Submanifolds and
Isometric Immersions
Man.:os Dajczcr
Ma"ricio Amo.."cci
(;jk", OIi'·C;"L
I'aulo Jj",,,-Filho
�"i 'Ii!icim
-
MATHEMATICS LECTURE SERIES
13
Series Editor
Richard S. Palais
I
,
Publish or Perish, Inc.
Houston, Texas
MATHEMATICS
LECTURE SERIES
13
SubDlanifolds and
Isornetric Irnrnersions
Marcos Dajczer
Mauricio Antonucci
Gilvan Oliveira
Paulo Lima-Filho
Rui Toj eiro
Copyright © 1990 by Marcos Dajczer
AlI Rights Reserved
Library ofCongress Catalog Card Number: 90-61664
Printed in the U nited States. of Arnerica
ISBN 0-914098·22-5
Publish or Perish, Inc.
Houston, Texas
I
I
Preface
These notes grew out of a course which I taught at IMPA - Rio
de Janeiro and at the State University of New York at Stony Brook
during 1 985 and 1 986. A first version was written by my students
Mauricio Antonucci, Gilvan Oliveira and Rui Tojeiro at IMPA, while
a second version was due to Paulo Lima-Filho at Stony Brook.
The guiding Principie of these notes is the use of the theory of flat
bilinear forms which was introduced by J. D. Moore as an outgrowth
of E. Cartan's theory of exteriorly orthogonal quadratic forms. Flat
bilinear forms are the natural tool to treat rigidity problems in the
theory of submanifolds. We devote the Iast four chapters to develop
the theory of flat bilinear forms and to present many applications.
Most of the results we prove are fairly recent, although we aIso
provide new proofs of some classicaI results. The first five chapters
are dedicated to present basic material on submanifolds. The back­
ground necessary for reading these notes is a working knowledge of
basic facts and concepts in Riemannian Geometry.
I am indebted to my colleagues Manfred do Carmo and
Lucio Rodrigues for many discussions and for their criticaI reading
of the original manuscript.
Marcos Dajczer
I M PA - Rio de Janeiro
Contents
1
The Btuie Equotions For Su6f1UJnifolds
1 .0
Introduction
.
.
.
.
.
. . . .. . . . . . . .1
. . . . .
1 . 1 The Fundamental Theorem for Submanifolds
.. 1
. 2
1 .2 Minimal and umbilical submanifolds
1.3 The Axioms of r-planes and r-sphe,res
Appendix
. . . . . . . . . . . . .. .
1.4 Riemannian Vector bundles . . . . . .
. . 8
12
14
14
.
.
.
.
.
.
.
.
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.
.
.
1.5 Proof of the Fundamental Theorem for Submanifolds . 1 7
Exerclses
2
.
Hypersurfoces.
2.0
Introduction
..
.
.
.
.
.
.
. .
19
..... . ..
23
. 23
2.1 The Fundamental Theorem for Hypersurfaces
24
2.2 Convex Euclidean hypersurfaces . . . . . . . .. . . . . 27
2.3 The classification of Einstein hypersurfaces . . . . .. . 32
.
.
,... . ...
.
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.....
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.
Exercises
J
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Su1nnanifoltls with Non-poritive CUf1loture
3.0
3.1
3.2
Introduction
..
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.
The Chern-Kuiper Theorem
.
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..
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..
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. . .
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36
39
39
40
.
The Jorge-Koutroufiotis Theorem
Exerclses
4
.
45
.
.
51
Reduction of Codimension . . . . . . . . . . . . . . . . . . . 53
4.0 Intoduction .. . ... . . .
.
..
.
4. 1 B asic facts . . . . . . .. .. . . . . . . . . . . . .. . . .
4.2 The parallelism of lhe first normal space .
.
..
4.3 An application
Exercises . .
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. ..
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53
54
57
61
62
viü
CONTENTS
5
Complete Su6mo.nilolds 01 Constam Sectiontll CUnJature.. 65
5.0 Introduction . . . .. . . . . .. ... . .
5. 1 Completeness of the relative nullity foliation
. ..
5.2 Isometric immersions between spaces of constant
curvature . . .
. . ..
....
.
.
.
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6
The Theory 01 Flat BilineaT Forrns aM Isometric Rigidity
.
.6.0
6.1
6.2
6.3
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82
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.
. . . . . . . . .. . . .. 106
Introduction ..
.. . . ..
. 106
Characterizations of conformally flat manifolds
. .. 107
Conformally flat submanifolds with low codimension. 1 17
Conformally flat hypersurfaces . . . . . . . . ... . . 126
Exercises
.
,.
.
.
.
.
.
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.
.
Real Kaehler Su6mo.nilolds
136
8.0 Introduction . . . .
.. . ........ . . . ..
8. 1 Complete real Kaehler submanifolds . .. . . ... . .
8.2 Holomorphicity of Real Kaehler Submanifolds . . .
Exercises
. .
... .. ..
. . .. . .
136
137
142
150
1 Hypersurfaces
152
.
.
9
71
80
. .
Conlormo.U, Flat Su6mo.nilolds
7.0
7.1
7.2
7.3
.
.
.
8
.
65
66
82
Intr(>duction
..
Flat bilinear forms . . . . . . . . . . .. ... . . . .. 83
Local isometric rigidity
. .
. . .. .
.
89
. Global isometric rigidity of hypersurfaces
. .. . . . . 96
Exercises
.
. .. .
.
.
. . 10 1
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7
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Exercises
.
.
Conlormo.l Rigidity
.
0
.
.. . ..
9.0 Introduction
. ..
9.1 Cartan's Conformai Rigidity .
Exercises .
. . .
.
.
Bi6liogrophy . .
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.
152
152
1 60
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.
.
.
.
16 1
.
Inda ........ .
. . . . 17 1
.,
.
Rigidity of
Subntanifolds
Marcos Dajczer
Mauricio Antonucci
Gilvan Oliverira
Paulo Lima-Filho
Rui Toj eiro
Chapter
1
The Basic Equations
For
1.0
Submanifolds
lntroduction
ln this chapter we introduce basic facts from the theory of sub­
manifolds which will be used in subsequent chapters. Initially, we
obtain the Gauss and Weingarten formulas and, based on them, we
derive the equations of Gauss, Codazzi and Ricci for submanifolds of a
Riemannian manifold. After that, we present the Fundamental Theorem
for Submanifolds which asserts that the above equations are suflicient
to determine uniquely the submanifolds of a Riemannian manifold
with constant sectional curvature. The proof of this theorem is given
in an appendix at the end of this chapter.
ln section 2, we obtain some basic results about minimal and um­
bilical isometric immersions. The chapter continues with a theorem
of E. Cartan, in section 3, which characterizes Riemannian manifolds
which possess many totally geodesic submanifolds as manifolds of
constant sectional curvature. We also show a generalization of this
theorem due to Leung-Nomizu. The language of vector bundles
used throughout these
notes is introduced in the appendix to this
.
chapter.
1.
2
1.1
THE BASIC EQUATIONS FOR SUBMANIFOLDS
The Fundamental Theorem for Submanifolds
Let
and Mm be differentiable manifolds of dimensions n and m,
respectively. We say that a differentiable map f:
-+ Mm is an
immersion if the differential f.(x):
-+
is injective for ali
x
The number p = m - n is called the codimension of f. Ao
immersion f:
-- M"+P·between two Riemannian manifolds with
metrics (
and ( , ) M' respectively, is called an isometric immersion
if
M"
EM.
TxM
Tf(x)M
M"
M"
, )M
(X, Y )M = (f.X,f.Y)M
We observe that iff: Mn -+
for every x E M and every X, Y
M n +p is an immersion, and ( , )M is a Riemannian metric on
we
can define a metric (
on
by setting at each point
E TxM.
, )M
50
M,
M
that the immersion f becomes an isometric immersion.
Let f: M"
M n +p be an isometric immersion. Around each
EM, there is a neighborhood UeM such that the restriction of
-+
x
f
to U is an embedding oo'to f(U). Therefore, we may identify U
with its image under f, that is, f is locally the inclusion map. Hence,
we may consider the tangent space of
at x as a subspace of the
tangent space to M at x, and write
M
TxM- TxM E9 TxM 1. ,
=
where
1. is the orthogonal complement of
in
this decomposition we obtain a vector bundle
1. =
called the normal bundle to
ln this way, the vector bundle
TxM
TxM TxM. From
TM UXEM TxM 1. ,
M.
TMlf(M)
ETM: 1I'(X) Ef(M), where 11': TM M is the projection }
is the Whitney sum of the tangent bundle TM with TM 1., that is,
TM- I f(M) TM E9w TM 1. .
=
{X
-+
=
With respect to this decomposition we have the projections
() T : TM- 1 f(M) T M
() 1. : TM- I f(M) TM1. ,
-+
-+
1.1. FUNDAMENTAL THEOREM FOR SUBMANI FOLDS
3
which are called tangential and normo.l, respectively.
Let M n +p be a Riemap.nian manifold with Levi-Civita connection
-+ M n + p be an isometric immersion . Given vector
fields X, Y E TM , we have that
V, and let f: M n
It follows easily from the uniqueness of the Levi-Civita connection
that (V) T is the Levi-Civita connection of M; we will denote it by Y'.
Thus, we obtain the
Gauss Formula
VxY
=
Y'xY
+
a(X,Y),
(1)
which defines a map a: T M x T M -+ T M .l called the seCOM fun­
damental form of f. We conclude immediately from the properties of
the Levi-Civita connections V and Y' that a is symmetric and bilinear
over the ring COO(M) of differentiable functions on M. ln particular,
for any point X E M and vector fields X, Y E T M, the mapping
ax: TxM x TxM -+ TxM.l , given by ax(X,Y) = a(X, Y)(x), de­
pends only on the values of X and Y at x.
Consider vector fields X of T M and ç of
the tangential component of -Vxç, i.e. ,
T M.l,
and denote by
A(X
Since for every
Y ET M we have
o =
X (ç, Y)
=
(V xç, Y) + (ç, V x Y),
the Gauss formula yields
(A(X, Y) = (a(X, Y),ç).
ln particular, the map A: T M x T M .l -+ T M given by A(X, Ç) =
A(X is bilinear over COO(M). Thus, the map A(: T M -+ T M is linear
over COO(M) and also symmetric, that is, (A(X,Y) = (X,A(Y) for
every X, Y E T M. The map A( is called the shape operator or by abuse
of language the second fundamental form in the normal direction ç.
1. THE BASIC EQUATIONS FOR SUBMANIFOLDS
4
It is easy to see that the normal component of V x�, which
we denote by V j � , defines a compatible connection on the normal
bundle TM.l.. We say that V.l. is the normo.l connection of f, and obtain
the
Weingarten
(2)
Formula
Now, using the Gauss and Weingarten formulas we derive the
basic equations for an isometric immersion, namely, the equations of
Gauss, Codazzi and Ricci. Let X, Y,Z ET M, then
VxVyZ
=
=
+
Vxa(Y,Z)
VxVyZ + a(X,VyZ) - Aa(y,z)X
VxVyZ
+
Vja(Y,Z),
(3)
where the first equality comes from ( 1 ), and the last equality follows
from (l) and (2). Similarly,
-
-
VyVxZ = VyVxZ
Again by (l) we have
+
a(Y,VxZ) - Aa(x,z)Y
V[X,y)Z = V[X,Y)Z
+
+
.l.
Vy
a(X,Z). (4)
a([X, Y] ,Z).
(5)
Subtracting (4) and (5) from (3), and taking tangential components,
we obtain the
Gauss
Equation
(R(X, Y)Z ,W )
=
( R (X ,Y)Z,W)
+ (a(X, W),a(Y,Z)} - (a(X, Z),a(Y,W)},
where R and R are the curvature tensors of M and M, respec­
tively. ln particular, if K(X,Y) = (R(X, Y)Y, X) and K(X,Y) =
( R(X,Y)Y,X) denote the sectional curvatures in M and M of the
plane generated by. the orthonormal vectors X,Y E TxM, the Gauss
'equation becomes
K(X,Y) = K(X, Y) + (a(X, X),a(Y, Y)} - Il a(X, Y)1 I 2
•
1. 1. FUNDAMENTAL THEOREM FOR SUBMANIFOLDS
5
On the other hand. taking the normal component of R(X, Y)Z.
we obtain the
Coda%%i
Equation
(R(X, Y)Z).l
=
(Vja)(Y, Z) - (Vya)(X, Z),
where by definition
(vja)(Y,Z)
=
Vja(Y,Z) - a(VxY,Z) - a(Y, VxZ).
'Ç7.l can be viewed as
a connection in the vector bundle Hom(T M x T M. T M.l).
Let R.l denote the curvature tensor of the normal bundle T M .i •
Observe that
that is,
for ali
V.la is COC(M) multilinear.
R.l(X, Y){ = vj Vy{
X, Y E TM
and {
E
-
Here
Vyvj{ - V�.YI{
T M.l.
It follows from the Gauss
and Weingarten formulas that the normal component of
R(X, Y){
satisfies the
Ricci Equation
A simpie calculation shows that the Ricci equation may also be written
as
(R(X,Y){, TJ)
where
=
(R.l(X, Y){, TJ)
X, Y E TM, {,TJ E
TM.l, and
-
([A{, A'I1X, Y),
[A{,A'I1
Similarly. the Codazzi equation can be written as
=
A{A'I - A'IA{.
where by definition
Next we write the equations of an isometric immersion
M:+P.
where from now on
Me
f: Mn
-+
denotes ,a manifold with constant
6
.. 1 . THE BASIC EQUATIONS
FOR
SUBMANIFOLDS
secúonal curvature c. ln this case the curvature tensor R of M is
given by
R(X, Y ) = c(X /\ Y )
for alI X, YE TM, where for every Z E TM,
(X /\ Y)Z = ( Y, Z)X - (X, Z) Y.
Then, for X, Y, Z, W ETM and {, ."E T M.l , the equaúons of Gauss,
Codazzi and Ricci are, respectively:
(i)
( R ( X, Y )Z, W )
=
c ((X /\ Y )Z, W)
+ (a ( X, W), a (Y, Z) ) - ( a(X, Z), a(Y, W)),
(ii) (V'f;a )(Y, Z) = (V'fa )(X, Z) , or equivalently,
(V'xA) (Y, { ) = (V'yA )(X, { ),
(iii) R.l ( X, Y ) {
=
a(X, A {Y ) - a (A {X, Y ) , or equivalently,
Notice that it folIows from (iii) that R;- = O if and only if there
exists an orthogonal basis for TxM that diagonalizes simultaneously
alI A {, {E TxM.l.
From now on, Q�+P denotes a complete and simply connected
(n + p)-dimensional Riemannian manifold with constant sectional
curvature c, i.e., the Euclidean sPhere S� + P, Euclidean space R n +p or
the hyperbolic space Hcn+ P .
It was just seen that the equaúons of Gauss, Codazzi and Ricci are
saúsfied for any isometric immersion f : Mn ---+ Mn+p . The theorem
stated below provides a local converse to this fact whenever M n +p =
Q� +p. Moreover, if M is simply connected the converse is global.
1.1. FUNDAMENTAL TH E ORE M
1.1
The orem
(Fundamental
The orem f or
Submanif olds)
7
FOR SUB MANI FOLDS
E
(i) Let Mn be a simply connected Riemannian manifold, 1T: �
M a Riemannian vector bundle of rank p with a compatible
connection 'V', and, let o: be a symmetric section of the homo­
morphism bundle Hom(T M x TM, E). Define, for each local
section � of E, a map Aç: TM � TM by
X, YE TM.
If o: and 'V' satisfy the Gauss, Codazzi and Ricei equations for the
case of constant sectional curvature c, then there is an isometric
immersion f: Mn � Q� + P , and a vector bundle isomorphism
j: E � TM.l along f, such that for every X, Y E TM and all
local sections �, '" of
E
(j(Ç), Í(",) ) =
jo:(X, Y)
=
j\1'x� =
(�, ",)
ó(X, Y)
\l� j(Ç ),
where ó and 'V.l are the second fundamental form, and the,
normal connection of f, respectively.
(ii) Suppose that f and gare isometric immersions of a connected
manifold Mn into Q� +p . Let TM , ai and 'Vt denote the
normal bundle, the second fundamental form and the normal
connection of f, respectively ; and let T M/ , ag and 'Vi be the
corresponding objects for g. If there exists a vector bundle iso­
morphism �: TM � T M/ such that, for every X, YE TM
and every �, '"E T Mi
l
l
(�(Ç ) , �("')) = (�, ", )
�al(X, Y) = ag (X, Y )
�'Vtx� = 'Vtx�(Ç ),
then there is an isometry T: Q� + P
g = TO f
�
Q� + P such that
and
The proof of this theorem, for the case c = 0, is given in the
appendix to this chapter, and may be omitted on the first reading.
1. THE BASIC EQUATIONS FOR SUBMANIFOLDS
8
1.2
Re1lUJrk
ln case our manifolds have indefinite (nondegenerate) metrics, the
so called pseudo-Riemannian manifolds, alI the previous definitions
make sense and the fundamental equations as welI as Theorem 1.1
are stilI valido The proofs are basicalIy the sarne. For details see
Greene [Gr].
A fundamental theorem of J . Nash [N] shows that every Rie­
mannian manifold can be isometricalIy embedded in some Euclidean
space. For a complete discussion on this subject as well as an
improvement of Nash's results see Gromov and Rokhlin [G-R], and
Gromov [Gro]. On the other hand, little is known about the lowest
codimension which makes the embedding possible. This is one of the
main problems in our subject, and will be considered several times
through these notes.
1.2
Minimal and umbilical submanifolds
We introduce the mean curvature vector of an isometric immersion
and present some related facts.
Given an isometric immersion f: M n ---+ Mn +P , we define the
mean curoature vector H (x) of f at x EM as
where o is the second fundamental form of f, and Xl •. . .• Xn ETxM
is an orthonormal frame. Noting that H(x) = � E}= l (traceA{){j
for any set of orthonormal vectors {lo . . . , {p ETxM l.., one concludes
that H(x) does not depend on the tangent frame.
We say that the isometric immersion f is minimal at x EM when
H(x) = 0, and that f is a minimal immersion when it is minimal at every
point of M. A special case occurs when the second fundamental form
vanishes identically at x EM. Then f is said to be totally geodesic at
x E M. We say that f is a totally geodesic immersion when it is totally
geodesic at every point of M. ln this case, it is an interesting fact that
the geodesics of M are geodesics of
which lie entirely in M.
M
1.2. MINIMAL AND UMBILICAL SUBMANIFOLDS
1.3
Proposition
9
Let f: Mn --+ Rn+p be an isometric immersion of a compact manifold
imo Euc1idean space. Then lhere is a point Xo E M n and a normal
vector � ETl:oM 1.. such that·.lhe second fundamental form A{ is positive
definite.
Proof. The function h: M --+ R defined by h(x) = ! IIf(x)112 is
differentiable, hence attains a maximum at some point Xo EM. Since
o
=
X(h)(xo )
=
(X,f(xo»),
for all
X ETxoM,
we conclude that f(xo) is normal to M at xo. Furthermore,
O ;?: XX(h)(xo)
So for �
=
=
(VxX,f(xo»)+IIXI12
=
2
(a(X,X),f(xo»)+IIXII .
-f(xo), we have
for every X E TxoM. I
1.4
There is no minimal compact submanifold of Euc1idean space.
Corollary
Recall that the Ricci tensor of a Riemannian manifold Mn
defined by
Ric (X, Y)
=
trace Z
f--+
IS
R (Z, X)Y, for all X, Y ETM,
and that the Ricci curvature in the direction of a unit vector X ETM
IS
Ric (X) =
�
n-
1
Ric (X,X).
1. THE BASIC EQUATIONS FOR SUBMANI FOLDS
10
1.5
Proposition
Suppose that f : Mn --+ Mcn +p is a minimal immersion at X o EM. Then
Ric (X) � c for every unit vector X E TxoM. Moreover, equality holds
identically if and only if f is tota/ly geodesic at xo.
Proof. Take a unit vector X E Tx oM, and let X =
TxoM be orthonormal vectors. By the Gauss equation
X..
.
.
.
, Xn E
Hence
Ric ( X)
= c + n : 1 (a.(X,X),H)
� t
. (X,Xj)112.
11a.
j=l
1
(6)
L 11a.(X, Xj)112,
(7)
-
n
Since f is minimal at Xo EM, we have
•
Ric (X)
=
c
-
n
�
1
n
j=l
and this proves the proposition. I
1.6
Corollary
If f : Mcn
Mcn, then
(i) c � é,
(ii) c = é if and only if f is tota/ly geodesic at Xo EM;.
We point out that for codimension one, Proposition 1 .5 can be
improvedin the following sense. If f : Mn --+ Mn+1 is a minimal
1.2 . MINIMAL AND UMBILICAL SUBMANIFOLDS
11
immersion at Xo EM, then there is an orthonormal basis Xt, , Xn
for TxoM such that K( Xi,Xj ) � í«xi, xj ) for i,j = 1, .. ,n, where
K and k are the sectional �urvatures of M and M, respectively. See
also [O-RI] and exercise 1 .6.
.
.
.
.
An isometric immersion f: Mn -- Mn+p is said to be umbilical
at Xo E M when A � = À�I for every � E TxoM 1.., where À� E R and
I is the identity map on TxoM. It is an umbilical immersion when
it is umbilical at every point of M. The following assertions are
immediately seen to be equivalent:
(i) f is umbilical at XoE M,
(ii) A � = ( H(xo ), � ) I, �E TxoM 1..,
(iii) a( X, Y) = (X, Y) H( xo ) , X, YE TxoM.
(8)
We say that f : Mn -- Mn+p has parallel mean curoature vector
when V'JiH = O for every X E TM. ln this case we have as a
consequence that IIHII is constant along M.
1.7
Proposition
If f : Mn -- M:+p n � 2, is an umbilical immersion, then f has parallel
mean curvature vector H and the normal curvature tensor R1.. vanishes
identically.
,
Proof.
TM,
Using the equivalence above we have for every X,Y, Z E
(V'Jia)(Y, Z) = V'Jia(Y, Z) - a(V' x Y, Z) - a(Y, V'xZ)
= V'Ji«(Y, Z ) H) - (V'xY, Z) H (Y, V'xZ ) H
= (Y, Z) V'JiH.
-
From the Codazzi equation for the constant sectional curvature case,
we get
(X, Z) V'fH = (Y, Z) V'JiH.
Hence, choosing Y = Z orthogonal to X, we condude that H is
parallel.
By the Ricci equation
R 1.. ( X,Y)� = a( X, A � Y) - a(A � X,Y)
=
( X, A � Y) H - (A{X, Y ) H = O,
for every X, YE TM and �E TM1.. , and thus R1.. = O. I
1. THE BASIC EQUATIONS FOR SUBMANIFOLDS
12
1.3
The Axioms Of r-planes and r-spheres
It is a well-known fact that for every point x E M and every tangent
vector X E TxM of a Riemannian manifold M, there is a geodesic
passing through x with tangent vector X. A natural generalization of
this fact would be: "Given a vector subspace L ofTxM with dimension
r � 2, there is a totally geodesic submanifold of M passing through
x, whose tangent space at x is L". It turns out that this is not true in
general. ln fact, the next result due to E. Cartan [Cas], shows that
the existence of such submanifolds is a quite restrictive condition for
it implies that M has constant sectional curvature .
A Riemannian manifold is said to satisfy the axiom of r-planes, for
some fixed r � 2, when for every x E M and every r-dimensional
vector subspace L C TxM, there is a totally geodesic submanifold S
of M passing through x such that TxS = L.
1.8
Theorem
Suppose that a Riemannian manifold Mn , n � 3, satis/ies Lhe axiom
of r-planes for some 2 ::; r ::; n 1. Then M has constant sectional
curvature.
-
We need the following.
1.9
Lemma
Consider a Riemannian manifold Mn, n � 3. Ifat some point x E M we
have (R(X, Y)Z, X) = O for every orthonormal X , Y, Z E TxM, then
alI the sectional curvatures of M at x are equal.
Y,
�(Y
Proof. Take X , Z orthonormal vectors. Hence X , Y' =
+
(Y - Z) are also orthonormal vectors, and therefore,
Z) and Z' =
�
O = (R(X, Y')Z', X)
=
� (R(X , Y)Y , X) � (R(X , Z)Z , X ).
-
K(X, Y ) = K(X, Z). This shows that the sectional curvatures
of any two planes P, P' which intersect orthogonally are equal. To
condude the proof, observe that if P, P' are any two planes there
80
1.3. THE AXIOMS OF R-PLANES AND R-SPHERES
13
is always a third plane P" which intersect both orthogonally. Thus
K (P) = K(P") = K(P' ) . I
Proof of 1.8. From Schur's lemma (see [Sp], I I p . 328), it is enough
to show that at each x E M the sectional curvatures of M are equal.
ln view of Lemma 1 .9 we just have to consider orthonormal vectors
X, Y, Z E TxM. Since M satisfies the axiom of r-planes for sorne 2 �
r � n -1, there exists a totally geodesic submanifold S of M passing
through x such that X, Y E TxS and Z E TxSl... Because S is totally
geodesic, it follows from the Codazzi equation that (R (X, Y)X)l.. = O,
and therefore (R(X, Y)Z, X) = -(R(X, Y)X, Z) = O. I
We say that an umbilical submanifold of a Riemannian manifold
is an extrinsic sphere (see [N-Y]) when it has parallel mean curvature
vector. Observe that, by Proposition 1 . 7, umbilical submanifolds of
manifolds with constant sectional curvature are extrinsic spheres.
We say that a Riemannian manifold Mn, n 2:: 3, satisfies the axiom
of r-spheres, for sorne fixed r 2:: 2, when for every x E M and
every r-dimensional subspace L C TxM, there is an extrinsic sphere
passing through x whose tangent space at x is L. The following
generalization of Theorem 1 .8 was obtained by Leung-Nomizu [L- N].
1.1 0
Theorem
Riemannian manifold M n, n 2:: 3, satisfies the axiom of r -spheres
for some 2 � r � n - I, then M has constant sectional curvature.
lf a
Proof. Entirely analogous to the proof of Theorem 1 .8 in view of
exercise 1 .9 (iii). I
1. THE BASIC EQUATIONS FOR SUBMANIFOLDS
14
Appendix
The purpose of this appendix is to introduce basic notions about Rie­
mannian vector bundles, and to prove Theorem 1.1 in the Euclidean
case. Some results will be stated without proof.
1.4
Riemannian
�ctor bundles
Let E and M be differentiable manifolds, and let 7r: E ---? M be a
differentiable map. We recall that 7r: E ---? M is a differentiable vector
bundle of rank k, or simply a vector bundle, when for each point x E M,
(i) 1I"- I (X) is a real vector space of dimension k,
(ii) there is an open neighborhood U of x in M, and a diffeo­
morphism cp: 7r-1(U) ---? U X Rk whose restriction to 1I"-I(y)
is an isomorphism onto {y} x Rk for each y EU .
Given a vector bundle 11" : E ---? M, and a subset F C E such that
the restriction 1I"1F: F ---? M is a vector bundle itself, we say that F
is a vector subbundle of if the inclusion i: F ---? E sends (7rIF)- I (X)
linearly into 1I"-1(X) for every x E M. The tangent bundle TM =
U TxM of a differentiable manifold M is a vector bundle, and the
E
xeM
normal bundle TM J.. of an isometric immersion f :
a subbundle ofTMIM.
Mn
---?
M n + P is
Let 7r: E ---? M be a vector bundle. For each x E M we calI
the space
= 7r-1 ( x ) the fiber of 11" over x. A local section over an
open set Ue M is a differentiable mapping �: U ---? E such that
11" o f = idu; if U = M we say that �: M ---? E is a global section, or
simply, a section of 11". It can be shown that for every e E E there is a
section � such that �(1I"(e» = e; in particular this shows that the set
r(1I") of sections of 7r is non-empty.
Ex
Let 11"1: E1 ---? M and 11"2 : E2 ---? M be vector bundles. We
define a projection 11": Hom(El, E2 ) ---? M by setting 1I"-1(X) =
Hom(E�, E;), so that the set Hom(El, E2 ) is the disjoint union
1.4.
RI EMAN N IAN VECTOR B U N DLES
15
of spaces of linear maps from E; into E;, x E M. Endowing
Hom( E1, E2 ) with the natural differentiable structure induced by the
projection it becomes a vectQ,r
The Whitney sum 11" 1 ffiw 11"2 of the vector bundles 11" 1 : E1
11"2: E2 � M is defined as the projection
given by 11"1 ffiw 1I"2«e 1 , e2»
�
M and
= 1I"1(et) = 1I"2(e2), where
E 1 ffiw E2 = {(eI, e2) E E 1
x
E2 : 1I" 1 (e1)
=
1I"2(e2)}.
It is easy to see that the Whitney sum is a vector bundle itself.
More generally, we can transfer to vector bundles certain op­
erations which are performed in the category of vector spaces and
linear maps, namely, those called continuous functors. As examples,
we have: E1 ®E2, Ak E, E(E1, E2 ) , E* , the tensor product of vector
bundles, the k -th exterior power, the symmetric bilinear maps fram
E1 x E1 to E2, the dual bundle of E, respectively; and many others.
For details see [Sp].
Given two vector bundles 11" : E1 � M1 and 11"2: E2 � M2 , and
1
a diffeomorphism <j>: M1 � M2, we say that a differentiable mapping
E1 � E2 is a vector bundle isomorphism along <j> if, for every x E M1 ,
we have
�:
o�
o
= <j>
(i) 11"2
11"1 and �(1I"1 1 (x» = 1I";1(<j>(x»,
(ii) the restriction
1I"1 1 (x) � 1I"; 1 (<j>(x» of
1I"1 1 (x ) is a vector space isomorphism.
�x:
�
�
to the fiber
It follows from the definition that is a diffeomorphism. Further­
more, for each section { of 11" we obtain a section
of 11"2 by defining
�(Ü = �o{o<j>-1.
A
1
Riemannian metric g on a vector bundle
�(Ü
11":
E
�
M is a map
bilinear over the ring COO(M) of differentiable functions on M, which
is symmetric and positive definite. It is well-known that any vector
bundle admits a Riemannian metric. A vector bundle 11": E � M
together with a fixed Riemannian metric is called a Riemannian vector
bundle.
16
1 . THE BASIC EQUATIONS FOR SUBMANIFOLDS
Let 7r: E -+ M be a vector bundle, and let X(M) be the set of
differentiable vector fields on M. A linear connection is an R-bilinear
map
V: X(M)
x
r(7r)
-+
r(7r)
(X,�) 1-+ Vx�
satisfying, for each f
properties
E
COC(M), X
(i) Vfx� = fVx�,
(ii) Vx(fü = X(f)�
+
E
X(M) and �
E
r(7r), the
fVx�.
From (i) it follows that the map X 1-+ Vx� is COC(M)-linear and,
consequently, the value of Vx� at x E M depends only on the value
of X at x. It is also easy to see that the operator Vx: r(7r) -+ r(7r) is
a local operator in the sense that the value of Vx� at x E M depends
only on the values of � in a neighborhood of x.
Let 7r: E -+ M be a vector bundle with a linear connection V.
We say that a section � E r(7r) is parallel when Vx� = O for every
X E X(M). A vector subbundle F of E is said to be parallel if. for
every section 7] of F and every X E X(M), we have that Vx7] is a
section of F.
Consider now a Riemannian vector bundle 7r: E -+ M. A linear
connection V is said to be compatible with the metric g when
Xg (� , 7])
for alI X
=
g(Vx�, 7]) + g(�, Vx7])
E X(M) and �, 7] E r(7r).
The curvature tensor of a vector bundle
connection V is the R-trilinear map
R: X(M)
x
X(M)
x
r(7r)
7r:
-+
E
-+
M with linear
r(7r)
defined by
R (X, Y)�
=
VxVy� -, VyVX� - V[X,YI�.
It is easily seen that R is trilinear over COC(M). W hen the vector
bundle is Rieman nian , we may associate with R another tensor
R: X(M)
x
X(M)
x
r(7r)
x
r(7r)
-+
R
1.5. PROOF OF THE FUNDAMENTAL THEOREM
17
given by R(X, Y, e, TJ) = g(R(X, Y)e, TJ), where g is the metric on E.
