere V is a disjoint union of balls
around 1,2,3,. . . . We obtain
where M n 1' ]>astlie same H' as a disjoint union of infinitely many copies
of S'; tliis sliows that H 1 ( M )is infinite dimensional (see Roblem 7 for more
infoi.niation about tlle cohomology of M). O n the other hand:
20. PROMSITION. If M has fiiiite type, then HP[ M ) and H;[M) are finite
dimensional for al1 k .
PROOF. BY iriduction on tlle number- of open sets i. in a ~iicccover; It is
clear for r = l . Suppose it is true for a certain r , and consider a nice cover.
{U', . . .,U,, U] of M. T l ~ e ntlle tlleor-em is true for V = Ur U . - .U U, and
for U. It is also true for UnV, sii-ice thisl~asthe nicecover { U n U i , . . .,UnU;j.
Now consider the Mayer-Vietoris sequence
The map o maps H k ( M ) onto a finite dimensional vector space, and the kernel
of o is also finite dimensional. So H k ( M ) must be finite dimensional.
Tlie pioof for H: (M) is similar,
For ai-iy mariifold M we can define (see Problem 8-31) t l ~ ecup product map
w
H ~ ( M x) H / ( M ) -+ H ~ + I ( M )
by
We can also define
by tlie same formula, since w A q Iias compact support if q does. Now suppose
that M" is connected a r ~ doriented: witli orientation ,u. Tliere is tlien a uriiqur
element of H:(M) repi-esented by any q E C , ( M ) witli
It is comler-iient to also use ,u to denote both tliis element of H:(M) and the
isomorphism H:(M) + W which takes this element to 1 E R. Now every
a E H" M) determines an element of the dual space H:-~ ( M)* by
We denote this clement of H:-'(M)'
that we llave a map
by P ! ( o ) , the "Poincaré duai" of o, so
One of tlle fundamental tlreorems of manifold tl~eorystates that P ! is always
an isomorpllisrn. We are al1 set up to prove this fact, but we shail restrict
tlle tl~eoi-emto manifolds of finite type, in order not to plague ourselves with
additional tecl~rlicaldetails. As witl~most big tl-ieorems of algebraic topology,
tlle maiil part of the p o o f is called a Lemma, and the theorem itself is a simple
corollan:
21. LEMMA. If M = U U V for open sets U and V and PD is an isomorpllism
for al1 k on U, V: and U n V: then PD is also an isomorphism for al1 k on M.
PROOF. Let 1 = I r - k. Consider the following diagram, in wl~icht l ~ etop row
is the Mayer-Vietor-is sequence, and the bottom row is the dual of the Mayer-
Vietoris sequence for compact supports.
By assuinption, al1 vertical maps, except possibly tlle middle one, are isomor-
pllisms. It is not hard to check (Problem 8) that every square in this diagram
commutes up to sign, so tllat by changing some of tlle vertical isomor.pllisins
to tlleir llegatives, we obtain a commutative diagiam. We now forget al1 abour
our manifold and use a purely algebraic result .
"TH E Fl VE LEh/lMA". Consider tlle followil~gcommutative diagram of vector spaces and linear- i-ilaps. Suppose that the rows are exact, and tllat #i, $2.
are isomorpl~isms.Then #3 is also ail isomorpliism.
44,
PROOF. Supposc 4 3 ( ~=)O for some x E V3. Tllcn P3#3 (x) = O, so #4a3(x) =
O. Hence a 3 ( x ) = O, sirlce #4 is an isomorpl-iism. By exactness at V3: tllere i5
1. E V2 witl-i A- = a2(y). Thus O = # 3 ( ~ - ) = #3a2 (y) = P2#2 (y). Hencc
#2()*) = r91 ( z ) for some 5 E Wl . Moreover, := 41 (w) for some w E VI. Then
wllicll implies that y = al (w). Hence
So $3 is one-onc.
Tllc ~ i ' o o tl-iat
f
#3 is onto is similar, and is left ro the reader. This proves thr,
original Lemma.
22. THEOREM (THE POINCARE DUALITY THEOREM). ~f M is a
connected or-iented n-manifold of finite type, then the map
PD: H ~ ( M +
) H:-~(M)*
is an isomorpllism for ail k.
PROOE By induction on tl-ie number I. of open sets in a nice cover of M. The
theorem is clearly true for r = 1. Suppose it ic true for a certain r, and consider
a nice cover {Ui,.. .,U,, U) of M. Let V = U1 U . U U,. The theorem is true
for U, V, and for U n V (as in the proof of Proposition 19). By the Lemma, it
is true for M. Tllis completes the induction step. 4%
23. COROLLARY. If M is a connected oriented n-manifold of finite type,
tllen H k ( M ) and N:-'(M)
have the same dimension.
PROOE Use tlle Theoren~and fioposition 19, noting that V* is isomorphic
to V if V is finite dimensional. 4%
Even though tlie Poincaré Duaiity Tlleorem holds for manifolds which are not
of finite type, Coi-ollary 23 does not. In fact, Problem 7 sliows that N i (IR2 - N)
and H,' (IR2 - N) have dúrerent (infinite) dimensions.
24. COROLLARY. If M is a compact corlr-iected orientable n-manifold, then
H k (M) and H " - ~( M ) have the same dimension.
25. COROLLARY. If M is a compact or-ierltable odd-dimensional manifold,
then x ( M ) = 0.
PROOF. In the expression for x ( M ) , tlle terms ( - 1 1 ~dim H'(M) aiid
cancel in pairs. ++:
A more iinrolved use of Poirlcaré duality will aleritually allow us to say much
more about tlle Euler characteristic of any compact connected oriented manifold M''. ]Ve begirl by consideiing a smootll k-dimensional orientable vector
bundle 6 = x : E + M over M. Orientatjons p for M and v for 4 give an
orientation p v for the (n + k)-mar~ifoldE. since E is locally a product. If
{Ui,.. . , U, 3 is a i~icecover of M by geodesically convex sets so small that each
bundle cIUi is ti-ivial: thei~a slight modification of the proof for Lemma 19
sliows that {ñ-' ( U ] ) ., . ., ñ l 1(U,)] is a nice cover of E , so E is a manifold of
finite type. Notice also that for the maps
ño s =
identity of M
s oñ
is smootllly homotopic to identity of E,
so ñ*: H ' ( M ) + H ' ( E ) is an isornorphism for al1 I . The hincaré dualih
tl~eoremshows that there is a unique class U E H ~ ( Esuch
) that
This class U is called tlle Thom class of t. Our first goal will be to find a
simpler property to characterize U.
Let Fp = ñ-' ( p )be the fibre of 6 over any point p E M , and let jp : Fp + E
be the inclusion map. Silice jp is proper, there is an element jp*U E H:(Fp).
0 1 1 the other hand, the orientation v for
determines an orientation vp for Fp:
and hence an element vp E H: ( F ~ ) .
<
26. THEOREM. liet (M, p ) be a compact connected oriented manifold, and
.f = 7~ : E
M an orienied k-plane bundle over M with orientation v. Theil
the Thom class U is the unique elernent of H , ~ ( Ewith
) the property that for
al1 p E M we have jpSU = vp. (This condition means that
wl~ereU is the class of the closed form o.)
PROOF. Pick some closed form o E C: ( E ) re~resentingU , and let r) E C n( M )
be a form representing F , so tllat
q = 1. OUTdefinition of U states that
Let A c M be an open set which is dfleomorphic to IRn, so that A is smoothly
conii-actible to any poini p E A . Also choose A so that there is an equivalence
This equivalente allows us to identify IT-' ( A )with A x Fp. Undes this identification, the map jp : Fp + n-' ( A ) corresponds to the map e H ( p ,e ) for e E Fp:
which we will continue to denote by jp. We \vil1 also use nz : A x Fp + Fp to
denote psojection on the second factor.
Let 11 11 be a norm on Fp. By choosing a smaller A if necessary, we can
assume that tl~ereis some K r O such that: ui~derthe identjfication of x - ' ( A )
with A x Fp, the support of wlrr-' ( A ) is coiitained in { ( q ,e ) : q E A , lle 11 < K j .
,support w
Usir-ig the fact that A is smoothly contractible to p, it is easy to see that there
is a smooth liomotopy U : ( A x Fp) x [O, 11 + A x Fp such that
we just pul1 tlie fibres aloiig the smootli liomoiopy which makes A contractible
to e . k r the H constructed in this way it follows t11ar
H ( e , r ) # support w if Ilell 2 K .
Consequentlx tlie form H*w on ( A x Fp) x [O, 11 has support contained in
{ [ q ,e , r ) : lle 1 < K ) . A qlance at the definition of 1 (page 224) shows that the
L.
form lH*w on A x Fp has support contained in {(q,e) : llell < K > . Theorem 7-14 sllows tllat
Now, on tlle one hand we have (Problem 8-1 7)
O n tlie otller llaild, we claim tllat tl-ie last integral in (3) is O. To prove tllis, i i
~.learlysuffices to prove tllat tlle integral is O over A' x Fp for any closed ball
A' c A . Since
x*p A cih = fd ( x * p A dh).
(5)
J Fp x*p ~ d =h S Fp d ( x * p ~ h )
A' x
A' x
a'pnh
=f
where n*p A A has
compactsupporton
A' x Fp by (2)
bv Stokes' Tl-ieorem
a A ' x Fl,
= o.
becaiisc tlle form x*p A h is cleai-ly O on a A' x Fp (sincc a A' is (n - 1)-dimeiisional).
Conlbii~ing(3), (41, (5) we see tllai
This shows that JF jpiois independent of p, for p E A . Using connectedness.
P
it is easy to see that it is inde~endentof p for al1 p E M, so we will denote it
simply by JF j'o. Thus
Con-iparing with equation (l), and utilizing partitions of u n i g we conclude tl-iat
wliich proves t l ~ efirst part of tlie theorem.
Now suppose we llave another class U' E H,k (E). Since
it follows tliat U' = cU for some c E R. Consequently,
Hence U' l ~ a sthe sarne propcrty as U only if c = 1. 4 3
The Tl-iom class U of $ = n : E + M can now be used to determine an
element of Hk(M). Let S : M + E be any section; tliere always is one (namel):
i11e O-sectior-i)and ar-iv two are clearly smoothly liomotopic. We define t l ~ Euler
e
class ~ ( $ E1 H k ( ~of) 6 h\.
R'otice that if 6 has a non-zero section S: M + E, and o E C;(E) repr-esents U, then a suitable multiple c . s of S takes M to the complement of
support o. Hence, in tllis case
T l ~ termiiiology
e
"Euler class" is connected witli the special case of the bundle
TM, wliose sections are, of course, vector fields ori M. If X is a vector field
on M wl-iich has aii isolaced O at some point p (tl~atis, X ( p ) = 0, but X(q) # O
for q # p iri a rieighborl-iood of p), tl~en,quite independently of our- previous
considei.atioi~s,\ve can define an "ii~dex" of X at p. Consider first a vectoi-
field X on an open set U c Wn with an isolared zero at O E U. We can define
a function fx: U - {O) + S"-' by fx(p) = X(p)/lX(p)l. If i: S"-' + U
is i(p) = EP, mapping S"-' into U, then the map fx o i : S"-' + S"-' has a
certain degree; it is independent of E, for small E , since the maps i ~ i2
, : S"-' +
U corresponding to E] and ~2 will be smoothly homotopic. This degree is called
the index of X at O.
index O
index O
index
-1
index 1
index I
index -2
index 2
iildex 1 in R"
index (-1)"
in Rn
Now consider a diffeom~r~hism
8 : U + V c Wn with h(0) = O. Recall thar
h,X is the vector field on V with
Clearly O is also an isolated zero of h , X
27. LEMMA. If h : U + V c R" is a diffeomorpllism with hIO) = 0, and X
has ai-i isolated O at O, tllen tlle ii-idex of Ir,X at O equals the index of X at O.
PROOF. Suppose first that h is orientation preserving. Define
U : IR" x [0,1]+ Rn
This is a smooth homotopy; to prove that it is smooth at O we use Lemma 3-2
(compare Problem 3-32). Each map U, = A- t+ H I x , t ) is clearly a dfleomorphism, O < r < 1. Note that U i E SO(n), since h is orientation preserving.
There is also a smooth homotopy {Ut], 1 _< I < 2 with each Ht E SO(n) and
Hz = identity, since SO(n) is connected. So (see koblem 8-25), tlle map h is
smoothly llomotopic to the identity, via maps wliich are dfieomorphisms. This
sliows that fh,x is smoothly homotopic to fx on a sufficiently small region of
IRn - { O ] . Hence the degree of _fh,x o i is the same as the degree of fx o i.
To deal witll non-orientation preseniing h, it obviously suffices to check the
theorem for h ( x ) = ( x l , .. . ,xn-l, -xn). In this case
which shows that degree h
, i~= degree fx o i. 4 3
O
As a consequence of Lemma 27, we can now define tlie iiidex of a vector field
on a manifold. If X is a vector field on a manifold M , with an isolated zero at
p E M , we clloose a coordinate system ( x ,U ) with x ( p ) = 0 , and define the
index of X at p to be the index of x,X at O.
28. THEOREM. Let M be a compact connected manifold with an orientation F , whicll is, by definition, also an orientation for the taiigent bundle
6 = IT: TM + M . Let X : M + T M be a vector field with only a finite
iiumber of zeros, and let a be the sum of the indices of X at these zeros. Then
PROOF. k t pl ,.. . , p, be t he zeros of X. Clloose disjoint coordiiiate systems
( U ] x, l ) , . . . ,(Ur,x r ) with xi(pi) = 0, and let
If w E CF(E) is a closed form representiilg the Thom class U of $, then we
are trying to prove that
f
\Ve can clearly suppose tliat X(q) # Support w for q # Ui Bi. So
tlius it suffices to prove that
X *(u)= index of X at pi.
(*)
It will be conueilient to drop the subscript i from now on.
We can assume tliat T M is trivial over 3: so that n f l (B) can be identified
with B x M,. Let j p aiid n2 llave tlie same meaning as in the proof of Tlieorem 26. Also clioose a iiorm 11 11 on M p . We can assume tllat uiider tlie
identification of n-' (3)~ i t hB x Mp, the su1ipo1-1of wln-' (3)is contained
iii {(y,u) : y E A , llvll 5 11. Recall fiom tlle proof of Theorem 26 tliat
Since we can assume tliat X(q)
support h for y E 8 3 , we llave
O n tlle maiiifold Mp we llave
jp* w = dp
p aii [ I T - 1 )-form oii Mp
(witll non-compact support).
If D c Mp is tlic unii disc (witli respect to tlle iiorni 11 11) and S"-' deiiotes
8D c M,, tliei-r
- 1.
-
by Tlieorem 26, and the fact
tllat support jP*w C D.
Now, for q E B - {p}, we can define
and f :aB + T M is smoothly homotopic to X : al3 + T M . So
(31
S, x*x2*(jp*u) J, x * x 2 * d p
=
X*x2*p
by Stokes' Theorem
From tlle defiilition of tlle iildex of a vector field, together with equation (2), it
follows tllat
( x 2 0 Z * ) p = index of X at p.
L B
Equations (1), (31, (4) togetlier imply (*). 43
29. COROLLARJ7. If X aiid Y are two vector fields witli only finitely many
zems on a compact orieiitable maiiifold?tllen the sum of the indices of X equals
tlie sum oi' tlie indices of Y.
At tlie monicilt, we do not even kilow that there is a vector field on M with
finitely nianv zeros, iloi- do we know what this constant sum of the indices is
(althougli our terniiilology certainly suggests a good guess). To resolve these
questions, we consider once again a triangulation of M . MJe can then find a
vector field X witll just oile zero in each k-simplex of the triangulation. We
begin by drawing tlie integral curves of X along the 1 -simplexes, with a zero at
cacli O-simplex and at oiie point in eacll 1-siinplex. \)Ve then extend this picture
to ir~cludetlie integral cunres of X on the 2-simplexes, producing a zero at oní
point in cach of tliem. We tlien continue similarly until tlie n-simplexes are
filled.
30. THEOREM (POINCARÉ-HOPQ. Tlie sum of tlie indices of t his vectoi.
field (and Iience of aiiy vector field) on M is tlie Euler characteristic X( M ) .
Tlius, for $ = n : TM + M we have x($) = x ( M ) p .
PROOF. At each O-simplex of tlie triangulation, the vector field looks like
with index 1.
Now coiisider tlle \lector field in a iieigliborhood of tlie place where it is zero
on a 1-simplex. The vector field looks like a vector field on W" = W 1 x IR"-'
wliicli points djrecOy iiiwairls on W 1 x {O) and directly outwards on {O) x W"-' .
t
(b) j? = 3
For 11 = 2, the index is clearly - l . To compute the index in general, we note
that fx takes the "north pole" N = (0, . .. ,O, 1) to itself and no other point
qoes to N. By Theorem 8-12 we just have to compute signN fx. NOWat N we
can pick projection on Wn-' x (O) as the coordinate system. Along the inversc
image of the x'-axis the vector field looks exacily like figure (a) above, where w
already know the degree is - 1, so fx. takes the subspace of S"-'N consisting of
tangent vectors to this cuive into the same subspace, in an orientation reversing
way. Along the inverse image of the x2-, . . . ,xn-'-axes the vector field looks likc
L
so fx. iakes the correspoiiding subpaces of S n - I N into themselves in an orientation preservjng way Thus signN fx = -1: which is tlierefore the index of
the vector field.
In general, aear a ze1.o within a k-simplex, X looks like a vector field on
W" = IRk x Wn-hrliicli points directly inlvards on JRk x (0) and directly outwaids
on (O) x IRn-'. The same argument shows that the index is ( - 1 )k.
Consequently, the sum of the indices is
a0
- al + a2 -
. = X ( M). 43
\Ve end tliis chapter with one more obsenlation, which we will need in the
last chapter of Volume V ! Let 6 = n : E + M be a smooth oriented k-plane
bundle over a compact conliected oriented 12-manifold M, and let ( , ) be a
Riemannian metric for 6 . Then we can form tlie "associated disc bundle" and
"associated sphere bundle"
D = (e : (e,e) 5 1 )
S = ( e : (e,e) = 1 ) .
+
It is easy to see tl-iat D is a compact oriented (iz k)-manifold, wjth aD = S;
mor.eover, tlie D constructed for any other Rieniannian metric is diffeorn~r~liic
to this one. We Iet no ; S + M be 171s.
31. THEOREM. A class a E H k ( ~satisfies
)
no*(a)= O if and only if a is a
multiple of ~ ( 6 ) .
PROOF. Consider the following picture. The top row is the exact sequence
Hk(M)
for ( D ,S ) @ven by Theorem 13. The map S : M +- D - S is the O-section:
while i: M + D is tlie same O-section. Note that everything commutes.
no*= i* o ( n1 D)*
S* = S* o C)
since no = ( n I D )o i;
since extending a form to D
does not affect its value on S ( M ) ,
and that
S* o ( n l D ) * = identity of H k ( M ) ,
since [n1 D ) o i is smootlily homotopic to the identity
Now lei a E H ~ ( Msatisfy
)
no*(a)= O. Then i*(n1 D)*a = O, so (nID)'a E
iinage L.. Since D - S is diffeomorpl~icto E, aiid every element of H: ( D - S )
is a niultiple of the TIiom class U of t, we conclude that
Tlie proof of the coiiverst: is similar.
Iixirn:rio?i i?r lhr. Ren hn of. A Lyth?uic 7Új)ologl.
C
PROBLEMS
1 . Find Hk(S' x .
dim H' = (;).]
x S') by induction on the number rr offactors. [Answei.:
2. (a) Use the Mayer-Vieioris sequence to determine Hk( M - { p ) )in terms
of H k ( ~ )for
, a connected manifold M.
(b) If M and N are two connected 11-manifolds, let M # N be obtained by
joining M and N as shown below. Find the c o h o m o l o ~of M # N in terms of
that of M and lb'.
(c) find x for ihe rt-holed torus. [Answer: 2 - 21n.J
3. (a) Find ~ ~ ( A 4 o b i strip).
us
(b) Find Hk(lP2).
(c) Iind Hk(P"). (Use Problem 1-15(b); it is necessary to consider wliether a
neighborliood of BW-'iii B" is orientable 01. not.) [Answer: dim H k ( P " ) = 1
if k even and 5 11: = O othenuise.]
(d) R n d Hk( ~ l e i nbot de).
(e) Fiiid tlie c o l i o n i o l o ~of M # (Mobius strip) and M # (Kleiri bottle) if M
is the n-l-ioled torus.
4. (a) Tlie fipr+eI~eIowis a triangulation of a rectangle. If we perform the
indicated ident&catioris of edges we do ?iol obtain a triangulation of the torus.
Why not?
(b) Tlie figure below does give a triangulation of the torus when sides are iden-
tified. Find ao, u ' , u2 for this triangulation; compare with Theorem 5 and
Problem l .
5. (a) For any trianplation of a compact 2-manifold M, show that
(b) Sliow that for triangulations of s2and tlie torus T~ = S' x S' we have
Find iriangrilatioiis for wlijch iliese incq uaiities are al1 equalities.
6. (a) Tiiid H! (S"x R"') 11y iiiductioii oii n, using tlie Mayer-Vietoris sequenci
for compact supports.
(h) Use tlie exact sequeiice of ille pair [S" x Rm, { p ) x R m ) to compute the
same vector spaces.
(c) Compute H ~ S x" snl-'
), using Tlieorem 13.
7. (a) The vector space ~ ' (
- N)3may be described as tlie set of ail scqueiices of real numbers. Using the exact sequeiice of the pair ( R 2 ,N ) , show
tliat H!(IR2 - N ) may l ~ ecorisidered as the set of al1 real sequeiices ( a , ) such
tliat a, = O for al1 but fiiiitely many 1 1 .
(h) Describe tlie map PD: H1(IR2- N ) + H,'(IR2 - N)* in terms of these
desci-iptionsof H' (LR2 - N ) and H: ( IR2 - N ) . aiid show thai it is an isomoi-phism.
(c) Clearly H:(LR2 - N ) has a countable basis. Show that H1(LR2 - N) does
iiot. Hiiil: If vi = { u i j ) E H 1 ( R 2- N ) , choose ( b 1 , b 2 )E R2 linearly iiidependent of ( a 1 1 , a 1 2 )tllen
;
choose ( b 3 ,b4,b s ) E R3 linearly independent of boih
( a l 3 a, l 4 ,a l S ) and ( í 1 2 3 , nz4, nz5);etc.
8. Show that the squares in the diagram in tlie proof of Lemma 21 commute,
except for the squaw
wliich commutes up to tlie sign ( - I ) ~ . (It will be iiecessary to recal] howvarious
maps are defined, which is a good exercise; the only slightly difficult maps are
tlie ones involved in the above diagram.)
9. (a) Let M = M i U M2 U M3 U. - be a disjoint union of oriented n-manifolds.