Finally we observe that, by abuse of language, it is common not
to refer to the mapping 'Ir: E -+ M when we are dealing with bundles
whose mapping is a natural one, but to the manifolds E and M. For
example, we say the "tangent bundle TM of the manifold M", when
we are actually alluding to the vector bundle 'Ir: TM -+ M, where 'Ir
is the natural projection. We also use the notation X ETM meaning
that X is either a tangent vector to M or a section of T M .
1.5
Proof of the Fundamental Theorem for Submanifolds
The proof we present here is restricted to the case of submanifolds
of Euclidean space, and is essentially the one in [Jzl. For a proof of
the general case the reader is referred to [Ei], [T], and [Sp].
Proolol1.1. We first prove (i). Let V be the Levi-Civita connection
on TM. Consider the Whitney sum É = T M EBw E endowed with
the orthogonal sum of the metrics on TM and E. Define V" by
Vx.y = VxY + a(X, Y),
vx.e ;; -A�X + Vxe,
X,Y ETM
XETM,�EE.
It is easy to see that V" is a compatible connection on É. Fur­
ther, using the fact that a and V' satisfy the Gauss, Codazzi and
Ricci equations for the case of constant sectional curvature zero,
it is straightforward to show that the curvature tensor of É van­
ishes identically. Choose a point x EM, and orthonormal vectors
el o . . . , en+p E Éx = 'Ir-1(x). Since M is simply connected and the
curvature tensor of É is zero, there exists unique global extensions
elo ... , en+p parallel with respect to V". These sections are pointwise
orthonormal because V" is compatible with the metric. Choose local
coordinates (xt. . . . , xn) in a neighborhood U of x. Hence, there are
functions aiv defined in U, such that
1 ::; i ::; n .
18
1. TH E BASIC EQUATIONS FOR SUBMANIFOLDS
Thus the coefficients of the metric on M are given hy
gij
Since the sections
=
(
-O
O
O
x]
XI
v=l
�v are parallel, we have
n+p O
� = L ajv �V.
"HZ; Ox]·
v=l OXi
V"a
Using the symmetry of a, that V is the Levi-Civita connection on
TM, and that
= 0, we have
[ô�j' ô�)
Oajv Oaiv
OXj - OXj·
_
Since closed l-forms are exact on U, there are functions Iv satisfying
= aiv. Define a mapping I : U ---+ Rn+p on a neighhorhood U of
X by I = (fI. ... ,fn+p ). Thus
�
1*
and for i,j
=
(o�J
=
(ail, ... , ajn+p),
l, . . . , n, we have
ln other words, I is an isometric immersion. We define an iso­
morphism � between the bundles TU EElw E and TRn+PI!(u) =
TI(U) EElw TI(U).l.. by �(�v) = ev, where ev, v = l, . . . , n + p, is
the restriction of the canonical frame of TRn+p to I(U). For the
.
Ô
+ p v."v,
tangent vectors ÔXj
= ",
ai , we h ave
L.Jvn=l
_
rp
( O) =
OXi
n+p
( )
n+p
O
=
v
�
L aj rp( v) L aivev = 1* OXj .
v=l
v=l
_
Hence � sends TM lu isomorphically onto TI(U). Being an isometry
in the fibres, it sends E isomorphically onto Tf(U).l... Moreover, since
EXERCISES
19
$ takes the paralIel orthonormal frame �1, . , �n+ pinto the paralIel
orthonormal frame
� E E,
e},
.
.
.
n+p' $
,e
.
.
satisfies for alI
X, Y
where V is the Levi-Civita connection of Rn+p.
components, and defining j = $IE' one gets
E TM and
Taking normal
ja.(X, Y) = ó(X, Y), j\l'xY = V't.xj(�).
If we had chosen different local coordinates (Yl, . . , Yn) we would
still end up with the equations
= aj/l' Since these equations
determine f up to a constant, the immersion is determined up to a
translation. If we had chosen a different initial frame, the isometries
would only differ by a rotation. So f is determined up to a rigid
motion. The fact that M is simply connected allows us to glue
together all those local isometries. For details see [Sp].
ln order to prove (ii), we have that the bundles T(Rn+P)lf(M) and
T(Rn+P)lg(M) both contain the tangent bundle T M, and that there
exists a bundle isomorphism which preserves metric and connection,
and is the identity on TM. Thinking of f as the one of (i) we see
that, on a neighborhood, they only differ by a rigid motion ofRn+p.
This concludes the proof. I
�
.
Exercises
1.1.
1.2.
Given an isometric immersion f : Mn � !VIn+p, show that TM.l is a
Riemannian vector bundle with compatible connection V.l.
Let f : Mn � Mn+p be an isometric immersion, and let,: [O, 1] �
M be a geodesic on !VI. Then, for each plane a C T"'((t)M such that
')" (t) Ea, prove that the Synge inequality
K(a) � K(a)
is satisfied.
1. THE BASIC EQUATIONS FOR SUBMANIFOLDS
20
1.J.
Show that the following maps are minimal immersions.
(i) The Clifford torus of dimension
I:
p(II. . . . ,ln) 1----+
sI
x ... x
sI
=
---+
given by
s;n-I
C
R2n
� (cos ..;n,I. sin ..;n,I. ... ,cos ..;n'n, sin ..;ntn).
vft
(ii) The Veronese surface I: S;/3
I(x, y, z )
n
C
(XYy'3' J3' y'3'
R
3
---+
S:
C
RS
defined by
x z yz x2_y2 X2 + y2_2Z2
,
2y'3
6
)
.
Furthermore, show that
(a) I induces an isometric embedding i:
s:' where
projective plane obtained from S;/3 by iden­
RPr/3
RPr/3
-+
is the
tifying antipodal points.
(b) (Rl.(X, Y)�,TJ} = 2/3, for alI orthonormal pairs X, Y
TS2 and �,TJ E TS21..
(iii) ThegeneralizedhelicQid I:
I(s, tI, . . . , l n )
where
=
Rn+I -+ Rn+k+I,
k
n-k
i=I
i=I
L tiei (S) +
E
n � k, defined by
L tk+iV2k+i
+
SbVn+k+h
+
VI. ... , Vn+k+I is the canonical basis of Rn k+l,
and b, ai E R, i = 1, . .. , k.
(iv) The spherical ruled minimal surfaces la: R2
a> Oby
-+
sl given
for
la(x,y) = (cosax cosy, sinax cosy, cosx siny, sinx siny).
Show that, with the induced metric, la is a ruled minimal
isometric immersion, that is, foliated by totally geodesic circles
in sl. Furthermore, the Clifford torus of dimension two belongs
to the family {falo
21
EXERCISES
1.4.
Give an example of a non-totally geodesic isometric immersion of Sr
into
Sln+l.
1.3
Hint: See exercise
1.5.
(i).
/: Mn
For an isometric immersion
S
=
c + n
n
_
1
-+
M: +P
1
II HI I2 - n( n
_
where s is the scalar curvature of
prove the relation
1)
I l al l 2 ,
Mn and II a l12 is the norm of the
second fundamental form defined by
n
I Ial12 = L Ila(Xi, Xj )112 ,
1.6.
where x}, ...
i,j=1
, Xn is an orthonormal frame.
A nonzero vector X E TzM is called asymptotic for an isometric
immersion if the second fundamental form satisfies
Show that
are
n
/:
Mn
-+
orthogonal asymptotic tangent vectors at
Hint: Use induction on dimension
1.7.
Let
/: Mn
-+
sn+p
a(X, X)
= o.
Mn+l is minimal at x EM if and only if there
n
.
x.
be an isometric immersion. The
cone
ove r / is
defined to be the immersion
(I, x)
1-+
I/(X).
Compute the second fundamental form ofF.
1.8.
Let
R(s,n-s)
be the vector space
Riemannian metric
Rn
(x}, ... , xn) , (Yl, ... , Yn ) } = (i) Show thatHn =
endowed with the pseudo­
n
L XiYi L XiYi·
s
i=1
+
{x ER(I,n): (x,x) = - l,Xl
i=s+1
> O} is a complete
connected and simply connected Riemannian manifold with
constant negative sectional curvature equal to
(ii) Consider the mapping
/:
-1.
Sr S�I,n+1) defined by
-+
/(x) = (1jJ(X), 1jJ(X), i(x»,
22
1.
THE BASIC EQUATIONS FOR SUBMANI FOLDS
where i:
-+ Rn +l is the inclusion, 1/J: sn
tiable function, and
Sr
S�l,n+l)(l) = {x ER(1,n + 2) : (x,x)
=
-+
R is a differen­
I} .
Show that f is an isometric immersion and compute its secolid
fundamental formo
(iii) Show that the map g: R(l,l) -+ H(1.2) C R(2 .2) given by
g(t,s) =
(
s2
_
2
t2
+ 1, t,s,
t2
- S2 )
2
'
where H(1.2) = {x E R(2,2) : (x,x) = -I}, is an isometric
immersion. Compute its second fundamental formo
1.9.
Show that if Mn is an extrinsic sphere in Mn +p, then:
(i) V.lQ = O, that is, the second fundamental form is parallel,
(ii) R(X, Y)Z = R(X, Y)Z + IIHII 2 (X" Y)Z, for every X, Y, Z
•
TM ,
(iii) R(X, Y)�
1.10.
1.11.
1.12.
1.lJ.
= R.l(X, Y)�, for every X, Y E TM
and �
E
E
TM.l.
Let f: M n -+ M n + p be an isometric immersion, and let N be a
subbundle of TM.l . Show that N is a párallel subbundle if and
only if for every normal section � which is parallel along a curve
"( : [a,b] -+ M, and such that �("«a» E N, we have H"(t» E N for
every t E [a,b].
Let f: Mn -+ Rn+p be an isometric immersion, and let � ETM.l be
a parallel normal vector field such that the shape operator A� is never
singular. Assume that h(x) = (f(x),�(x)} = constant. Conclude that
f(M) is contained in a sphere sn +p-l centered at the origin ofRn + p.
Let f : M n -+ Q� + P be an isometric immersion. and suppose there
exists a parallel unit normal vector field { ETM.l such that the shape
operator A� has n distinct eigenvalues. Show that f has ftat normal
bundle.
+l -+ R!n(n+1) +n+ l given by
Verify that the map t.p:
c Rn
Sr
provides an isometric immersion of the real projective n-dimensional
space Rpn into R !n (n +1) + n .
Chapter
2
H ypersurfaces
2.0
Introduction
Our aim in this chapter is the study of hypersurfaces, that is, isomet­
ric immersions with codimension one. Hypersurfaces in Euclidean
space Rn constitute a natural generalization of surfaces in R3 and,
consequently, several of their properties extend to hypersurfaces.
Initially, we discuss the basic equations and the Fundamental
Theorem for Submanifolds for the special case of hypersurfaces.
Later, in section 2, we prove a theorem due to Hadamard on the
convexity of compact hypersurfaces with nonzero Gauss-Kronecker
curvature. We dose this section classifying the umbilical hypersur­
faces of Eudidean space. The chapter concludes with the local and
global dassification of Einstein hypersurfaces in Euclidean space.
2. HYPERSURFACES
24
2.1
The Fundamental Theorem for Hypersurfaces
We proceed to the determination of the basic equations for hyper­
surfaces. Let f : M" --+ JVt"+l be an isometric immersion, and let
x E M. We can always consider locally a differentiable unit normal
vector field, i.e. , a differentiable vector field � in T M.l. defined in a
neighborhood U of x such that ( �Y' �y ) = 1 for ali y E U. ln fact,
there exist only two pos&ible choices for f Given X E TxM and
YE T M it is easy to see that the Gauss formula becomes
On the other hand, since � is a unit normal vector field, we have
(V x�,�) = O, hence V'�� = O for every X E T M . Therefore, the
Weingarlen formula becomes
Using the fact that a(X, Y) = (A{X, Y) � , we see that the Gauss
equation may be written as
�
The Codazzi equation now is
where by definition
ln the case that JVt"+l has constant sectional curvature c, the
equations of Gauss and Codazzi are, respectively,
and
2.1. FUNDAMENTAL THEOREM FOR HYPERSURFACES
25
Let f: M n ---. Mn+ 1 and g : M n ---. Mn+1 be two connected
hypersurfaces in Mn+l, and suppose that Mn+1 is orientable. Under
these conditions we daim that there exists an isometric vector bundle
isomorphism �: TM/ ---. TMg1.. ln order to construct such isomor­
phism we initially fix an orientation of M n + I. Then, for every x EM,
we choose an ordered basis XI, . . . , Xn in TxM, and a unit normal
vector {! E TxM/ , such that the ordered basis f. Xt, . . . , f.Xn, {! is
positively oriented in Tf (x ) M . Hence, there is a unique unit normal
vector field {; E TxMg1. such that the basis g.XI, ... , g.Xn , {; is
positively oriented in Tg(x) M. We define the vector bundle isometry
as the map which satisfies �({D = {;, and is linear on the fibres.
Clearly, � and -� are the only such maps.
We now state the Fundamental Theorem for Submanifolds in the
case of hypersurfaces.
2.1
Theorem
(i) Let Mn be a simply connected Riemannian manifold, and let
A: TM ---. T M be a symmetric tensor satisfying the Gauss and
Codazzi equations in the case of constant sectional curvature c.
Then there is an isometric immersion f: M n ---. Q:+ such that
A = A{ for some unit normal vector field { E TM1., where A{
denotes the second fundamental form of the immersion f.
(ii) Let f: M n ---. Q:+l and g : M n ---. Q:+I be connected hyper1.
1.
surfaces, and let r/J: TMI ---. TMg be one of the two vector
bundle isomorphisms. Suppose that
I
or
for every X, Y E TM, where aI and ag denote, respectively,
the second fundamental forms of f and g . Then there exists an
isometry
such that g
=
7
o
f, and 7.
-
= r/J or 7.
=
-r/J on TM .
-
1.
26
2. HYPERSURFACES
Emmple.
We use the preceding theorem to obtain non-trivial
isometric immersions f : R n � R n + I . Consider a non-negative
differentiable function k : R � R. A sim pIe calculation shows that,
with respect to a canonical basis e I , . . . , e n of R n and coordinates
(t I , . . . , tn ), the symmetric tensor A : R n � R n defined by
satisfies the Gauss and Codazzi equations for zero sectional curvature.
From Theorem 2. 1 (i) there exists an isometric immersion f : R n �
R n +I whose second fundamental form is A. On the other hand, we
know that, up to isometries of R2, there is a unique smooth curve
"( : R � R2, parameterized by arc length, with curvature equal to k .
Now, let j: R n � Rn +I be given by j(t I ' . . . , t n ) = ("((tI), 12, . . . , ln ).
It is easy to see that j is an isometric immersion whose second
fundamental form is A. Hence, from Theorem 2. 1 (ii), we conclude
that f and j are equal up to isometries of R n +I . We will see, in
Chapter V, that any isometric immersion from R n imo Rn +I has this
form, that is, it can be written as:
"(
x
I:
R x Rn- I
R n - I = R n +l,
R n - I R n - I i s the identity map.
�
R2 X
�
where "( i s a plane curve and I :
Such immersion is called a cylind;er over the curve "(.
Let us consider a hypersurface f : _M n � li1 n + t , and let � be a
unit normal vector field defined in a neighborhood of a point x E M.
We define the princiPal curvatures of f at x to be the eigenvalues of A{x '
and the princiPal directions to be the corresponding unit eigenvectors.
The product of all principal curvatures K = À I . . . . . Àn is called
the Gauss-Kronecker curvature of f. It is not difficult to verify that for
hypersurfaces in Q� + I , the Gauss-Kronecker curvature is invariant
under isometries if n is even, and invariant up to sign if n is odd (see
exercise 2.3).
Given an orientable hypersurface f : M n � R n + 1 of Euclidean
space Rn + l , we choose a global unit normal vector field � E T M .L .
The (normal) Gauss map is defined as
l/> : M n
�
Si
x � �x ,
2.2. CONV E X EUCLIDEAN HYPERSURFACES
27
where Sf C Rn+l is the canonical unit sphere, and Çx E Sf denotes
also the parallel translation to the origin in Rn+ l of the vector Çx E
.L
TxM . Observe that for each x E M n , the vector spaces Tx M and
Tt/I(x)Sf are parallel in Rn + l . Hence, there is a canonical isomorphism
between Tx M and Tt/I(x)Sf which allows us to identify these two spaces.
2.2
Proposition
Let f : Mn --+ Rn + l be an orientable hypersurface with Gauss map
cp : M n --+ Sf . Then, for each x E Mn, we have
Proof. Given X E Tx M , ler "I : ( -f, f) --+ M be a differentiable curve
such that "1(0) = x and "I' (O) = X . Then, we have
The result follows from the Weingarten formula. I
2.2
Convex Euclidean hypersurfaces
Given an immersion f : M n --+ R�+ l , we say that f is locally convex at
a point x E Mn when there exists a neighborhood U of x at M , such
that f (U) lies on one si de of the tangent hyperplane of M at x in
Rn + l . We say that the immersion is strictly locally convex at x when f(x)
is the unique point in f(U)nf. (Tx M). The cylinder over a circle and
the sphere are, respectively, examples of convex and strictIy convex
hypersurfaces at any point.
28
2. HYPERSURFACES
2.3
Proposition
Let f : Mn ---+ Rn + l be a hypersurface with definhe second fundamental
form at a point Xo E M . Then f is strict/y locally convex at Xo . ln
particular, any compact hypersurface M n ofR n + 1 is strict/y locally convex
at some point.
.L
Proof. Let �xo E TxoM , and let h : Mn ---+ R be the height function
given by
h (x ) = (f (x ) - f (x o ) , �xo ) .
Then, for every X E Txo M ,
X (h )
=
( X, �xo )
=
O,
and from the Gauss formula
Since Xo is a criticaI point for h, where A{zo is definite, h has a strict
local minimum or maximum at Xo . Hence f is strict1y convex at Xo .
The second assertion follows from Proposition 1 .3;- 1
We say that an embedded hypersurface f : M n ---+ Rn + l is a con­
vex hypersurface when it is the boundary of a convex body B C R n + l .
By a convex body we mean an open subset B of Rn + l such that, given
two points p, q E B , the line segment joining p to q is contained in
B.
The second fundamental form and the Gauss-Kronecker curva­
ture are dosely related to the convexity of hypersurfaces, as shown
by the following result due to Hadamard [H].
2.4
Theorem
Let f : M n ---+ Rn + l be a compact hypersurface. The following assertions
are equivalent:
(i) The second fundamental form is definite at every point of M,
(ii) M is orientable and the Gauss map is a dilfeomorphism,
(iii) The Gauss-Kronecker curvature K is non-zero at every point.
2.2. CONVEX EUCLIDEAN HYPERSURFACES
29
Furthermore, any of the above conditions implies that the hypersurface
is a convex hypersurface.
Proof. (i) => (ii). Choose at every point x E M n a unit normal vector
{x so that Açx is negative definite. Since the second fundamental
form is definite everywhere, we conclude that such vector field {
exists and is continuous on M . So M is orientable. Since the map Aç
is non-singular, we have from Proposition 2 . 2 that (q;* )x is injective
for every x E M n . Hence q; is a local diffeomorphism. Actually q;
is a covering map, because Mn is compacto We conclude that q; is a
global diffeomorphism from the fact that Sr is simply connected for
n
� 2.
(ii) => (iii). Since q; is a diffeomorphism , (q;* )x is injective at every
x E M n , and hence everywhere non-singular. Using Proposition 2.2,
we conclude that the Gauss- Kronecker curvature is non-zero at any
point.
(iii) => (i) . From Proposition 1.3 we know there is some point
Xo E M such that Açzo is definite. Since the Gauss-Kronecker
curvature K is different from zero, we have that the second funda­
mental form is everywhere nondegenerate. It follows that the second
fundamental form is definite at ali points of M .
ln order to prove the last assertion we will first show that f i s an
embedding. Since M is compact, it suffices to show that f is one-to­
one. Suppose there are X l , X2 E Mn such that f (xt) = f (X2)' Now
.L
choose a unit normal vector field { E T M so that Aç is negative
definite on M n . Consider the height function h : M ---+ R given by
Then h(xd = h(X2) = O. From previous arguments (proof of
Proposition 2.3) it follows that X l is a local strict maximum of h.
We claim that, actually, X l is the unique strict maximum of h. For
if y E M is a local strict maximum of h we have that, for every
Z E Ty M ,
which implies that {X l
=
±{y , and that
Z Z (h) = (AÇZI Z, Z )
<
O,
30
2. HYPERSURFACES
which implies �X l = �Y ' i.e., the Gauss map satisfies l/J(XI) = l/J(y).
Therefore, X l = Y since l/J is a diffeomorphism. Being the unique
local maximum, X l is in fact the global maximum of h. From h(XI ) =
h(X 2 ), we obtain X l = X 2 .
Since f is an embedding, it follows from the Jordan-Brouwer
separation theorem that f(M) divides Rn+ l into two arcwise-connect­
ed components. Both components have f(M) as boundary, and one
of them, say B , is bounded. We conclude the theorem by showing
that the interior of B is a convex body.
Consider arbitrary points p, q E int B . There are points p =
Yo , Yt. . . . , Yr = q in int B such that the segments YOYI , Y I Y 2 , . . . ,
Y r-I Y r form a polygonal path entirely contained in int B . We want
to prove that the segment pq itself is contained in int B . Suppose, by
contradiction, that there exists some 1 < j ::::: r such that PY i C int B ,
1 ::::: i ::::: j - 1 , but P Y j ct i nt B Let f3 : [0, 1] -+ int B be given by
f3( s ) = sY j + (l - S )Yj l and define as : [0, 1] -+ Rn+ l by as (t ) =
tf3(s ) + (1 - t )p. Since f(M ) is closed, we have Zl = as 1 (tl ) Ef (M),
where S I = sup{s E [0, 1] : as ([O, 1]) n f(M ) = l/J}, and t I = inf { t E
([0, 1]) : aS1 ([0, t]) n f(M) -:f l/J}.
Let X l E M be such that f(X I) = Zl . Choose a unit normal vector
field � so that A{ is negative definite. From a previous argument, the
function h(x) = (f(x) - f(X t}' �X l ) has a unique global maximum
which is X l . On the other hand, by construction, �Xl points inward,
and this allows us to find À > ° such that
.
-
'
since f(M ) is compacto Thus h(X 2) = À > 0, and this is a contradic­
tion since Xl is the maximum of h and h(XI) = O. I
There is a result similar to Theorem 2.4 for the case of compact
orientable hypersurfaces of the sphere s n + l , obtained by do Carmo­
Warner [C-W] . Under the weaker assumption of non-negative Gauss­
Kronecker curvature, Chern and Lashof [Ch-L] proved that a com­
pact surface in R3 with non-negative Gaussian curvature (which
coincides with the Gauss-Kronecker in this case) is convexo They also
give an example of a compact, non-convex hypersurface in Rn+ l ,
n 2:: 3, with non-negative Gauss-Kronecker curvature. ln the case
2.2.
CONVEX EUCLI DEAN H YPERSURFAC ES
31
of surfaces i n R3 , the compactness assumption may b e replaced by
completeness. It is shown in Stoker [Sto] that complete surfaces
in R3 with positive Gaussian curvature are convexo Furthermore,
they are homeomorphic to the sphere when compact, and to the
plane when non-compact. ln a more general setting, Sacksteder
[Sd shows that complete hypersurfaces in R n + l with non-negative
sectional curvature are convex if there exists at least one point where
all the sectional curvatures are positive. For related results see also
[W] , [Ca-L] , [Rod, [ Küh], [Cu], [Tr], [AI], and [M].
We now tum our attention to the classification of all umbilical
hypersurfaces of Euclidean space.
2.5
Proposition
Let f : M n -+ Rn + l be an umbilical isometric immersion ETOm a con­
nected Riemannian maniEold Mn into Rn + l . Then f(M) is an open
subset of either an afline hyperplane or a sphere.
Proof. Choose a point x E M , and a unit normal vector field {
defined in some neighborhood U of X. Since f is umbilical, there is
a function À : U -+ R such that A{ = >.I in U , where I is the identity
tensor. ln particular À = � trace A{ is differentiable. Given vector
fields X, Y E T U , it follows from Codazzi's equation that
Taking X and Y linearly independent, we conclude that À is constant
on U.
If À = O , the Weingarten formula shows that V x { = O for every
vector field X E TM l u . Hence { is constant in Rn + l . Now, given any
i E U, and any differentiable curve T [0, 1] -+ U joining x to i , we
have
d
( f o 'Y)(t), {('Y(t))) = (f"'Y'(t), {) = O.
dt
This shows that ( f 0'Y)(t), {) is constant. Therefore f (U) is contained
in the hyperplane passing through f(x) and normal to {.
If À =I O in U, we have
2.
32
H YPERSURFACES
for every vector field X E TU . Then, there exists a point c E Rn+ l
such that I(y ) + À - l �y = c for every y E U . ln other words, I( U ) is
contained in a sphere with center c and radius I À I - 1 •
We have just proven that the set of points in Mn whose image
under I belongs to a hyperplane or a sphere, is open in Mn . An easy
argument shows that this set is also dosed in M n . The result follows
from the connectedness of Mn . I
2.3
The classification of Einstein hypersurfaces
Our next goal is to dassify the Einstein hypersurfaces in Rn+l . Recall
that a Riemannian manifold is an Einstein manifold when the Ricei
tensor satisfies
Ric (X, Y) = p(X, Y)
for ali tangent vectors X, Y, and some constant p E R. Riemannian
manifolds M n with constant sectional curvature c are the simplest
examples of Einstein manifolds, where p = ( n - l)c" Conversely,
for n = 3, every Einstein manifold has constant sectional curvature.
However, in dimension 4, there are Einstein manifolds with non­
constant sectional curvature, (see exereise 2.2). We say that an
immersion I : Mn --+ Mn+ l is an Einstein hypersurface when Mn is
an Einstein manifold with the induced metric. The following result
is due to Thomas [Thd, Fialkow [Fi], and Ryan [RY ll
2.6
Theorem
If I : Mn --+ Rn+l, n
is non-negative, and
� 3, is a connected Einstein hypersurface, then p
. (i) ii P = O then Mn is locally isometric to Rn,
(ii) ii P > O then I(Mn ) is contaÍned in a sphere sn .
Proof. Let x E M n be an arbitrary point, and let A = A{ be the
second fundamental form of I . From the Gauss equation and the
definition of the Ricei tensor, we obtain
Ric (X, Y)
=
r(AX, Y) - (AX, AY)
2 . 3 . Cl.ASSI FICATION OF EI NSTE I N HYPERSURFACES
33
for every X, Y E Tx M , where r = trace A . Since f is an Einstein
hypersurface, the above equation becomes
r {A X, Y } - {AX, AY } - p{X, Y}
=
O.
TxM
Now, choose an orthonormal basis Xl , " " Xn E
= ),j X , 1 � j � n . The last equation appears as
j
AXj
),� - r ),j + p
=
1
O,
�j�
such that
n.
( 1)
We have for 1 � j � n , that the principal curvature ),j is a root of
the quadratic equation x2 - r x + p = O . After reordering, we may
suppose ),1 = . . = ),p and ),p +1 = . . = ), n for some 1 � P � n .
.
.
x (x
r) = O. Suppose ),1 =
ln this case (n - 1»,2 = O. So f is totally geodesic and
thus, locally isometric to R n • ln case O = ),1 = . . = ),p :I ),p + l =
. . = ), n = )" we have (n - p
1»,2 = O, hence p = n 1 . Thus
only one principal curvature is non-zero. It follows · from the Gauss
equation that Mn has zero sectional curvature, therefore it is locally
isometric to R n •
.
.
.
If p
=
= ), n
=
O, the equation becomes
-
), .
.
.
-
-
If p > O, we claim that f is umbilical but not totally geodesic.
ln fact, if at some point x E
we have ),1 = . . . = ),p = v :I
.
.
JL = ),p + l = . = ), n , it follows from the quadratic equation ( 1) that
v + JL = r = pv + (n - p)JL, or equivalently, (p - 1)v + (n - p - 1)JL = O,
and that vJL = p > O. The last equality shows that JL and v have
the sarne sign, and from the first equation, we conclude p = 1 and
n = p + 1 . This is a contradiction for n
The immersion is not
totally geodesic because p > O . From Proposition 2.5 it follows that
f(M) is contained in a sphere.
M
� 3.
ln order to conclude the proof we need to show that p � O. This
is more delicate. Suppose p < O. As before, we hav e ),1 = . . . = ),p =
v and ),p + l = . . . = ),n = JL, for some 1 � p � n , where JL :I v at
every point. and satisfy
(p - l)v + ( n
-
p - l)JL
=
O,
and
It follows that (p - 1)v2 + (n - p - 1)p = O. If p
which is impossible . Thus, p > 1 and
(n - p - 1)
p.
(p - 1)
vJL
=
=
p < O.
1 , then (n - 2)p
=
O,
(2)
34
2. HYPERSURFACES
Since r = trace A is differentiable, the distinct roots p. and v of the
equation X 2 - r x + p = O are also differentiable functions. From
relation (2) we conclude that v is constant. Therefore p. and p are
also constant.
Next, we consider the two orthogonal distributions Dv and Dp
defined by Dv(x) = {X E TxM _ : AX = vX}, and Dp (x) =
{X E Tx M : AX = p.X}. First we show that these are smooth
distributions. Let Xo E M , and let Xl , . . . , Xn be differentiable vector
fields in a neighborhood of Xo such that Xl ( xo), . . . , Xp (xo) and
Xp + l (xo), . . . , Xn (XO) are, respectively, bases for Dv(xo) and Dp(xo).
Define differentiable vector fields Yi = (A - p.I)Xi, 1 � i � p,
and Yj = (A - vI)Xj, P + 1 � j � n , where I is the identity
tensor. Since (A - vI)Yj = O, 1 � i � p ; (A - p.I)Yj =
O, P + 1 � j � n ; and Yj (xo) = (v - p.)Xi(xo), 1 � i � p ;
Yj(xo) = (p. - v)Xj (xo), p + 1 � j � n , w e conclude _ that
Y1 . . . . , Yp and Yp + l , . . . , Yn are, respectively, bases for Dv and Dp
in a neighborhood of Xo.
Now we claim that Dv and Dp. are parallel distributions. l n order
to prove this assertion, consider vector fields X E Dv and Y E Dp.
Since v and p. are constant, it follows from the Codazzi equation that
(A
-
p.I)'\l x Y = (A - vI)'\ly X.
By observing that Im(A - p.I) E Dv, and that Im(A vI) E Dp, we
obtain that 'V x Y E Dp., and that '\ly X E Dv. Furthermore, for every
differentiable vector field Z E Dv, we have
-
0 = Z ( X, Y) = ( '\lzX, Y) + ( X, '\lzY ) .
Since '\lzY E Dp., it follows that ( '\lz X, Y) = O. As Y E Dp is
arbitrary, we have '\l zX E Dv. This shows that Dv is parallel, and
then, so is Dp.. ln particular, for X E Dv and Y E Dp, we have
(R( X, Y)Y, X ) = O. On the other hand, from the Gauss equation
(R(X, Y)Y, X ) = vp. < O. This is a contradiction, and shows that
p � 0· 1
The global classification reads
2.3. CLASSIFICATION OF EINSTEIN HYPERSVRFACES
2. 7
Theorem
35
The complete Einstein hypersurfaces in Rn+ l are spheres or cylinders
over complete plane curves.
Proof. The fiat case follows from Theorem 2.6 (i) and the Hartmann­
Nirenberg theorem in Chapter V. For the spherical case, use The­
orem 2.6 (ii) and a standard argument of covering spaces for n � 3.
For n = 2 and negative curvature the result follows from a classical
theorem of Hilbert ([Sp], I I I p. 373) which says that there are no
complete surfaces of constant negative Gauss curvature in R3 . The
positive curvature case is Hilbert's theorem ([Sp], I I I . p. 349) of
rigidity of the sphere in R3 . I
We point out that the case in which the ambient space has non­
zero constant sectional curvature was also considered in Fialkow [Fi]
and Ryan [RY l]' See also [RY2 l
2.8
Remark
Observe that, in the proof of Theorem 2.6 (i), we obtained that
if A has exactly two distinct eigenvalues, then those eigenvalues,
together with their distributions, are smooth. More generally, it can
be shown [Nod that if À is a continuous eigenvalue of A with constant
multiplicity, then À and its distribution T>. are both differentiable.
Moreover, the hypothesis on the continuity of À is always satisfied
when A : T M � T M is any symmetric tensor, as shown in [Ryd. See
also [C-R] for other related facts.