H: ( M i ) , this "direct sum" consisting of al1 sequences
Show that H! ( M )
( a l ,a 2 , a 3 , . . . ) with aj E H:(Mi) and al1 but finitely rnany ai = O E H$ ( M i ) .
(b) Sliow that H k ( M ) %
H k ( M ~ ) this
: "direct product" consisting of nll
sequences ( a l ,a2, a3, . .. ) with ai E H k ( M i ) .
(c) Show that if tlie Poincaré duality theorem holds for each Mi,
then it holds
for M .
(d) The figure below shows a decomposition of a triangulated 2-manifold into
tliree open sets U*,U1,
and U2.Use an analogous decomposition in n dimensions to prove that Poincarii duality holds for any triangulated manifold.
ni
U. is union of shaded
(-1)
i\
U2is union of shaded .:.i;i:'
,,;,
10. Let 6 = rr : E + M and 6' = rr' : E' + M be oriented k-plane bundles.
over a compact orieiited manifold M , and (f,
f ) a bundle map from t' to t
\vliicli is an isomorphism on each fibre.
(a) If U E H: ( E ) and U' E H: ( E') are the Thom classes, then f * ( ~ =
) U'.
(b) . [ * ( x ( c ) )= ~ ( 6 ' )(Using
.
tlie iiotation ofProbIem 3-23, we Iiave f * I x I E ) ) =
xrf*rt>>.)
11. (a) Let 6 = T : E + M be an oriented k-plar-ie bundle over an oiiented
rnalijfold M , witli Tliom class U. Using Poincaré duality, prove tlie Tliom
Isomorpliism Theorem: Tlie map H i ( E ) + H:+~( E ) @ven by a t+ a v U is
an jsomoi-phism for a11 i .
(11) Since we can also coiisider U as being in H k ( E ) , we can form U v U E
H : ~ ( E ) .Using anticominutativity of A, sliow that tliis is O for k odd. Concludr
tliat U represents O E H ~ E ) so
) , that ~ ( =
t 0.) It follows, in particular, tliai
~ ( =t O)wheii 6 = rr : TM + M foi- M of odd dimension, providing anothel
proof that x ( M ) = O in tliis case.
12. IS a \rector field X has aii isolated singularity at p E M n , show that thc
iiidex of - X at p js (-1)" times tlie index of X at p. This provides ariothelproof that x ( M ) = O for odd n.
13. (a) Let pr, . . .,pr E M. Usir-ig Problem 8-26, sl-iow that there is a subsei
D c M diffeomor.pliic to the closed ball, sucli tliat al1 p; E interior D.
(h) If M is compact, tlien tliere is a vector field X oli M with only one siiiyularitx
(c) is a fact that a Cm niap j.: S"-' + Sn-' of degi-ee O is sinootlily Iioniotopic to a coiistant map. Usii-ig this, show tliat if x ( M ) = O, theii tliere is a
lio~rliereO vector field on M .
(d) lf M is connected ai-id not compact, then there is a nowl-iere O vector field
ciii M . (Begiri wjth a tria1i~gu1atioi-i
to obtaiii a vector field with a discrete set oj
zeros. join tliese by a ray goirig to infirlity, enclose tliis ray in a cone, and pus11
ever-ything off to irifiriit\:j
(e) If M is a coririected riianjfold-witli-boiriidary.with aM # 0, tlien tliere is a
rioivher-e zero vector field on M .
14. This Problem proves de Rliam's Theorem. Basic knowledge of singular
col~omoIogyis rcquired. l e wiil denote tlie group of singular k-chains of X
by Sk(X). For a manifold M, we let S r ( M ) denote the Cm singular k-chains,
aiid let i : S r ( M ) + S k ( M ) be the inclusion. 11is not hard to show that theiac
is a cliair-i map r : S k ( M ) +
M ) so that r o i = identity of
M), wliile
i o r is chain homotopic to the identity of Sk(M ) [basically, r is approximatioi~
by a C m cbain]. Tliis mearis that we obtain the correct singular cohomology
of M if we consider the complex H o m ( S r (M), R).
Sr(
Sr(
(a) If w is a closed k-fo~-mon M, let Hl(w) E H o m ( S r [ M ) , W) be
Show tliat Rli is a cllain iiiap from ( c ~ ( M ) ) to ( H o m ( S ~ ( M ) , R ) ) .(Hitii:
Stokes' Tlieol-em.) It follows t hat t here is an iriduced map Rii from the de Rliam
cohomology of M to tlie singular cohomology of M.
(b) Sliow tliat Rii is aii isomorphism on a smootlily coritractible manifold (Lemmas 17, 18, and 19 will not be necessary for this.)
(c) Imitate the proof of Theorem 21, using the Mayer-Vietoris sequence for
singular. col-iomology, to show t hat if Rii is an isoniorphism for U, V , and U n V:
tlieti it is an isomorphism for U U V.
(d) Coiiclude that Rii is aii isomorpliism if M is of finite type. (Using the
method of Problem 9: it follo\vs that Rii is an isomorpl-iism for any trianLgulated
manifold.)
(e) Check that tlie cup product defined using A corresponds to tlie cup product
defined in singular cohomology.
APPENDTX A
CHAPTER 1
Following the sugpestions in tl-iis chapcer: we wiII no-\vdefine a manifold to b i
a topological space M such that
(1) M is Hausdofi
(2) For eacli x E M there is a neighborliood U of x and an integer n > O
such that U is homeomorphic to IR".
Condition (1) is necessary, for there is eveii a l-dimensional "manifold" which is
i-iot Hausdorff. l t consists of R U (*) where * #
~ 4 t htlie following topo lo^-\.:
A set U is open if and only if
(1) U n R is opeii:
(2) If * E U. thcn (U n R) U (O: is a neighborliood of O (in R).
Thus tlie neigliborlioods of * look just like neigl~borhoodsof O. Tliis space mav
also be obtained by idei~tif~ring
al1 points except O in one copy of R with thr
correspondirig point in anotlier copv of R. Alt l-iougli non-H ausdofl manifold?
are important in certain cases, we ~rillnot consider tllem.
We llave just seen that the Hausdoflproperty is not a "local property", but
local compactiiess is, so every maiiifold is locally compact. hgoreover, a Hausdorfflocally compact space is regular. so every manifold is regular. (By the wa)i
this argunient does i ~ owor-k
t
fol. "infiiiite dimensional" manifolds, which are loc a l ] like
~ Banacli spaces; these iieed iiot be regular even if they are Hausdofl.)
O n tlie otlier liaild, there are manifolds whicli are not iiormal (Problem 6). E\:ery mariifold is also clearly locally coniiected, so e v e q component is open, and
ihiis a mariifold iiself. Before exhibitjiis noii-rnetrizable manifolds, we first note
tl-iat alnlost al1 "nice" propertie.; of a manifold are equivalerit.
THEOREhl. The following pr-operties are equivalei-itfor any manifold M:
(a) Each cornponent of M is a-compact.
(b) Eacli coinpoiieilt of M is secoild coui-itable (lias a countablc base for t he
t o p o l o ~. j
(c) M is metrizable.
(d) M is paracompact.
(In particiilai: a compact mariifold is metriza ble.)
FIRST PROOF. (ai a (b) foIlo\m immediately fi-om tlie simple propositioil tliai
a o-compact IocaIlv second countable space is second countable.
(11) a (e) follo-\vsfi-om tlie Urysohn metrization tlieorem.
(cj 4 (d) because any metric space is paiaacoinpact (Kelley. Ckieraf 7Üpofog.i..
pg. 160). TIie second pr-oof does iiot rely on this dificult theorem.
(dj + (a) is a consequeiice of the follouliii?,
LEMMA. A coiiiiecied. locally coinpact, ~~ai.;icoiiipact
space is o-compaci.
Ao@ T1iei.e is a locally fiiiite covei' of tlie space by open sets with coinpact closure. If U. is oiie ofthese. then U. can interseci oiily a finite numl~ei-L'i, . . . ,L',,,
of ilie otbers. Siiiiilai.ly ÜoU Üi U - - U U,I, iiiiersects only Un,+i.. . . ,U,,: and
so on. The uiiioii
is clcarl!; opeil. It is also closed, foi- if A- is i i l ilir closure! then x níiisi be i i i
the closure of a finiie iiilion of these Ui. becausc s has a neighboi.liood \vliich
iiitei.sects oiily fiilitel!~niany. Tlius x is iii the uilioii.
Siilce the spacc js coililecied, ii equals this couiitable unioii oi' coiilpact sets.
Tliis proIres tlie Leninia aild the Tlieorcm.
SECO.ArR l'R001': (a) d (b) 3 (c) aiid (d) d (a) as befoi-e.
(c) a (a) is Theorem 1-2.
. ~ Ci is conlpact. Clcai-lv C1 ha'
(a) d (d). Le1 M = C1 U C7 U . . \ v l ~ c ieach
an opeii iieiglil~orlioodUi \r?iili compaci closure. Tlieii Ül U C2 lias a11ol~eil
iiriyliborhood fi wjtll ('OiiilIact closui*e. Coniiniiiiig iii thjs way, we obtaiii open
seis Ui witli Ui roiiqlaci aiid G. c Ui+i. whosr uiiioii coniaiiis al1 Ci. aiid lieiicr
is M. It ir; easv to sliow li-onl thjs tliat M is pai-acoml~act.43
a
a
It iuriis out thai t l i ~ iaiLr
- ~ eveii 1 -mailifolds wliich a1.e noi paracoinpact. Thrcoilst ruction of ihrse exain~~lcis
I-equires t11e ordiilal iiumbers, whicli are bricfl~.
cxplaiiied herc.. (Oi.diiial iiiiiii11ei.s will iioi 11c needed for a 2-diineiisioiial example io come latei:
ORDINAL NUMBERS
Recall tliat a n orderine < on a set A is a i~latioilsucli tliai
(1) a < b and h < í-implies a < c for al1 a , b, r, E A (transitivit?.~
(2) For al1 a , b E A : one aild onlv one of the follo1~41ig
holds:
(i) a = h
(ii) a <
(iii) b < a (also written cr > h q
(trichotom?~).
An ordered set is iust a pair ( A : i ) wliere < is an orclering;on A . 1-1410 oi-dered
sets ( A , <) aiid ( 3 , + ) are order isomorphic if there is a one-one onto hiictioii
,f' : A + 3 sucli that a < b implies / ( a ) < .f(b): t11e map $ itself is called al1
order isomorphism. aiid ,f - 5 s easih: seeii to be an oi.dei- isomorphism also.
An orderine < on A is a well-ordering if every non-eimpty subset 3 c A
has a,prs/ clement. tl~atis. ail element b such tliat b 5 b' for al1 b' E B, Somc
well-orda-ed sets are illustrated below: in tliis scl~emewe do 1101 list anv of the <
rrlations whicl-i are co-iiseqiieiices of the olies already listed.
0
(01
O<I
(A = (0,l))
O<]<?
( A = (0, 1,3)!
Oi1<3<.7
etc.
0<1<3<.7<...
O<l <3<.-.<w
(w is some set # O,], 2,3.. . .,
(W
O<l <3<4..<w<w+I
+ 1 is? fol. the pi-esei~i.
just a set distjnct from
tliose already meiitioii ed 1
An! subset of a IY~II-ordered
set is. of course, also a well-ordered set wjtli tl-il
same 01-derjiig. 111particulai: a subset B of a 1ve11-oi.dered set A is called aii
(initial) s e p e n t if b E B and a < h imply a E B. It is easv to see that if B is a
serinent of A , then eitlier 3 = A 01. else there is some a E A such thai
B = (a' E A : a' < a )
in fact. a is tl-ie first element of A - B. Notice tliat each set on our list is ci
s c ~ ~ i ~ofe the
i ~ tsucceeding oiies. It is not liard to sic tl-iat 1-10two sets oii our lis1
are 01-derisomorpliic. For example.