For manifolds with constant scalar curvature, Cheng and Vau
[C-V] have shown that complete hypersurfaces in Rn+ 1 , with non­
negative sectional curvature and constant scalar curvature, are gen­
eralized cylinders SP x Rn-p, O � P � n. ln fact, [C-V] also
considered the case in which the ambient space has non zero constant
curvature. Recently, Ros [Rosd proved that the sphere is the only
compact hypersurface with constant scalar curvature embedded in
Euclidean space. See aiso [8s], [ROS2] , [Ko], [M-R] , and [Vi].
2.
36
HYPERSURFACES
Exercises
2.1 .
2.2.
2.3.
Show that any 3-dimensional Einstein manifold has constant sectional
curvature.
Show that the Riemannian product Mt x M�- P of two manifolds
of constant sectional curvature is an Einstein manifold if and only if
(p - l)CI = ( n - p - 1 )c2 .
Let M n be a hypersurface in Rn + l with principal curvatures k1 . . . . ,
k n.
(i) Show that the set o f (;) numbers {kjkj : i < j } i s intrinsic, i.e.,
it is independent of the isometric immersion.
(ii) Conclude that the Gauss-Kronecker curvature is intrinsic if n is
even, and invariant up to sign if n is odd.
(iii) Show that (i) remains valid if the ambient space has non-zero
constant sectional curvature.
Hint: The above set of numbers is the set of eigenvalues of the
endomorphism of the space of 2-forms n2(Tx M) defined as follows:
Let Xl , . . . , Xn be a basis of Tx M with dual basis Xt, . . . , X; , and
consider the map n2(Tx M) � n2(Tx M) given by
xt
2.4.
Â
XI'
f-----+ R(Xj , Xj).
This map makes sense since R ( X, Y) = -R(Y, X) and is indepen­
dent of the choice of basis.
We say that an isometric immersion f : M n � Q�+P is ruled if M
admits a continuous codimension one foliation such that f maps each
leaf (ruling) onto a totally geodesic submanifold of Q�+p . Show that
the following hypersurfaces in R n + l are complete and ruled.
(i) Assume that the curve c : R � R n +l has a Frenet frame ê =
e l , e 2 , . . . , e n + l . Then consider the hypersurface F : Rn � Rn+ l
given by
F(S, tl , . . . , t n-l)
n
=
(ii) Consider the graph G : Rn
c(s) + L tjej+1 (s).
j=2
�
Rn + 1 defined by
37
EXERCISES
where f/Jj E COO( R ),
2.5.
A
1�j�
n
-
1.
rotation hypersurface Mn i � Rn + l can be locally parameterized by
F(t, S l , . . . , S n- l ) = (e(t ), tf/J(Sl, . . . , S n - l )),
where e(t)
>
S1n -l lO
· Rn .
0 , and f/J
=
f/J(s) is a parameterization of the unit sphere
(i) Show that the principal curvatures satisfy
(ii) Conclude that the minimal rotation hypersurfaces (generalized
catenoids) are given by the solutions of
(iii) Show that the solutions for
e'
n
� 3 of
= ± (at n - 2 - 1)- !
give rise to complete non ftat rotation hypersurfaces with van­
ishing scalar curvature.
2.6.
2.7.
Suppose that the hypersurface f : M n - Q�+P , n � 3, has a
principal curvature of constant multiplicity k , 2 � k � n . Show that
the corresponding eigenspaces form a smooth integrable distribution
such that each leaf is umbilical in the ambient space.
Let f : M n _ Q� + l be an orientable hypersurface, and let ç be a
unit normal vector field to the hypersurface. For each t > 0 , define
f, (x) E Q�+ l to be the point on the geodesic starting from f(x) in
the direction Çx , which has geodesic distance t from f (x). That is,
f, (x)
f, (x)
=
=
f (x ) + tçx ,
if
cos tf(x) + sin tçx , i f
c
= 0,
c >
O.
Let gt , At denote the first and second fundamental forms of the
parallel hypersurface f" respectively.
(i) Compute gt .
2 . HYPERSURFACES
38
(ii) Verify that
( I - tA ) - l A,
At = (cot tl - A)- l (cot tA + I ) ,
At
=
if c = O
if c > O.
(iii) Show that !t has constant mean curvature for each t if and only
if f has constant principal curvatures (isoparametric hypersurface).
(iv) Verify that the unit normal bundle of the Veronese surface in
S4 is an isoparametric hypersurface.
(v) Given any surface in R3 (or S 3 ) with non-zero constant mean
curvature, show that the family of parallel surfaces contains
another surface of constant mean curvature.
2.8.
Let Mn be an n-dimensional oriented Riemannian manifold, and let
f : Mn f--+ Sl + 1 C Rn+ 2 be an isometric immersion. Then the unit
normal vector field of Mn in Sl+ 1 induces a mapping v : Mn f--+ Sf+ 1 ,
called the spherical Gauss map of the immersion f .
(i) Prove that v i s an immersion provided that the second funda­
mental form A of f is everywhere non-singular.
(ii) Compute the metric induced by v, and show that its second
fundamental form is A - 1 .
(iii) If n = 2, and f i s a minimal immersion, conclude that v i s also
a minimal immersion except at the points where the Gaussian
curvature satisfies K = 1 .
Chapter 3
Submanifolds with
Non-positive Curvature
3.0
Introduction
It is a classical fact that every compact surface in R3 has an "elliptic
point" , that is, a point where the Gaussian curvature is positive. ln
particular, if M2 is a compact Riemannian manifold with non-positive
Gauss curvature everywhere, then M2 cannot be isometrically im­
mersed in R3 . More generally, Proposition 1 .3 shows that, if Mn is a
compact Riemannian manifold, and f : M n ---+ Rn+p is an isometric
immersion, then there is a point x E M n , and a normal direction
� E Tx M � , such that the second fundamental form A( is positive
definite. However, for n � 3, the existence of such � does not reflect
in general any strong intrinsic property of M n . Therefore, contrary
to the bidimensional case, we do not obtain immediate restrictions on
the existence of isometric immersions. We will show, however, that in
sufficiently low codimension, the existence of isometric immersions
imposes strong restrictions on the curvature tensor.
There is a long series of results along this line, initiating in 1 939
with Tompkins, and passing through the works of Chern-Kuiper,
O'Neill, Stiel and Moore, among others. A more general result was
obtained in 1 98 1 by jorge-Koutroufiotis.
3.
40
S U B MAN I FOLDS
WITH
NON-POSITI V E CURVATURE
AIthough we couId have proven the ]orge-Koutroufiotis theorem
directly, and derived other resuIts as corollaries, we chose to expIain
the techniques used in some intermediate works because of their own
interest.
3.1
The Chern-KuiPer Theorem
The basic tooI for this chapter is the aIgebraic Iemma below, which
was conjectured by Chern-Kuiper [C-K), and proved by Otsuki [Ot) .
3.1
Lemma
Let V and W be real vector spaces endowed with positive definite inner
products and dimensions n and p, respectively. Let a : V x V ---+ W be
a symmetric bilinear form such that
(i) (a(X, X ), a( Y, Y ») - I l a(X, Y) 11 2 � À ,
(ii) Ila(X, X) I I > ,.fi,
for some real number À 2:: O, and every orthonormal pair X, Y E V.
Then p 2:: n .
Prooj. Suppose p < n, and set S = {X E V : I I X I I = I}. Let Xo
be a minimum for the smooth map f : S ---+ R defined by
f(X) = Il a(X, X) 1 1 2
For a unit vector Y E TXoS , the curve "f : R
E
S
•
---+
S given by
"f (t ) = cos tXo + sin tY
satisfies "f (0)
O
=
=
Xo, "f' (0)
Y (f )(Xo)
=
=
2
Y. We have
( :t a("f(t ), "f(t » l t=o ' a(Xo, Xo) )
= 4 (a(Xo , Y), a ( Xo , Xo ») .
(1)
3. 1 . TH E CH ERN-KUI PER THEOREM
41
Using '}'''(O) = -Xo, i t follows that
o
� YY(f )(Xo)
=
8 1I a(Xo, Y) 11 2 - 4 1I a( Xo, Xo) 11 2
+
4 (a(Y, Y), a(Xo, Xo)) .
(2 )
Now, consider the linear map L : TxoS
a(Xo, Y) . Equation ( 1 ) implies
( L (Y), a(Xo, Xo))
=
W given by L ( Y)
---->
=
O.
Hence dim 1m L � p - I, since I la (Xo, Xo) 11 > V>. 2:: O. It follows
that the kernel of L contains a unit vector Yo orthogonal to Xo . From
equation (2) and the hypothesis, we obtain
o �
(a(Yo, Yo), a(Xo, Xo)) - I l a(Xo, Xo) 11 2
<
which is a contradiction. I
3.2
Corollary
,\
-
(.J>..f
= O,
Let V and W be real vector spaces of dimensions n and p . respective1y,
endowed with positive definite inner products. Let a : V x V ----> W be
a symmetric bilinear form such that
(a(X, X), a(Y, Y)) - Il a (X, Y) 11 2
for every orthonormal pair X, Y E V . Then p 2::
n
<
O
- 1.
Proo/. I f a (X, X ) t- O for every X E V, the result follows from
Lemma 3. 1 . Suppose p < n - 1, and that there is a non-zero vector
Xo E V such that a(Xo, Xo) = O. Denote by U the orthogonal
complement to Xo in V, and consider the linear map L : U ----> W
defined by L(Y) = a(Xo, Y) . Since dim U = n 1 > p , there is
a unit vector Yo E U such that L ( Yo) = O . This fact, together with
a(Xo, Xo) = O, contradicts the assumption. I
-
Remark. Assuming the hypothesis 'of Lemma 3. 1 and, further, that
,\ = O, we can obtain a stronger result as follows. Extend a to
42
3. S U B MANIFOLDS WITH NON-POSITIVE CURVATURE
a complex symmetric bilinear form in the complexification of the
spaces involved, and suppose p < n. The equation a(Z, Z) = O is
equivalent to p quadratic equations al (Z, Z) =
= ap (Z, Z) = O
in n variables. It is a well-known fact that p < n implies the existence
of a non-trivial solution Z (see [Uar], p.48), which cannot be real by
assumption. If Z = X + iY, we have
.
O
=
a(Z, Z ) = a(X, X) - a(Y, Y)
+
=
a(Y, Y) :f O ' and
.
2ia(X, Y ).
Thus, we obtain a pair of non-zero vectors X, Y
a(X, X )
.
E
V
a(X, Y)
so that
=
O.
This yields a contradiction. The existence of such a pair was shown
to be very useful, d. O-K].
The following result, due to Otsuki [Ot], is a direct consequence
of Corollary 3 . 2 .
Let f : Mn ...... Mn + p be an isometric immersion. If there exists a point
Xo E M, and an m-dimensional subspace B xo C Txo M, with m � 2 ,
such that K(a) < K (a) for every plane (J C B xo , then p � m 1 .
3.3
Theorem
-
Prooj. Apply Corollary 3.2 to the restriction of the second funda­
mentai form of f to Bxo x Bxo ' I
The assumption on the dimensions in the above result cannot
be improved, even globally, as shown by the n-dimensional Clifford
torus ln s 2 n - l .
M
For what follows, we will need some definitions. Let f :
n ......
be an isometric immersion, and let x be a point of M . The
subspaces of TxM given by
Mn+ p
r(x)
=
{X
E
Tx M :
R(X, Y) = (R(X, Y) IT.r M )T ,
for every
Y
E
TxM }
3 .1.
TH E CHERN-KU I PER TH EORE M
43
and
.6.(x)
=
{X E Tx M : Q(X, Y)
=
Y E Tx M } ,
O, for every
are called the subspace of nuUity of f at x , and the subspace of relative
x , respectively. The dimensions p.(x ) of r(x ) and v(x)
of .6.(x) are, respectively. the index of nullity and the index of relative
nullity of f at x . Whenever M = Me, the subspace r(x ) , unlike
.6.(x), is intrinsic, i . e., does not depend on the isometric immersion.
The following result, obtained by Chern-Kuiper [C-K], shows that the
above indices are related.
nullity of f at
1.4
Proposition
Given
an
isometric immersion f : Mn
-
Mn+p, we ha ve for all x
v(x ) � p.( x) � v(x )
+
E M,
p.
Proof. The first inequality is obvious, for .6. (x )
C
r(x) by the Gauss
equation. ln order to prove the second inequality. let L denote the
orthogonal complement of .6.(x ) in r(x ). We may assume dim L 2 1.
I t follows fTOm the Gauss equation that
(a( X, X) , a(Y, Y)) - I l a(X, Y)1I 2
=
O
for alI X E L and Y E Tx M . Since X f!Í. .6.(x ) , a(X, Y) =f O for some
Y E TxM . We conclude that a(X, X) :f O for any non-zero X E L.
Then we can apply Lemma 3 . 1 to the restriction of a to L x L to
obtain p 2 d i m L . Therefore p.(x ) = v(x) + dim L � v(x) + p . 1
We know that if f : M n - Rn+p is an isometric immersion of a
compact Riemannian manifold into some Euclidean space, then there
exists a point Xo E M , and a normal direction ç E TxoM .l , such that
the second fundamental form A{ is positive definite. ln particular,
vexo) = O, and from Proposition 3.4 we have p 2 p.(xo). This proves
the following:
44
3.
3.5
Proposition
SUBMAN I FOLDS WITH NON-POSITIVE
CURVATURE
If Mn is a compact Riemannian manifold, and f : Mn � Rn+p is an
isometric immersion satisEying ",, (x) � I, for every x E Mn, then p � I .
Since the index of nullity of any flat submanifold of Euclidean
space is equal to n, the above result generalizes the theorem of
Tompkins [To] referred in the introduction . ln fact, as a consequence
of Lemma 3. 1 , we easily obtain a more general result, due to Chern­
Kuiper.
3.6
Theorem
Let Mn be a compact Riemannian manifold such that, for every x E M ,
there is a m -dimensional subspace Bx C TxM, with m � 2, satisEying
K (a ) :::; O for any plane a C Bx . IE f : Mn � Rn+p is an isometric
immersion, then p � m .
Proof. Since Mn i s compacto by Proposition 1 .3, there is a point
Xo E M with the property that n( X , X) =I O for every non-zero
X E TxoM . Furthermore, for every orthonormal pair X, Y E Bxo '
we have
( n ( X, X), n ( Y, Y ») - Il n (X, Y) 11 2
=
K(X, Y)
:::;
O.
The theorem follows from Lemma 3. 1 by restricting n to Bxo
x
Bxo . 1
Historically, Chern-Kuiper [C-K] proved Theorem 3.6 for di­
mensions n = 2, 3. Later on, Otsuki [Ot] proved Lemma 3. 1 with
.À = O and, consequently, obtained Theorem 3 . 6 for ali dimensions.
On the other hand, Chern-Kuiper provided a proof of Proposition
3.4 independent of Otsuki's Lemma. ln fact, the Chern-Kuiper
inequalities 3.4 as well as Tompkin's theorem follow easily from
the symmetric version of Corollary 6.6, which was first obtained by
E. Cartan (see [C8.4] p. 1 54).
Theorem 3.6 has been subsequently generalized by O'Neill
[O'N], Stiel [Stj , Moore [M03], and finally by jorge-Koutroufiotis O-K]
whose result we present in the next section.
3.2. THE ]ORGE-KOUTROUFIOTIS TH EOREM
3.2
45
The ]orge-Koutroufiotis Theorem
A complete, simply connected Riemannian manifold, with non-pos­
itive sectional curvature is called a Hadamard manifold. It is a standard
fact for Hadamard manifolds that the exponential map at any point is
a global diffeomorphism (see [Sp] , IV p. 330). Recall that a subset A
of a Riemannian manifold is bounded if it is contained in some (metric)
ball of finite radius. Note that an unbounded set cannot be compacto
We are now in a position to state the jorge-Koutroufiotis U-K] result.
3. 7
Theorem
Let Mn be a complete Riemannian manifold with scalar curvature
bounded below, and M n + p a Hadamard manifold with p :-:; n 1 . If
f : M n ---+ Mn+p is an isometric immersion, and
:-:; K(a) for every
point x E Mn and every plane a C TxM , then f(M ) is unbounded.
K(a)
-
The version of the ]orge-Koutroufiotis theorem presented here
is a simplified one which avoids some technical details of the general
case. Before giving the proof of the theorem, we need to establish
two lemmas and some formulas. Our basic· tool will be the following
result due to Omori [O], which we do not prove here.
3.8
Lemma
Let M be a complete Riemannian manifold whose sectional curvature is
bounded fTOm below, and suppose that the smooth function h : M ---+ R
is bounded Erom above. Then, for every Xo E M and every E. > O, there
exists x E M satisfying:
(i) h ( x) � h(xo) ,
(ii) I l grad h (x) 11 < E. ,
(iii) Hess h(x)(X, X)
<
E. I I X I1 2 , for all X ETx M , X t- O.
Here grad h ( x) denotes the gradient of h at x E M , defined by
(grad h(x), X ) = X(h )(x ),
X E Tx M,
46
3.
SUB MAN I FOLDS WITH NON-POSITIVE C URVATURE
and Hess h(X ) is the hessian of h at x E M, defined as the symmetric
bilinear form given by
HeSS h(X, Y)
=
(V'x grad h, Y)
XY(h) - V'xY(h ),
=
X, Y E Tx M.
Suppose that f : M -4 !VI is an isometric immersion, and that
g : !VI -4 R is a smooth function. We wish to compute the gradient
and the hessian of the function h = g o f : M -4 R . We have at
xEM
(grad h, X )
=
(grad g , X )
for every X E Tx M . Hence, i f we write
gradg
=
grad h + (gradg) .1.,
where (grad g ) .1. is perpendicular to TxM, we obtain for every X, Y E
Tx M,
HeSS h(X, Y ) = (V'x grad h, Y) = ( fl x grad h, Y)
= X (grad h, Y) - (grad h , fi x Y)
= X (gradg - (grad g ) .1., Y) - (grad g - (grad g) .1., fl x Y)
= ( flx gradg, Y) + (grad g , a(X, Y») .
Thus
HeSS h(X, Y)
=
Hess g (X, Y) + (gradg, a(X, Y » ) .
(3)
We are concerned with the case in which !VI is a Hadamard
manifold, and g : !VI -4 R is the smooth function given by
g(x)
=
1
2
2 d(x o , x) ,
where X o E !VI is a fixed point and d is the Riemannian distance
function on !VI . Using classical comparison arguments we obtain the
following estimate.
3.2. THE JORGE- KOUTROUFIOTIS TH EOREM
3.9
Lemma
47
Let M be a Hadarnard rnanifold, and let Xo E M be a fixed point. Take
any point x E M, and let 'Y : [O, i] ---+ M be the unit speed geodesic
joining X o to x . For any uni{ vector X E TxM perpendicular to 'Y'(i),
the hessian of the function g satisfies
Hess g (X, X) � 1.
Proo! Take a unit vector X E TxM perpendicular to 'Y'(i), and a
geodesic {3 : ( - é, 6) ---+ M so that (3(0) = x and (3'(O) = X . Lift (3 to
Txo M through expxo ' and let expxo (iv(s» = (3(s), S E (-é, é). Now
we have a smooth variation [ : ( - é, é ) X [O, i] ---+ M of 'Y(t) given by
[(s, t) = expxo (t v(s» .
The variational vector field l(t) along 'Y(t)
l(t)
=
� (O, t) = (d eXPxo )t-y'(O) (t v' (O»
is a Jacobi field, and satisfies
1(0)
=
O, lei) = X,
(l(t), 'Y ' (t») = O,
where the last equality follows from the assumption that X is per­
pendicular to 'Y'(i). Restricting the function g to the geodesic {3 we
easily see that
(g o (3) " (0) = Hess g (X, X).
On
2
2 1
[
]
a[
a[
o f3)(s) = - f I - I dt = -i f i - I dt = -iE(s),
the other hand, we have
(g
1
e
2 lo at
1
e
2 lo at
2
where E(s) denotes the energy of the geodesic 'Ys : t ---+ [(s, t). Thus,
(g o (3)"(0) = !iE"( O). Since the variation has fixed left end point,
and (3 is a geodesic, we have that ! E"(O) = fe(l, ]) , where fe is the
index form of 'Y, defined by
48
3 . S U BMAN IFO LDS WITH NON-POSITIV E CU RVATURE
for a vector field V along 'Y with covariant derivative V'. It follows
that Hess g (X, X) = iIt (J, J ) . Now we proceed as in the proof of
Rauch comparison theorem (see [Sp], IV p. 348 or [C-E] p. 29).
Note that the absence of conjugate points imposes no restrictions on
the geodesics involved.
Pick a point po E R m and a geodesic r(I), I arc-length, with
r(O) = po and r (i) = p. Let { V l (I), . . . , Vm (l)} be an orthonormal
frame parallel along 'Y, with vm(t) = 'Y' (I); and let { V1 (t), . . . , Vm (t )}
be the corresponding frame along r. If J ( I ) = Ej= l Àj (t)Vj (t),
define ( fjJJ) (t ) = Ej=l /Xj (t)Vj (I). Using (J, 'Y') = O = ( fjJJ, r'), and
the curvature assumption, we see that the index forms I of 'Y and Í
of r, satisfy It (J, J ) � Ít ( fjJJ, fjJJ). Now we have to estimate k ( fjJJ, fjJJ).
Consider the Jacobi field V(t) along r defined by
,
V (t)
=
I
[ W( / )
where W(I) is the parallel translate of ( fjJJ )(i) along r, from li to Po.
Since V (O) = O = ( fjJJ )(O) , V(i) = ( fjJJ )(i), and (V (/ ) , r ' (/ ) } = O , we
conclude from the Index Lemma (see [C-E], p. 24), that Í((V, V) ::;
Ít (fjJJ, fjJJ). Therefore,
He ss g ( X , X) � i · Ít(V, V) .
Computing Ít(V, V) we obtain the desired estimate. I
Prool 01 3. 7. Suppose that f(M n ) C M n + p is bounded, that is,
there exist Xo E M and À > O such that f (M) is contained in the ball
B (xo , À) with center Xo and radius À. Define h : M � R by h = g o f .
Under these conditions the function h is bounded. The idea of the
proof is to find a point y E M where the second fundamental form
of f satisfies I l a(X, X ) I I > O, X E TyM , X =I O. This, together with
Lemma 3. 1 , yields a contradiction.
ln order to find a point as above we apply Lemma 3.8 to the
function h as follows. The hypotheses that the scalar curvature of
M is bounded from below and that
::; k ::; O, imply that the
sectional curvature of M is bounded. Pick a point Z E M so that
f e z ) = i =I xo. Given a positive f < 1 there exists a point y E M so
that h(y) � h( z ) > O, and
K
Hessh (Y)(X, X) < f I I X I 1 2
(4)
3. 2. TH E JORGE-KOUTROU FIOTIS TH EOREM
for every non-zero X
49
E
2
Ty M .
Set x = f(y), and let "( : [ 0, 00) ---+ M b e the unique unit speed
geodesic such that "( O) = X ó and "( s ) = x . Observe that g (x) = ! S ,
and that grad g (x) is parallel to "(' (s ) since grad g (x ) is perpendicular
to the hypersphere {y E M : d(y, xo) = s } . Th u s we obtain
grad g (x ) =
S"( ' (S).
From equation (3) we have.
- (grad g(x ), a(X, X ) )
=
Hes sg (x)(X, X) - Hessh (Y )(X, X)
(5)
for any X E Ty M . Decompose X as
where Xp is perpendicular to
gradg(x ) , that is
Xc
grad g (x ),
and Xc is collinear with
= (X, "(' (S)) "(' (S).
Hence
Hess g (x)(X, X)
=
Hess g (x)(Xp , Xp ) + 2 Hess g (x)(Xp, Xc)
(6)
+ Hess g (x)(Xc. Xc).
Computing the last two terms of (6), we obtain
Hess g (x)(Xc, Xp )
=
=
(Vxc grad g , Xp ) = I I Xc l 1 ( V-Y'(s) grad g, Xp )
II Xc l 1 ( "(' ( S ) , Xp ) = O.
and
Hess g (x)(Xc , Xc) = I I Xc l 1
2 (V'- -Y'(s) grad g, "(
'
(s ) )
=
2
II Xc l1 .
50
3. SUBMANIFOLDS WITH NON-POSITIVE CURVATURE
We have using (4), (5), (6) and Lemma 3.9,
- (grad g (x ), a(X, X)) = H ess g (x) ( Xp , Xp )
� ( 1 - f) II X I1 2 .
+
I I Xc l 1
2
-
Hess h (y)(X, X)
This concludes the proof. I
There exist many examples of complete surfaces in R3 with Gauss
curvature K satisfying -00 < H :::; K :::; O, and , thus unbounded.
This is the case of many complete minimal surfaces. See [K-V] for an
example of a complete bounded surface in R3 with non-positive Gauss
curvature. On the other hand, it is not known whether there exist
complete bounded minimal surfaces in R3 . Ao example of a complete
minimal surface lying between two parallel planes was obtained in
u-X).
A classical theorem of Hilbert asserts that the hyperbolic plane
2
H cannot be isometrically immersed in R3 (see [Sp] , I I I p. 373). The
existence of a (global) isometric immersibn of the hyperbolic space
Hn into R 2 n-1 still remains an open problem for n � 3. Locally such
immersions exist as first shown by Schur [Sch] (see exercise 3.2),
and must have flat normal bundle as proved by Cartan [C84] (see
exercise 6.4). For additional information on this subject, we refer to
[He], [Am], [Az], [X], and [PJ.
Hilbert's theorem was generalized by Efimov in two directions.
ln [Ed he showed that there is no complete surface in R3 with Gauss
curvature K :::; 6 < O. ln [E2] he proved the non-immersibility of
the hyperbolic half plane. More recently, Smyth and Xavier rS-X]
considered the question of whether there are complete hypersurfaçes
in Rn+ l with Ric :::; - 6 < O. They obtained a negative answer for
n = 3, and also for n � 4 under the additional assumption that the
sectional curvature of M does not assume every real value.
Finally 'je point out that Omori's result, Lemma 3.8, has many
other applications in submanifold theory, cf. U-K], [Di], and [H-K].
51
EXERCISES
Exercises
3.1.
3.2.
'.
Show that Theorem 3 . 7 is false if the ambient space itn+p is not
simply connected.
Choose non-zero real numbers aj, 1 :::::; i :::::; n - 1 , so that E ar = 1 ,
and define an immersion from
into R2n- l by the equations
Y2j-l = ajeXn cos(xda j )
Y2j = aje Xn sin(xdaj)
Y 2n - l =
3.3.
3.4.
3.5.
lx n
(1
-
1 :::::; i :::::; n - 1
e 2U ) ! du o
D
Show that the induced metric on is of constant negative curvature
but it is not complete.
Show that if the isometric immersion f : Mn - itn+p is minimal at
x E M , then 6(x) = r(x).
We say that an isometric immersion 'IjJ : M n - RN has d Fee directions
if 'IjJ(M) C Cd;r = B N -d ; r X Rd, where B N -d ; r is an open ball of
d
dimension N - d and radius r in RN - , and d is an integer such that
° :::::; d :::::; N . Show that if M n is a complete Riemannian manifold
whose sectional curvatures are nonpositive and bounded from below,
then there is no isometric immersion 'IjJ : M n _ Rn+k with d free
directions if n - d � 2 and k :::::; n - d - 1 .
Hint: Suppose that 'IjJ : M n _ Rn+k has d free directions, and that
n - d � 2. Define p : Cd;r - R by p(x) = ! IIx l 1 2 - ! E1=1 (X, hj ) 2 ,
where {hb . . . , hd} is an orthonormal basis ofthe free part Rd. Apply
Omori's Lemma 3.8 to the function f = p o 'IjJ, and use formula (4)
to show that I l o:(X, X)II > (1 - é)r-2 1 I X I1 2 for every é E (0, 1) and
d
every X E Tq(e)M n (R )..L , X -=f 0, where q (é) E M is given by
Omori's lemma. Conclude from Otsuki's Lemma 3. 1 that k � n - d .
The divergence of a vector field X E T M and the Laplacian of a smooth
function h : M - R are defined, respectively, by
d iv X
=
trace(Z
1--+
VzX)
52
3. SUBMANIFOLDS WITH NON-POSITIVE CURVATURE
and
6.h
(i) If f : Mn
- RN
=
trace Hess h
=
div grad h.
is an isometric immersion show that
6.f
( 6.ft, . . . , 6.fN )
=
=
nH,
where H is the mean curvature vector of f.
(ii) Use the identity
6. (h2 /2)
=
h 6.h + Il grad h l l 2
and Stoke's theorem to prove Hopf 's theorem: I f M is compact
and 6.h � O, then h is constant.
(iii) Use (i) and (ii) to provide another proof of the fact that there
is no minimal isometric immersion of a compact manifold into
a Euclidean space.
C R N + 1 , show that
(iv) For an isometric immersion f : Mn -
Sf;r2
6.f =
n
-2
f + nH.
r
(v) Prove Takahashi's theorem: If f : Mn - RN + 1 is an isometric
immersion such that 6.f = - )..f , À > O, then f (M ) C
r = (n/À)I /2 and f ' Mn _ S N is minimal
.
,
Sf;r2•
1/r2
•
Hint: First observe that by (i) f is normal to the immersion,
and conclude that IIt II i s constant. Using the identity
6. (
� I I f 11 2 ) = n + ( 6.f, f )
show that II f ll = (n / À) 1 /2 . The result follows from (iv).
and R N + l , respec­
(vi) Let 6. 1 and 6. denote the Laplacians on
N
1
tively, and let f : R + - R be a smooth function. Show that
SN
6.f lsN
� I SN
= .6,1 ( f l s N ) + �
ar
+
N
I '
af
a r SN
where :, denotes radial derivatives. ln particular, the re­
strictions to S N of the harmonic homogeneous polynomials of
degree k � O, are eigenfunctions of _ .6, 1 corresponding to the
eigenvalue (n + k - l)k .
(vii) Use (v) and (vi) to show that the Veronese surface of exer­
cise 1 .3 (ii) is minimal.
Chapter 4
Reduction of Codimension
4.0
Intoduction
ln this chapter we deal with the problem of reducing the codimension
of an isometric immersion into a space of constant sectional curva­
ture. ln the first section, we show that if the normal bundle of an
immersion f : Mn � Q� + P has a parallel subbundle of rank q < p ,
which contains the first normal spaces of f everywhere, then there
exists a proper totally geodesic submanifold Q� + q of Q� + P such that
f(Mn) C Q� + q , i.e. , f admits a reduction of codimension to q .
ln section 2 , w e restrict ourselves to isometric immersions such
that the first normal space form a subbundle of the normal bundle.
Under this regularity assumption, we obtain necessary and suflicient
conditions for the parallelism of the first normal space in terms of
the normal curvature tensor and the mean curvature vector. We also
show that parallelism of the first normal space may be obtained by
imposing conditions on the s-nullity or on the type number of the
immersion. The s-nullity is a generalization of the index of relative
nullity, while the type number extends the notion of rank of a linear
transformation.
ln last section, we apply the previous results to study minimal
immersions with ftat normal bundle betw.een spaces of constant sec­
tional curvature.
4. REDUCTION OF CODIMENSION
54
4. 1
Basic facts
Given an isometric immersion f : Mn ---+ Mn+p , we define the first
normal space of f at x E M as the subspace N1 (x) C Tx M.i spanned
by the second fundamental form ll! of f at x, i.e. ,
N1 (x) = span {ll!(X, Y) : X, Y E Tx M } .
A sim pIe computation shows that
We say that the immersion f is l-regular if the dimension of
N1 (x) is constant along M . ln this case, N1 is a subbundle of
TM .1. (see exercise 4. 1 ). Observe that the notion of l-regularity is
a differentiable concept, that is, it is independent of the metric of M .
Through this chapter we assume M n connected. An isometric
immersion f : M n ---+ Q� + P admits a reduction of codimension to q if
there is a totally geodesic submanifold Q� + q in Q� + P , with q < p ,
such that f(M) C Q� + q . The immersion f i s substantial i f the
codimension of f can not be reduced. The smallest codimension that
an immersion f can be reduced to is called the substantial codimension
of f .
The following proposition i s the basic result on reduction of
codimension.
4.1
Proposition
Let f : M n ---+ Q� + P be an isometric immersion, and suppose there exists
a parallel subbundle L Df the normal bundle, Df rank q < p , satisJYing
N1 (x) C L( x) for all x E M . Then the codimension of f can be reduced
to q .
ln particular, we have
4. 1 .
4.2
Corollary
BASIC FACTS
55
Let f : Mn -+ Q�+ P be a l -regular isometric immersion. If N1 is a
parallel subbundle ofrank q < p, then f has substantial codimension q .
Prool 01 4.1. Case c = o. Take a n arbitrary point Xo i n M . We
will show that f(M) C Txo M ffi L(xo). Let TI be a vector in the
orthogonal complement of L(xo) in Txo M .l , and let TIl be the parallel
transport of TI along an arbitrary smooth curve "( : I � M , through
Xo . Since L is a parallel normal subbundle, so is L.l , and hence
TIl E L("( t)) .L , t E I. Thus from the Weingarten formula, and the
assumption that N1 (x) C L(x), we obtain
Therefore, TIl = TI is constant in Rn+p , and
d
f( t)) - f (xo) , TI) = (f* "(' ( t ), TI)
dt ( "(
=
O.
We conclude that (f("( t)) - f (xo), TI) = O for t E I. Since the curve
"( and the vector TI E L(xo).l were chosen arbitrarily, we have that
f(M) is contained in Txo M ffiL(xo), which is a totally geodesic (n + q)­
dimensional submanifold of Rn+ p .
Case c > O . Consider the isometric immersion j: Mn � Rn+p+ 1
given by f = i o f , where i : S: + p � Rn+ p + l is the canonical
inclusion of the sphere into Euclidean space. It is clear that tx M .l =
TxM .l ffi span {f(x)}. We have N1(x) C N1 (x) ffi span {f(x)}, and
hence N1 (x) C L(x) ffi span {f(x)} = L(x). On the other hand, the
orthogonal complement of L in t M .l is equal to the orthogonal
complement of L in T M .l , which is parallel with respect to the
normal connection V'.l = V.l I TM .L of f . Since f(x), as a normal
vector field, is parallel in the connection V.l , we conclude that L is
parallel with respect to V .l . U sing Case c = O, it follows that j admits
a reduction of codimension to q + 1, namely,
for sorne Xo E M. Notice that a; + q + l is a linear subspace. Thus
which is the desired resulto
56
4. REDUCTION OF CODIMENSION
Case c < O. This is analogous to the Case c > O, by considering the
isometric immersion j : Mn -+ Ln+p+ l , given by j = i o f, where
i : H: + P -+ Ln+p+l is the canonical inclusion of the hyperbolic space
H: + P into the Lorentz space Ln+p+l : = R( 1 · n + p ) . Details are left as
an exercise. I
The assumption of l -regularity in Corollary 4.2 is necessary as
shown by the example below.
Example.
Let "( : R
-+
"( t)
R3 be the smooth curve given by
=
{ I ?),
(I, e - ii" , O) , for t > O
(O, O,
for t = O
(t, O, e - ii" ), for t < O.
The first normal space has dimension one for t '" O and is parallel,
but the substantial codimension is two.
Next we reduce the classification of the umbilical submanifolds
with arbitrary codimensions to the case of umbilical hypersurfaces.
ln Euclidean space, the latter were classified in Proposition 2 . 5 .
4.3
Proposition
Let f : Mn -+ Q�+P , n � 2, be a non-totally geodesic umbilical isometric
immersion. Then f has substancial codimension one.
Proof. We see from equations ( 1 .8) that the first normal space of
f is generated by the normal curvature vector H, which is parallel
with respect to the normal connection, by Proposition 1 .7 . The result
follows from Corollary 4.2. I
4.2. PARALLELISM OF THE FIRST NORMAL SPACE
4.2
57
The parallelism of the first normal space
ln this section we give necessary and sufficient conditions for the
paralIelism of the first normal space of a l-regular isometric immer­
sion. We also introduce the type number and s-nullity, and prove
that some conditions on these numbers imply that the first normal
space is parallel. The folIowing result was obtained by Dajczer [D. ],
and generalizes a result in do Carmo-Colares [C-C] .
4.4
Theorem
Let f : M n ---+ Q�+P be a l -regular isometric immersion. Then N1 is
parallel i[ and only if
(i)
(ii)
V.lR.l I N) .L
V .lH E N• .
=
O,
Proof. Suppose that (i) and (ii) hold. Let "I E N1 .l be a unit normal
vector field. Since A 1J = O, the Ricci equation implies that
for alI X , Y E T M . Hence, from
O = ( V' z R .1 )( X, Y, "I)
V' z R .1 (X, Y )TJ - R .1 ( V z X, Y ) TJ - R .1 (X, V z Y)TJ
.L
=
.L
-
.L
R .l ( X , Y ) V zTJ,
we obtain for all X, Y, Z E T M
.L
.L
R ( X, Y )V' z TJ
=
O.
U sing the Ricci equation again, we have for all Z, W E T M
This implies the existence, at any Tx M , of an orthonormal ba­
sis Zl, . . . , Zn which diagonalizes {Av.L 1J : Z E Tx M } simultane­
z
ously. It suffices to show that N1 .l is parallel, which is equivalent to
58
4. REDUCTION OF CODIMENSION
(a (Zj , Zj), V�k 1'/) = O for all 1 ::; i, j, k ::;
basis Zl, . . . , Zn, we have
n.
From the choice of the
if i :f j. Suppose i = j :f k . It follows from the Codazzi equation
and 1'/ E N1 .L , that
Therefore,
If i
=j
=
k , the assumption V.L H E N1 and the above, imply that
This completes the proof that conditions (i) and (ii) are sufficient.
For the converse, observe that condition (ii) is immediate from
the definition of H, and (i) follows from the Ricci equation and the
"
fact that N1 .L is parallel. I
Now we proceed to define the s-nullity of an isometric immer­
sion. This was first done in [C-D4J, Let V, W be real vector spaces
of dimensions n and p, respectively, where W has a positive definite
inner product. Let {3 : V x V -+ W be a symmetric bilinear formo For
an integer s , 1 ::; s ::; p, and each s-dimensional subspace US C W,
define {3us : V x V -+ US by
(3us (X, Y ) = '!ru s
o (3(X, Y ),
where '!ru s is the orthogonal projection '!rus : W
qf {3 is defined as
Vs
-+
US • The s-nullity
Vs = max {dim N({3us n,
us c w
where N ( ) denotes the nullity space of the enclosed bilinear form,
namely,
N({3us ) =
{X E V : (3us (X, Y ) = O for all Y E V}.
4.2. PARALLELISM OF THE FIRST NORMAL SPACE
59
Le t ! : M n -+ M n +p be an isometric immersion. We define the
s nullity vs (x) of! a t x E M as the s n u ll i ty of its second fundamental
form a at x. I f we restrict our subspaces US in the definition of vs (x )
to subspaces of NI (x ), we obtain t h e s - nulli ty of ! on the first normal
space, to be denoted by v; (x ). Notice that vZ (x), k = dim NI (x), is
the usual index of relative nullity.
-
-
The following proposition, obtained i n, [D-R3] , uses the s -nullity
to give con dition s for the p ar allel is m of the first normal space NI •
4.5
Proposition
Let ! : M n
n 1 , and
-+
Q: + P be a l -regular isometric immersion.
If dim
NI :::;
-
v; (x ) < n
-
s,
for each x E M. then N} is parallcJ.
Proof.
Let x E M, and let TJ E N} .l be a unit vector field defined
in a neighborhood of x . Define <p", : Tx M -+ N} (x ) by
<p", (X )
=
.L
1I"(Vx TJ) ,
11" :
Tx M.l -+ N} (x ) is the orthogonal projection. Since TJ
N} .l , we have from Codazzi's equation that
where
E
for ali X, Y ETxM . ln particular,
(a (ker <P1/' Tx M ), 1m <P1/ )
=
O.
If r = dim 1m <p"" then O :::; r :::; dim N} (x ) :::; (n - 1). Let us s u ppose
that r "I O, then by the above v; (x) � dim ker rp", = n r. However,
by assumption, v; (x) < n s for 1 :::; s :::; dim N} (x). This is a
-
O, and thus Nt is parallel. I
-
contradiction, therefore
r =
Let V be an n-dimensional real vector space, and let T}, . . . , Tr
be en dom orphisms of V. We define the type number of {TI , . . . , Tr }
to be the largest in te ger T for which there are T vectors VI. • • . , Vr
60
4. REDUCTION OF CODIMENSION
in V such that the rr vectors T; (vj), 1 � i � r, 1 � j � r,
are linearly independent. Observe that the type number of {T} is
equal to rank T. Let V and W be real vector spaces with positive
defini te inner products, and let {3 : V x V - W be a bilinear formo
If �l . . . , �, is a basis of W, and B{j : V - V, 1 � j � r, is given by
( B{j X, Y) = ({3(X, Y), �j ) , we define the left type number of {3 as the
type number of {B{I " ' " B{. } . The right type number is defined in a
similar way. Notice that the left (right) type number of {3 does not
depend on the basis �l, . . . , �, . If (3 is symmetric, the left and right
type number coincide.
The type number r(x) of an isometric immersion f : Mn Mn+ p at a point x E M , is the type number of its second fundamental
form o: at X. The type number was introduced by Allendoerfer in
[AI]] . We observe that if r(x ) � 1, then the first nomIaI space N1 (x)
of f at x is equal to Tx M .l . As in the definition of the s-nullity, we
define the type number r* (x) of f on N1 (x), by taking onIy normal
vectors �i E N1 (x) in the definition of r(x). The following resuIt
shows that s-nullity and type number are related.
Let f : Mn - Mn+p be an isometric immersion. If r* (x) � r, then
v; (x) � n - rs for all s = 1, . . . , dim N1 (x) .
4.6
Proposition
ProoJ.
Suppose r � 1, for if r = O, the resuIt hoIds trivially.
Take linearly independent vectors �l, . . . , �s E N1 (x), and Iet us =
span {6, . . . , �s } . Since r* (x ) � r , there are X.. . . . , X, E TxM such
that W = span {A {i Xj , 1 � i � s, 1 � j � r} C Tx M has dimension
r s. Thus dim W .l = n - r s. It suffices to show that
N(1rus
0 0: )
= {Y
E
TxM : (o:(Y, Z), �j )
=
O, 1 � j � s, for all Z E Tx M }
C
W.l.
But, if Y E N (1ru s 0 0: ) , then (A{j Z, Y) = O, 1 � j � s , for ali
Z E. TxM. Therefore Y E W .l , and the result follows. I
4. 7
Corollary
Q� + P is a l-regular isometric immersion with type number
r* (x) � 2 for all x E M, then N1 is parallel.
If f : Mn
Proof.
-
Immediate from Propositions 4.5 and 4.6. I
4 . 3 . AN APPLICATION
61
Notice that the assumption T* ::::: 2 imposes more restrictions on
the dimensions of the spaces involved than the assumption v; < n - s ,
1 :::; s :::; dim N1 . For example, let f : M n ---+ Q � + P be an isometric
.L
immersion such that N1 (x) = TxM . If T(X ) = 2, then n ::::: 2p ,
whereas vp (x ) < n - p only implies that n ::::: p + 1 .
4.3
An application
As an example of how to use the techniques developed throughout
this chapter, we classify the minimal immersions between spaces of
constant curvature with flat normal bundle. We will make use of the
following result, due to Moore [M02].
4.8
Theorem
Let f : Mcn
---+
Q� n - l be a minimal isometric immersion. Then, either
(i) f is totally geodesic,
or
(ii) c = O and f (Mô) is part of a Clifford torus in the sphere.
Proaf.
See exercise 6.5.
Observe that the Clifford torus has flat normal bundle. We prove
the following result due to Dajczer [Dd.
4.9
Theorem
Let f : Mcn ---+ Q ; + p be a minimal isometric immersion with R i.. = O.
Then, either
(i) f is totally geodesic,
62
4. REDUCTION OF CODIMENSION
or
O and ! (Mô ) is part ofa Clifford torus in some (2n
dimensional sphere.
(ii) c =
-
1)­
Proof. It follows from Corollary 1 .6 that c � é, with equality holding
if and only if ! is totally geodesic. Assume c < é. Then�it follows
from Corollary 3.2 that dim N1 (x) � n 1. Since R1.. = O, Ricci's
equation implies, as in the proof of Theorem 4.4, the existence of an
orthonormal basis Zl" , . , Zn at each TxM such that a(Zi , Zj) = O,
i t= j. However, the relation
-
assures that dim N1 (x) � n 1. Thus rank N1 = n 1 and, from
Theorem 4.4, N1 is parallel. Moore's theorem together with Corollary
4.2 completes the proof. I
-
-
For further results on the subject of this chapter, we refer the
reader to [Ce], [Dd , [D2 ] , [D4], [II ] , [D-R3] , [R-T] , and [T-T].
Exercises
4. 1.
Let ! : Mn � Mn+p be a l -regular isometric immersion. Show that
N1 and Nl 1.. are subbundles of TM 1..
•
4.2.
Show that the substantial codimension of a minimal isometric immer­
sion ! : Mn � Rn+ p with R1.. = O is at most n 1 .
-
4.3.
Hint: Use the fact, that a minimal isometric immersion ! i s real
analytic.
We say that an isometric immersion is m-regular if the k -th normal
space of ! at x E Mn, given hy
EXERCISES
63
has constant dimension, for 1 ::; k ::; m. Show that Nk is a parallel
normal subbundle if and only if (i) (V' lf R 1.. I Nk -L = O, and (ii)
4.4.
(V'1.. ) k H C Nk .
Let f : Mn ---+ Rn+p be a l -regular isometric immersion of a con­
nected Riemannian manifold without ftat points. If dim N1 = 1, show
that f has substantial codimension 1 .
4.5.
4.6.
Provi de a direct proof of Corollary 4.7.
Let 'Y : R ---+ R3 be a curve with curvature and torsion different from
zero, everywhere. Determine the dimension of N1 of the immersion
f : Si C Rn+l ---+ R n+ 3 given by
4. 7.
Let f : Mn ---+ Mn+p be an isometric immersion with parallel second
fundamental form, that is V' -L Q = o. Prove that:
(i) V'1.. H = O,
(ii) f is l -regular.
Assume, in addition, that M n+ p = Mcn + P , and prove that:
(iii) V R = O, that is, M is locally symmetric,
(iv) V 1.. R1.. = O.
4.8.
ln particular, f admits a reduction of codimension.
Let f : M n ---+ Q�+ p be an isometric immersion, and let '" E T M 1.. be
a unit umbilical vector field, i.e., AIJ = À/, À =f O. Assume Vs < n - s
for all 1 ::; s ::; p - 1 .
(i) Show that '" is parallel in the normal connection.
Hint: At x E M, consider the linear map rp : TxM ---+ Tx M 1..
defined by rp ( X) = V'�",. Then dim 1m rp = r ::; p - 1 , and
dim ker rp = n - r. Use the Codazzi equation to show that
Y( À) = O for all Y E ker rp. Now take X E ker rp 1.. , Ó = rp(X ) Xf') "" and use the Codazzi equation again to verify that Aó Y =
O for ali Y E ker rp. Conclude that r = o.
(ii) Show that f(Mn) is contained in a umbilical hypersurface of
Qn+p
c .
4.9.
Let f : Mn ---+ Q�+ P , P ::; n - 1, be an isometric immersion, and
let L C T M-L be a normal subbundle of rank k . Suppose that f is
umbilical with respect to L, i.e. , for all '" E L, AIJ = ÀIJI, and that L is
64
4.
REDUCTI ON OF CODIM ENSION
parallel. Show that f (M ) is contained in an
umbilical submanifold of
4. 1 0.
Let f :
Mn
-+
Rn +p
manifold, and let ."
Q�+p .
(n
+
p
-
)
k )-dimensional
be an isometric immersion of a connected
E T M.l
be a p araUel norm al vector field such
that the supportfunction (f, .,, ) is a non-zero constant. If rank
everywhere, conclude that f (Mn) i s contained in a sphere.
AI'J
= n,
C hap te r 5
Complete Submanifolds of
Constant Sectional Curvature
5.0
Introduction
The main purpose of this chapter is to classify isometric immersions
with low codimension between spaces of constant sectional curvature.
This study is carried out through a careful analysis of the relative
nullity distribution of the immersion.
ln fact, we will be considering the quite more general situation of
immersions f : M n ---+ M:+P with everywhere positive index of rela­
tive nullity. ln section 1 , we show that the relative nullity distribution
in any open subset with constant index of relative nullity, is smooth
and integrable, and the leaves are totally geodesic submanifolds in
both M and Me . ln the particular case where the open set consists of
points with minimum index of relative nullity, these submanifolds
are complete whenever M is complete. We apply these facts, in
section 2, to obtain several interesting results. For example, we show
that an isometric immersion f : Mn ---+ Q�+P of a complete M n with
' Ric M � c, and positive index of relative nullity, is totally geodesic if
c > O, or a cylinder with the relative nullity as a Euclidean factor if
c = o.
66
5. SUBMANI FOLDS OF CONSTANT CURVATURE
5. 1
ComPleteness 01 the relative nullity loliation
Consider a Riemannian manifold M n, and a smooth distribution D
defined on an open subset U e M , which is involutive and has totally
geodesic leaves. Define the distribution Dl.. on U by assigning to each
x E U the orthogonal complement (Dx ) l.. of Dx in Tx M . We associate
to each X ED a map Cx : D l.. � D l.. defined by
Cx Y = -P(V'yX),
where P : TU � Dl.. is the orthogonal projection. The map C : D x
Dl.. � Dl.. given by C(X, Y) = Cx Y , X E D and Y E D l.. , is a
tensor since
C(hX, Y)
= ChX Y = -P(V'y hX)
=
-P(Y(h)X + hV'y X ) = hCx Y = hC(X, Y)
and
C(X, hY) = Cx hY = - P(V'hy X)
= -P(hV'y X) = hCx Y = hC(X, Y),
where h E COO (U) . Notice that D l.. is involutive if and only if Cx
is symmetric for alI X E D . l n this case, Cx is precisely the shape
operator in direction X of the inclusion of the leaves of Dl.. in M .
Let Z E D and Y E Dl.. . Since V'z W E D for alI W E D , we
have
0 = Z (Y, W) = (V'zY, W),
and therefore,
V'zY ED l.. ,
for alI Z E D
and
Y ED l.. .
l n particular, we can define the covariant derivative of Cx by
(V'z Cx )Y = V' z (Cx Y) - Cx V' zY.
(1)
5 . l . THE RELATIVE N U LLITY FOLIATION
5. 1
Proposition
67
The operator C"(, along a geodesic , contained in a leafof D satis/ies the
following differential equation:
,
Dr
dt
'""'( '
=
2
C"(, + P ( R( , ,I )f I ).
(2)
Proof. Let Y E T M and Z E D . Using ( 1 ) we see that V' z P(Y ) E
D l. . Also V'z ( Y - P(Y» E D. Therefore,
P(V'zY )
=
P(V'z (Y - P(Y» + V'zP(Y»
Let X E D be a local extension of ,' , and let Y
have using (3)
( V'x Cx )Y
=
=
V'zP( Y ).
=
E
Dl.. Along " we
V' x ( - P(V'y X » + P(V'vx y X )
-P(V'x V'y X ) + P( V'vxyX)
= P (R( Y, X) X - V' y V'x X + V'[Y.X)X)
= P (R(Y, X ) X ) + P ( V'Vyx X) .
The last equaJity follows from P(V' y V'x X )
V' x X = O along ,. Since
= -
(3)
C
+
P(V'vx y X )
vx xY
=
O
because
we have obtained
(� C"(, )
Y = C�, Y + P(R( Y, ,/) f/),
as desired. I
Vo
Let f : M " --+ M"+ P be an isometric immersion. We denote by
the index 01 minimum relative nullity of f given by
Vo
= min v (x ) ,
XEM
where v(x ) is the index of relative nullity of f at x
ter I I I .
E M,
cf. Chap­
5. SUBMANIFOLDS OF CONSTANT CURVATURE
5.2
For an isometric immersion f : Mn
Proposition
(i)
(ii)
'The relative nullity distribution x f-+ � (x ) is smooth on any
open subset where v is constant,
The set O = {x E M : v(x) = vo} is open.
Proof.
U
of
M.
given Xo
(i) Assume dim
Since
E U,
�(x)
there exist
TxoM1.. , such that
=
TM and TM 1.. , respectively.
I ::::;
C
j
U
::::; n
of
-
xo,
m for alI points in the open subset
X.. . . . , Xn -m E TxoM,
Take smooth local extensions of
V
- Mn+p, we have:
X.. . . . , Xn-m
and �t ,
. . . , �n -m E
. , �n -m in
{A{j Xj } ,
and {t , . .
By continuity, the vector fields
m, remain Iinearly independent in a neighborhood
and therefore
smooth distribution on
span � 1.. .
U, hence also
�.
We conclude that � 1.. is a
(ii) follows immediately from the above argument.
I
The next theorem is the basic resuIt of this chapter, and has been
proved by several authors. See e.g. ,
5.3
Theorem
[FIJ.
Let f : Mn - ifcn+ p be an isometric immersion, and let e C M be an
open set where the index Df rela tive nullity v is equal to some constant
m . Then, on e we have:
(i)
(ii)
(iii)
The rela tive nullity distribution � is smooth and integrable, and
the leaves are totalJy geodesic in Mn and Mcn +p,
If "I : [O, b] - M is a geodesic such that "1([0, b» is contained
in a leaf Df �, then v("I(b» = m ,
The leaves ofthe minimum rela tive nullity distribution are com­
plete whenever M is complete.
5. 1 . THE RELATIVE NULLITY FOLIATION
69
Before proving the theorem, let us discuss sorne simpie exam­
pIes.
Examples.
( 1 ) Let f : M2 � R3 be a flat surface which is nowhere totally
geodesic, that is, v = 1 on M . Then the distribution
b. is integrable and has totally geodesic leaves, which are
lines. This is in accordance with the local dassification of flat
surfaces in R3 as cylinders, cones and tangent surfaces (se e
[Sp], III p. 205).
(2) Similarly, let f : M 2 �
be an isometric immersion of
a surface with constant Gauss curvature K = 1 into Sf,
without totally geodesic points. Thus the surface is foliated
by segments of great cirdes. ln this case the surface can
never be complete, otherwise, the liftings of the leaves of the
foliation to the universal isometric covering Sr of M 2 would
be entire great cirdes which would have intersections.
Sf
Prool 01 5.3.
(i) Let
X, Y
E
b. and Z ETM .
(V'io)(X, Y) = Vi o(X, Y)
-
Then
o(VzX, Y) - o(X, V'zY)
=
O.
Using the Codazzi equation, we obtain
o
= ( V i o)( Z, Y)
=
-o (Z, vx Y) .
Thus V x Y E b.. This implies that b. is involutive with totally
geodesic leaves in both M and M , since
Vx Y
=
Vx Y
+
o(X, Y) = Vx Y E
b. .
(ii) Let L be the leaf of b. which contains '}'([O, b)), and let Z be
a parallel vector field along '}' such that Z('}'(b)) E b.('}'(b)). It
is sufficient to show that Z('}'(O)) E b.('}'(0)). If this is so, then
v('}'(O)) � v('}'(b)), and thus v('}'(O)) = v('}'(b)). We will make use
of the following:
70
5. S U B MAN I FOLDS OF CONSTANT CURVATURE
5.4
Fact
For each W E .6. .1 (1'(0» , there exists a unique vector field Y along 1' l [o,b)
such that
( 1 ) Y(O) = W,
( 2 ) � Y + Cy' Y = O,
O ::; t
<
b,
and Y extends smoothly to t = b.
The existence and uniqueness are immediate, since this is a
Cauchy problem for an ordinary linear differential equation of first
order. Now, taking a second derivative, we have
D
D2
0 = - Y + - C ,Y
dt 2
dt "/
D2 (D )
D
= - Y + - Cv' Y _ C"/2, y
dt I
dt2
2
Y + cY,
=
dt 2
where the last equality follows from Proposition 5 . 1 . Therefore, Y
is a solution of a ordinary second order linear differential equation
with constant coefficients in [O, b), and thus extends to t = b. This
proves the Fact.
Let X be an extension of "1' in .6. and let Y be as in Fact 5.4. We
have
L
V',,/, a ( Y, Z )
L
(D )
(� )
(� )
� )
=
(V' x a)(Y, Z ) + a
=
( V'� a)(X, Z ) + a
Y, z
=
-a( V'yX, Z ) + a
y, z
(
= a C,,/, Y +
Y, z
dt
Y, Z
= O.
ln particular, lI a(Y, Z)II is constant along l' and vanishes at 'Y(b), since
Z('Y(b» E .6. ('Y(b» . Hence a(Y('Y(O» , Z('Y(O» ) = O, and therefore
Z('Y(O» E .6.("1(0» , as we wanted to show.
(iii) is immediate from (ii). I
71
5.2. ISOMETRIC IM MERSIONS
Theorem 5.3 has been extended b y Abe-Magid [Ab-M] to the case
of isometric immersion between manifolds with indefinite metrics.
Completeness of umbilical foliations has been discussed in Reckziegel
[R]. See also exercise 5.4.
5.2
Isometric immersions between spaces of constant
cUnJature
ln this section we discuss existence and u niqueness of isometric
immersions of complete manifolds into spaces of constant sectional
curvature with positive index of relative nullity. We start with the
case of codimension one.
5.5
Proposition
Let f : M:
--+
Mp+ l , n � 3, be an isometric immersion. Then, either
(i) c = é and II �
or
(ii) c
>
n
-
1,
é and f is umbilical.
Proof. Choose a basis Xl , . . . , Xn for TxM consisting of principal di­
rections. and let À1, , Àn be the corresponding principal curvatures.
cf. Chapter I I . It follows from the Gauss equation that
•
.
.
Then we have ÀiÀj = ÀiÀk . for alI 1 � i, j, k � n . Since n > 2, ir
c =I é. this implies À I = . . . = À n = À, and c é = À2 > O . If c = é .
we have ÀiÀj = O, i =I j, 1 � i, j � n . Therefore, if one of the
principal curvatures is non-zero. all the others are zero. I
-
Remark. For a rather complete discussion of the case n
([Sp]. IV p. 1 34).
=
2 see
5. SUBMANIFOLDS OF CONSTANT CURVATURE
72
Next we discuss the case of complete hypersurfaces Mcn with
constant sectional curvature c in space forms Q� + l . Here 5.5 (i)
applies.
(i) Case c > O. Let [ : Mcn - S�+ l be given. Suppose that [ is not
totalIy geodesic, and let L I , L2 be two distinct leaves belonging to
the foliation of the open set O = {x E M : v( x ) = Vo = n - I}
b y complete, totally geodesic submanifolds. Since Mcn i s isometricalIy
covered by S; , the liftings of LI and L 2 to S; are disjoint totalIy
geodesic ( n - l )-spheres in S; , which is absurdo Therefore [ has
to be totalIy geodesic. ln fact, the sarne argument remains valid for
immersions [ : Mcn - S; + P , with p � � . ln this case, the Chern­
Kuiper inequality 3.4 implies v � n - p � � . After passing to a
universal cover of Mcn , the relative � ullity foliation would provide
two non-intersecting totally geodesic spheres in S� , of dimension � � ,
which is impossible. This i s the argument in [O-S].
(ii) Case c = O. ln this case, we have the folIowing result obtained by
Hartman-Nirenberg [H-N] .
5.6
Theorem
Let Mn be a complete flat Riemannian manifold, and let [ : Mn _ Rn + l
be an isometric immersion. Then [(M ) is a cylinder over a plane curve.
Proof. Mter passing to the universal cover 11" : Rn
suppose that Mn = Rn , for [ (M) = [ o 1I"(Rn).
, ,
_
Mn, we may
If v = n, then [ is totalIy geodesic, and [ (M) is a cylinder over
a straight line. Suppose this not to be the case. Then, the non-empty
open subset O = { x E Rn : v (x ) = n - I} is foliated by complete
hyperplanes which are, consequently, paralIel. Now we can extend
the foliation to alI of R n . Fix Xo E O, and let r, be a line in Rn passing
through Xo and perpendicular to the leaves Lx of the foliation. Take
Y E .6(xo), and let Y, be its paralIel transport along r, in R n • It is
dear that Y, E .6 (r, ) . Hence, we have
5.2. ISOMETRIC IMMERSIONS
73
and thus Yt is constant in Rn+ l . This fact and Theorem 5.3 (i)
imply that the images [(Lx) are parallel, ( n l )-dimensional affine
subspaces of Rn + 1 . Since r, is perpendicular to the leaves, we
conclude that 'Y, = [(r, ) is the desired plane curve. I
-
(iii) Case c < O. This case is far more complicated than the others,
even for immersions [ : H: ---+ Hcn+l without umbilical points. The
complexity is caused by the existence of many distinct totally geodesic
foliations of H" , as follows: Fix a unit speed curve 'Y : R ---+ H" which
has curvature k � 1 . Let N('Y) denote its normal bundle, and let
')' : N('Y) ---+ TH" be the natural lifting. Ferus, in [F21, has shown that
the map exp o ')' : N(-y) ---+ H" is a diffeomorphism which induces
a totally geodesic foliation F('Y) of H", whose complete leaves are
images of the fibers of N ('Y) under exp o ')' . This shows the existence
of many such foliations. Conversely, any complete totally geodesic
foliation arises in this way by considering 'Y as a trajectory of a unit
vector field normal to the foliation. Furthermore, given any function
À : R ---+ R {O} , there exists an immersion [ : H" ---+ H"+ l , without
umbilical points, whose nullity foliation is F('Y), and whose second
fundamental form A satisfies A'Y' = À'Y' along 'Y. ln particular, one
concludes that every foliation of H" by totally geodesic hypersur­
faces arises as the nullity foliation of a suitable isometric immersion
[ : Hn ---+ Hn+1 without umbilical points.
-
Remarks. 1) The idea of describing the isometric immersions of H"
into H"+ l by orthogonal trajectories to the leaves of the foliation was
first used by Nomizu [N02 1 .
2) There is an alternative description of the umbilic-free isometric
immersions Hcn ---+ Hcn+1 more in the spirit of the Euclidean cylinder
theorem, due to Alexander-Portnoy [A-P] . They show that any such
immersion takes the form of a hyperbolic ( n 1 )-cylinder over a
uniquely determined parallelizing curve.
-
Now we proceed with the discussion of isometric immersions
into space forms with arbitrary codimension. I nstead of considering
submanifolds with constant curvature, as we did in the codimension
one case, we turn our attention to the broader class of submanifolds
with positive index of relative nullity everywhere, i.e. , those with
Vo > O. The following theorem, due to Dajczer-Gromoll [D-G d , is
5. SUBMANIFOLDS OF CONSTANT CURVATURE
74
a generalization of earlier results of O'Neill-Stiel [O-S] , Ferus [Fd,
Abe [A2], and Rodríguez [Ro21
5. 7
Theorem
Let f : M n - S� + P be an isometric immersion of a complete manifold
with v > 0, everywhere. Then the numbers of positive and negative
principal curvatures, in each normal direction, are equal at any point
where v = vo is minimal.
Proof. Let O = {x E M : v(x) = vo}, and let "'( : [0, 00) - M be
a geodesic contained in a leaf L of the nullity foliation of O. Take
� E Ty(o) M l.. , and let �t be its parallel transport along "'( I ) in the
normal bundle of f, i.e., V'�(t)�t = O. Notice that the composition of
f with the inclusion of S�+P into R n + p + l verifies V "(/(t)�t = 0, where
V stands for the connection in Rn+ p + l . We claim that A{, I �-1- satisfies
the differential equation
(4)
-1-
Let Xt E 6. ("'( /» be a vector field along "'(. Using the Codazzi
-1equation and A{(6. -1- ) C 6. , we obtain (omitting I),
and the claim follows.
Since A(, satisfies equation (4), it has the form A(, = A {o o
C"(/(r) dr) and, consequent1y, has constant rank. ln particular,
exp(Ir:
the numbers of positive and negative eigenvalues remain
along "'( t).
According to Theorem 5.3, f imbeds the leaf L onto
geodesic sphere Silo . Hence, the antipodal map I = - id :
S�+P induces an involution r on O satisfying f(r(x» =
Therefore, the second fundamental form of f satisfies
constant
a totally
S�+ p -
I (f (x» .
5.2. ISOMETRIC IMMERSIONS
75
We have seen above that any parallel normal field � along a
geodesic 'Y C L, is constant along 'Y in Rn+p + 1 . ln particular, when 'Y
joins x to r (x ) , we have �(T(X» = - I* � (x ) It follows that
.