O<l <..-<o
and
O<l < . - . < o < # + ]
al-c 1101 ordei- isomoi-pliic because the second Iias botl-i a last and a ~iextto lasi
elei-ileiit. while tlie firsi does 1101. But thei-e is a mucli inore geiieral proliositioii
whic11 wjll settle al1 cascs ai oi-~cc:
1. PROPOSITION. If B # A A
s a seLmentof A . theil B is iiot ordei isomoiplijc to A . 11-1fact, tlii oiily oi-der isomoi.]ilijsni kom B '10 a ~egirleniof A is tlii
jdeiititv.
I'ROOF. If J ' : B + 3' c A is ai-i oi.dei. jsornoi.pliism aiid 3' is a seLmei~i
of A . tlien fol. tlie fii-si rlcn-iei-it b of B (aiid Iieiice of A ) we cleai-Iy I ~ U S Ilia\:c.
J ( h ) = b. Tlien J'(b1)n~ustbe b'. where b' is tlie second element. And so oii.
i\re~ifoi-the " w ' ~ "eleiiieiit (tlie fii-SI oiie after tlie Iirst, second, tliird, etc.)! TIir
.\va!- 1 %pi-OIT
~
t liis iigoi.ou.sly is alnaziiigly simlile: 11- J'Ib) # b for some b E B.
just coiisider tlie first eleinent of ( b E B : f ( b ) # b ) : aii ouiriglit contradjctioii
aplicai+salmost immediatel\r. Q
Pi.oliogtioii 1 lias a c o m ~ ~ a ~ i i wliicli
o i i . makes tlie study of well-ordei-ed set?
simplv delightful.
2. PROPOSITION. If ( A , <) aiid (3,<) are ~rell-01-deredsets, theli oiie il;
oider- iso~norphicto a segment of the other:
PROOF. W7e match tlic hrst elemeiit of A m i t h tl-ie first of 3, the second witli
tlie secoiid, . . . , the "oth"aitli tlie "oth",
eic., uiiiil we ].un out of one sri.
To do iliis rigorously, coi-isjder order isomorphjsms fi-om segments of A oiito
sepnieiiis o f B. Ir is easy to S ~ O Mtliat
~ any two sucli oi-del. isomor]iliisins ag.rcs
o11 tlic sniallei. of tlieii. t\vo domains (just co~isider-the smallest element \vliei.r
tliev don't). So al1 sucli ordei. isomorpliisms caii be yut together to give anotliei:
~vhicliis clearlv the largest of all. If it is defined oil al1 of A we are done. l f ir
js iiot: tlien its raiige must be al1 of B (or we could easjlv extend it) and iue ai-c
still do11e. *:*
Suppose wc defiiie a relation < bebveen iuell-oi-dei-ed sets by stipulating tliar
( A . <) < ( B , < ) wlieii ( A . <) is oi.der isoinorphic to a proper segment oí
(B. <). Ti-aiisitivihr of < is ob\lious, and Propositions 1 and 2 show tliat we alinost liave tricliotom. "Niiiost", because the coiiditioii "(A, <) == (B, <)" musr
be i4eplacedby "(A, <) order isomorphic to (B. <)". TOobviate this djficult?.
we need oiilv nrork witli 01-dei-isomorplijsm classes of nrell-ordered sets, instead
of witli tlie weII-oi.dei.ed sets theniselves. These ol-del- isomorphism classes arc.
called ordinal numbers. They are beautiful:*
3. PROPOSITION.
is a i~rell-orderiiigof tlie ordinal numbers.
I'ROOF. Giireii a non-empty set A of ordiilal ntimbers, let ( A , <) be a welloi-dered set repi-esciiting one of its elcmerits a . To pi.oducc a smallest elemenr
of A \ve can obviouslv ipiiore elemeilts > cx. E\~ervelement < cx is repi.esented
lxj aii 01-dei-ed set wliicli is order i.soinorp1iic io soine proper segmeii t of A :
eacli of tliese is tlie sepileiit coi-isisting of eleiileiits of A less tlrat some u E A .
Coi-isider tlre least of tliese a's. It detei-mines a segmeiit which represents some
fi E ,A. Tliis #isl tlie snldlest elemeilt of A, 4 4
Kotice tliat if a is aii oi-dinal xiumber, i-epreseiited by a well-ordered set
( A . <), tlieri tlie \vell-01-dered set of al1 oi-diiials fi < cx lias a particularly simple
i.epi.esentation: it is ordei- isomorphic to the set { A . <)! Rouglily speaking: AII
oi.dinal irumhi- is oidei. jsoxiioi-pliic to tlie set of al1 oi-diilals less tlian it.
If a is an ordinal iiumbei; we will deriote by cx 1 tlie smallest ordii~alafter cx
(if cx is iaepresented by dre well-01-dered set (A, <). tlien cx 1 is represented
I n ; a \krell-ordered set witli just one moi-e eleniait. larger. thail al1 member.~
of A). Notice tliat some ordjnals are not of tlie forin a
1 for any a ; these
are called limit ordinals, wl-iile those of tlie form cx
1 are cailed successor
+
+
+
+
*OilIi. oiie feature niars the beauty of tlie ordina1 iiuiiilriers as preseiited here. Eacl-i
oiliiiá~nurnber js a horribly larg set; it would he mur11 iiicei. to rhoose one specific welloi.dei-eclser fmn-ieacli oi.dei-jsoinorphisrn rlas, aild drfine these specjfic sets to he tI3c
01-cliilal iiumhcrs. Tliei-e is a panic.ularJy eIegaiit way to do t h i ~due to LronNairnaiiii,
~ b h i c hcan be fouiid in tlie Appei~dixto Kelley, C;EncralIÓpolog~.
ordinals. \Ve \vil1 also denote some ordinals by the svmbols appearing before:
0, 1,2.3....,w,w
1,... , etc.
Oui- list of well-ordered sets only begins to suggest the complexity which wellordered sets caii acliieve. \dTjtli a little tliouglit, one caii see how the syrnbolc
w w . . . . would appear (symbols hke w 3 w2 . 3 w 4 6 would be used
soinewliere between w3 aild w4): after al1 tliese oiie ~ l o u l dneed
+
+
and after al1 tliese the synbol
+
+
pops up. After
one comes to
and this is oi~lythe begiiiiiiiic!
Al/ tlic wcll-oi.dcied sets iiieiit ioiied so fai- are rori~iinb/r.Thei-e are indeed aii
~1io1.1110~~
nuinl~erof couii talde well-ordered sets:
4. PROI'OSI TlOn'. Let R be tlir collectioii of al1 countable ordiiials (ordinalc
1.v-esrilted bv a countable wcll-oi-del-edset). Theii R js iii~countablc.
I-'ROOE BV P~.opositioii3. ( R . <) is a well-ordei-ed set. l f jt were countable. ji
~vouldi-epresent a countal~lc01-djlialcx E R . By tlie remark after Pi-oposition 3.
thjs would nieaii iliat 52 is ol.clri. isoinorpliic to tlie collcctioii of orclinals < a .
i.c.. to a prolxr scgmeiit of jtsclS. conti*adicting*Proposition l. e:+
\Vr hm;c tlius e~tablishedtlie existente of an ulicouiitable ordinal. Oui. spt-ciíic rxainplc. i.cpl.csellted l ~ yR , js clcai-lg tiic fii'st ulirouiitable oi.dii~al;aiiy
mrmbei- of R is countable, and consequentl\~has onlv countablv mariy p1.rclec.essors. (lt is liopdess to ti.? to "reacli" R bv co iitiiiuin~the listing of wellordei-ed sets begiln abmle. Soi. oiie ~ ~ o u liave
l d to go u~icountablyfar, aild cilroLiiitrl- S C ~ Swith an uiicouiitaI>le ~iuinberof degrees oi- complexity. A leap oí
fai tli ir; required.)
Altl~oii~li
tli(%r-oiiiit;iMc oi.dnials exliibjt ulícountabl\: manv degrees of coiill~lcxit,; tliey are eacli simple jii one wa.1
* 13y drlrtii~qtlle words coiiiiial~lr+
aild iinroui~taI~le
ili this ~-ii.ool'one
obiaiiis thc "BuraliFo i-t i l'iii.~iclos": thr s e i Ord ol' al1 oi-cliiial nurnhers js well-oi-dei-ed,so it represeilts ñn
oi-dii-iala E M . aiicl hencr is oi-clvi isoinorpl-iir to ail iiiitjal sCgiiirilt of (M. ]coii-esoliiiioii of iliis pai-ados, ser Kclley's Appei~clís.
5. PROPOSITION. If a E R is a limit ordinal, tlien there is a sequence #?t <
#?2 < P3 < - . . < a, such that every #? < a satisfies #? < #?, Sor some n (we sal.
that {#?,) is "cofinal" in a).
PROOF. Sirice a is countable: all its members cari be listed (in not-necessarilv
iiicreasing order) y,, y2, y ? : . . . . Let #?t = yt and let #?,+i be the first y in the
list wliicli comes after #?,.
+
6. COROLLARY. If a E R. then cx is represeii ted by some well-ordered subset
of R . However, iio subset of R is order isomorpliic to R .
I'ROOF. Suppose the1.e were orie, and heiice a smallest, cx E S2 not repi-esentcd
bv some subset of R. l t carinot happen that a = #? + 1, Sor then #? would bc
1-epi-esentedby a subset of R t thus also by a subset of (-ca,O) and a could
b\: 1-epi-esented by a subset of R. So by Proposition 5, there is a sequence
jl1 < #?* < P3 < - ' - < a cofinal in a . Tlien Di is represented by a suhset oi
(-m. i). and we can easilv arrailqe that the subset representing Di is a s e L ~ e i - i i
of tlie subset represeritiiig #?, fol. i < j. Tlie union of al1 these sets would then
represent a , a contradiction.
If a subset of R wei-e order isomorpliic to R ? tliei-i there would be uncouiltablv
niariv disjoint inteivals in R, iianieiy tliose betlveeii tlie poiiits r-epresentin~a
aiid a 1 Sor al1 a E R . Thjs is impossible. O
+
The first example ol' a noil-metrizable mar-iifold is defined in terms of S2.
Consider R x [O, l ) ? witli tlie order < defined as follows:
b , ~<)( # ? J )
ifcx < #? or ii'cx = #? aiids < t .
'Tliis can be pictured as follows:
Tlie set R x [O, 1 ) \vitli tlre order topology (a suM3ase coilsists of sets of tlie forin
(.Y : < xo)aiid {S ; .v > -yO:) is called tlie closed longray (witl-i"origin" (O, 0)).
aiid L+ = R x [O, 1) - ((0,O)) is t l ~ e(open) long ray. Tlie disjoint union of two
copies of the closed long rav with tlieii- origins identified is tlie long line L. 7-0
distiiiguish L+ and L. the iiames "half-loiig liric" aiid "long h e ' ' may also bc
irsed. Tlie Coro11ai-v to Pi-oposition 5 ii-ilplies easil!. tliat the long ray and tlir
lorig liiie are 1-din~ensionalrnatiifolds; aside fi-om tlie line and the circle, t1iei.c
are iio otlier coniiected t -manifolds.
Quite a femr new Zmaiiifolds can now be constructed:
L+ x S'
L+ x W
(hall-long cvlinderj,
(half-Ion9 strip,.
L x L
(big ~lanc;.
L+ x L+ (big quadrani).
L x S'
L x Jli.
(long c~linder).
(loilg strip),
L x L+ (big hall-plane),
ldeiitifying al1 points ( ( 0 ,O ) , 8) iii the product of ihe closed long ray and S'
pi.oduces another 2-manifold. w11jch might be called the "big disc".
Tliere is aiiotliei- way of producing a non-metrizable 2-manifold which d o e ~
not use R at ell. \Ve begin 14th tlle open upper hall-plane R: = ( ( x y, ) E IE2 :
J . > 0 ) and another copy W 2 x ( O } of the plane; we will denote thjs set by W i .
aiid denote the poini (.Y, }.,O) by (A-,
Define a map / o : (w:)+ +
b!.
IR:
IP;;
Coiisider the disjojiii unjoii of IE: and IE;, wiih p E (IR:)+ and h ( p ) E
ideiitified. Tliis is a Hausdorfl' manifold; the following djagram sliows two
open seis liomeomorphjc to W 2 . The manifold itself js in fact, homeomorphir
io IR2: wc could Iiave thrown away R: to begin \vi111 siiice it is idcnijfied ly a
lion~íomorphismwith ( W i ) + .
Bui roiisidrr iiow, for eacli a E W , another copy of W 2 , say W 2 x { a } , whicli
we \vil1 denoie by W z . Deliiie f.: (P:)+ + W: b!.
In tlie disjoint union of IR: and !o! IR:, a E W mrewish to identify each p E (IR:)+
with f,(p) E RI.:
\+e may dispense irrith IR: complerely, and in the djsjoini
uiiion of al1 IR: ideniify each ( x , g ) , and ( ~ ' , p ' )for
~ which y = 11' > O and
x!.+a
x ' ~ ~ + bThe
. equivalence classes, of course, are a space homeomorphir
io RI,:
so we will consider IR: a subset of tlie resulting space. This space is
still a Hausdorff manifold, but it cannot be: second countable, for it has aii
uncouri table discrete subset, riamely tl-ie set ((0,O),:. This manifold, the Prüfer
manifoId, and related manifolds? have some ver' strange properties, developed
in tl-ie problems.
-
PROBLEMS
1, (a) A well-ordered set cannot contain a deci-easing infinite sequence A-t >
X2 > Xg > . . . .
(b) If Irre denote (cx
1 ) 1 by cx 2: (cx
3) 1 by cx 3, etc., theii any cx
iquals /3 I I for a unique limit ordinal /3 axid iiiteger ,I >_ O. (Tl-ius one can
define even and odd 01-diiials.)
+
+ +
+
+ +
+
2, Let c be a "clioice functioil", i.e., c ( A ) is defined for each set A # 0, and
c ( A ) E A for al1 A . Giveii a set X,a well-oi-dei-iiig < on a subset Y of X \vil1
be called "distjnguisl-ied" if for al1 y E Y,
(a) Shomr that of aiiv two distinguished well-orderings, one is an extension of
tl-ie othei:
(b) Shomr that there is a ~vell-orderingon X. (Zorn's Lemma may be deduced
from t his fact fairly easil!r.l
(c) Given two sets, sliow tliat one of them is equivalent to (can be put in one-one
coi-1-espoiidencewith) a subset of the other.
(d) SIiow that on any iiifinite set tha-e is a well-ordering whicl-i represents a
limit ordinal.
(e) From (d), aiid Problem 1, show that jf X aiid Y are disjoint equivaleni
infinite sets, then X U Y is equivaleni to 1:
3. (a) L+ and L are iiot metrizable. .
(b) II' xl 5 x 2 5 .y3 5 . ' . is a sequelice ii-i L+. tlien (x,) converges to some
poiii t. Cor~sequently,aiiy sequence I-ias a coiivergerit su bsequence (but L + is
i ~ o compact!).
t
(c) If {A-,] and {}r,j are sequences in L+ with x, 5 y, 5 x , + ~for al1 n: tlien
hoth sequences converge to the same point.
(d) L+ (and also L) are normal. (Use (c)).
(e) hilore ~enerally,anv order topology is normal (completely dflerent proofj.
(f) If f : L + + R is contii~uous,and 7' > S, then one of the sets f-' ( [ - m ,S ] )
and .f -'(Ir,m ) )is countablt.
(g) If .f : L+ + R is contjnuous, then f is eventually constant.
4. (a) L' is iiot contractible. N&: Given W : L + x [O,11 + L+ with W ( x ,O ) =
x for a11 x ! show that for ever-y I we have { H ( ~ , I=) L
) +.
(b) ni ( L + )= nl ( L ) = O. Siniilarly for L+ x R, L x R, L x L, L x L+, L+ x L' .
(c) q ( L + X S I ) = n1(L x S i )= Z.
5. (a) L+ and L are not Aomt~omorphic.Nini:Imitating ProbIem 1-19, defini
"paracompact ends".
(b) L + x
and L x R are iiot homeomorphic; L' x S' and L x S' are iioi
h omeomorphic.
(c) Of the 2-manifolds constructed from L+ or L with ni = O and one paracompact end, oi11y L+ x R Iias the homotopy type of L + .
(d) Tlie ~ione-CechcompactHcations of L x L , L+ x L, L+ x L+, and thca
l i g disc arc al1 distjiict. (Usji-ig Rnblem 3(g), one can explicitly cor-istruct thesc
~ i o n e - ~ e c~om~actificationr.
h
6. (a) S l ~ o wtl~atthe Prüier manifold P js Hausdorff:
(b) P does not llave a countable dense subset.
( c ) Let U l ~ aii
e open set in
whjch js ihe uiiion of "wedges" ceiitered ai
(cr.0)foi e w q r ir~.ationalu . Show tliat U iiicludes a whole rectangle of the forni
( u .b ) x ( O . & ) . N ~ TLet
~ IA:, = { u : the wedge centered at a has width >_ 1 / 7 7 ) .
Silice R = $ U
A,. son-ie A , is not nowhere denst.
U,
(d) Let C1,C2 c P be
-
( ( 0 ,O ) , : a irrationai]
C2 = ( (O, O ) , : a rational) .
Ci
Show tliat Cl and C2 are closed, but that tliey are not contained in disjoint
open seis.
(e) Define W : P x [0,1]+ P b y
Sliour that H is well-defined and that H ( p , 1 ) E
U ( ( 0 ,O),) for al1 p E P.
Conclude that P is contractible.
(f) P - ((x, y ) , : y O ) is a manifold-with-bouildary P', whose boundary is a
disjoint union of uncountably many copies of R.
(g) The disjoint uiiion of t w copies
~
of P', uitli con-esponding points on thc
boundary identified, is a manifold which is not metrizable, but which has a
countable dense subset. Its fundamental group is uncountable.
7. It is known that every second countable coiitractible 2-manifold is
or IR2.
Hence ihe i-esuli of constiucting the Prüfer manifold using only copies W: for
i-ational u must be hoineomorphic to P2. Desciibe a homeomorphism of thiz
manifold onto P2.
8. Let M be a connected Hausdofl manifold which is not a point.
(a) If A C M has cardinality c (the cardinalitv of IR)' then the closure 2 lis‘
cardinaiity c.
(b) If C c M is closed and has cardinality then C has an open neighborliood
witli cardinality r.
(e) Let p E M . There is a function f : R + (set of subsets oí" M ) such that
-f( a ) has cai-diiialits c foi. al1 rr E R , and suc11 thar
f(0) = (pl
/ ( a ) is an open iieighborhood of tlie closure of
f(B).
(Consider functions defined oii iiii tia1 segnients of R wit1.i tllese same properties,
and apply Zorn's Lemma. Alteriiatively, oile caii I-equireJ ' ( c x ) to be the result of
applying tlie elloice fuiiction to tlie set of al1 open iieigl~borhoodsof the closurc
of UBo f ( B ) ~ i t cardinality
h
C. Then there is a unique f with the required
properties. This is an example of defining a function by "transfinite induction".)
(d) A function f : Q + (set of subsets of [O, 11) with the properties of the function in part (c) is eventuaIIy constant.
(e) M has cardinality r, (Given p' c M : consider an arc from p to p'.)
9. (a) A coniiected 1-maiiifold whose topology is the order topology for some
order, is Iiomeomorphic to either the real line, the Iong line, or the half-long
Iine.
(h) Every 1-manifold M contains a maximal open submanifold N whose topology is the order topology for some ordei:
(c) If M is connected and N # M, then M is homeomorphic to S'.
CHAPTER 2
The long ray L+ can be given a Cm structure, and even a CW structure.
To see this we need the result of Problem 9-24-anv Cm [or C W ]structurc
on a maiiifold M homeomorphic to R is d8eomor;hic to R with the usual
structure. Thjs implies that it is also dSeomorphic to (O, l ) , and consequently
that tlie structui-e on M can be extended if M is a proper subset of L+. An
easy applicatioii of Zorn's Lemma then shows that C m and CWstructures exist
on L+.
1 do not know whether al1 C m structures on L+ are diff'eomorphic. It is
klio~.ntliat tliei-e ale uncountably many inequivalent C Wstructures on L+, If
p E L+. and L+p deiiotes al1 points 5 p , then L+ - L+p is clearly homeomorpliic to L+. If O is a CWstructure for L+, then it yields a CW structure foi
L+ - L+p' and Iience for L+. These are al1 distinct, in other words, there is
110 C Wmap
f : L + - L ~ +L + - L + ,
Y
f'
~ 4 t ha CWinverse. In fact, wre must Iiave f (q) > q, and then jt is easy to ser
that we must also have J'(f ( Y ) ) 3 f (Y), f (f(f( Y ) ) ) 3 f (f(Y)), etc. The
iiicreasing sequence y, J'(y ), f (j'(q)), . . . has a limit point xo E L+ - L+p.
and .f[xo) = xo. Now j' cannot be the identity on al1 points 3 xo (for then
it would be tlie ideiitity werywliere, siiice it is CW).So foi. some ql 3 xo we
have j'(qi ) # ql; we can assume J'(ql) > ql, since we can consider f -' in
tlie coiltrary case. Reasoni~igas befoi-e, we obtain xl 3 x.0 with f ( x l ) = xl.
Colitinujng iil tliis wax we obtain xo < xl < x2 < . - with f (x,) = x,. This
sequence has a limit jn L+ - L+p: but this implies that .f[x) = x for al1 x, a
contradictioii.
A C Wsti.uctui-e exists oii tlie Prüfer manifold; this follows immediately from
tlir fact iliat ilie iiiaps /.. used for ideniifying poiiiis in various (R:)+ wiili
points in R;, are al1 C W .1 do not know whether eveqr ?-manifold has a Cm
stiuctui-c.
Using the C Wstructui.e on L+, we can get a CWstructure on L+ x L+. However, tlie ine thod used foi-obtainiiig a CWstructui-e on L+ will not yield a comphx
ariaiiyiic ,rirwciu?r oii L+ x L+; tlie prohlem is that a complex analytic structure
on R2 niav bc con for nially eq uivalent to the disc, aiid lieiice extendable, bu t it
may also be equivalent to the complex plane, and not extendable. In fact, ii
is a classical t11eo1-emof Rado that every Riemannian surface (2-manifold mith
a complex analvtic structure) is second countable. O n the other hand, a modification of the 'Priifer manifold yields a non-me~izablemanifold of complex
dimension 2. Refei-ences to these matters are to be found i i ~
CaIabi and Rosenlicht, Comklex Anabfic Manfoldr widiour Coutifable Base, Proc.
Arner. h4ath. Soc. 4 (19531, pp. 335-340.