Thus, the number of positive eigenvalues of A{(x) is equal to the
number of negative eigenvalues of A{(r(x)) . The theorem follows from
the fact that A{(')'(t)) has constant rank along "f(t). I
5.8
Theorem
Let f : Mn - S� +p bc an isometric immersion ofa complete connected
manifold with /J > O, everywhere. Suppose there exists a point x in the
open set where /J is minimal, where the Ricci curvature satis/ies for any
unit vector X ETxM that
Ric ( X ) � c .
Then f is totally geodesic.
Proa!.
By equation 1 .6, we have
n-l
n
. (X) - c )
( a(X, X) , H) = - (RIC
+
l n
2
- L Il a( X, Xi ) l I ,
n i=
l
where H is the mean curvature vector of f at x and X = X. , . . . , Xn
is an orthonormal basis o (TxM . If f is not totally geodesic at x , the
above equation and the assumption Ric ( X) � c imply that H =I O
and that AH I L>..L is positive definite, which contradicts Theorem 5.7.
Hence f must be totally geodesic. I
As an immediate consequence of the above theorem, we have that
if /J > O is constant and f is not totally geodesic, then Ric ( X ) < c
everywhere. Examples for this situation are the minimal (homo­
geneous) isoparametric hypersurfaces with three distinct principal
curvatures, cf. [Mi].
We obtain, as a consequence ofTheorem 5.8, the following result
due to Borisenko [Do] and Ferus [F3 J.
76
5. SUBMANIFOLDS OF CONSTANT CURVATURE
5.9
CoroUary
Let f : S� --+ S� + p be an isometric immersion with 1 � P � n - 1 . Then
f is totally geodesic.
Proof.
v(x)
Since J.l(x) = n , the Chern-Kuiper inequality 3.4 implies that
p > O for alI x E S� . The result follows from Theorem
� n
5. 8 · 1
-
There are non-totally geodesic immersions f : Sr
Example.
l
rJ>
.
sln+ Take : Rn + l --+ R2 n+ 2 defined by
"'(x b . .
'I'
· ,
X
n+ l )
=
1
Jn+1
(
e i Vri+1 XI ,
...
,
e i Vri+1 Xn+l
)
--+
,
and note that rJ>(Rn+ l ) C Sln+ l C R2 n + 2 . Let t : Sr --+ Rn+ l be the
natural inclusion, and define f : Sr --+ sln +l C R2n+ 2 by f = rJ> o t.
This is a non-totally geodesic isometric immersion. The existence of
non-totally geodesic isometric immersions f : S� --+ s?;n remains an
open problem for n > 2. A positive answer for n = 2 was obtained
in Ferus-Pinkall [F-P]. For other related results see [D-F] and [D-R.].
Complete submanifolds of positive relative nullity of Euclidean
space with non-negative Ricci curvature have the folIowing nice
description, obtained by Hartman [Ha].
5. 10
Theorem
Let M n be a complete Riemannian manifold with non-negative Rica
curvature, and let f : M n --+ Rn+ p be an isometric immersion. Assume
that the index Df minimal rela tive nullity satis/ies Vo > O. Then f is a
vo -cylin der.
We say that f : Mn --+ R n + p is an m-cylinder if there exists
a Riemannian manifold N n-m such that M n , Rn+p , and f have
factorizations Mn = Nn- m x Rm , Rn+ p = Rn+p-m x Rm , and
f = g x id, where g : N n - m --+ Rn+ p - m is an isometric immersion
and id : R m --+ R m is the identity map.
5.2. ISOMETRIC IMMERSIONS
5.11
Corollary
77
Let I : Mn --+ Rn+p , n � 2 and 1 � p � n - 1, be an isometric
immersion ofa complete flat Riemannian manifold. Then f is a ( n - p)­
cylinder:
This corollary follows immediately from the Chem-Kuiper in­
equality and Theorem 5. 1 0. For p = 1, it reduces to the Hartman­
Nirenberg theorem 5.6.
The main tool for the proof of Theorem 5. 1 0 is the following
result.
5.12
Lemma
Let I : Rn --+ Rn+p be an isometric immersion such that I (Rn) contains
m linearly independent lines. Then I is a m -cylinder:
Proof. The assumption I (Rn) contains m linearly independent lines
L!, . . . , Lm means that L!, . . . , Lm intersect at a point and span an
affine m-subspace of Rn+p . We may suppose n = 2 and m = 1 . Let L
be a line in I (R2) C R2+p . Since I is an isometric immersion, there
exists a line L! in R2 which I maps isometrically onto L. Choose
coordinates (u, v) = ( u !, . . . , u ! +P, v ) E R2+p such that L : u = 0,
and choose coordinates (r, s) in R2 such that L! : r = O, and that if
I(r, s) = (u(r, s), v er, s» , then
ueo, s)
=
O and
v (O, s)
=
s.
It suffices to show that u(r, s ) = u (r) and v (r, s) = s , thus
I (r, s) = (u( r ), s),
Let Ir =
r, s E R.
íJfr, Is = Vs · We have
Il/r I I = Il /s I I = 1 ,
(/r. /s ) = O.
(5)
This implies 11/( r, s) - 1 (0, s ) 1I :::; I r l , that is,
Il u( r, s) 11 2 + I s - v er, s ) 1 2 :::; r 2 .
ln particular
( 6)
78
5.
SUB MAN I FOLDS OF CONSTANT CURVATURE
whieh does not depend on s . Therefore,
i:[I -
vt (r, t)] dt � 2( l v( r, 0) 1 + I r l ) ·
(7)
Fix r. The funetion h(s) = s - v er, s ) is bounded and satisfies
h'(s) = 1 vs (r, s) � O, by (5) and (6). Sinee h is non-deereasing, it
suffiees to show that
-
i:
h2 (s) ds
< 00
(8)
to get that h ( s) = o, and henee v er, s ) = s . The relation 1 = l I /s 11 2 =
( us , us ) + 1 then shows that u s (r, s) = O, i.e. , u( r, s) = u(r) .
We have
i:
i:
i: 1 1'
� i: l'
l i:
[ v er, s ) - S ] 2 ds =
[ v e r, s) - v(O, s)f ds
Vr (T, S) dT
=
Irl
= ,r'
'
I
2
ds
I Vr (T, s ) 1 2 dTds
I Vr (T, s ) 12 dsdT
(9)
by Sehwarz's inequality. From (5)
or, equivalently.
v , = - ( u r , us ) + ( 1 - vs )v , .
Henee
-
I v , I 2 � ( 1 l u, ll l l us l l + 1 1 vs l l v r i) 2 � I I u s l 1 2 + ( 1 - vs f
= 1 v ; + (1 vs f = 2( 1 - v s).
-
-
Now (8) follows from (7), (9) and ( l O). I
( 1 0)
5.2. ISOM ETR IC I M M ERSIONS
79
The above lemma holds if Rn is replaced by a complete fIat
Riemannian manifold. ln that case it is sufficient to argue for the
universal covering.
Prool 01 5.1 0. It follows from Theorem 5 .3 that M contains Vo
linearly independent lines through each point where the index of
relative nullity is minimal. Recall that a line in a Riemannian manifold
is a complete geodesic such that every subarc is minimizing. By the
splitting theorem of Cheeger-Gromoll [C-G], Mn splits like Mn =
N n -vo x RVo , and we may consider f : N n -vo x RVo -+ R n + p . Fix
a point Xo E N, and let 'Y= R -+ N n -vo be any geodesic such that
')'(0) = xo. Consider the immersion f'Y : R x RVo -+ R n + p given by
f'Y(t, y) = f(')'(t), y ) . By Lemma 5. 1 2, we have a splitting Rn + p =
Rn + p - vo x RVo so that
f'Y(t, y) = (h(t ), y).
It is clear that this splitting does not depend on the geodesics starting
at xo, hence the result follows. I
Remarks. ( 1 ) Hartman proved the result for non-negative sectional
curvature using Toponogov's theorem [Top] since the Cheeger­
Gromoll theorem was not available at that time.
(2) The theorem holds if the assumption on the relative nullity is
replaced by the hypothesis f (Mn) contains vo-linearly independent
lines.
,
An analogous result to Theorem 5.6 in the case of Lorentzian
manifolds was obtained in Graves [G].
80
5. SUBMANIFOLDS OF CONSTANT CURVATURE
Exercises
5.1 .
5.2.
I f the tensor C : D x D l.. ---+ D l.. , associated to a foliation D as i n § 1 ,
is identically zero, show that M is locally a Riemannian product.
Given an isometric immersion f : M n ---+ M:+ P with constant index
of relative nullity, show that the following equations hold.
(i) Let V'v (resp. V'h ) denote the b" (resp. b"l.. ) component of V'.
Then
(ii) For S, T E b"
V'S CT = CTCS + CVsT + c (S, T)/.
5.3.
Let c : / ---+ Sf be a smooth regular curve in the unit sphere of R n + 1,
parametrized by arc-Iength, and let "( : / ---+ R be an arbitrary smooth
function. Define a map F : / x R n - I ---+ Rn+ l by
n- l
F(s, 11, . . . , l n - I ) = "( s )c (s ) + "(' ( s ) c' (s) + L tj e j,
j=l
where e}, . . . , e n -l span the normal bundle o f c i n
are taken in the Euclidean space.
sn, and the sums
(i) Show that, away from singular points, F is a flat hypersurface
in R n + l without totalIy geodesic points.
(ii) Compute the singular points, the Gauss map N , and the sup­
port function "( = (F, N) .
(iii) Show that any ftat Euclidean hypersurface free of totally geo­
desic points can be 10calIy parameterized as above.
5.4.
Let f : M n ---+ Q� + P be an isometric immersion, and let U be a
umbilical distribution on M , that is, there exists a normal vector field
q(x) such that
a(X, Y) = (X, Y) q(x )
for alI X E U(x) and Y E TxM . Show that U is smooth and
integrable, and that the leaves are umbilical submanifolds of Q� + p .
81
EXERCISES
5.5.
5.6.
Let f : Mn --4 S� + l , n � 5, be an isometric immersion of a complete
manifold with sectional curvature KM � c. S h ow that f is totally
geodesic.
Verify that the map g : R2 --4 R (I ,3) , given by
g ( s, t)
=
(1
+
s2j2) cosh t,
s cosh t , (s 2 j2) cosh t,
sinh t)
provides a n umbilic-free isometric immersion o f H� l into H� l '
Chapter 6
The Theory of
Flat Bilinear Forms and
Isometric Rigidity
6.0
Introduction
We say that an isometric immersion f : Mn --+ Q � +P is rigid if, given
any other isometric immersion g : Mn --+ Q�+P , there is an isometry
p
p : �+P
Q
--+ Q �+P , such that g =
o f . The purpose of this chapter
is to study the local and global isometric rigidity of submanifolds.
Surfaces in R3 are locally non-rigid, and very few results on the global
situation are known; we refer to ([Sp], V chapter 1 2 ) for a discussion
of this subject. The submanifolds here considered have always at least
three dimensions.
The first important result on isometric rigidity, which we present
in secoon 2, is Allendoerfer's theorem which assures local rigidity
when the type number T of the immersion satisfies T � 3, every­
where. This result generalizes a classical result of Beez-Killing for
hypersurfaces. ln order to prove Allendoerfer's theorem, we develop,
in section 1 , the theory of ftat bilinear forms, introd uced by Moore as
an outgrowth of Cartan's theory of exteriorly orthogonal quadratic
forms ([C84] §20). See also ([Sp], V Chapter 1 1 ), and exercises 6.2
6. 1 . FLAT B I Ll N EAR FORMS
83
to 6.4 of these notes. This theory will play an essential rôle in all
subsequent chapters. The last section is devoted to a global rigidity
result for compact hypersúrfaces, due to Sacksteder. It asserts that
whenever the set of totally geodesic points does not disconnect the
manifold, the immersion is rigid.
6. 1
Flat bilinear forms
ln this section we present the basic results of the theory of flat bilinear
forms.
Let W be a real vector space of dimension n. Consider an inner
product ( , ) : W x W � R, i.e., a real-valued, symmetric bilinear form
on W which is nondegenerate. The signature p ::; n is the maximal
dimension of a subspace of W where ( , ) is negative definite. When
W has an inner product with signature p, and q = n - p, we
say that W is of type (p, q) and write W (p,q ) . A subspace V C W
is degenerate rela tive to ( , ) if the restriction of ( , ) to V x V is
degenerate. A degenerate subspace V C W is isotropic if (u, v) = O
for ali u, v E V ; an isotropic vector is also called light-like. Notice
that V is nondegenerate iff V n V..L = {O}, where V ..L denotes the
orthogonal subspace to V , relative to ( , ) ; and if V is degenerate
then U = V n V ..L is the non-zero isotropic subspace of maximal
dimensiono Moreover, if R C V is a subspace such that V is the
direct sum V = U EB R, then the sum is necessarily orthogonal, but
not unique.
The vector space Rn + p endowed with the nondegenerate inner
product ( , ) : Rn+p x Rn+p � R defined by
p
( ( X l , . . . , X n + p ), (Y I, . " , Y n+p ) ) = - L X i Y i
i=l
+
n+p
L X j Yj
j= p+ 1
is a vector space of type (p, n ) , which we denote by R (p,n ) . ln
particular, when p = 1 , we obtain the Lorentz space L n + l = R ( l, n ) .
Example. Consider in L3 the subspace S = span {e l + e 2 , e 3 } , where
el, e 2 , e 3 is the canonical basis of R3 . Then S ..L = span {el + e 2 } =
84
6. ISOMETRIC RIGIDITY
S n S .l. . So S is a degenerate subspace, and has isotropic orthogonal
complement S.l. .
VI.
A subset
of elements of
.
. , Vr, V , . . . , Vr, U I. . . . , U + - 2 r
1
p q
p
)
)
W(P,q is called a pseudo-orthonormal basis .of W( ,q when
.
(Vi, V
( Vi,
(i)
i ) = ( Vi, Vi )
(ii)
Vi ) = óij ,
(iii) (Ub U Ú = ±Ók l ,
for all
1
� i, j �
r,
1
= ( V i, Uk )
= ( V i, Uk ) = 0 ,
� k, I � p + q - 2r.
-e 1 + e2), e 3 form a
For example, the vectors
(e1 + e 2 ),
3
pseudo-orthonormal basis of L .
If
V1, . . . , Vr, UI . . . . , Up + q - 2r is a pseudo-orthonormal
basis of W(p,q) , and
0
0(
V1, . . . , Vr,
u+
= span { w i , . . . , w: } ,
u-
= span { w l' . . . , w;- } ,
- Vi), then ( , ) is
where wt =
+ Vi), and wi- =
positive definite on U + x U + , and negative definite on U- x U - .
l n particular, r � min { p , q } . N otice also that a subspace of the type
S = span { vil ' . . . , vik ' vil , . . . , vik ' Uil , . . . , Uik } is nondegenerate.
6.1
Proposition
0 ( Vi
0 (Vi
Let V C W(P,q ) be a subspace, and set U = V n V.l.. Let R C V be
a subspace such that V is the direct sum V = U E9 R. Let �I . , �k
be a basis for U . Then there exist isotropic orthogonal vectors �I. . . . , �k
in R .l. such that (�i ' �i ) = óii, 1 � i , j � k . Consequen dy, � l , . . . , �k
can be extended to a pseudo-orthonormal basis � 1 ,
, �k , �I. . . . , �k ,
, 1}p+ q -2k for W(p, q) . ln particular, k � min{ p , q } .
1}1 ,
.
.
.
.
.
.
•
•
•
For the proof of Proposition 6. 1 we shall make use of the follow­
ing algebraic facts.
6.2
Proposition
C
W(P,q) be a vector subspace. Then
(i) dim V + dim V.l. = p + q .
(ii) V.l..l. = V.
Let V
•
85
6. 1 . FLAT B I LlNEAR FORMS
Proof of 6.2.
by
Define a linear map cp : Wj V .l --+ Hom(V, R) = V"
cp (w + V .l ) ( v )
=
( w, v ) .
Since cp is injective, we have
dim(WjV .l ) = dim W - dim V.l ::; dim V*
=
dim V.
Now, define another linear map 'I/J : V .l --+ Hom(Wj V, R) = (WjV)*
by
'I/J( v ) (w + V) = ( v, w) .
Since ( , ) is nondegenerate, 'I/J is injective and
dim V.l ::; dim(WjV)* = dim W - dim V.
This proves (i). From the definition V C V .l.l , and using part (i), we
obtain that dim V = dim V .l .l . Therefore V = V .l .l . I
Proof of 6.1 . If k = O, this is trivially true. Suppose, by induction,
that it is true for k - 1 . Define Vo = span {ÇI, . . . , Çk- t l EB R. It
is clear that Çk fi. Vo , and Çk E Vl . Also Uo = Vo n Vl =
span {{t, . . . , Çk - t l. Since Çk fi. Vo = Vl.l , by Proposition 6.2, there
exists TJ E Vo.l such that (çk . TJ) :f:. O. Consider the nondegenerate
subspace P = span { çk . TJ } , which is a Lorentz plane, since Çk is
isotropic. Consequently, there exists �k E P so that (�k . �k ) = O,
and ( çk. �k ) = 1. Since P C vl , we have p.l :::) Vl.l = Voo
Therefore, by applying the induction hypothesis to Vo C p.l , we
obtain vectors �I , . . . , �k-I in p .l which are orthogonal to R and such
that (�i, �j) = O, (Çi, �j ) = Óij, 1 ::; i, j ::; k - 1 . It is clear that
�I , . . . , �k has the desired properties. The last statement now follows
using the Gram-Schmidt processo I
Let V , W be finite dimensional real vector spaces, and let (3 : V x
W be a bilinear formo We denote by S({3) the subspace of W
spanned by the image of (3 , that is,
V
--+
S({3 )
=
span { (3 ( X, Y ) : X, Y
E
V} ,
86
6. ISOMETRIC RIGIDITY
and we denote by N (f3) the subspace
N(f3) = {n E V : f3(Y, n ) = O, for all Y E V }
called the (right) kernel of f3. We may define, similarly, the left kernel,
which agrees with N (f3) when f3 is symmetric.
For each X E V , let f3(X ) : V ---+ W be the linear map defined
by f3(X)(Y) = f3(X, Y). We denote by ker f3(X) and f3(X, V) the
kernel and the image of f3(X), respectively. We say that a vector
X E V is a regular eLerrumt of f3 if
dim f3(X, V) = max dim f3(Z, V).
ZEV
The set of regular elements of f3 is denoted by RE(f3).
Before proceeding, we observe an elementary fact from linear
algebra. Let V be a finite dimensional real vector space, and let
Vt. . . . , Vn , Ut. . . . , Un be vectors in V. Define
Xf = Vi + t Ui,
1 ::; i ::; n,
t E R.
If either V1,
, Vn or U1,
Un are linearly independent, then
except for a finite number of values of t, the vectors X: ' . . . , X� are
also linearly independent.
•
6.3
Proposition
•
•
• • • ,
The set RE(f3) is open and dense in V .
Let X E RE(f3) , and let Zl, " " Z, E V be so that
f3(X, ZI ) , . . , f3(X, Z, ) are linearly independent, and
Proo/.
.
f3(X, V ) = span {f3( X, Zi ),
1 ::; i
::;
r} .
For any Y in a neighborhood of O E V , the vectors f3(X + Y, Zj),
1 ::; j ::; r , are linearly independent. This implies that RECO) is
open. Given Y E V, since
.
we may choose a sequence {tk } of real numbers, converging to zero,
so that for all k we have that Y + tk X E RE(f3). Thus RE(f3) is
dense· 1
6. 1 . Fl.AT B ILINEAR FORMS
87
We say that a bilinear form fJ : V x V
to a non degenerate inner product ( , ) : W
-
-
X
W is fiat with respect
W - R if
(fJ(X, Y), fJ(Z, W)} = (fJ (X, W), fJ( Z, Y)}
for all X, Y, Z, W E V . We say that fJ is null if
(fJ(X, Y), fJ(Z, W)} = O
for all X, Y, Z, W E V . Therefore, null b ilinear forms are Hat.
6. 4
Proposition
Let fJ : V x V - W be a flat bilinear formo If fJ(X, V) is an isotropic
subspace of W for any X in a dense subset D of V, then fJ is nul1.
Proof.
Since D is dense, by con ti n ui ty we have that
,
(fJ(X, Y), ,6(X, Z)} = O
(1)
for all X, Y, Z E V . N ow, using that ,6 is Hat and ( l ), we obtain
O = (,6 ( X + W, Y), ,6 ( X + W, Z) }
(,6(X, Y), ,6(W, Z)} + (,6(W, Y), ,6(X, Z )}
= 2 (,6(X, Y), ,6(W, Z) }
=
for all X, Y, Z, W E V. I
Next we establish the ma in result of this section, due to Moore
which will be extensively used from now on.
[M04],
6.5
Proposition
Let fJ : V
we have
Proof.
x
V
-
W be a flat bilinear formo Then, for any X E RE(,6),
,6(V, ker ,6(X))
C
,6(X, V) n fJ( X, V ).L .
Let Y, Z E V and n E ker ,6(X). Since ,6 is Hat, we get
(,6(Y, n), fJ(X, Z) } = (fJ(Y, Z), fJ(X, n)} = O,
6. ISOMETRIC RIGIDITY
88
and hence f3(V, ker f3(X» C f3(X, V ) l. .
Now let Zl , . . . , Zr b e vectors i n V such that
and that
f3(X, V)
=
span {f3(X, Zj) ,
r =
dim f3(X, V),
1 � j � r} .
We know that the vectors f3(X + tY, Zj ), 1 � j � r , are linearly
independent except for a finite number of values of t. We conclude
that given Y E V, there exists f. > O such that the family of vector
subspaces f3(X + tY, V) of W varies continuously with t, and that the
function cp(t) = dim f3(X + tY, V), t E R, verifies cp(t) = cp(O) = r
when I t l < f.. Now, if n E ker f3(X), then f3(X + t Y, n) = tf3(Y, n).
Therefore, f3(Y, n ) E f3(X + tY, V) for t =I O. By continuity, this
holds for t = O, i.e., f3(Y, n ) E f3(X, V). Hence f3(V, ker f3(X » C
f3(X, V). I
6. 6
Proposition
Let f3 : V x V � W be a flat bilinear form with respect to the p ositive
definite inner product ( , ) : W x W � R. Then
dim N (f3) � d i m V - dim W.
Proo/. Let X E RE(f3). We assert that N(f3) = ker f3(X). By
definition, we have N (f3) C ker f3(X). On the other hand, if Y E V
and n E ker f3(X), it follows from Proposition 6 . 5 that f3(Y, n) E
f3(X, V) n f3(X, V).L = {O}, since ( , ) is positive definite. Thus n E
N(f3) , i.e., ker f3(X) C N (f3) . Therefore,
dim N(f3) = dim ker f3(X) = dim V - dim f3(X, V) � dim V - dim W. I
For f3 symmetric the above result was first obtained hy E. Cartan
[CÀ!.4], and can be proved easily using Otsuki's Lemma 3 .1. For W
with Lorentzian signature see Corollaries 2 and 3 of [Mo4J.
6.2. LOCAL ISOMETRIC RIGIDITY
6.2
89
Local isometric rigidity
This section is devoted to the proof of a rigidity theorem, due to
Allendoerfer [Ali], by means of the theory of flat bilinear forms.
6. 7
Theorem
Let f : Mn ---+ Q� + P be an isometric immersion with type number T 2:: 3,
everywhere. Then f is rigid.
When the codimension is one and c O, the above result reduces
to the classical Beez-Killing theorem [B ] , [K]. For c t- O, see [Ei], p .
=
2 1 2.
For the proof of the theorem, we will make use of some results,
each of which is interesting in its own right. The first one is due to
Nomizu [N03].
6.8
Proposition
Let f : Mn ---+ Q� + P be an isometric immersion such that Nt (x ) =
Tx M 1. for all x E M . Then lhe normal connection 'V 1. is lhe only
connection in T M 1. which is compatible with lhe metric and satisfies
Codazzi's equation.
Let V be a connection on T M 1. which is compatible with
the metric and satisfies Codazzi's equation, i.e., for alI X, Y, Z E TM
the folIowing equation holds:
Proof.
( V x a)(Y, Z ) = ( V y a)(X, Z).
E TM , define a map
K(X) : TM l. TM l. by
K (X)� = V'� � - V x � .
Clearly K (X ) is linear over C'O (M). Also K(X) is skew-symmetric,
because for alI �, 1/ E T M we have
( K (X) � , 1/ ) ( 'V� � - V x�, 1/)
X ( �, 1/) - (� , V'� 1/ ) - X (� , 1/ ) + ( �, V x 1/)
For each
X
---+
1. ,
=
=
=
-
( � , K (X )1/ )
.
6. ISOMETRIC RIGIDITY
90
Since both V.l and V satisfy the Codazzi equation, we get
K(X)a( Y, Z )
=
K( Y )a(X, Z)
(2)
for ali X, Y, Z E TM. For every Xt , X2 , X3, X4, XS E TM , we have
equations
(K(XI)a(X2, X3), a(X4, Xs))
(K(X2)a(X3, X4), a(XI, Xs))
(K(X3)a(XI, X2), a(X4, Xs))
- (K(X4)a(XS, Xt), a(X2, X3))
- (K(XS)a(XI, X2), a(X3, X4) )
+ (K(XI )a(X4 , Xs), a(X2, X3)) = O
+ (K(X2)a(XI , Xs), a(X3, X4)) = O
+ (K(X3)a(X4, Xs), a(Xt, X2)) = O
- (K(X4)a(X2, X3), a(Xs, Xl )) = O
(3)
- ( K(XS)a(X3, X4), a(XI, X2)) = O .
Summing up identities (3), and using (2), we get
Since N (x ) = Tx M .l everywhere, this means that K(X)
I
X E TM , that is V.l = V . I
6.9
Corollary
=
O for any
/
Let f, j : M n -) Q� + P be isometric immersions, and let ep : TM -)
'
T M be a vector bundle isomorphism preserving the metrics and the
t
. second fundamental forms. lf N( (x ) = TxM , for alI x E M , then
preserves the normal connections.
/
ep
Define V x ç = ep - I [ Vf; (epO ] , where V.l is the normal
connection of í, X E T M , and ç E T Ml- It is easy to see that
V defines a compatible connection on TM/ . Furthermore,
Proof.
( V x a)(Y, Z) = ep - I [( Vf; õ )(Y, Z )] = ep -I [(VfÕ) (X, Z)]
( Vy a)( Y, Z).
=
Proposition 6.8 implies that V
=
V.l , hence Vf; ( epO
=
ep( Vf;Ç ) . 1
The last step needed for the proof of Theorem 6.7 is the folloW1
ing result obtained in [D-R21
6.2. LOCAL ISOMETRIC RIGIDITY
6.1 0
Proposition
91
Let a, á : V x V --+ U be bilinear forms, where V, U are finite dimen­
sional real vector spaces of dimension n and p, respectively, and positive
definite inner products. Assume that
(a(X, Y ), a ( Z, W »)
- (a (X, W), a (Z, Y »)
=
(á(X, Y), á (Z, W» ) - ( á(X, W), á(Z, Y » ) (4)
for all X, Y, Z, W E V . If the left (right) type number T of a satisfies
T � 3, then there exists a linear isometric isomorphism T : U
U such
that á T o a .
--+
=
Proaf.
Let W = U ffi U . Define an inner product (( , )) : W x W
of type (p, p) hy
--+
R
(((�, � / ), (1",/'» )) = (�, "I) - (� / , "I' ),
and a hilinear form fJ: V
x
fJ (X , Y)
V
=
--+
W hy
( a (X, Y), ã ( X , Y » .
I t is clear from (4) that fJ is fl.at with respect to (( , )) . It suffices to
prove that fJ is null, for in this case
(a( X, Y) , a(Z, W » ) = ( ã (X, Y ) , ã ( Z, W » ) ,
and thus
Il a(X, Y ) I I = l I ã (X, Y) I I
for all X, Y, Z, W E V , which immediately implies that ã
defines the desired isometry.
Let us prove that fJ is null. For each X
U (X )
Let ko
=
=
fJ(X, V)
min{dim U(X ) : X
RE* (fJ )
=
{X
E
E
n
E
=
T
o
a
V , set
fJ(X, V) -L .
RE(fJ)} , and define
RE(fJ) : dim U (X ) = ko } .
(5)
92
6. ISOMETRIC RIGIDITY
6. 1 1
Fact
RE· (13) is open and dense in V .
First, observe that Yo E RE·(f3) if and only if Yo E RE(f3) and
there exist Zl. . . . , Zq-kO such that q = dim f3(Yo, V) and det( Cij) :f:. O,
where
For X in a small open neighborhood CJ of Yo, CJ C RE(f3) (cf. Propo­
sition 6.3), we have det( ((f3(X, Zi), f3(X Zj ))}) :f:. O. Since ko is the
minimum, it follows that RE· (f3) is open. ln order to verify density,
let X E RE(f3) be arbitrary, Yo E RE" (f3), and take f > O such that
XI = X + t Yo E RE(f3) for It l < f. Set
,
Then
Thus, det(bij (t)) is a polynomial in t of degree 2(q ko), having
det(cij) as its leading coefficient. So it has a finite number of zeros.
Consequently, there exists O < f' � f such that det bij (t) :f:. O for
-
O < It l < f', showing that RE· (f3) is dense in RE(f3). The Fact
follows from Proposition 6.3.
To conclude the proof, it suffices to show that dim U (X) = p
for X E RE" (f3), because since U(X) C fJ(X, V), and U(X) C
fJ(X, V) -L , Proposition 6.2 implies that f3(X, V) = f3(X, V)-L , and
thus 13 is null by Proposition 6.4.
Let us show that ko = p. Suppose ko � p - 1. Since T � 3, there
exist vectors Xl, X2 , X3 E V such that {A�j Xi, 1 � i � 3 , 1 � j � p}
are linearly independent. Here 6, . . , {p is a basis of U, and A� is
defined by
(A�X, Y) = (a(X, Y), { ) .
.
Furthermore, we can assume Xl. X2 , X3 E RE"(f3). Now, the subspace
s = { Z E V : a (Xj, Z) = O, j = 1 , 2 , 3 }
satisfies
6.2.
LOCAL I SOM ETRIC RIGIDITY
and therefore,
dim S
= n
93
-
(6)
3p.
Consider the map {3 (Xt} : V W, defined by (3(XI )( Y) = f3 ( XI, Y).
From
dim f3 (XI , V) � 2p dim U (Xl )
�
-
we get that
dim ker (3( XI ) �
n
By Proposition 6.5, f3(X2 )(ker (3 (XI »
linear transformation
-
C
2p + ko.
U (X1 ) , and therefore, the
satisfies
and
dim ker ,6(X2) � dim ker (3 (X1 ) - dim U( Xt} � n
-
2p.
Similarly, the linear transformation
satisfies
3
ker ,6(X3)
=
n ker (3(Xj)
j= l
and
dim ker ,6 (X3) � dim ker ,6 ( X2 )
�n
�n
-
-
-
ko
2p ko
3p + 1
-
which contradicts (6), since n� = l ker f3( Xj)
C
S. I
We observe that in the above result, the bilinear forms a, õ: are
not necessarily symmetric. The symmetric case is due to Chern (see
[Sp] V p. 364).
94
6. ISOMETRIC RIGIDITY
Let f, j : M "
Q� + P be isometric immersions with
type number T I 2: 3, everywhere. From Gauss' equation, the second
fundamental forms 0:, Õ of f and j, respectively, satisfy (4) at every
x E M. Now, Proposition 6. 1 0 implies the existence of a vector
bundle isometric isomorphism T : T M/ TMi defined by
Prool oI 6. 7.
--+
--+
T(o:(X, Y))
=
õ ( X, Y)
for X, Y E Tx M and x E M . By definition T preserves the sec­
ond fundamental forms, and therefore, it also preserves the normal
connections by Corollary 6.9. The result follows from Theorem 1 . 1 . I
It has been conjectured by Dajczer [C-Dd that the hypoth­
esis on the type number in Allendoerfer's theorem can be replaced
by an assumption on the s-nullity, namely, Vs < n 2s, 1 ::::: s ::::: p ,
everywhere. A positive answer for p ::::: 5 has been given by do Carmo­
Dajczer [C-D4] (see also exercise 9.2).