H. Kneser. A~iaJírfiscJieSfruckfu~und Ab<aJilbark&f, Ann. Acad. Sic. Fennicae Series A: 1 25 1i5 (19581, pp. 1-8.
10. Prove thai íor y > p there is no non-constant C Wmap / : L+ - L.+, +
L+ - L+ 4 11. Let (Y, p) be a n~etricspacc a11d let .f : X + Y be a continuous locall!.
one-one map. where X is Hausdorff: connected, Iocdly coni~ected,and locallv
compac t.
(a) EWI-ytwo pojilts A-,]? E X are coi~tainedin a compact connected C c X.
(h) Let d(a,J.) be the greatest lower bouild of the diameters of f ( C ) (in the
p-metric) Sor a11 compact connected C containing A- and J.. Show that d is a
n ~ e t ~on
i r X which @ves the same topology for X.
12. Of tbe 1-al-iousmanifolds mnltioned in the previous section, try to determine w11ich can be immersed in which.
CHAPTER 6
Problem A-G(g) describes a non-paracompact 2-manifold in which two open
half-planes are a dense set. \Ve will now describe a 3-dimensional version with
a twist.
Let A = {(x, y, z ) E W 3 : j~# 0): and for each a E W let W
: be a copy of W3.
points in P
: being deiloted by (x, y, r),. In the disjoint union of A and al1 P
:
.
a E R we identify
+ yx, y , r + a )
y, L), for y < O with (a + yx, y, z - a )
(x, y, E),
for y > O
with
(a
(A-,
TIie equivalente classes form a 3-dimensional Hausdoflmanifold M. O n this
and the sets r = constant form a
manifold tbere is an obvious function "í",
foliation of M by a 2-dimensional manifold N. The remarkable fact about t h i ~
2-dimei~sionalmanifold N is that it is connected. For: tlie set of points (x, y, c) E
A with y > O is ideniified ~ i t tlie
h set of points (x, y. r - a), E W: with y > 0.
Now the folium contairling {(A-, y, c - a),) contains the points (x, y, c - a),
with y < 0. and these are identified with the set of points (x,y,c - 20) E A
with y < O. Since we can choose a = c/2, we see t l ~ a al1
t leaves of the foliatioi~
are the same as the leaf coiltaiiling ((x, y, O) : y < 0) c A .
This example is due io M. Kneser, Baipiel ail~idiinensio~zserllohendenanal)iíicclieii A lrbildutig ptvlsclieíi ülrerabpalilbal-en A4a1i?li&alt$hztm. Archiv. Math. 11 (1 960);
pp. 288-281.
C H A P T E R S 7, 9, 10
l. 1% llave seen tl~atany pai-acompact Cm manifold has a Riemannian metric.
The converse also holds, sjnce a Riemannian metric determines an ordinanmetric.
2. Problem A-1 l jmplies that a manifold N immersed jn a paracompact manifold M js paracompact, but a much easjer p o o f is now available: Let ( , ) bc
a Riemannian metrjc on M; if f :N + M is an immersion, then N has the
Riemannian metric f* ( , ).
\+e can now dispense witll the argument in the proof of Theorem 6-6 whicli
was used to show that each folium of a distributjon on a metrizable manifold j >
also metrizable, for the folium is a submanifold, and hence paracompact.
3. Since there is no Riemannian metric on a non-paracompact manifold M.
the tangent bundle TM cannot be trjvial. Thus the tangent bundle of the long
line is i ~ o trivial,
t
ilor is the tangent bundle of the Prüfer manifold, even thougli
the Prüfer manifold is contractible. (On the other hand, a basic result about
bundles says that a bundle ovei- a paracompact contractjble space is trivial.
Compare pg. V. 2 72 .)
4. The tangent bundle of tlre long line L is clearly orientable, so there can-
no, be a iioivhere zero 1-form w o11 LI for w and the orientation would dr-
termille a i~owherezero vector field, contradicting tlle fact that the tangeni
bundle is not trivial. Thus, Theorem 7-9 fails for L. Notice also that if M
is non-pai-acompact, then TM is definitely not equivalent to T*M, since an
eqiiii~aIei~ce
ivould determine a Riemanilian metric. So there are at least two
ii~e~uiiraleil
t non-trivial bundles over M.
5. Altliougli the results in the Addendum to Chapter 9 can be extended to
closed. not iiecessarily compact. submanifolds, they cannot be extended to nonpai-acompact manifolds, as can be seen by consjdering the O-dimensional submanifold ( ( O . O ) a j of the Prüfer manifold.
6. A Lje group is automatjcally paracompact, sii~ceits tangent bundle is trivial.
Adore gei~eralh;a locally compact connected topolocgical group is a-compaci
(Problein 10-4;.
-1 . 11 is not clear that a non-paracompact mailifold cannot have an ji~deíii~jich
m e t i c (a non-degenerate iniler product o11 each tangent space). Tl-iis will bc
proved in Volume 11 (Chapter 8, Addendum 1).
PROBLEM
13. 1s there a nowhere zero 2-form on the various non-paracompact 2-mailifolds which have been described?
CHAPTER 1
CHAPTER 2
A'
C'
C0
C"
C"
D i f (0)
GL(n, R)
R(n)
(Rn, U i
R)
CHAPTER 4
?I
d.v '
End(V)
,f*
c
CHAPTER 3
Hom(V, W)
T*M
T@S
116
2 68
285
29 1
264, 291
2 64
264
263
248, 285
285
260
246
2 39
243, 245, 246, 248.
J
sin11
tanh
M/L
Xl ( v )
5t
61~)
S I
¿-1
r
r!.
1.1
I . )
r
r
3
3
I
)<
)-,
)I
r 257, 259, 288
CHAPTER 9
cosh
cosh
-'
d ( p .q )
ds
d t/
Euc( V )
Eucít)
exp
f*( , )
(giJ)
(U,13
Lt
(
)*
CHAPTER 10
1,
i-0
Lu
J(G)
al3)
oín)
P
P-1
8,
SO(n)
-
A'
A'
C
@*
Y'
A
v)
40 1
374
379
376
388
376
41 1
41 1
374
373
376
3'79
380
395
403, 410
o (natural g-valuecl
I-form,
403
1-1
111 A A1
376
40.';
CHAPTER 11
Ahelian Lie alpehra, 376, 382, 395
Adams, J. E, 100
Adjoint T* of a linear transformatioi~
T , 103
Ado, J. D., 380
Alexander's Horned Sphere, 55
Algehra. Fundamental Theorem of.
285, 293
Algehraic inequalities, principie of
irrelevante of, 233
Alternatin~
covariant tensor field, 207
multilinear function, 201
Alternation, 202
Analytic mai-ijfold, 34
Annihilator, 228
Annulus, F:
Antipodal
map, 27l:
point, 11
Arclength, 31 2
fuiiction, 313, 332
Arcwise coi~nected,20
suhgroup of a Lie group, 409
Area, generalized, 246
Associated
disc huiidle, 451
sphere bui-idle, 45 1
Atlas, 28
maximal, 24
Auslai-ider, L., 106
Rariach space. 145
Rase spacc, 7 1
Basi!
dual, 1 O7
for Mp*:208
for R~ ( p ) , 208
Beloiip to a distributioii, 191
Besic.ovitc11, A. S., 174
Rie
djsc, 460
half-l~lai~e.466
plane, 460
quadrant, 466
Bi-invarian t metric, 40 1
Boundary, 19, 248, 252
Bouiided manifold, 1
Boy's Surface, 60
Bracket, 154
in gr(n,R), 3TE;
in o(n, R), 359
Bundle
cotancentr 1 O?
dual, 108
fihre, 309
ii-iduced, 101
map, 73
n-plane, 71
i-iormal, 344
of contra\nariant teasors, 120
of covariant tensors, 1 1'7
tangerit, / j
trivial, 72, 210
\lector, 71
Burali-Forti Paradox, 464
--
Calahi, E., 472
Calculus of variations, 31 6
Cartan, Elie, 39, 348, 361)
Cartan's Lemma, 230
Cauc hy-Riemani-i equations, 200
Cayley iiumhers, 100
Chain, 248, 285
Chain Rule, 35, 38
Change, infiiiitely small, 1 11
Chart, 28
Choice, 283
Choice fui-iction, 46;
Circle, 6
Closed
form, 21 8, 253
geodesic, 367
half-space, 19
long ray, 465
manifold, 19
suhgroup of a Lie group, 391
su hmanifold, 4ci
Cl osed (conli~rued,
up to first order: 16f;
Cofinal, 465
Col1omology, 41 9
de Rliam, 263
group of M tttith real coefficients.
263
of a complex, 421
Coinmutative diagrain, 65, 420
Comrnutati\7e h e al_q-bi.a. 371:
Complete, geodesically. 341
Complex, 421
ai-ialytic structure, 4 7 1
i~uml)ersof norm 1. 37 3
Conjugate, 358
Constants of structure, 391)
Coi~tinuoushomomorplijsm, 387
Coi-itractihle, 220, 235. 236
Coi-itraction, 121, 139. 227
h m m a , 139
Contravariani
fuiictor, 130
tensor field, 120
lector field, 1 13
Conves
~eodesjcally,363
polyhedroi-i, 429
Coordiaate lii-ies, 159
Coordinate systeim, 28. 158
Coordinates, 28
Cotangent bundle. 1 09
Co\yai.iai-ii
functor, 130
icnsor field, 1 1;
vector field, 1 13
C;O\~PI
locallv finite, .5O
point-finite, 60
rcfinement of, S(1
Ciamer's Rule, 372
Critica1 poini, 40
j11 the calculus of \.ariatioi-is, 3120
Critica1 value, 40
Crass section, 227
Cross-cap, 14
Ci-oss-product, 299
Cul~e,sin-gular, 24ú
Cup product, 299, 439
Curl, 238
Cginder, 8
Cr manifold, 34
C0 manifold, 34
C"
distrihution, 179
form, 207
fuilction, 32
manifold, 29
manifold-with-houndary, 32
Riemaiinian nietric, 308
structure on TM, 82
Cm-related, 28
Darbou':
integrahle, 283
integral, 283
DarBoux's Theorem, 284
Dehaucti of iildices, 39, 123
Decomposahle, 228
Definitioi-i, invariant, 2 t 4
Delormatjon retractioii, 259
Degenerate, 286
Degree, 275
mod 2, 295
Dei~sinevei; sealar, 133, 209
odd scalar, 133, 259
relative scalai-, 23 1
scalar, 133
Derj\xtjoi~,39, 78
of a riiig, 83
Derived set, 25
Descartes-Euler Tlieoi-em, 429
Detei-miilant, 232
Diffkomorphic, 30
Diileoinoi-pliism, 30
one-parameter .y-oup of, 148
Djflkrentiable, 27, 28, 31, 39
at a point, 3 1
mai-ijfold, 29
structure, 30
oil the loi-ig line, 47 1
011 P",33
Dflerent iahle (cm/inirru'
(structure conlinrc~u',
on IR", 21)
on S", 30
Dflerential, 2 10
equation, 136, 164
deperidjng on parameters, 169
linear, 165
forms, 201
of a functjon, 109
Djmensioil, 4
Direct sum, 421
Disc bundle, associated. 45 1
Djscrjminant, 233
Djsjoint union, 4, 20
Djstrihution, 179, 181
ideal of, 2 15
on torus, 180
Divergente, 238
Theorem, 352
Domajn, 3
Du Bois Reyniond's Lemma, 355
Dual
hasis, 10;
space, 105
vector bundle, 108
Ejnsteiii summatjoi~coi~\lciitjon,39
Elemeiits of norm 1 , 308
Elliptjcal non-Eucljdeai~geometry, 367
Emhedding, 49
Ei-id, 23
paracompact, 466
Ei~domorplijsm, 12 1
Eiiergy, 324
Enwlope, 358
Equa tioiis depeiidjiip o i ~paranle ter>.