Remark.
-
Given a simply connected Riemannian manifold Mn, one may
try to produce a local isometric immersion into Euclidean space,
with a specific codimension, via the Fundamental Theorem of Sub­
manifolds 1 . 1 . ln general, this is a very difficult task, even for
codimension one. ln this case one has first to solve the algebraic
problem by finding a "second fundamental form" which satisfies the
Gauss equation, and then consider the differential problem given
by the Codazzi equation. However, by a theorem of Thomas [Th2],
for manifolds of dimension at least four, generically any solution of
Gauss' equation will automatically satisfy the Codazzi equation.
A generalization of Thomas' result was given by Allendoerfer
[AlI]. H� established that any solution of Gauss' equation for a
given codimension. with type number T 2: 4, everywhere, will also
satisfy the remaining equations needed for an isometric immersion
i.e., the Codazzi and Ricci equations. ln view of Theorem 6.7, the
isometric immersion obtained this way is rigid. For a modem proof
of Allendoerfer's result we refer to Chem-Osserman [C-O].
We conclude this section by proving a very useful rigidity result
due to Moore [Mo} ].
6 . 2. LOCAL ISOMETRIC RIGIDITY
6. 12
95
Suppose that MI . . . . , Mp are connected Riemannian manifolds, and that
Proposition
is an isometric immersion of the Riemannian product. If the second
fundamental form a has the propertr that
a(X, Y)
=
O,
Y
for all X E TMj,
then f is a product immersion, i.e., f = ft
x
E TMj,
...
x
i t- j,
fp , and f l M - = fj ·
1
Proof.
It suffices to consider p = 2. We claim that if X E
T( m l . m2 ) (MI x {m2 } ) and Y E T( m ; . m � )( { m D x M2), then
(X, Y)RN = O.
ln order to prove the claim it is clearly sufficient to show that if (J"t is
a curve joining m 2 to m� in M2 , and X is now considered as a vector
field aIong mI x (J"" then X is constant aIong mI x (J"t . To see this,
just observe that
by assumption.
Now, given (mI, m2) E MI x M2 , consider this point to be the
origin in RN . Then, we have the orthogonaI decomposition
where RI (resp. R2) is the subspace of RN spanned by the tangent
vectors to MI x {mÚ, for alI m2 E M2 (resp. { m I l x M2 , for alI
m I E MI ), and ao = (RI ffi R2)� . Let Po, PI , P2 be the orthogonaI
projections Pj : RN ----+ Rj , i = 0, 1, 2. Given m2 E M2 , we define
fm2 : MI ----+ R N by fm2 (m t } = f(mI . m2), and fI = PI o fm2 ' We
claim that fI is independent of the. choice of m2. For given m� E M2 ,
and a curve (J"t joining m2 of m� in M2 , then
6. ISOMETRIC RIGI DITY
96
To conclude the proof we just have to show that lo = Poo f is constant,
hence
l(mI. m2) = (const, /t (mt ), f2 (m2 »'
But, given two points (mt, m2), (m�, mD of Mt x M2 , take a curve UI
joining them which first goes from (mI. m2) to (mt, m�) in { m t } x M2 '
and then from ( m I . m � ) to (m�, m � ) in Mt x { m; } . Clearly j, (Po o
I(u, » = O . •
interesting global version of Proposition 6. 1 2, which we will
not prove, is the following result of Moore [Mod.
An
6.13
Theorem
Let Mb . . . , Mk be compact connected Riemannian manifolds, ni =
dim Mi � 2, 1 ::; i ::; k . Then, any isometric immersion f : Mt x
X Mk - Rnl + · · · +nk + k is a product ofthe Mi as hypersurfaces.
•
•
.
Inspired by Moore's proof, Alexander and Maltz [A1-M] general­
ized the above result to the case where the manifolds Mi are complete,
non Bat, and do not contain an open submanifold isometric to the
product Rni - t x ( - f, f ) , where ni = dim Mi .
6.3
Global isometric rigidity of hypersurfaces
ln this section we prove a rigidity theorem for compact hypersurfaces
of space forms due to Sacksteder [S2J, The proof given here is
basically the one in Ferus [F4].
6.1 4
Theorem
Let I : Mn - Q�+1 be an isometric immersion of a compact (resp.
complete) Riemannian manifold with n � 3 and c ::; O (resp. n � 4
and c > O). If the set B of totally geodesic points does not disconnect M,
then I is rigid.
We will make use of the following.
6 . 3. GLOBAL ISOMETRIC RIGIDITY OF HYPERSURFACES
6. 15
Lemma
97
Let f : M n --+ Q� + 1 be an isornetric irnrnersion with shape operator A,
and let U e M be an open subset where the index of reJative nullity
satis/ies v = m. Let "f : [a, b J --+ M be a unit speed geodesic such that
"f([a, b» C U is contained in a leaf of the reJa tive nullity distribution.
Then the rnap Cy' has a srnooth extension C-y' to [a, b] which satis/ies the
differential equation on [a, b]
(7)
Proof. Note first that P o "f smoothly extends to b, where (P o
"f)(t) is the projection onto the orthogonal complement of ker A ("( t»
for alI t E [a, b]. Consider, for t E [a, b], the endomorphism field
à : T-y(t)M --+ T-y(t)M given by
Ã
=
(A
+
I
- P) o "f,
where I is the identity map. Observe that
X E T-y(t)M , we have
ÃX
=
ÃoP
=
A o P. For
A X + (X - P X ).
Since A X E (ker A)l.. and (X - P X ) E ker A, we have that à X = O
if and only if X = O. Hence à is invertible. Now define Cy'(t) along
"f(t) by
- -1
o V'-y, A .
C-y' = (A)
Then, using Codazzi's equation for
C-y'
=
=
=
t E [a, b)
- -1
V'-y,A = (A) - 1 A C-y'
(A)
(Ã)- I APC'I' (Ã)-IÃPC-y'
-
=
C-y"
Therefore C-y' is the desired extension. I
Proof of 6.14. Let f, j : M n --+ Q� + 1 be isometric immersions, and
consider the following subsets of M - B , where B is the set of totally
geodesic points of f.
Mr = {p E
Mr = {p E
M
- B v(p) �
M
-
:
B : iI(p) �
- r} ,
n - r} .
n
98
6. ISOMETRIC RIGIDITY
By the Beez-Killing rigidity theorem, each connected component of
M3 is rigid, so the second fundamental forms A, Ã of 1, 1, respec­
tively, verify à = ±A at any point of M3' Also, using the Gauss
equation, it is not difficult to show that the subspace of nullity r(p),
which is intrinsic (see Chapter III), satisfies r(p) = ker A (p) at each
point of Mk , k � 2. ln particular, Mk = Mk for k � 2, and the
relative nullity foliations of I and 1 are the sarne on the open set
W = M2 -M3 = M2 - M3• We claim that à = ±A on each component
of W .
Case c � O . Since M i s compact, the leaves of the relative nullity
foliation of W cannot be complete. Therefore, given p E W , we may
choose a geodesic "(: [O, b] � M with 'Y(O) = p, 'Y([O, b» C L p , but
'Y(b) fi. W. Here L p stands for the leaf of the foliation containing
p. Since 'Y(b) E M2 by Theorem 5.3 (ii), we have 'Y(b) E M3, and
therefore à = ±A at 'Y(b). From the uni queness of solutions of the
ordinary differential equation (7) satisfied by A , Ã in [O, b] with a
given initial condition, we conclude that A = ±Ã at p . This proves
our claim for c � O. N otice that C'f' is intrinsic to the nullity foliation.
Now we consider the open set W' = M1 n M1 - M2 = M1 - M2
since M1 C M1 by definition. ln order to conclude, hy the above
argument, that à = ±A at any point of W', it suffices to show that
the relative nullity foliations of I and 1 are the sarne on W'. Consider
a geodesic 'Y : [O, b] � M such that 'Y(O) = p , 'Y([O, b » C L p n Lp ,
but 'Y(b) fi. W'. We conclude that 'Y(b) E M2 , and therefore à = ±A
at 'Y(b). Consequently, ker A = ker à at 'Y(b), which implies that
L p = Lp . ln particular, Ã = ±A at M1 n M1 •
Next we show that the set W" = M1 - M1 is empty. Since
M1 , we have lJ = n - 1 , ÍI = n at W". Given p E W", take
a geodesic 'Y : [O, b] � M such that 'Y(O) = p , 'Y([O, b » C Lp , and
'Y(b) - fi. W". Then lJ = n - 1 , ÍI = n at 'Y(b). On the other hand, since
'Y(b) E MI . we obtain that 'Y(b) E M1 n M1 • By the above, A = ±Ã
at 'Y(b), which is a contradiction.
M2
C
Let q E M1 - M1 • From M1 C M1 C MI . it follows that
E M1 n M1 , and thus à = ±A at q, which is a contradiction.
Consequently, M1 = M1 , and thus the set ÍJ of totally geodesic points
of 1 satisfies ÍJ C B . ln a similar way B C ÍJ , and therefore A = ±Ã
q
6.3. GLOBAL ISOMETRIC RIGIDITY OF HYPERSURFACES
99
on M . Since M B is connected, we conclude, from Theorem 2. 1 ,
that j is congruent to f .
-
Case c > O. Notice that in the previous case the fundamental fact was
the non-completeness of the leaves of the relative nullity foliation.
The following result proves this case by using the sarne argument as
before.
6.1 6
Lemma
be an open subset where v == n 1 Dr
2. Then no leaf Df the relative nullity foliation is complete for
Let f : Mn
v == n
n � 4.
-
---+
S;+l, and U
e M
-
Proo!. Let x E L c U , where L is a leaf of the relative nullity
foliation Let us first show that if either v = n 1 or v = n 2 and
n � 4, then there exists X E Tx L such that Cx has a real eigenvalue.
For v = n 1 the assertion is clear. Let v = n 2, and suppose that
Cx has no real eigenvalues for ali X ETx L . If Xl , " " Xn --2 is a basis
of Tx L , then for Z E 6..1(x) and a , a I . . . . , a n - 2 E R, the equation
.
-
-
-
-
O
=
aZ +
n-2
?= aiCxi Z = aZ
1=1
+
CEi ai Xi Z
implies a = a I = . . . = a n - 2 = O. H ence Z, CXI Z, . . . , Cxn_2Z are
linearly independent in 6.(x ).1, which is a contradiction if n � 4.
The proof will be completed if we show that there is no geodesic
'Y= [O, 00 ) ---+ L such that 1' (0 ) = x and 1" (0 ) = X , where Cx has a
real eigenvalue. As sume that such a geodesic exists, and let
CX Yo
=
ÀYo
for some À E R and Yo E 6.(x) . Denote by Yo(t) the parallel
transport of Yo along 1'(t). By Fact 5 .4, the unique solution of
.L
D
dt Y
is a solution of
+
Cy' Y
= O,
Y(O) = Yo
(8)
100
6. ISOMETRIC RIGIDITY
The solution of this equation with initial conditions Y(O) =
�r (O) = -Cy,Y(O ) = -ÀYo is
Y(t)
=
(cos .jCt -
Yo,
Jc sin .jCt)Yo(t).
But then Y(t) has a zero, which contradicts the fact that a solution
of (8) never vanishes. I
6.1 7
Remarks
( 1 ) If the set B separates M in several components, the theorem is no
longer true. Consider a surface which consists of two parts which are
individually kept rigid, but which are glued together in two different
ways along a totally geodesic plane curve.
(2) The theorem remains true, by a similar argument, for t : Mn �
Q�+ 1 , with M complete, assuming that M does not contain a com­
plete ( n - 2)-dimensional submanifold L, for which t iL is totally
geodesic.
(3) The theorem is false for c > O and n = 3 . The universal covering
of the three-dimensional hypersurface in S4 given in exercise 2.7 (iv)
is a non-rigid (see [D-G2], p. 9) complete minimal hypersurface with
v = 1 , everywhere.
(4) Locally deformable Euclidean hypersurfaces were classified by
Sbrana [Sb], and later on by E. Cartan [Cad.
Sacksteder's theorem has been generalized in two different di­
rections. Rigidity of complete Euclidean hypersurfaces has been
discussed by Dajczer-Gromoll in [D-G41 Roughly speaking, it is
shown that for n � 4, deformable n-dimensional hypersurfaces are
ruled by complete Euclidean spaces of codimension one. See also
Dajczer-Tenenblat [D-T]. The case of compact submanifolds in higher
codimension was considered by do Carmo-Dajczer [C-D3]. For other
rigidity results see [B-B-G], [D-�], [M06], [D-�], and [S-X]. The
existence of local isometric deformations in the real analytic case was
considered in [h]. See also [Uan].
Finally, one may also consider rigidity for submanifolds satisrying
an additional strong condition like having constant mean curvature,
fiat normal bundle, parallel second fundamental form or being ruled,
among others. See, [Y], [De], [Te], [Fs], [B-D-Jt 1 , [B-D-J2]' and [B-D].
EXERCI SES
101
Exercises
6. 1 .
6.2.
Use Corollary 6.6 to give another proof of the Chern-Kuiper inequal­
ity 3.4, and Tompkin's theorem: If f : M n --+ R n + p is an isometric
immersion of a compact ftat Riemannian manifold M n , then p � n.
Let f3 : V x V --+ W be a ftat symmetric bilinear form with respect
to a positive definite inner product ( , ) : W x W --+ R such that
S(f3) = W , and dim N(f3) = dim V - dim W .
(i) Show that f3 is a direct sum of one-dimensional bilinear forms
where each f3i : V
and n = dim W .
x
V
--+
U'i is ftat, dim U'i
=
1 , U'i 1. Wj if i t- j
Hint: Observe first that we may assume dim V = dim W, and
that for any regular element X E V , the map f3(X) : V --+ W
is an isomorphism. Now fix Xo E R E(f3), and use the fact that
for alI Y E V, the linear endomorphisms B(Y) of W defined
by
B (Y )
=
f3(Y) o f3(XO) - l
commute and are symmetric with respect to ( , ) . Now take a
common diagonalizing basis � 1 , . . . , �n of alI B(Y) , Y E V , and
let lYi span{�i } .
(ii) Show that the subspaces lYi , 1 � i � n , are uniquely deter­
mined up to permutations.
=
Hint: Let /l-i be linear functionals on V such that B(Y)l wi =
/l-i(Y)I for every Y E V, where I is the identity endomorphism
on lYi . Then, it suffices to show that there exists Yo E V
such that /l-i(YO) t- /l-j(Yo) if i t- j . Suppose this not to be
the case, and show that there would exist i t- j such that
/l-i = /l-j , which leads to a contradiction in the folIowing way: If
f3(XO)Vi = �i , f3(Xo)Vj = �j , use the fact that f3 is ftat and f3(Xo)
is surjective to show that f3(Vi, V;) = f3(Vj, Vj). Then conclude
from f3(V; , V;) E lYi , and f3(Vj, Vj) E Wj , that both vectors must
be zero. FinalIy use ftatness of f3 again to obtain that.
6. ISOMETRIC RIGIDITY
1 02
6.3.
Let [ : Mcn - M; n be an isometric immersion.
(i) Show that at any point x E Mcn where v(x) = O, there exists a
basis e l , . . . , e n of TxM of unit-Iength vectors, an orthonormal
.L
basis {I, · · · , {n of Tx M , and positive numbers À 1 ,
, À n , such
that the second fundamental form a at x satisfies
•
•
.
.
Hint: Use parts (i) and (ii) of exercise 6.2.
(ii) Prove that the basis e 1 , . . , e n is orthonormal if and only if we
have that R.l (x) = O.
(iii) Assume that M n is simply connected, and that v = O every­
where. Show that the frames e}, . . . , e n , and {I, . . . , {n , and the
functions Àl' . . . , Àn , are uniquely determined (up to signs and
permutations) and differentiable.
(iv) Further suppose that [ has ftat normal bundle. Verify that the
Codazzi equations for [ are equivalent to the following set of
differential equations
'\lejej
=
Àjej
(;j) ej,
'\l t {j
=
Àj ej
( ;j ) {j,
1 � i :f j � n.
Conclude that the functions Àj, 1 � j � n, are constant if and
only if [ (M ) is a product of circles.
6.4.
6.5.
Let [ : M:' _ Q� n - 1 ,
R.l = O, everywhere.
c
< ê, be an isometric immersion. Show that
Hint: Com pose [ with a totally umbilical isometric immersion of
l
Q� n - into Q�n , and use exercise 6.3.
Prove Theorem 4.8.
Hint: Let g : M:' - Q� n be an isometric immersion obtained by com­
l
posing [ with a totally umbilical imbedding of Q�n - into Q� n , and
consider the orthonormal smooth frames e l ,
, e n , and 6, . . . , {n ,
and the smooth functions À1,
, À n , given by exercises 6.3 and 6.4.
Verify · that a normal vector { E TMcn spans an umbilical direction if
and only if it is a multiple of the vector 1] = L:7= 1 f{j. Then observe
that the mean curvature vector H of g satisfies H � (é c)1] since [
is minimal. Now use that
1 n
1 n
···
•
.
•
-
H
=
- Lj =1 a(ej, ej) - Lj = l Àj{j,
n
=
n
EXERCISES
6.6.
1 03
and obtain that the ,xi are alI constant
and equal. To conclude the
.
proof u:; e exercise 6.3 (iv).
Let f : Ncn Q�n be an is()metric immersion of a simply connected
Riemannian manifold with lJ = O and R.L
O, everywhere. Let
e l , . . . , e n and �I . . . . , �n be the orthonormal smooth frames given by
exercise 6.3.
(i) Show that there exists an isometric immersion g : Ncn Q� n - l ,
for some c > c , such that f is the composition o fg with a totalIy
umbilical inclusion of Q�n - l into Q�n if and only if �7= 1 1 /,x7 =
l /c - c.
Hint: Prove that the above equality holds if and only if the
umbilical direction '" = � t �i is parallel in the normal bundle.
Then use exercise 4.4.
(ii) Prove that these exists a coordinate system ( u I . . . . , un ) in a
neíghborhood of any point of Ncn such that 8/ 8 ui = f' ei ' I �
i � n.
Hint: Use that, given n linearly independent vector fields
Xl , . . . , Xn E TNcn , there exists a coordinate system (UI, , U n )
in a neighborhood of any point of Ncn with 8/ 8 Uj
Xj ,
I � j � n, if and only if [Xi, Xj] O for alI I � i, j � n.
�
=
�
•
.
.
=
=
Let f : Mcn Q�n be a isometric immersion with lJ = O and R.L = O,
everywhere, where Mcn is complete.
(iii) Show that the third fundamental form I I I of f defined by
�
I11(X, Y)
=
n
L (a(X, Xi), a(Y, Xi)),
i=l
where a denotes the second fundamental form of f and Xl ,
. . , Xn is an orthonormal basis of TxM , provides a ftat metric
on Mcn , which is also complete if there exists 6 > O such that
,xi > 6, I � i � n .
(iv) Conclude that there i s no isometric immersion of a complete
M; into Q� n - t , c < c, if M; does not admit a complete ftat
metric. This is the case if c > O, or if c < O and M; is a
Fuchsian space form (see [P]).
.
6. 7.
Let TlM .L be the unit normal bundle of an isometrÍc immersion
S� + p . Consider the map F : TlM.L
S�+ P defined by
f : Mn
�
�
1 04
6 . ISOMETRIC RIGIDITY
F(x, O = �. Then
singular points.
F(M)
is a hypersurface of sn+p , possibly with
(i) Determine the singular points of F, and compute the second
fundamental form at the regular points.
(ii) For n = 2, show that if f is minimal then F is a minimal hy­
persurface with two non-zero principal curvatures at all regular
points
(iii) Conclude that the unit normal bundle of the Veronese sur­
face in si is a compact minimal hypersurface without singular
points.
(iv) Construct examples of minimal hypersurfaces of Euclidean
space with only two non-zero principal curvatures.
Hint: Use exercise 1 .7.
(v) Show that any simply connected minimal hypersurface Mn of
Q�+ l with two non-zero principal curvatures, admits only a
l-parameter family of non-congruent minimal isometric defor­
mations . .
Hint: Use exercise 3 . 3 and Theorem 1 . 1 .
6.8.
Let f : Mn � Rn+ l be an isometric immersion, and let S e M be a
connected component of the subset of totally geodesic points.
(i) Show that S is contained in an n-dimensional affine subspace
in Rn+ 1 tangent to f along S.
Hint: Use the fact that any smooth functíon 'I/J : Mn � R with
'I/J. = O on S, must be constant because 'I/J(S) is an interval in R
which must contain regular values of 'I/J unless it is a point.
(ii) Assume that Mn is compact, and that M - S has only two
components. Conclude that all possible isometric immersions
of M n into Rn + 1 are given by ( l ) of Remark 6. 1 6.
6.9.
Let f : Mn � Rn+ l be a simply connected complete ruled hypersur­
face (cf. exercise 2 .4) without fiat points.
(i) Show that there exists an orthonormal tangent frame el , . . . , en
such that e2 , . , e n span the rulings, and so that the second
.
.
EXERCISES
1 05
fundamental form has the form
A=
[� � J
(ii) Prove that the set of alI ruled isometric immersions of M n
into Rn+ l are in one to one correspondence with the set of
differentiable functions on R.
Hint: Show that the second fundamental form of any other
ruled immersion is, with respect to e l , · · · , e n , of the form
where 1> is any solution of the differentiable equation
Chapter
7
ConformaIIy FIat
SubmanifoIds
7.0
Introduction
Conformally flat manifolds are defined to be Riemannian manifolds
locally conformaI to Euclidean spaces. The simplest examples are sur­
faces and manifolds with constant sectional curvature. ln section 1 ,
we characterize the simply connected conformally flat manifolds as
hypersurfaces in the light cone of the Lorentzian space LN , transver­
sal to the generators of the cone. As a consequence, we obtain the
classical characterization of Schouten in terms of the Weyl curvature
tensor. Another application is Kuiper's theorem, which says that a
compact simply connected conformally flat manifold is conformaI to
the Euclidean sphere.
We start section 2 with a result, due to Moore, which asserts
that conformally flat submanifolds of spaces of constant curvature
with low codimension must have umbilical subspaces at every point.
If the manifold is compact, this result together with Morse theory,
provides some topological obstructions for the existence of isometric
immersions in Euclidean space. Next, we use Moore's result to
obtain a characterization, due to Cartan-Schouten, of the confor­
mally flat hypersurfaces in terms of the multiplicities of the principal
curvatures. We conclude section 2 , by showing that conformally
7. 1 . CHARACTERIZATIONS
107
ftat manifolds are the only Riemannian manifolds which can be
locally isometrically immersed as hypersurfaces in two manifolds with
distinct constant sectionai curvatures.
We dose the chapter with section 3, where we present a (local)
dassification of the conformally ftat hypersurfaces of Eudidean space
obtained by E. Cartan.
7. 1
Characterizations of conformally flat manifolds
We say that a Riemannian manifold M n is conformallyfiat if each point
of M lies in a neighborhood which is conformally diffeomorphic to
an open subset of Eudidean space Rn , with the canonical metric.
Examples.
(i) Any surface is conformally ftat because it admits local isother­
mal coordinates (see [Sp], IV p. 455).
(ii) Eudidean spheres sn C Rn+ l are conformally ftat since the
stereographic projection is a conformaI diffeomorphism. ln
fact, we will prove later, that any Riemannian manifold with
constant sectional curvature is conformally ftat.
(iii) Rotation hypersurfaces or tubes around smooth curves. This
will follow from Theorem 7 . 1 8.
Let M n be a Riemannian manifold. We define the Weyl tensor or
conformai curvature tensor C of M by
( C(X, Y ) Z, W )
for X, Y, Z, W
\
=
E
(R (X, Y ) Z, W ) - L(X, W) ( Y, Z ) - L (Y, Z) ( X, W )
+ L(X, Z) (Y, W ) + L ( Y, W) ( X, Z )
T M , where L is the tensor defined by
L (X, Y)
= n
�
2 ( Ric (X,
Y)
- ! n s (X, Y ) ) .
7. CONFORMALLY FLAT SUBMANIFOLDS
1 08
Here Rie and s are the Ricei tensor and the scalar curvature of M ,
respectively.
Now we state a charaderization of conformally flat manifolds
obtained by Schouten [Se].
7.1
Theorem
Let Mn, n � 3, be a Riemannian manifold. Then M is conformally !lat
ifand only if the following conditions are satis/ied:
(i)
(ii)
C = O,
L is a Codazzi tensor, that is, for all X, Y, Z
E
TM,
(V'x L ) (Y, Z ) = ( V'y L ) (X, Z ) .
Moreover, (i) implies (ii) when n � 4, and C
7.2
Corollary
= O if n = 3.
Any Riemannian manifold ofconstant sectional curvature is conformally
!lat.
Proof of the CoroUary. Let c be the constant sectional curvature of
Mn. Then for X, Y E TM , we have
Ric ( X, Y)
Therefore,
and thus
=
( n - l ) c (X, Y )
and
s
=
c.
c
L(X, Y ) = 2 (X, Y) ,
L is a Codazzi tensor. Also
( C (X, Y ) Z, W) = c ( X 1\ Y) Z, W) - 2 [(X, W ) (Y, Z) + ( Y, Z) (X, W)
- (X, Z) (Y, W ) - (Y, W) ( X, Z )] = O
c
for all
X, Y, Z, W E TM , and the result follows from Theorem 7 . 1 . I
7. 1 . CHARACTERIZATIONS
7.3
Corollary
1 09
Any conformally flat Einstein manifold has constant sectional curvature.
Proof. By assumption, Ric ( X, Y) = P (X, Y ) for all X, Y E T M .
(X, V) . Since M is conformally fiat, the
Therefore L(X, Y ) =
Weyl tensor is identically zero, hence
�
=
( R (X, Y)Z, W)
n
� 1 «(X, W) (Y, Z) - (X, Z ) (Y, W) ) .
This shows that the sectional curvature is constant and equal to
6·
I
We will review some facts about Lorentzian geometry which will
be used in the proof of Theorem 7. 1 .
Consider the Lorentz space Lm + 1 , i.e. , Rm+ 1 endowed with the
inner product
(X, Y)
=
- xoY o + X1Y 1 + . . . + x m Y m ·
We define the light cone Vm of Lm+ 1 as the submanifold
vm
= {X
E
Lm + 1 : (X, X )
= O, xo > O}.
It is clear that the induced metric on V m is semi-definite, where only
the tangent vectors to the generators of the cone are isotropic. We
will make use of the fact that the isometries of Vm are the restrictions
of isometries of Lm+ 1 which leave Vm invariant.
ln what follows, the Euclidean unit sphere S;"-l will be consid­
ered as a submanifold of the light cone Vm , m � 2, via the isometric
imbedding i : S;"- l C Rm ---+ Vm C L m + 1 , given by
i(x 1 . . . . , x m ) = (1, X l , . . . , x m ).
7.4
Proposition
There is a one to one correspondence between isometries ofthe light cone
and conformal transformations ofthe sphere s m - l .
Proof.
by
Given an isometry G : vm
F(x) =
1
1fJ (x)
G(x),
---+
1fJ(x )
Vm , define F :
=
-
(G(x ), eo )
S;"-1
>
0,
---+
S;"- l
1 10
7.
CON FORMALLY
FLAT
SUBMAN I FOLDS
where G(x) is considered as a position vector in Ln +1 , and e o =
(1, O, . . . , O) E Lm+ 1 . We see that F is conformaI because (G, G ) = O
impIies
(G, G * X) = O, for all X E rv m ,
and consequently,
for all Y, Z E TS'('- 1 .
Now, Iet 1r : V m C L m + 1
y /Yo . We have
---T
C
V m be defined by 1r (Y )
=
1
2" (X, eo ) y .
Yo
Given a conformaI diffeomorphism F : S'('-1
S'(' - 1 with conformaI
factor À , we define G : v m
v m by
Yo
G (y ) = q,(y)F(1r(Y » ,
q, (y )
À ( 1r (Y »
m
Let X, Z E Ty V • Since
1r* (y) X
=
1
S'('-1
-X
Yo
+
---T
---T
=
G. X
=
X ( q,)F(1r(Y » + q, (y)(F o 1r )*X,
we have
(G * X, G*Z)
=
=
q,2(y) ( F o 1r )* X, (F o 1r)* Z )
q,2(Y )À2 (1r(Y » (1r* X, 1r * Z)
1
= q,2 (Y )À2(1r (Y » 2" (X, Z)
=
(X, Z) .
Yo
Therefore G is an isometry of V m , and this concludes the proof. I
An immersion ! : Mn ---T M n + p between Riemannian manifoIds
is said to be conformai if there exists a nonwhere vanishing function
À E COC(M) , called the conformai factor, so that
(!* ( x ) Y, f* (x) Z) M = À2 (x) ( Y, Z) M
for all x E M and all Y, Z E TxM .
7. 1.
7.5
Proposition
CHARACTERIZATIONS
111
Let Mn be a Riemannian manifoId. Then, there exists an isometric Ím­
mersion g : M n vm ifand only ifthere exÍsts a conformaI immersion
I : Mn ...... sm- l . Moreov€r, g is unique (up to isometries of Vm) if and
only if I is unique (up to conformaI diffeomorphisms of sm - l) .
�
Proo!. Given an isometric immersion g : M n ...... V m , define I : M
...... S;n - l by
I (x )
=
n
(g x , eo ) g (x).
( )
-1
A sim pIe calculation shows that I is a conformaI immersion.
Re­
ciprocaIly, given I : Mn ...... S ;n - l conformaI, we define an isometric
immersion g : Mn ...... vm by
g (x)
=
1/
x (x),
À(
where À is the conforma� factor of I .
To prove uniqueness, let II , h : M n ...... s;n-l be conformaI
immersions with respective conformaI factors Àl' À2 . Define isometric
immersions gj = -f:lj : M n � Vrn , j = 1 , 2, and suppose that
g2 = G o gl , where G is an isometry of Vm . Let F : S;n - l ...... S;n-l be
G(x). Then F is a conformaI transformation
given by F(x ) =
of sm-l and
}
(G (�J,eo)
The converse is similar and Ieft to the reader. I
7.6
Corollary
A Riemannian manifold Mn is conformally fiat if and onIy if it can
be Iocally isometrically immersed in the light cone Vn+ 1 of Ln +2 as a
hypersurface.
Proa!. This follows easiIy from Corollary 7.2 and Proposition 7.5. I
7. CONFORMALLY FlAT SUBMANIFOLDS
1 12
The characterization of conformalIy flat manifolds, given in The­
orem 7 . 1 , will follow as a consequence of the next result, proved by
Asperti-Dajczer [A-Dt ] .
7. 7
Theorem
Let M n, n ::::: 3, be a Riemannian manifold. Then M n can be loca1Jy
isometrically immersed into Vn+1 ifand only if the following conditions
are satisfied:
(i) The Weyl curvature tensor C vanishes identically,
(ii) L is a Codazzi tensor.
Moreover, any such immersion is rigid. Furthermore, if M is simply
connected and (i), (ii) are satisfied, then the immersion is globally defined.
Proo! Suppose there is an isometric immersion [ : M n
Ln+ 2 . Then ([, f) = O implies that
O
=
-+
Vn+ 1
C
X ([, f) = ([* X, f )
for alI X E T M . ln other words, if we think of [ as the position vector
of M in Ln+ 2 , then [ (x) E Tx M .1. for alI x E M . Furthermore, since
where V is the connection in Ln+ 2 , then V' � [ = O, i.e., [ is a paralIel
normal vector field along M , and A f(x) = - I for every x E M . ln
particular, R .1. = O.
Fix X o E M . Then, it is easy to see that there exist orthogonal
vectors {xo ' 1/xo E Txo M .1. , so that
Since R .1. = O, there exist local extensions of {xo ' 1/xo to parallel vector
fields {, 1/, which can be globally defined if Mn is simply connected.
Since TxM .1. is a Lorentzian plane, we have
[ = <p({ - 1/),
with <p(xo)
O=
=
<p E COO(M ),
1 . But, for any X E T M ,
�
V' x [ =
X ( <p)({ - 1/) + <pV' x ({ - 1/) = X (<p)({ - 1/),
�
7. 1 .
1 13
CHARACTERlZATIONS
O, hence ip == 1 .
Now choose an orthonormal basis elo . . . , e for
n
multaneously diagonalizes A{ and A1J' that is,
so
X ( ip)
=
TxoM, that si­
and
for some Àj, /Jj E R, i
=
1,
.