169
Equatjons of structure. 404
Equi\laleiice (01' vector bundles), 72
weak, 96
Eu cljdeai~
inetric, 305, 315
motjoii. 374
11-space, 1
Euler, 429
characterjstjc, 428
class, 445
Euler's Equatioii, 320
Eveii
ordinal, 467
relative scalar, 23 1
relative tensor, 134, 231
scalar density, 133, 209
Exac i
form, 21-8
sequence, 419, 422
of a pajr, 433
of vector bundles, 1 0 3
Exponential map, 334, 383
Exponential of matrices, 384
Extension, 432
Extrema], 320
Fajth, leap of, 464
Fihre, 64, 68, 5 1
Fjnite
characteristic, 205
qTe, 438
First elemeiit, 461
Fjrst varjatioii, 3 19, 32;
Fjve Lemma, 440
Fjxed point, 139
Foliatioi~,194
Foljum, 194
Force field, 240
Form, 207
differei-itial, 20 1
left invariant, 374
righ t ii-ivarjaiit, 400
.f-related, 190
Frol~ei~ius
Intep-ahility Theorem, 192,
215
Fuhiiii's tl~eorem,254
Fuiictor, 130
Functorites, 89
Fui~damei-italTheorem of Algebra.
285, 293
Fui~damentalTlieorem of Calculus,
254
Gauss's Lemma, 33,
General linear group, 61, 372
Gei-ieraljzed area, 246
Geodesic, 333
closed, 36;
reversiiig map, 401
Geodesjcallv complete, 34 1
~eodesjcallkcoi-i\lex. 363
Geodesy, 333
Germs of k-forms, 432
Glol~altheorv of integral mai-iifoldi
194
Gradient , 23;
Grarn-Schinidt ort11oiioi.iiialjzatioi-i
process, 304
G I P ~ 8-1
.
Gioup
Lie. 371
n-iatrix, 373
op~osite,407
or~thogonal,372
topolo~ical,3 7 1
Guillemiri, M!, 106
Hal-in-Rai~acl-itheorem, 14 5
Hair-, 69
H al f-1oi-i~
cvliiider. 466
liile, 46.5
strip, 466
Half-space, 19
Handle, H
Hardy, G. H., 179
Has oi-ie eiid, 23
Heiiilcin, Robert A., 84
HausdodT, 459
Ho~no~eiieou
S, 7
Homomoipl-iism
conti~iuous, 387
of Lje al~ebras,380
Homotopic, 104, 277
Hon~otoly,104, 277
Hopf, H., 342, 450
Hopf-Rii~ow-deRliam Tl~eor.em,342
Hyperholir
cosine, 356
sine, 356
tai-igent, 356
Ideal of a Lie algehra, 410
Jdentificatioi~, 10
Jmbeddinc: 49
topological, 14
Immersed submai-iifold, 47
Jmrnersjoii, 46
iopolo@cal, 14, 46
Inipljcit fuiictjoi~theorem, 60
Jndefiniie metric, 350
Jndependerlt ii-ifii-iitesimals, 3 14
Iiidex of jni-ier ]>roduct, 349
Ii-idex of \rector field
on a manifold, 447
on Rn, 44 G
Ii-idices
dehauch of, 39, 123
raisii-ig and lowering, 35 1
Ii-iduced
l~uiidle,101
orie~ltation,260
Jr-ieqiialit ies, principle of irrele~ranceoI
algebraic? 235
Jnertia, Syl~lester'sLaw of, 349
Jnfinite volunie, 312
Ji-ifiniiely small cl-iarige, 1 1 1
Ji-ifinitely small displacemenis, 3 14
Ji-ifinjtesimal generator, 148
Jnfinitesimals, iriclependent, 314
Ji-iitial conditions, 136
of integral clirve, 136
Jnjtjal segmcnt, 462
Jnner product, 227, 301
preserviiig, 304, 372
usual, 301
lnside, 21
Jntegahility conditioiis, 189
Jntegrahle disti.il>utioi.i, 192
Jiiiegable functioii
Darhoux, 283
Riemann, 283
In tepral
c u m , 136
Darhoux, 283
Ijne, 239, 243
manifold, 179, 181
maximal, 194
of a djfferentjal equatioi-i, 136
Riemann, 283
surface, 245
Intepration, 136, 226, 339
Tnvariance of Domain, ?
Invai-iant, 128, 232
definjijon, 2 14
Trrelevance of algebraic inequaljtiea,
prii-iciple of, 23 3
Isometry, 340
Ison-iorphic Lie groups. locally, 382
lsomorphjcm, natural. 10P
lsotopjc: 294
Jacobi jdei-itity, 155, 3TG
for tlie hi-ac.kei in ai-iyl i'ji~g, 37 8
'jacobjan mairjx, 40
Jordan C u m Tl-ieoren~,21, 435
Kelley, J., 460, 463, 464
Kji~k,366
Klej11 bottle, 18, 435
Kneser, H., 472
Kneser, M., 473
Lanc, S., 145
Laplace's expai-ision, 2311
Laplacian, 58
1,aw of 1 nertia, Sylvesier's, 34 9
Leaf. 194
Leap of faith, 464
~ e f jiwai-jaii~
;
form, 394
n form, 400
vector field, 374
Left trar~slation, 374
Lei-igh, 243, 305, 312
of a curve, 59
Lie algebra: 376
ahelian, 376, 382, 395
coinmutative, 376
homomorphism of, 380
ideal of, 410
opposi te, 40 T
Lie derivaiive, 150
Lie group, 37 1
arrwjse connected suhgmup of. 409
closed si~bgroupof, 391
local, 415
ilormal subgi-oup of, 410
topologically isom orp1.ijc, 388
Lje suhproup, 373
Lie's fui~damei~tal
theorem
firsi, 414
second, 415
third. 41 6
Limh
ordj~~al,
463
set. 60
Lii-ie integral, 239, 243
Linear differential equations, 165
svstems of, I T 1
~jilkartrai-isformatjoi~
adjojilt of. 103
contractjoi~of, 121
positive defiiiite, 104
positiw seinj-defii-iite, 104
Linkji-iFi~umber,296
Ljpschitz condition, 138
Littlewood, J. E., 159
Lives at points, 1 19
Lobacl.ie\~sliia~-i
non-Euclideaii geometr)i; 368
Local
ffow, 144
L e g'oup, 41 5
one-parameter group of local dfleomorphism, 148
spani-ied locally, 179
u-jvjaljty, 7 f
Local ilieory of inte~ralmanifolds, 190
Locall!
compact, 20
coiii-iected, 20
finite cover, 50
isomorphic Lje groups, 382
Lipschitz. 1 39
oilc-oi1e, 13
pathwise connecied, 20
Long
cyfinder, 466
ljne, 465
ray, 462
closed. 46.7
opeil, 465
Lo\ver sum, 283
MacKenzie. R. E., 1O(;
Magic, 2 14
Mailjfold, 1. 459
aiia1ytic. 3-1
atlas foi-. 28
l~oui~dary
of, 19
l>oui~ded.19
closed. l!i
C', 34
CO,34
C W , 21'
rlifl'erciitial)le, 3!i
dimei~sjoiiof. 4
imheddji~gin IRN. 52
ji~tegi.aI. 159, 181
niasjmal, 19.3
noii-nictrizalJe: 465, 4 GTi
oi-jciitation of. Ht,
smooil~. 29
Manjfold-witli-l~ounda~r.
19
C", 32
Map
I~etwreiiconiplexes. 43 1
1)uiidlc. 73
rai~kof. 4 ( i
Massev. 1V.S.. 3
hlaii-jx groiips. 37:
h4asii11:il integral mai~ifold,194
Mayer-Vietoris Sequence, 424
for compact su pports, 43 1
Measure zero, 40: 41
hksh, 239
Meiric
hi-invariant, 401
Euclidean, 305, 315
indefiníte, 350
Riemannian, 308, 31 1
usual, 3 12
spaces, disjoii-it union of, 4, 20
Milnor, J. M!, 43
h4od 2 degree, 29.5
Mohius sirjp, 10
~eiieralized,1 Ofl
h4ulii-index, 208
M uItiIjnear fui-ic~ioi~.
1 15
Munkres, J. R., 34, 106
11-dimetisioiial, 4
11-forms, left iiivariai-it, 400
n-holed torus, 9
11-maiiifoId, 4
11-plane buiidIe, 3 1
n-sphere: 7
17-torus. 7
Natural Q-valued 1-forni, 403
Natural isomorpl~isin,108
Neighhorhood, tul~ular,345
Newmail, M . H. A.. 3
Nice cmrer, 438
NOII-houi~ded,1')
Noii-deperierate, 30 1
NOII-Euclidean geoineti.?.
elljptical, 36;
LoI)achevskiai~. 368
Noi-i-meirizable inanifold, 465, 466
No~~-orje~~iablchui~dle,86
manifold, 8h
Norrn, 303
pi-ese~jilg,304 , 372
Normal
bui~dle,344
Normal (cmlinzud}
space, 459
s u b p u p of a Lie gmup, 410
outward unit, 351
Nowl~erezero section, 209
Odd
ordinal, 465
relative tensor, 134, 288
scalar density, 133, 259
One-dimensional distribution, 179
One-dimensional sphere: 6
01ie-parameter p u p
of diffeomorphisms, 148
of local diffeomorphisms, local. 148
One-parameter subgmup, 384
O pe i-i
loiig ray, 465
map, 60
submanifold, 2
Opposiie
gmup, 407
Lie algebra, 407
Order
isomorphic, 461
isomorphism, 461
topology, 465
Ordered set, 461
Ordering, 460
Ordinal iiumhers, 463
Orientablr
I)uiidIe, 86
manifold. 86
0iieritatioi-i
of a bundle, 85
of a manifold, 86
of a vector space, 84
presenring, 84, 85, 88, 105, 248
reversiiig, 84, 88, 248
Orthogonal gmup, 61, 352
Orthoiiormal, 304, 348
0i.tl-ioiiormalizatioii pmcess, GiaiiiSclimidt, 304
Os~ood'sTheorem, 284
Outside, 21
Outward pointing, 260
Outward unit iiormal, 351
Palais, R. S., 100, 225
Paracompact, 210, 459
end, 468
Parameter curves, special, 165
Parameterized by arclength, 3 13
Partid derivatives, 35
Partition, 239, 245
of ui-ijty, 52
Pathwise coni~ected,20
Piece~ljsesmootli, 3 12
Pig, yellow, 434
Poincaré, H., 450
Poincaré dual, 439
Poincaré Duality Theorem, 441
Poincaré-Hopf Theorem, 450
Poincaré Lemma, 225
Poiricaré upper half-plane, 367
Poini
inward, 98
ourward, 98, 260
Point-derivation, 39
Poiiikfiniie cover, 60
Polar coordinates, 36
iiitegration in, 266
Polarization, 304
Pollack, A., 106
Positive definite, 104, 301
Positive element of norm 1, 308
Positive senii-definite, 104
Pmduct
of vector huiidles, 102
tensor, 1 16
Pmjection, 5, 30, 32
Pmjectivr
plaiie, 11, 435
space, 19, 88
PI-oper map, 60, 275
Prü fer nianifold, 46 7
Pseudometric. 95
Quaternions, 100
of norm 1, 373
Radial function, 435
Rado, T., 473
Rai-ik
of a b r m , 229
of a map, 40, 98
Rectifiahle, 59
Refinemeni of a cwer, 50
Regular
point, 40
space, 459
value, 40
Related vector fields. 190
Rclative
scalar, 134, 23 1
tensor, 134, 231, 288
Reparameterizatiori, 244, 248
Retraction, 264
deformation, 259
Revolutioii, surface of, 8, 321
de Rham, G., 343
de Rliam cohom01o~gyvector spaces.