.
. , n. Since
( 1)
we obtain
/Jj
= 1 + Àj
which, together with the Gauss equation, implies that
K(ej , ej)
A
=
- Ài Àj + /Jj/Jj
=
1 + Àj + Àj .
sim pie calculation shows that
in other words,
L( X, Y ) = 2 (X, Y) + (A{ X, Y ) ,
1
X, Y E TM.
(2 )
From ( 1 ) and (2) we obtain
1
(A1J' ) = L + 2 (
,
).
(3)
U sing the above expressions, and the fact that {, 1J are parallel,
the reader may verify that conditions (i) and (ii) of the theorem
are precisely the Gauss and Codazzi equations for this immersion
of Mn into Ln+2 . Uniqueness is an easy consequence of (3), and
Theorem 1 . 1 adapted to the Lorentzian case, cf. Remark 1 .2.
Conversely, suppose M satisfies conditions (i) and (ii). Consider
the trivial Lorentzian vector bundle M n x L 2 ove r M. Endow
this bundle with a compatible connection, say '\l', which makes the
sections {(x) = (x, e l ) and 1J(x) = (x, e 2 ) parallel, where el = ( 1, O)
and e 2 = (0, 1). Define A{ and A1J as in (3), and set
1 14
7. CONFORMALLY FLAT SUBMANIFOLDS
Conditions (i) and (ii) are equivalent to the fact that 'V' and (3 satisfy,
respectively, the Gauss and Codazzi equations for constant curvature
zero. Since A{ and Af/ commute, they verify the corresponding Ricci
equation. Therefore, Theorem 1 . 1 asserts the existence of a local
isometric immersion G : U C Mn --+ Ln+ 2 , whose normal bundle,
normal connection and second fundamental form are, respectively,
U x L2 , 'V ' and (3. This immersion is globalIy defined if Mn is simply
connected. Now, set h = G - (� - 1] ) From
.
'\7 z h = G .. Z + A{_f/Z = Z - Z
=
O,
we see that h is constant. Therefore, f = G - h defines an isometric
immersion of Mn into the light cone Vn+ 1 , since
(f, ! ) = ( � - 1], � - 1] ) = O.
To conclude the proof we observe that rigidity folIows from (3), and
the fact that the normal vector fields �, 1] must be parallel. I
We are now in position to present the
Proof of 7.1 . It follows from CorolIary 7.6 and Theorem 7.7 that
M is conformally flat iff (i) and (ii) are satisfied. It remains to be
shown that when n 2: 4, then (i) implies (ii). This will folIow from
the second Bianchi identity (see [Sp], IV p. 256)
(\7xR)(Y, Z, V, W) + ('VyR)(Z, X, V, W) + (\7zR)(X, Y, V, W)
for . alI X, Y, Z, V, W E T M .
calculation shows that
('V x R)( Y, Z, V, W)
=
=
O
If the Weyl tensor is zero, a direct
(\7 x L)( Z, V) (Y, W ) + (\7 x L)( Y, W) (Z, V)
- (\7xL)(Y, V ) (Z, W) - ( 'Vx L)(Z, W ) (Y, V) .
7. 1 .
CHARACTERlZATIONS
1 15
V sing the second Bianchi identity and permutation of the vectors
X, Y, Z, we obtain the following equation
[ ( \7xL)( Z, V) - (\7zL)(X, V) ] (Y, W ) + [(\7x L)(Y, W) - (\7yL)( X, W)] (Z, V)
+ [ (\7yL)(X, V) - (V'xL)(Y, V)] ( Z, W ) + [(\7zL)(X, W)
- (\7xL)( Z, W)] (Y, V) + [ ( \7yL) ( Z, W) - (\7zL)(Y, W)] (X, V)
+ [(\7zL) (Y, V ) - (\7yL)(Z, V)] (X, W ) = O.
(4)
{ Xl , . . . , Xn } be a local orthonormal frame in M. If we take Y
W = Xi , X = Xj , Z = Xk and V = Xl in the above expression,
Let
=
with the indices i, j, k , I all distinct, we obtain
It suffices to show that (5) holds when j =f k = I . Now, choosing in
(4) successively: X = Xj , Y = W = Xi , Z = V = Xk ; X = V = Xi ,
Y = Xj , Z = W = Xh ; and X = W = Xh , Y = V = Xk > Z = Xj ,
we get the following equations .
[(V'xj L )(Xk> Xk ) - ( \7Xk L) (Xj , Xk )] + [(V'Xj L )(Xi , Xi ) - (V'xj L)(Xj, Xi) ] O
[ ( V' Xj L )(Xi , Xi) - (\7 xj L)(Xj , Xi)] + [(\7 Xj L)(Xh, Xh ) - (\7 Xh L)(Xj, Xh)] O
[ ( \7Xj L )(Xh , Xh) - (V'xh L )(Xj, Xh)] + [(\7Xj L)(Xk > Xk ) - ( V'xk L)(Xj, Xd] = o.
=
=
If we subtract the second equation from the sum of the other two
equations, we obtain
and this completes the proof that L is a Codazzi tensor. The proof
that (i) holds for n = 3 is straigthforward, and is left to the reader. I
7.8
COTollary
Let Mn , n 2: 3, be a simply connected Riemannian manifold. Then
M is conformally flat ifand only if it can be isometrically immersed as a
hypersurface in the light cone.
n
7.
1 16
CON FORMALLY
FLAT
S U B MAN I FOLDS
Observe that the induced metric on a submanifold of the light
cone is Riemannian if and only if the submanifold is transversal to
the generators of the cone. Otherwise, the submanifold would have
some isotropic tangent vectors.
The manifold Mn + m = S� x H::!c is conformally flat
Example.
for any dimensiono ln fact, the image of the product of standard
embeddings
lies isometrically in the light cone v n + m + 1 as a hypersurface.
The following result is due to Kuiper [Ku] .
7.9
Corollary
Let Mn be a simply connected conformally flat manifold. Then M n
can be conformally immersed into the Eudidean sphere s n and this
immersion is unique, up to conformal transformation of s n . ln particular,
if Mn is compact, then Mn is conformaI to s n .
Proof. It follows immediately from Proposition 7 .5 and Theorem
7.7, that Mn is conformally immersed into the Euclidean sphere s n
in a unique way. If Mn is compact, then the immersion f is a local
diffeomorphism between compact connected spaces. It is well known
that f is a covering map, and since s n is simply connected for n � 2,
f is a diffeomorphism. I
7.2. LOW COD I MENSION
7.2
1 17
Conformally jlat submanifolds with low codimension
ln this section, we first discuss some metric restrictions for con­
formalIy ftat submanifolds of space forms, and then give a char­
acterization of the conformalIy ftat hypersurfaces due to Cartan­
Schouten. Next, we obtain some topological restrictions for compact
conformalIy ftat submanifolds of Euclidean space. We conclude the
section studying submanifolds of two spaces with different constant
sectional curvature.
We say that Mn is a conformally flat submanifold of a Riemannian
manifold Mn+ p if there exists an immersion f : Mn --t Mn+p such
that Mn is conformalIy ftat with the induced metric. When the
codimension p is one, we say that Mn is a conformally flat hypersurface
of Mn+l .
Let f : Mn --t Mn+p be an isometric immersion. We calI a
subspace U C TxM umbilical if there exist { E TxM .l , II { II = 1 ,
and >. E R, such that the second fundamental form a satis fies
a(Z, X)
=
>. (Z, X) {,
for alI
Z
E
U,
X
E
TxM.
(6)
The next result, due to Moore [Mo5 ] , asserts the existence of
umbilical subspaces of conformally ftat submanifolds of space forms,
with low codimension.
7.10
Theorem
Let f : Mn --t Mcn + p be an isometric immersion with p � n - 3. If
Mn is conformal1y fiat, then for each x E M there exists an umbilical
subspace U(x) C TxM such that
dim U(x) � n
-
p
� 3.
As a consequence of this result, which will be proved in a
while, we obtain the following characterization of the conformalIy
ftat hypersurfaces of spaces forms due to Cartan [Ca2] and Schouten
[Se].
7.
1 18
7.1 1
Theorem
CON FORMALLY FLAT SUBMAN I FOLDS
Let f : Mn --+ Mcn+ 1 , n 2:: 4, be an isometric immersion. Then Mn is
conformally flat if and only if f has principal curvatures of m ultiplicity
1, n - 1 or n only.
Proof. Suppose Mn is conformally Bat. By Theorem 7. 1 0 , there is
an umbilical subspace U(x) C TxM , with dim U(x ) 2:: n - 1 , for each
x E M . Therefore, there is a principal curvature with multiplicity at
least n - 1 .
l n order to prove the converse, we just need to show that C
Let x E M and define a linear map T : Tx M --+ Tx M by
Ric (X, Y)
( T X, Y ) ,
=
X, Y
E
=
o.
TxM.
If we introduce this identity in the definition of L(X, Y), a simple
calculation shows that the Weyl tensor may be written as
=
C(X, Y)
R( X, Y )-
1
n
2
_
(T X J\Y + X J\ TY) + (
n
_
tr T
l (
) n
_
2) X J\Y.
ln the case that M n is a hypersurface of M;+ 1 , we obtain from the
Gauss equation
TX
=
( n - l)cX
+
(trA )AX - A 2 X,
X E Tx M,
and thus
C(X, Y)
=
AX J\ AY +
1
n
_
2
(A 2 X J\ Y + X J\ A 2 y)_
(t rA f - t rA 2
t rA
( n 2 AX J\ Y + X J\ AY) + ( n - l )( n 2) X J\ Y.
_
/
_
Since the principal curvatures have multiplicities equal to 1 , n - 1 or n,
we may choose an orthonormal basis Xl , . , Xn for Tx M , satisfying
.
AXj
=
ÀXj ,
j
=
1, . . . , n - 1 ,
.
and AXn = J.lXn .
Therefore,
t rA
=
(n - l )À + J.l,
and t rA 2
=
(n - 1)À2 + J.l2 ,
7.2. LOW CODIMENSION
1 19
from which we conclude after a sim pie calculation that C
= o.
I
Remarks.
( 1 ) Theorem 7. 1 1 is not valid for n = 3. ln fact,
Cartan [Ca2] (see also [L]) and later Lancaster [La] obtained examples
of conformally fiat hypersurfaces of R 4 with 3 distinct principal
curvatures. See also exercise 7 . 3 .
( 2 ) Consider a conformally fiat hypersurface f : M n --+ Q� + l , with­
out umbilical points, and having principal curvatures À and J1, with
multiplicities ( n - 1) and 1 , respectively. It was shown by do Carmo­
Dajczer [C-D2 ] that if J1, = J1,(À), then f (M ) is contained in a rotation
hypersurface. ln particular, a conformally fiat minimal hypersurface
of Euclidean space R n+ 1 , without totally geodesic points, is contained
in a generalized catenoid (see exercise 2.5).
ln order to prove Theorem 7 . 1 0 we need the following result,
obtained by Moore [Mo4 ], which is a Lorentzian version of Corollary
6.6.
7.12
Lemma
Let V and W be Enite-dimensional real vector spa ces, and let (3 : V x
V --+ W be a symmetric bilinear form flat with respect to a Lorentzian
inner product. Suppose
(i) dim V > dim W,
(ii) (3(X, X) =I O for all X E V, X =I O.
Then there exist a non-zero isotropic vector e
bilinear form cp : V x V --+ R such that
dim N«(3 - cpe)
2
E
W, and a real valued
dim V - dim W + 2.
Proo! Take X E RE«(3) and 1] E K er(3(X), 1] =I O. By Propo­
sition 6.5, the non-zero vector e = (3(1], 1]) is isotropic. Since W
is a Lorentzian space, we may choose a pseudo-orthonormal basis
e = el, e 2 , · . . , e n such that (e l , e l ) = (e 2, e 2 ) = O, (e I , e 2 ) = 1 and
(ei, e j ) = Dij , i = 3, . . . , n, j = 1, . . . , n. With respect to this basis,
7. CONFORMALLY FlAT SUBMANIFOLDS
120
fJ has the form fJ = Ej=l qJ e j , where the
valued bilinear forms. However,
qJ : V
x
V ---t R are real­
2
<jJ (y, Z ) = (fJ (Y, Z ) , e l ) = (fJ (Y, Z ) , fJ( 1], 1])) = (fJ (Y, 1]) , fJ (Z, 1])) = O,
where the last two equalities follow from Proposition 6.5, and the fact
that fJ is symmetric and Hat. Since the bilinear form
is Hat with respect to the positive definite inner product on W, we
obtain from Corollary 6.6
dim N(fJ - <jJle) � dim V - dim W = dim V - dim W
+
2. I
Proof of 7.10. Let f : M n ---t Mcn +p be an isometric immersion, and
let x E M. Define a Lorentzian inner product on W = REBREBTxM �
by
((( a , b , 1]) , (a' , b' , 1]' ))) = - aa' + bb' + (1], 1]' ) .
Let fJ : TxM x TxM ---t W be the symmetric bilinear form defined in
terms of the second fundamental form a and the tensor L , given by
fJ(X, Y)
= ( L (X, Y)
+
1
1
2 (1 - c ) ( X, Y) , L (X, Y) - 2 (1
+
c ) (X, Y) , a( X, Y)).
Then fJ satisfies fJ(X, X) ::f O, for all X ::f O, and it is Hat. The second
assertion follows easily from the Gauss equation and the fact that the
Weyl tensor vanishes. Let e and <jJ be given by Lemma 7. 12. Then
dim N(fJ - <jJe)
� n - p � 3.
Choose a basis e l , e2, . . . , e p+ 2 for W , such that el = (1, O, . . . , O),
e2 = (0, 1, O . . . O) and e3, . . . , e p+ 2 is an orthonormal basis for TxM � .
Write e as
Let n E N(fJ - <jJe) and Y E Tx M . We have
7.2. LOW CODIMENSION
12 1
This means that
L(n, Y)
and
1
+ 2( 1 - c ) ( n , Y) = l{>(n, Y )
1
L(n, Y) - 2 ( 1
+
c ) ( n, Y )
= b2 l{>(n, Y) .
(7 )
(8)
By taking the difference between (7) and (8), we obtain
( n , Y ) = (1 - b2 )l{> (n , Y),
which implies that 1 - b2 =I O, and
l{>(n, Y) =
1
1
_
b2
( n , Y) .
Finally, from the definition of (3, we have that
Thus N«(3 - l{>e) is an umbilical subspace of Tx M satisfying dim N«(3 l{>e) � n - p · 1
We now show that the existence of umbilical subspaces, for
isometric immersions of compact manifolds into Euclidean space,
provides topological information about the submanifold itself. ln
particular. we obtain some topological properties of conformally ftat
submanifolds of Euclidean space. The following result is due to
Moore [Mos].
7. 13
Theorem
Let f : Mn --+ Rn+ p be an isometric immersion ofa compact manifold.
Assume that for each point x E M, there exists an umbilical subspace
U (x) C TxM, such that
dim U ( x ) � k > O.
1 22
7.
CONFORMALLY FLAT S U B MAN I FOLDS
Then M has the homotopy type ofa CW -complex structure with no cells
of dimension r , where n k < r < k. ln particular, we have for the
homology groups of M that
-
n
-
k
< r <
k,
for any coefficient group G.
Proof. Let v E Rn+ p be a fixed unit vector, and consider the height
function hv : M - R given by
hv(x )
Let X, Y E Tx M
.
A
=
direct calculation shows that
X (hy)
and
(f (x), v ) .
=
(X, v )
Hess hy(X, Y) = (a( X , Y ), v ) ,
(9)
where a is the second fundamental form of f . Observe that x E M is
a criticaI point of f iff v E TtM.l. For each point x E M , choose an
orthonormal basis Xt, . . . , Xn for TxM , so that X1 , . . . , Xk E U (x).
Then, for 1 ::; i ::; k and 1 ::; j ::; n ,
where ex = À (�x, v ) , cf. (6).
If hv has only nondegenerate criticaI points, then ex =1= O, and
the index of each criticaI point will be either 2: k if ex < 0, or
::; n
k if ex > O. ln this case, a well-known result in Morse theory
([Mi] Theorem 3 . 5 ) asserts that the compact manifold M has the
homotopy type of a CW -complex structure with no cells of dimension
-
n
-
k <
r <
k.
It remains to be shown the existence of a vector v E Rn+ p such
that hv is a Morse function, i.e., hv has only nondegenerate criticaI
points. To see this, we introduce the generalized Gauss map <j> : N
sn+ p - l , defined by
----7
<j>(x, w)
=
w,
where N stands for the unit normal bundle defined as
N
=
{ ( x, w)
E
M
x
Rn + p w E T M .l and Il w ll = I }.
:
x
7.2. LOW CODIMENSION
7.1 4
Lemma
1 23
The generalized Gauss map 1> : N
----t
s n + p- l satis/ies
We conclude from the above result and Sard's theorem (see [Sp],
p.
55) that, for almost ali W E s n + p- l , hw is a Morse function. This
II
concludes the proof of Theorem 7. 1 3. I
Proof of 7. 1 4. We will compute 1>* at an arbitrary point (x, w) E N,
by choosing a suitable coordinate system in N . Let (X b . . . , X n ) be a
coordinate system in a neighborhood V C M of x , and let 1/1, . . . , 1]p
be an orthonormal frame in TM.L l v , so that 1/p (X ) = w. Consider a
coordinate system
where W is a neighborhood of the origin, such that
'/f' (0)
=
Define
(O, . . . , 0, 1)
'/fi : V
x
W
----t
'/fi (Xl" ' " X n , 1 1 , . . . , lp _ I }
l S: i, j S: p - l .
and
=
N by
(
Xl" ' " Xn ,
)
t '/f'j (/l, "
" Ip - l )1/j (Xl, . . . , x n ) .
)=1
This is a coordinate system for N in a neighborhood of (x , w) , where
the generalized Gauss map can be written as
1>(Xl, " " X n , 11, · · · , lp _ I }
Therefore, we have, 1>(x , O )
P
=
L '/f'j (/ l ,
"
Ip - l )1]j ( X l , . . . , x n ) .
=w
p 8
8
'/f' "
- 1>(x, O) = � -) (O)1/j (x) ,
8I "
� 8' "
j=l
I
"
j=l
I
i
=
1 , . . . , P - 1,
7. CONFORMALLY FLAT SUBMANIFOLDS
1 24
and
- 4>(x, O) - - (x) _ m,p
8X k
8
8X k
_
- Aw
(-)
8
8X k
+ V'
.L
B
�
T/p ,
k
=
1 , . . . , n.
Since we are identifying T(x,w)N with TwSn+p-I via parallel transport
in Rn + p , we obtain
-
..
ln other words, the matrix of 4> (x, w) in the basis
. . . ' ôt
Ô
p- l
.
IS
Ô l'
,
� • • • Ô�n ' Ô�l '
The proof follows using (9). I
From Theorems 7 . 1 0 and 7. 1 3 we obtain topological restrictions
for a compact submanifold of Euclidean space, with low codimension,
to be conformally Ralo
7. 15
Theorem
Let f : Mn ---+ Rn+p be an isometric immersion with p � n - 3. If M is
compact and conformally flat, then M has the homotopy type of a C W ­
complex structure with no cells of dimension r, for p < r < n - p . ln
particular, the homology gTOUpS of M must satisfy
Hr(Mn, G)
= O,
P <
r
< n
-p
for any coellicient group G .
Next w e consider the situation where a Riemannian manifold
M n can be isometrically immersed into spaces of constant curvature
M;+ 1 and M; + P , with c < é and low codimension. ln this situation
we have the following result, obtained by do Carmo-Dajczer [C-DIJ.
7.2. LOW CODIMENSION
7.1 6
Theorem
125
Let Mn be a Riemannian manifold. Assume that Mn can be isometrically
immersed in both Mcn + 1 and M;+ P , c < ê , and p :S n - 3. Then,
for each x E M, there exists a subspace Ux C TxM umbilical for both
immersions, with dim Ux ;:::: n - p .
Proof.
Let fI : M n ----; M;+ l and h : Mn ----; Mén + p b e the two
isometric immersions referred to in the statement, and denote by
( , ) , ( , ) 1 ' ( , h the Riemannian metrics of M n , Mcn+l , M; + P , respec­
tively. Fix a point x E M throughout the proof, and let
be the second fundamental forms of !t, h , respectively. Let Nj be the
first normal space of J; , i = 1, 2, and set W = NI EB R EB N2 . Observe
that NI =I O by Theorem 3.3. We define a Lorentzian inner product
(( , )) on W by requiring that (( , )) = - ( ' ) 1 in NI , (( , )) = ( ' ) 2 in N2,
and that the direct surnmands of W are pairwise orthogonal. Define
a symmetric bilinear form f3 : TxM x TxM ----; W by
f3(X, Y )
=
(1l 1 ( X , Y ), Vê - c (X , Y) TJ, 1l2 (X, Y » ,
where TJ is a generator for R with I TJ I = 1. Then f3(X, X ) =I O for alI
non-zero X E TxM , and it follows from the Gauss equation that f3 is
ftat. By our hypothesis, dim TxM > dim W , and thus Lemma 7 . 1 2
applies. Therefore, there exists a non-zero isotropic vector e E W,
and a real valued bilinear form cp : Tx M x TxM ----; R such that
dim N (f3 - cpe)
;:::: n - p .
( 1 0)
Take an orthonormal basis (I , . . . , (p for (Tx M ) so that e = N +
cos CPTJ + sin cP(t , where N is a generator of N1 · For any v E N (f3 - cpe)
we have
f3( v, Z) = cp( v , Z)( N + cos CPTJ + si n cp(t )
�,
for alI Z E Tx M. Hence
1l 1 (V, Z) = cp( v, Z)N
Vê - c ( v , Z) = cos cpcp(v, Z)
(1 1)
( 1 2)
and
( 1 3)
1 26
7.
CONFORMALLY FLAT SUBMAN I FOLDS
It
folIows from ( 1 2) that COS I{J ::f O. From ( 1 1 ), ( 1 2) and ( 1 3), we have
Q' 1 (V, Z) =
Jt - c
cos I{J
(v, Z) N
and
for alI v E N (f3 - </Je) and Z E Tx M . Therefore, N (f3 - </Je) is an
umbilical subspace for both fI and /2 , and the statement follows from
( l O). I
7.1 7
Corollary
Let Mn, n � 4, be a Riemannian manifold. Assume that Mn can be
isometrica11y immersed in both Mcn+ 1 and Mp+ l , t ::f c. Then M n is
conformally Bat.
Proo!. Follows directIy from Theorem 7. 1 6, and the characteriza­
tion of conformally flat hypersurfaces 7 . 1 1 . I
Theorem 7. 1 6 was used in [A-D2] by Asperti-Dajczer to describe
the n-dimensional submanifolds of both Rn+ l and sn+ p .
7.3
Conformally jlat hypersurfaces
This section, essentially from do Carmo-Dajczer-Mercuri [C-D-M],
is devoted to the local classification of the non-umbilical and non­
flat conformally flat Euclidean hypersurface s. It turns out that these
,
hypersurfaces can be described as envelopes of a one-parameter
family of spheres (see exercise 7.8). More precisely, they are given by
the following construction.
7.3. CONFORMALLY FLAT HYPERSURFACES
127
Let c : (a , b) � Rn + l be an immersed curve, and let r : (a, b) �
R be a positive real differentiable function with I r ' (I ) 1 < I l c'(t) l l ,
t E (a, b). Set
For each 1 , consider the affine hyperplane of Rn + l orthogonal to c ' (t)
and passing through
"f(I)
=
c(t ) - S(I ) ' c'(t),
and in this hyperplane consider an ( n - 1)-sphere 2:�- 1 with radius
R(t ) and center "f(I ) . As t runs in (a, b), 2:�- 1 describes a set that is
the image set of the map g : (a, b) x Sf- l � Rn + l given by
g (t, x) = c(I ) - S(t )c'(t) + R(t)1>(t, x ),
t E (a , b) ,
x
E
Sf-l . ( 1 4 )
Here 1> : (a, b) x Sf- l � Rn + l is any diflerentiable map which, for a
fixed l , is an immersion 1>t of Sf - 1 in Rn + l that satisfies (1)I 1 C'(t)) = O.
It is clear that for any 1> satisfying the above conditions, the image of
g will remain the sarne. Furthermore, any change of the parameter
I of the curve c to s = s(t) (with s' (t ) =f O) does not affect either the
condition I r ' l < I l c ' ll or the image set of g .
We will see i n a short while what conditions g must satisfy to be
an immersed hypersurface.
7.1 8
Theorem
Assume that g in (14) is an immersed hypersurface for n ?: 4. Then g is
a conformally flat hypersurface without umbilical points whose principal
curvature À with multiplicity n - 1 , is everywhere non-zero. Conversely,
any conformally flat hypersurface f : M n � Rn + l , n ?: 4, without
umbilical points and with À =f O everywhere, is locally of the form g;
furthermore if M is orientable, f (M ) is contained in the image of a
hypersurface of the form g .
Proof. We will first prove the converse. Consider an arbitrary point
x E M , and a unit normal vector field 'fJ defined in a neighborhood
of x . Denote by D). and DIJ the distributions associated to the
128
7. CONFORMALLY FLAT SUBMANI FOLDS
eigenvalues À and I" of A1J' respectively. These distributions do
not depend on the choice of 1] and, as it was seen in the proof of
Theorem 2.6 (i), they are differentiable. Since DJl is one-dimensional,
it is involutive.
We claim that D>. is also involutive, and that À is constant along
the leaves of D>.. ln fact, given X, Y E D>., it follows from the Codazzi
equation that
X(À)Y - Y(À)X
= (A1J - M)[X, Y].
The right hand side of the above equality lies in DJl' while the left
hand side lies in D>.. Hence they are both zero. Thus D>. is involutive
and À is constant along D>..
From the above considerations we see that coordinates (u I . . . . ,
un -I. t) can be chosen around x so that the coordinate vectors éJ�i =
Vi , f, = T, satisfy
where V is the canonical connection of R n + l . Thus, since Rn +l is Hat,
Hence, since À =I 1",
-
V'T Vi
=
- -- Vi
À'
À - I"
+
Vi ( I")
-- T ;
À - I"
here a prime denotes derivative in t .
Now consider a leaf L >. c M, and denote by i : L >. ----> M its
inclusion map. Then, by setting I I T I I = i , and denoting by Ã1J and
Ã.rr the shape operators of f o i relative to 1] and fJT, respectively, we
obtain
( 1 5)
It follows from equations ( 1 5) that the leaf L >. is umbilical relative to
the immersion f o i : L >. ----> Rn + l . Thus we can find an orthonormal
pair of vectors {I . {2 in the plane span{T, 1]}, such that the corr-e­
sponding shape operators of f o i satisfy
7.3.
CONFORMALLY FLAT
where
{32 1- O
HYPERSURFACES
129
is the (constant) sectional curvature of L).. . l n fact, it is
easily checked that
and
Çl
1
= -
{3
(--
)
6À'
6T + À11 ,
À-P
Consider the function
c (t, u)
=
( - -6>.' )
-
1
6 = - ÀóT
(3
À
P
11 .
1
f (t, u ) + >.,11(t, u),
where we have, perhaps after a change of orientation, that
positive. Then, since
8c
8uj
that is,
c (t, u)
=
À
À
is
is constant along the leaves of D).. , we obtain
V; - >.,AI)V; = O,
1
depends onIy on
c
t.
1
� i � n - 1,
Further,
-( )
' _
À'
À-p
-- T - 11·
À2
À
Hence, if we denote by , the function
t , we have
( 1 6)
Therefore,
Il c' 11
2 >
, 12 , and thus the curve
function " satisfy the required conditions.
that is,
and the positive
On the other hand,
the definition of the vector field Çt , we have
and
c (t)
using
130
7. CONFORMALLY FlAT SUBMANI FOLDS
J (;��2
If we introduce in the above expression fJ =
and use the obtained value o f ! ! c' ! ! , w e conclude that
+
À2 , À
=
!: '
1
r12
rr'
TJ =
c' + r 1 2
2
>.
I I c / 1l 6 ·
I I c' 1 1
Therefore, i n the coordinate system ( uI . . . . , Un -I . t ), the isometric
immersion f is given by
1
f (t, u ) = c(t) - >. TJ(t, u ) = c(t) - S(t )c/(t ) + R (t)ljJ(t, u ) ,
where ljJ(t, u ) = -Çl (t, u ) . A simple inspection of the expressions for
Çl and c' in the orthogonal basis T, TJ shows that ( ljJ, c /) = O. This
proves the converse of the theorem.
Now assume that g is an immersed hypersurface. Since (ljJ, c' ) =
O, we get I I g - c l l 2 = r 2 • Choose orthogonal local coordinates
(UI, . . . , u n -d for Sf - l , and differentiate I I g - c l l 2 = r 2 in t and Ui ,
1 � i � n - 1, to obtain
\ ::/ g )
)
/ ag - c , , g - c = r r .
\ at
From the above, and the fact that (g - c, c / ) - r r', we obtain that
-c
=
O,
\ �� g )
,
It follows that TJ given by
-c
=
=
g - c = -rTJ
�
I
O.
lfJ;
( 1 7)
is a unit normal vector to g . Since
= -r
, we conclude that g
is a conformally flat hypersurface with À = !: > O as the eigenvalue
of A1J with multiplicity at least n - 1 .
It remains to prove that g i s non-umbilical. To see this, we
observe that the image of g is naturally foliated by spheres; thus we
can choose locally an orthogonal coordinate system ( u I . . . . , Un- l , t )
for g adapted to this foliation. By what we have seen in the first part
of the proof, this gives rise to a curve a(t) = g + À - I TJ, which by ( 1 7)
agrees with c (t) . Thus by ( 1 6) we have that À :f JJ, since I I c ' II 2 > r ,2 .
This completes the proof. I
ln what follows, we describe the necessary and sufficient condi­
tions for the map g , given by ( 1 4), to be an immersion. We do not
give the proof here, but refer the reader to [C-D-M] for details.
7.3. CONFORMALLY FLAT HYPERSURFACES
7.1 9
Proposition
131
Let g be given as in (14). A point (t, q ) is singular for g if and only if
both conditions below are satis/ied.
(i) 1 - S ' = IIc�1I2 (R (4), c") + S ( c' , c" ) ) ,
(ii) R ' = S ( c " , 4» .
Condition (i) implies condition (ii), and if S =f O (i. e., ri =f O),
condition (ii) implies condition (i).
The last results can be used to construct a large number of
examples of conformally fiat hypersurfaces. For instance, if c is a
straight line, so is the curve 1(t) = c - Sei, and we obtain a rotation
hypersurface. On the other hand, if r is a small constant, then
c(t ) = 1(t), and we obtain a tube around c, i.e., the total space of
a normal sphere bundle of c with radius r. We can also obtain
complete conformally fiat hypersurfaces that are neither tubes nor
rotation hypersurfaces. See [C-D-M] for details.
Remarks. ( 1 ) Theorem 7. 1 8 is essentially contained in E. Cartan
[Ca2], where he proves with his own methods that a conformally
fiat hypersurface of Q�+ l is locally an envelope of a one-parameter
family of umbilical submanifolds of Q�+ l . Cartan only considers such
envelopes that are immersed hypersurfaces, and does not go into the
question of singularities treated in Proposition 7. 1 9. He also do not
describe explicitly the above method of construction of conformally
fiat hypersurfaces.
(2 ) The compact conformally fiat hypersurfaces of a conformally fiat
manifold have been classified by do Carmo-Dajczer-Mercuri [C-D-M] .
The results obtained in this section are part of the local tools for that
classification, which runs as follows. Diffeomorphically, a compact
conformally fiat hypersurface Mn is a sphere s n with b1 (M ) handles
attached, where b1 (M) is the first Betti number of M . Geometrically,
it is made up by (perhaps infinitely many) non-umbilical submanifolds
of Rn+ l that are foliated by complete round (n - 1 )-spheres, and
are joined through their boundaries to the following three types of
umbilical submanifolds of Rn+ l : (a) an open piece of an n-sphere or
an n-plane bounded by a round (n - 1 )-sphere, (b) a round (n - 1)­
sphere, (c) a point.
(3) Conformally fiat submanifolds in low codimension have also been
considered by Moore-Morvan [M-M] , and Noronha [Nor] .
7. CON FORMALLY FLAT S U BMANIFOLDS
132
Exercises
7.1.