263
with compact supports, 268
de Rham's Theorem, 263, 455
Ricmaiii~
integrahle, 283
integral, 283
sum, 283
Riemaiinian meiric, 308, 31 1
iisual, 31 2
Righ t iiivariant n-fom, 400
Rigl~ttranslatioii, 3 74
Riiioy M!, 343
Roman surface, 17. 26
Rosciiliclit, M., 472
Rotatioii ~ r o u p .62
Sard's Theorem, 42, 294
Scalar, relative, 134, 231
Scalar density, 133
Schwarz, H., 354
Schwarz inequaljty, 303, 362
Second countahle, 459
Section of a vector bundle, 73
zem, 96
Segment, initial, 462
Se1f-adjoint linear trarisformation, 104
Semi-definite, positive, 1 04
Separate points and closed sets, 95
Sequence
exact, 419, 422
of vector bundles, 103
Mayer-Vietoris, 424
fbr compact supports , 4.31
of a pair, 433
SIirjnking Lemma, 5 1
Shrinking Lemma, 60
Shuffle perrnutatioii, 227
Simplex
of a triaiigulation, 427
singular, 285
Simply-connectcd, 287
Lie p u p , 382
Siiigularcuhe, 246
simplex, 285
Skew-svmmetric, 201, 378
Slice, 194
Slice maps, 54
Smooth, 28
homotopy, 277
maiiifbld, 29
piecewise, 3 12
Smoothly
contractihle, 220
llomotopic, 277
isotopic, 294
Solid angIe, 290
Space filling curve, 58
Spar-ined locally, 179
Special liiiear gmup, G l
Spccial ortliogoiial gmu p? 6'1
Sphere, 7
Si11iei.e bundle, associated, 45 1
Standard
n-simplex, 426
singular cuhe, 246
Star-shaped, 22 1
Steiner's surface, 17, 26
Steriiherg, S., 42, 106
Siokes' Theorem, 253, 261, 285, 352
~ i o i i e - ~ e cmmpactification,
h
468
Srructure constants, 396
Suhalgehra of a Lie algebra, 37 cl
Suhhundle, 198
Suhcover, 50
Suhgmup
Lie, 373
or-ie-parameter, 384
Sul~manifold,49
cm,49
closed, 4:)
immersed. 4 7
open, 2
Su ccessor ordir-ial, 464
Sum of vector hundles, M'hitney, 101
Support, 33, 147
Surface, 5
area, 354
integral, 239
of revolution, 8, 321
Sylvester's Law of Inertia, 349
Svmmetric hiliriear form, 301
Spstem of linear differei-itial equatioiis.
171
o-cornpact, 4, 459
Tangelit bundle, 57
Tangerir space of IR", 64
Tai-igent vectoiir-iward pointing, 98
of a manifold, 76
of IW" 64
ou tward poii-itirig, 98, 260
to a curve, 63, 66
contravariaiit, 120
covariant, 1 1 3
even relative, 134, 231
odd relative, 134, 288
Tensor field
classical definition of, 123
contravariani, 120
covariant, 1 13
mixed, 121, 122
Tensor produ ct: 1 1 6
Thom class, 442
Thom Isomorphism Tlieorem, 456
Topological
group, 371
imhedding, 14
immersion, 14, 46
Topologically isomorphic Lie groups,
388
Torus, 5, 8
wholed, 7, 9
Total space, 7 1
Totally disconiiected, 25
Transitivity, 4 60
Translation
left, 374
right, 374
Triangle ii-iequality, 303
Triangulation, 426
simplex of, 427
Trichotomy, 461
Trivial vector bundle, 72
Tubular neighborhood, 345
Two-holed torus, 8
Vick,j. W., 3
Wedge pmduct, 203
Whitney, H., 106
\Yhitney sum, 101
T
hese books wei-e typeset using Doi~aldE. Knutli's
typesetting system,
together ~ 4 t l - Berthold
i
H orn's DV3PSONE PostScript driver. The figures
were produced with Adobe Illustrator. and new or modified fonts were created
using Fontographei.
The text font is 1 I point h/lonotype Baskenrille-though the em-dash has beei-i
modified-together with its italic. The elegant swashes of the italicy and f
cause problems in words like iopology and apolog31, so a special gy ligature was
added; special and gj ligatures w r e also required.
Altliough a Baskenrille bold face is ui-iliistorical, bold ty-pe was useful in special
circumstances-niainly for indicating defined terms. The bold face supplied by
Monotype: even the "semi-bold", is obtrusi\?ely extended, so a non-extended
version was created.
Tl-ie somewliat bold appearaiice of cl-iaptei.l-ieadings, in 1 6 point ty-pe, results
from the linear scaling, as well as the fact that t l ~ eupper case Baskerville letters
are of somewliat Iieavier weight tlian tl-ie lowei- case. O n the other hand, the
tal1 initiai letters beginning eacl-i cl-iapter were designed specially, since simple
scaling would llave made them u npleasantly heavy
A tl-iicker set of iiumerals was constructed for use witl-i tlie upper case lettering
in chapter headings and statements of theorems, and special parentheses and
other punctuation symbols were also required. Numerous otlier modifications
of this sort, including additional kerns and alterations of set widths, were made
for V ~ ~ O UpuSrposes.
The niathematics fonts are a variation of the Mutl?7Ime fonts: now based on
the h4onotype Times New Roman famdy, togetliei. with the Monorype Times
N R Seven and Times Small Text famdies-presumably these tliree families
are based on the original designs for Times New Roman, whicl-i was created in
tlii-ee essential sizes: 9, 7 and 51 point.
Tl-ie italic foi-its of tliese tl~reefamilies were used as the basis for creating the
three separate "math italic" fonts-for use at ordinary size, in superscripts, and
in second order superscripts. The proportions and weights for these were then
used for tl-ie tl-iree sizes of the s)fmbol font and the other mathematics fonts,
including bold s~mbols,script letters. and additionai special symbols, as well as
for the extension font and its bold version.
The bold letters for matl-iematics come directly from the bold fonts of the
Times families: wl-iile blackboard bold le t ters were made by hollowing out these
bold letters. Tlie Adobe h/lathematical Pi 2 font was used for the ordinary sized
German Fraktur letters, with suitably modified versioiis used for superscripts.
The covers, paiiited by the author iil his spare moments, are loosely based
on Samuel Taylor Coleridge's poem
Rime ofilre Attcimt A4an%e;l.
nte
CORRECTIONS FOR VOLUME 1
pg. 3, line 3-: change di(x, y ) < 1 to di (x, i 1.
pg. 14: relabel the lower left part of the central figure as
pg. 19: replace the next-to-last paragraph with the following: .
The set of points in a manifold-with-boundary that do not have a neighborhood homeomorphic to R" (but only one homeomorphic
to W") is caiied the boundary of M and is denoted by aM. Equivaiently, x E a M if and only if there is a neighborhood V of x and
a homeomorphism t$ : V + Wn such that t$(x) = O. If M is actuaily a manifold, then a M = 0, and a M itself is aiways a manifold
(without boundarjr.
pg. 22: Replace the top lefi figure with
pg. 43: Replace the last line and displayed equation with the foliowing:
Since rank f = k in a neighborhood of p, the lower rectangle in the ma&
pg. 60, Problem 30: Change part (f) and add part (8):
(f) If M is a connected manifold, there is a proper map f : M -,R; the function f can be made Cm if M iS a CM manifold.
(g) The same is true if M has at most countably many components.
pg. 61, Problem 32: For clarity, restate part (c) as foiiows:
(c) This is faise if f : MI + R is replaced with f : Mi -, N for a disconnected manifold N.
pg. 70: Replace the last two lines of page 70 and the first two lines of page 7 1 with the following:
theorem of topology). If there were a way to map T(M,i), fibre by fibre, homeomorphicaiiy onto M x IR2, then each vp would
correspond to (p, v(p)) for some v(p) E R2, and we could continuously pick w(p) E R2, corresponding to a dashed vector, by using the
criterion that w(p) should make a positive angle with v(p).
pg. 78: the third display should read:
pg. 103, Problem 29(d). Add the hypothesis that M is orientable.
pg. 117: After the next to last display, ~ ( X I. ., .,Xk ) ( p ) = A (p) (XI(p), . .. ,Xk (p)), add:
If A is Cm, then Ais Cm, in the sense that ~ ( x I.,. ,.Xk) is a Cm function for al1 C m vector fields X i , . . . ,Xk.
pg. 118: Add the following to tlie statement of the theorem: If A is Cm, then A is also.
pg. 119: Add the foliowing at the end of the proof:
Smoothness of A foiiows from the fact that the function Ail...ik is A(a/axrl,. . . , a/axik).
pg. 131, Problem 9: Let F be a covariant functor from V , . . . .
pg. 133. Though there is considerable variation in terminology, what are here calIed "odd scalar densities" should probably simply be
called "scalar densities"; what are c d e d "even scaiar densities" might best be c a k d "signed scalar dmsities".
In part (c) of Problem 10, we should be considenng the h of part (a), not the h of part (b)! Thus conclude that the bundle of signed
scalar densities (mt the scalar densities) is not triviai if M is not orientable.
pg, 134. Extending the changed terminology from pg. 133, we should probably speak of the bundle of "signed tensor densities of type
(f) and weight w" (though sometimes the term relative tensor is used instead, restricting densities to hose of weight l), when the
transformation mle involves (de{ A)W,omitting the modifier "signed" when it involves [ det Alw.
pg. 143. The hypothesis of Theorem 3 should be changed so that it reads:
Let x E U and let a l , a2 be w o maps on some open i n t e d I such that al(¡),
(
and
Y
)
i = 1,2
= f ( ( Y(t))
ai(to) = a2(t0)
(Y*(¡) C U,
for some t~ E 1.
And the first sentence of the proof shouid be deleted.
pg. 177. Problem 17, part (d) should begin:
N, and suppose that f., = 0. For X,, Y, E M, and . .. .
(d) Let f : M
pg. 198. In Problem 5, we must also assume that each Ai @ Aj is integrable.
pg. 226. In the comutative diagram, the lower right entry should be "1-forms on N".
pg. 233. The reference "pg. V375" refers to pg. 375 ofVolume V.
pg. 237. In Problem 26, replace par& (b) and (c) with:
(b) Determine the ih component of vl x .
x u,-1
in terms of the (n - 1) x (n - 1) submatrices of the matrix
In particular, for lR3, show that
U X UI = (u2w3 - u3w2, u3w1
- u1w3, u1w2- u2w1).
pg. 292. In Problem 20, the condition Ui n U, # 0 should be Ui n Ui+l # 0.
pg. 408. Problem 16 (b) should read: "For any Lie group G, show that ...".
pp. 408-410. For consistency with standard usage, Aut should be replaced with Aut , and then replace End with End. In part (g) of
Problem 19, add the hypothesis that H is a connected Lie subgroup.
pg. 41 1. The display in Problem 2 1, part (c) shouM read:
(-I)~"[w A [q A A]]
+ ( - ~ ) ~ ' [ qA [A A&]]+ (-1)lm[h A [oA ql] = 0.