L et
gl
and
manifold
R
g2
M n.
such that
be (pseudo-) Riemannian metrics in a differentiable
Suppose there exists a differentiable function >. :
gl
connections of
for alI
7.2.
X, Y
Mn
E
=
gl
).g2.
if V I
Show that
and
v2
M ......
are the Levi-Civita
and g2, respectively, then
TM .
...... (Mn+P, gl), /2 : Mn (M n +p , g2) be isometric immer­
sions such that ft(x) = /2(x) for every x E M , and suppose there
exists a differentiable function >. : Mn+p - R such that g} = ).g2.
Let /} :
_
(i) Let a } and a 2 denote the second fu ndamental form of ft and
h.
a
}
respectively. Show that
(X, Y)
= a
2
(X, Y) -
1
.1
2>' g2( X, Y) (grad2 >') ,
(ii) Show that the normal curvature tensors
R�
X, Y
and
/2, respectively, verify
.L
Ri (X, Y)�
for alI
7.3.
X, Y
E
TxM
and � E
=
E
R�
TxM.
of ft and
.L
R2 (X, Y)�
Tx M .
.L
(Mn, g) be a manifold of constant sectional curvature.
(i) Show that the "warped product" (M n x R, <f}(t )g
Let
+
dt 2 )
is
conformally fiat for any non-vanishing real function <p.
(ii) Conclude that the cone i n
R4 over a surface
in
S3
with constant
Gaussian curvature is conformally flat.
(iii) Verify that any hypersurface in
R4
whose second fundamental
form has at most two eigenvalues is conformally fiat.
(iv) Use (ii) to show that the condition in (iii) is not a necessary
condition.
7.4.
Show that Theorem
7. 1 1
remains valid if the ambient space
replaced by an arbitrary conformally fiat manifold.
Mcn+1
is
133
EXERCI SES
7.5.
Mn
Let
be a Riemannian manifold, and let
T : TxM - TxM
be
defined by
=
(T X, Y)
Ric (X, Y),
( i ) Verify that the Weyl tensor
C(X, Y )
=
R(X, Y) - n
Mn
(ii) I f
1
_
2
C
X, Y
E
TxM.
can b e written as
trT
(TX i\Y + X i\ TY ) + (n _ l (n
)
is a hypersurface of Á(n+
TX = (n - l)cX
+
1,
_
2) X i\Y.
then
(trA)AX - A2 X
and
C(X, Y)
=
1
A X i\ AY + n 2 (A2X i\ Y + X i\ A2y)
trA
(trA)2 - trA2
- (n _ 2 (A X i\ Y + X i\ A Y ) + ( n l)(n 2) X i\ Y.
)
_
_
7.6.
Let
Mn, n
_
� 4, be a Riemannian manifold. Show that the following
are equivalent:
(i)
M
is conformally fIat.
(ii) At every point x
E M,
and for every 4-dimensional subspace
C TxM, there exists a constant C(L) such that for any two
mutually perpendicular 2 -planes aI , a2 spanning L,
L
(iii) At every point x
vectors
E M,
and for every quadruple o f orthogonal
7. CONFORMALLY FLAT SUBMANIFOLDS
134
7. 7.
7.8.
Assume that the isometric immersion f : Mn ----t Rn+ 2 , n 2': 3, has
an umbilical non-totally geodesic normal unit vector field � E T M l..
which is nowhere parallel, i.e. , for alI x E M we have that V��(x ) =I °
for some Z ETxM . Conclude that M is conformally ftat.
An envelope of the one-parameter family of spheres with center a (t ) and
radius ')'(t) is a solution of the system
(X - a ) 2 - ')'2 = 0,
7.9.
7.1 0.
7.1 1 .
7.12.
(X - a, a ' )
+ ')'')'
'
= o.
Verify that the hypersurfaces given by ( 1 4 ) are of this type.
Show that any conformally ftat hypersurface Mn of Mcn+1 , n 2': 4,
can be locally isometrically immersed in any other space of constant
curvature MF+1 with é < c.
We say that a Riemannian manifold Mn , n 2': 3, satisfies the axiom
of conformally fiat hypersuifaces if for every point x E M and every
hyperplane section H C TxM, there exists a conformally ftat hyper­
surface S of M, passing through x , such that Tx S = H. Show that
if Mn , n 2': 4, admits an isometric immersion as a hypersurface into
the Lorentz space L n+ 1 , then M n satisfies the axiom of conformally
ftat hypersurfaces.
Let M n , n 2': 4 , be a connected Riemannian manifold. Assume that
M n can be isometrically immersed in both Q�+ 1 and Qg+ 1 , c =I é,
with constant mean curvature H and fI, respectively. Conclude that
M is a space Nk of constant curvature k or contained in a Riemannian
product Nk x R.
Let f : M n ----t Rn+p , p :::; n - 3, be an isometric immersion of a
conformally ftat Riemannian manifold. Given x E M , let U(x ) C
TxM be the umbilical subspace given in Theorem 7 . 1 0.
(i) Verify that for all X
E
TxM, Y E U(x )
L(X, Y ) = 2 À2 ( X, Y ) ,
1
where n (X, Y)
=
À
(X, Y ) f
Hint: Use the formula
Ric( X , Y) = (n( X, Y ), nH )
n
-
L ( n (X, Xi ), n (Y, Xi ) )
i=l
135
EXERCI SES
to show that
Ric (X, Y) = ( n
-
1 ) À2 (X, Y)
X
for all
E
Tx M,
Y
E
U(x ) .
(ii) If RJ.. = O, show that f is quasi-umbilical, i.e. , there exists an
orthonormal basis 6, . . . , çp of TxM J.. such that each Açj has a
principal curvature of multiplicity at least n 1 .
-
E
Hint: First show that for all X, Y
holds:
L(X, Y)
For
any
-
À ( a(X, Y), Ç)
+
Tx M the followin g relation
1 2
2 À (X, Y )
Y E U this foJlows fram (i). For X, Y
ZEU
L ( X , X ) + L (Z, Z )
=
K (X, Z)
=
Conclude that the bilinear form [3
defined by
[3(X, Y)
=
a(X, Y)
-
E
=
O.
U J.. use that for
( a(X, X) , a(Z, Z) )
:
Tx M
À (X, Y) ç
x
Tx M
-+
.
TxM J..
is fIat. Now use the hypothesis that RJ.. = O and Corallary
6.6, to show that dim N ([3 ) = n dim S ( [3 ) , and apply exercise
6.2 (i).
-
,
,
Chapter 8
Real Kaehler Submanifolds
8.0
Introduction
ln this chapter we present sorne results on real Kaehler submanifolds,
that is, isometric immersions of Kaehler manifolds into Euclidean
space. ln section 1 , we first review definitions and basic facts about
Kaehler manifolds. Then we show that a complete real Kaehler
submanifold in low codimension, must be unbounded.
ln section 2, we consider the question of whether a real Kaehler
submanifold is a holomorphic submanifold with respect to sorne com­
plex structure of Euclidean space. We prove that if the type number
T is greater or equal to three, everywhere, then the imrnersion has to
be holomorphic. To obtain this result, we make use of the existence of
a l-parameter associated family of isometric immersions to a minimal
real Kaehler submanifold. The holomorphicity question is reduced
to the triviality of the associated family.
8. 1 . COMPLETE REAL KAEHLER SUBMANIFOLDS
8.1
137
ComPlete real Kaehler submanifolds
ln this section we recall basic facts about Kaehler manifolds. We
refer the reader to [K-N] for further details. We also present a
Kaehlerian analogue of the J orge-Koutroufiotis theorem, d ue to Fwe
and Hasanis.
An almost complex structure on a real differentiable manifold M is
a tensor field J of type (1, 1) satisfying J 2 = -I, where I denotes the
identity tensor field. A differentiable manifold with a fixed almost
complex structure is called an almost complex manifold. It is clear that
an almost complex manifold M is even dimensional, and that each
tangent space TxM has a basis of the form Xl , J Xl , . . . , X , J Xn . Any
n
two such bases differ by an isomorphism with positive determinant;
from this one easily concludes that an almost compie x manifold is
orientable.
The space c n = { (Zl , . . . , zn ) : Zk = Xk + iYb Xk , Yk E R} carries
a natural almost complex structure J defined by
k = 1, .
. .
, n.
It is easy to see that a map f : U C cn � cm is holomorphic if
and only if f* o J = J o f* , since this condition is equivalent to the
Cauchy-Riemann equations for each coordinate function.
A complex manifold M of complex dimension n, is a 2n-dimen­
sional real differentiable manifold admitting an open cover {Ua } and
coordinate maps 'Pa : Ua � Cn such that 'Pa o 'Pi l is holomorphic on
'Pf3(Ua n U(3) C Cn for alI Q, f3. A complex manifold M can be given
naturally an almost complex structure JM via the coordinate maps,
i.e. , in each Ua define
where J is the almost complex structure of Cn • This definition is
independent of the map 'Pa, and thus JM is globally defined.
An almost complex structure J on a manifold M is called a
complex structure if M is an underlying differentiable manifold of a
complex manifold which induces J in the way just described.
8. REAL KAEHLER SUBMANIFOLDS
138
A map f : M --+ M between two complex manifolds is said to be
holomorPhic if its representation in local coordinates is holomorphic.
This tums out to be equivalent to the condition
f* o J = j o f* ,
where J and j are the almost complex structures of M and M ,
respectively.
We define a Kaehler manifold M as an almost complex manifold
endowed with a Riemannian metric ( , ) such that the almost complex
structure J of M is an orthogonal tensor which is parallel with respect
to Levi-Civita connection; in other words, the following properties
are satisfied
(J X, JY) = (X, Y)
and
(V'xJ)(Y) = V'xJY - JV'xY = O
for alI X, Y E TM . l n view of a deep theorem of Newlander­
Nirenberg, the almost complex structure of a Kaehler manifold is
a complex structure.
As sim pie examples of Kaehler manifolds we have: the complex
space c m with the canonical metric ds 2 = � j dzj dzj , and an ori­
entable surface with any metric, since there is a canonical complex
structure given by the oriented rotation of angle 7r/2.
8. 1
Proposition
Let M be a Kaehler manifold with curvature tensor R. Then, for every
E M and X, Y E TxM , we have
(i) R(X, Y) o J = J o R(X, Y),
(ii) R(J X, JY) = R(X, Y),
(iii) Ric (JX, JY) = Ric (X, Y).
x
Proof. This is a straightforward computation. I
For every plane P C Tx M invariant by the complex structure J
of a Kaehler manifold M , we define the holomorphic sectional curvature
of P by
K(X, J X ) = (R(X, J X)J X, X) ,
8. 1 . COMPLETE REAL KAEHLER SUBMANIFOLDS
139
where X is any unit vector in P . There are several results, analogous
to the real case, about spaces with constant holomorphic sectional
curvature. For example, the unique complete simply connected
Kaehler manifolds with constant holomorphic sectional curvature are:
the complex projective space Cpn with the Fubini-Study metric, the
open complex unit ball Dn(C) = {z E Cn : Il z ll < I } with the
Bergman metric, and cn for positive, negative or zero curvature,
respectively.
Now we start our study of isometric immersions of Kaehler
manifolds.
8.2
Proposition
Let M 2n and M2m be Kaehler manifolds with complex structures ]
M2m is a holomorphic isometric
and j, respective1y. If f : M2n
immersion, then the second fundamental form a of f satisnes
---4
a(X, J Y) = ja(X, Y )
=
a(J X, Y)
(1)
for all X, Y E T M . ln particular, f is a minimal immersion.
Proo!. Let X, Y E
we have
TxM .
Since f is holomorphic, and M is Kaehler,
Now ( 1 ) folIows from the Gauss formula and the symmetry of a .
The equality a(X, JY) = a(] X, Y) for all X, Y E Tx M ,
equivalent to
Aç o ] = -J o A ç
1S
for alI ç E TxM 1.. . This easily implies that every odd symmetric
function of Aç is zero. ln particular, f is minimal. I
N ext we tum our attention to real Kaehler submanifolds, begin­
ning with the following result, due to Hasanis [Has].
8. REAL KAEHLER SUBMANIFOLDS
140
8.3
Theorem
Let M 2n be a complete Kaehler manifold with seccional curvatures
bounded /Tom below, and let f : M 2n --t R2n+p be an isometric immer­
sion. If p < n then f(M) is unbounded.
Proof. Suppose that f(M) is bounded, and let h : M
differentiable function defined by
h(x )
=
--t
R be the
1
2 (f(x), f(x» ) .
It follows from Lemma 3.8 that there exists y E M such that
for all non-zero Y
E
TyM . On the other hand, we have that
Hess h (Y )(X, Y ) = XY h(y) - V' x Y h(y)
=
=
(VxY, f(y ») + ( X, Y ) - ( V'xY, f (y» )
( X, Y ) + (a (X, Y) , f(y» )
for every X, Y E TyM . Therefore,
(a (X, X) , f(y» )
< O
(2)
for all non-zero X E Ty M .
Consider the vector space W = TyM1. EB TyM 1. , with the inner
product (( , )) of type (p, p ) given by
Define
a
bilinear form f3 : TyM
f3(X, Y)
=
x
TyM
--t
W(p , p) by
( a (X, Y ) , a (X, J Y» ,
X, Y
E
Ty M.
It follows from the Gauss equation and Proposition 8. 1 that
((f3(X, Y ) , f3( Z, W»)) = (a (X, Y), a ( Z, W» ) - (a (X, JY ) , a (Z, J W»)
= ( R(X, Z )W, Y) + (a (X, W ) , a (Z, Y » ) - (R (X, Z)JW, J Y )
- (a ( X, JW), a (Z, J Y» )
= ((f3(X, W ) , f3( Z, Y» ))
8. 1. COMPLETE REAL KAEHLER SUBMAN I FOLDS
141
for ali X, Y, Z, W E Ty M . ln other words, f3 is a fiat bilinear form
with respect to (( , )) .
Clearly, for X E Ty M the subspace Ker f3(X) = { Y E TyM :
f3(X, Y ) = O} is invariant by the complex structure of M , and
dim Ker f3(X) � 2(n p) > O. ln particular, when X E RE (f3),
it follows from Proposition 6.5 that
-
for Zt, Z2
E
Ker f3(X) and Y1, Y2
E
TyM ; that is
Thus there exists an isometric linear isomorphism Í : U --+ U, where
U is the vector subspace generated by the set a(TyM, Ker f3(X» ,
satisfying
Ía(Y, Z )
=
a(Y, J Z )
E
TyM and Z E Ker f3(X ) . Hence, for every unit vector
Z E Ker f3(X ), the unit vector J Z satisfies
for alI Y
a(JZ, J Z )
=
Ía(JZ, Z)
=
Ía(Z, JZ)
=
a(Z, J2 Z)
=
-a(Z, Z ) ,
which contradicts (2). I
As an immediate consequence we obtain the following result of
Fwe [Fw].
8.4
Corollary
There is no isometric immersion of a compact Kaeh/er manifold M 2n Ín
R2n+p with p < n.
A local classification of the real Kaehler hypersurfaces of Eu­
clidean space was obtained by Dajczer-Gromoll [D-G3l The case
where the hypersurface is complete was first considered by Abe [AI].
8. REAL KAEHLER SUBMAN I FOLDS
1 42
8.2
Holomorphicity 01 Real Kaehler Submanilolds
ln this section we establish some conditions for an isometric immer­
sion of a Kaehler manifold into Euclidean space to be holomorphic.
An isometric immersion f : M2n --? M2n +p from a Kaehler
manifold whose second fundamental form a satisfies
a(X,JY) = a(JX, Y), for all X, Y E T M,
is called a circular immersion. We conclude, as in the proof of
Proposition 8.2, that circular immersions are minimal. Surprisingly,
the converse is true when M is a Euclidean space, as shown by the
following result, obtained by Dajczer-Rodriguez in [D.R2].
8.5
Theorem
Let M2n be a Kaehler manifold. An isometric immersion f : M2n
R2n + P is minimal if and only if it is circular.
--?
Proof. Let x E M be an arbitrary point, and consider an orthonor­
mal basis Xl , . . . , X2n for TxM, such that X2 j = JX2 j - l , j = 1 , . . , n.
Since f is minimal, it follows from equation 1 .7 that
.
Ric M(Xi , Xi) =
Ric M(JXi,JXi) =
2n
-
-
L I l a(Xi, Xj)11
j= l
2n
2
L Il a(JXi , Xj)112 .
(3 )
j=l
On the other hand, from the Gauss equation and Proposition 8. 1 , we
obtain
Ric M(Xi,Xi) = L (R(X j ,Xi)Xi,Xj )
j�
2n
=
-
L (R(Xj , Xi)J Xi,JXj )
j�
L ( a(Xj , JXi), a(Xi, JXj» ) ,
j=l
=
(4 )
8.2. H OLOMORPHICITY
1 43
where the last equality follows from J 2 = - I and the choice of the
basis.
Let V be the direct sum of 2n copies of TxM J. , i.e. ,
V
and let
(( , )) : V
x
V
--7
=
2n
EB (TxM-L) j,
j=l
R be the inner product defined by
(( , ))
2n
=
L
( , ).
=l
j
V i, wi E V be the elements
Vi = (a(Xi, JXI ), . . . , a (Xi , JX2n )),
wi (a(X1, J Xi), " " a(X2n , J Xi» '
For each 1 ::; i ::; 2n, let
=
By the Cauchy-Schwarz inequality, we have
Ric M ( Xi, Xi) 2
(( V i. Wi)) 2
::; ((V i, VÚ) ((Wi, W i ))
2n
2n
= L Ila(Xi, J Xj)11 2 L Il a( Xj , J Xi ) 1 1 2
j=1
j=l
2n
2n
= L Il a (Xi, Xj ) 1 1 2 L Il a (Xj, JXi ) 1 1 2
j=l
j=l
Ric M (Xi, Xi) Ric M (JXi . JXi) .
But from Proposition 8. 1 , we get Ric M ( Xi • Xi )
Ric M (JXi, JXi ) ,
=
=
=
hence we have equalities in the above expression. This implies Vi =
±Wi . Suppose V i = -Wi for some 1 ::; i ::; 2 n . From (4) , we obtain
Ric M (Xi• Xi ) = - ((Vi, Wi )) = (( Vi, Vi )) '
Since Ric M (Xi, Xi) ::; 0, we have Ric M ( Xi • Xi ) = 0, and thus Vi
Wi = ° by (3). The case Vi = Wi for all 1 ::; i ::; 2n is the aSSf'rtion
=
that f is circular, and this concludes the proof. I
The following result, obtained by Dajczer-Gromoll in [D-G3 ],
asserts the existence of a l-parameter family of minimal immersions
associated to a minimal real Kaehler submanifold. This generalizes
the classical associated family to a minimal surface in R3 (see [Sp], I V
p . 40 1 ).
144
8. REAL
8.6
Theorem
KAEHLER
SUBMANIFOLDS
Let M 2n be a simply connected Kaehler maniEold, and let I : M 2n R2 n + p be a minimal isometric immersion. There exists a l -parameter
Eami1y le : M 2n - R2n +p, () E [0, 1f) oE minimal isometric immersions
such that lo I .
=
-
Dajczer Gromoll theorem
Proo/. Consider for each () E [0, 1f) the tensor Te : T M - T M
defined by
To = cos ()l + sin OJ,
where J is the complex structure of M , and I is the identity tensor
field. It is clear that To has the following properties:
(i) To is parallel with respect to the Levi-Civita connection of M ,
(ii) To i s an orthogon al tensor field,
(iii) To o Lo = l.
Let ao : T M
x
TM
- TM
�
ae (X, Y )
be the bilinear form given by
=
a (TeX, Y ) ,
where a is the second fundamental form of I . For each ç E T M � ,
denote by A� the linear transformation associated to a(J, i.e.,
(A�X, Y)
=
(ae(X, Y), ç) ,
for ali
X, Y
E
TM.
If A{ denotes the second fundamental form of I in the direction of
ç, we claim that
Since I is circular hy Theorem 8.5, we have
)
(A�X, Y = cos O ( a ( X, Y ), Ç) + sin () ( a (J X, Y ), Ç)
cos O (a ( X, Y ) , Ç) + sin O ( a ( X , JY ), Ç)
= cos O (A{X, Y) sin O (JA{X, Y)
= (T_eA(X, Y)
=
for alI X, Y
E
TM .
-
We leave the other equalities as a n exercise.
8.2. HOLOMORPHICITY
1 45
We assert that A� is self-adjoint. Given X, Y E T M , it follows
from the above equalities and the properties of To that
(A �X, Y ) = (A{ToX, Y ) = (ToX, A{Y )
= (X, T_oA{Y ) = ( X, A �Y ) ,
which proves the assertion.
Now we claim that, with respect to the normal connection of
I, the tensor a(J satisfies the Gauss, Codazzi and Ricci equations.
The Gauss equation is a straightforward computation. The Codazzi
equation is clear because To is parallel and A� = T_(JA{ . For the Ricci
equation, we have
( [A � , A �] X, Y ) = (A �A �X , Y ) - (A �A � X, Y )
= (T_(JA{T_(JAT/X, Y ) - (T_ (JA T/ T-oA{ X, Y )
= (A{ToT-oAT/X, Y ) - (A{ToT_(JAT,X. Y )
= ([A{, A T/]X, Y ) ,
and this proves the claim.
Under these conditions, Theorem 1 . 1 implies the existence of
an isometric immersion lo : M2n -+ R2n+p , which is unique up to
isometries of R2n+p , and whose second fundamental form is a(J .
Clearly lo = I. It remains to be shown that lo is minimal. Since
I is circular,
ao(J X, Y) = a(ToJ X, Y)
= cos (Ja(J X, Y) + sin (Ja(J 2 X, Y)
= cos (Ja(X, J Y ) + sin (Ja(X, J2 y)
= a(J( X, JY)
for ali X, Y ET M. This conc1udes the proof by Theorem 8.5. I
The family lo obtained above is called the associated family to
the minimal isometric immersion I . Although the sarne construc­
tion applies to circular isometric immersions into real space forms
I : M 2m -+ Q; m + p , it can be shown that if c '" O, then m = 1 . Hence
8.
1 46
REAL KAEHLER S U B MAN I FOLDS
the only circular isometric immersions into non-flat space forms are
minimal surfaces. For details see [D-�].
Now we relate the associated family with the problem of holo­
morphicity of the immersion. The following result was obtained by
Dajczer-Gromoll in [D-G3].
8. 7
Theorem
Let I : M 2n -+ R 2n +p be a substantial minimal isometric immersion of a
simply connected Kaehler manifold, and let fe, fJ E [0, 11"), be its associated
family. If I is holomorphic then, for each O E [0, 11"), lo is equal to I up
to isometries of Rn+ p . Converse/y, ifthere are 01 t (h E [0, 11") such that
fel and 102 differ by an isometry of Rn+p, then I is holomorphic.
Proof. Suppose I is holomorphic. This means that there exists a
complex structure j on R 2n+p such that I.. o J = j o f.. , where J is the
complex structure of M . Define, for each O E [0, 11"), an orthogonal
tensor to : R 2n + p -+ R21T+ P by
to = cos O]
+
sin oj,
where ] is the identity map. From Proposition 8.2 and Theorem 8.6,
we conclude that the second fundamental form of the immersion fe
is given by
ao(X, Y)
a(TI}X, Y)
= cos Oa(X, Y) + sin Oa(J X, Y)
= cos Oa(X, Y) + sin oja(X, Y)
= tl}a(X, Y)
=
for alI X, Y E T M. l n other words, tI} preserves the second
fundamental formo
It is immediate that tI} is paralIel with respect to the Levi-Civita
connection of R 2n+ p . We claim that tI} is parallel with respect to
the normal connection 'V� of TM� . Indeed, given X E TM and
� E T M � , it follows from the Weingarten formula that
-L
-
'Vx (TI}Ü
-
-
'V x (TI}Ü + At9 � X
= tl}('er xÜ + TI}A�X
tl}('\7xÜ + tI}A�X
= tl}('V�ܷ
=
=
8.2.
H OLOMORPH I CITY
147
From the above and Theorem 1 . 1 , we conclude that f and fe differ
by an isometry of R2 n + p .
ln order to prove the converse, we may suppose that (h = O
and denote (h by o. The assumption that fo and f differ by an
isometry of R2n + p implies, by Theorem 1 . 1 , in the existence of a
parallel orthogonal tensor 5 : T M.L --; T M .L , such that CY.() = 5 CY. .
Therefore, the adjoint S = 5' of 5 is also an orthogonal tensor,
parallel in TM .L , and satisfying
As{
for alI ç
E
=
A{ o T()
T M .L . Consider S() : TM .L --; T M .L defined by
sin OSe = S - cos 01.
Then, for each X
E
TM and ç E TM.L , we have
sin o V'i (S()Ç)
=
=
=
Vi (Sç - cos OÇ)
S (V'i O - cos OV' i ç
sin OSe(Vi O ,
that is, SI) is parallel with respect to V' .L .
We claim that S� = - I , and that Se is orthogonal. Note that
ASeç = Aç o J , since
sin BAseç
=
ASç - cos BAç
=
Aç
o
To
-
cos OAç
=
sin OAç o J.
Therefore, A (s: +l )ç = O for ali ç E TM .L . Suppose that S� -:f - I
Since S� + I is a parallel tensor, thus the orthogonal complement
in T M.L of the image of S� + I is a parallel normal subbundle,
which contains the first normal space of f, since A (SJ + I )ç = O. From
Proposition 4. 1 , f admits a reduction of codimension, which is a
contradiction. Therefore si = -I and, in particular, p is even.
ln order to show that SI) is orthogonal, first observe that, from
the definition of S() , we have that
.
S2
since si
=
-I.
-
2 cos BS + I
=
O,
Now we apply SI to this expression, and get
S
+
Si
=
2 cos O I.
8. REAL KAEHLER SUBMANIFOLDS
1 48
Hence
sin20SoS� = SSI COS O(S + SI) + COS2 0!
= ! - 2 cos20! + cos20!
2
= s i n 0! ,
-
that is, So is orthogonal. This proves our claim.
It remains to define a complex structure on R2n+p which makes
f holomorphic. This is accomplished through the use of So and I as
follows. Define a tensor Ío : T M E9 T M.l --t T M EB T M .l by
This is an orthogonal tensor satisfying Íi = -!. We claim that Ío is
parallel in R2n+p . Observing that AS8{ = A{ o I, and S� = -Se, we
have that Sea(X, Y) = -a(X, IY) and Ass{ = -IA{ . Using these
last equalities, we get
VxÍe(Y
+
0 = Vx (IY - SeO
= VxIY + a(X, JY) + ASs{X - V'�Se�
= IV x Y - Sea(X, Y) - IA{X - Se V'��
= ÍeVx (Y + O,
which proves the claim. We have obtained that Ío is parallel in R 2n+p
along M. Now extend Íe to alI of R2n + P by parallel transport. It is
clear that f is holomorphic with respect to this complex structure. I
The last result of this chapter, obtained by Dajczer-Rodriguez in
[D.R2 ], relates the type number of the immersion to the question of
holomorphicity.
8.8
Theorem
Let f : M2n --t R2n+ P be an isometric immersion of a Kaehler manifold
M 2 n into R2n+ p . If Lhe type number T satis/ies T(X) 2:: 3 for all x E M,
Lhen f is holomorphic.
8.2.
HOLOMORPH IC ITY
149
Proof. It suffices to show that 1 is circular. ln fact, for any simply
connected open neighborhood U of an arbitrary point x E M, the
restriction I l u admits an associated family 19 by Theorem 8.6. Since
T 2:: 3, it follows from Theorem 6.7 that the family 19 is trivial, hence
I l u is holomorphic by Theorem 8.7.
ln order to prove that 1 is circular, consider the map ã : TxM x
TxM ---t TxM 1- , defined by
ã(X, Y) = a(J X, Y ),
X , Y E Tx M,
where J is the complex structure of M . It follows from Proposition
8. 1 and the Gauss equation that
(a(X, Y), a(Z, W) ) - (a(X, W), a( Z, Y )) = ( R (X, Z)W, Y ) = ( R (J X, J Z )W, Y)
= ( a (JX, Y), a(J Z, W) ) - (a(JX, W ), a (J Z, Y))
= ( ã (X, Y), ã( Z, W )) - (ã(X, W ) , ã( Z, Y)) .
We conclude from Proposition 6. 1 0, that there exists an isometric
linear isomorphism T : TxM1- ---t TxM 1- such that T a(X, Y ) =
a(J X, Y) . Therefore,
a(J X, Y) = Ta(X, Y ) = Ta(Y, X) = a(X, JY ) ,
and the result follows. I
Remark.
Calabi [C] has established that holomorphic isometric
immersions into complete simply connected Kaehler manifolds with
constant holomorphic sectional curvature are rigid regardless of the
codimension.
For other results re1ated to Theorem 8.8, we refer to [D3 ], [D-R5],
[D-Th] , and lU].
8. REAL KAEHLER SUBMANI FOLDS
150
Exercises
8.1.
8.2.
8.3.
Provide examples showing that the condition p < n in Theorem 8.3
is the best possible.
Let I : M 2n ---> Q;n + p be a circular isometric immersion. If c i O
conclude that n = 1 .
Assume that a real Kaehler submanifold M 2n of R 2n +p satisfies
a(I X, Y)
8.4.
=
-a(X, JY)
for alI X, Y E T M . Conclude that the second fundamental form a is
paralIel , i.e. , V.La = O.
Let I : Mn ---> Rn+p be an isometric immersion.
(i) Show that the l-form 1* T on M with values on R n + p is closed
jf and only if T satisfies
o
for alI ( E T M .L , where T : T M ---> M is an orthogonal paralIel
tensor field.
(ii) Assume that M is simply connected, and that I is a minimal
real Kaehler submanifold. Show that the associated family lo is
explicitly given by the line integral
fo(x)
where
xo . is
=
j 1*
X
Xo
o
To,
any fixed point in M .
Hint: Use that
which shows that lo is isometric, and that the tangent spaces of
I and lo at x are parallel in Rn+ p for alI x E M , i.e., lo has the
sarne Gauss map as I .
8.5.
We say that two immersions I, g : M n ---> RN make a constant angle
if for alI X E TxM , the angle in RN between the vectors 1* ( x) X ,
g* (x)X depends only on the point x E M . Verify that any two
associated minimal real Kaehler submanifolds make a constant angle.
EXERCISES
8.6.
151
Let L be a holomorphic curve in C2 without fIat points, and let >.(x )
be the positive principal curvature of the second fundamental form
Aç for any unit normal vector � E Tx L..L . We define the focal loeus S
of L in R4 = C2 as the hypersurface
(i) Compute the singular points of the induced metric on S .
(ii) Verify that the scalar curvature o f S vanishes everywhere.
Ch apte r 9
ConformaI Rigidity of
H ypersurfaces
9.0
Introduction
ln two of his Iess known papers, E. Cartan studied the conformaI
deformations of hypersurfaces of the Euclidean space Rn+ 1 , n � 5 .
As a consequence of his work, h e obtained a (local) sufficient condition
for conformaI rigidity.
9. 1
Cartan's Conformai Rigidity
We say that a conformaI immersion f : Mn - Rn+p is conformally
rigid if given any other conformaI immersion g : M n _ Rn+p , there
exists a conformaI diffeomorphism T of Rn + p such that g = T O f .
Recall that by Lioville's theorem (see [Sp] , II p . 3 1 0 ), the conformaI
diffeomorphisms of open sets in Euclidean space are compositions of
rigid motions, homo�eties and inversions.
The proof of Cartan's result which we present here, is basically
the one given in do Carmo-Dajczer [C-D4], where Cartan's theorem
(see [Ca2] p. 1 0 1 ) has been generalized to higher codimensions.
9. 1 .
9.1
Theorem
CARTAN'S CONFORMAL RIGID ITY
153
Let f : Mn - R n+ l , n ;::: 5 , be a conformal immersÍon of a connected
Riemannian manifold. Assume that, with the induced metric, f has no
principal curvatures ofmultiplicity at least n - 2 at any point. Then f is
conformaJJy rigid.
( 1 ) It is easily checke d that the multiplicity of a
9.2 Remarks.
principal curvature is invariant by conformai transformations ofRn + l .
(2) The above result is sharp in the following sense. ln [Ca3], Cartan
has shown that for n = 4 , there exist hypersurfaces that have distinct
principal curvatures at each point and are not conformally rígido
ln this case, there exists only one other non-congruent conformai
immersion. On the other hand, in [Ca2], Cartan classified 10cally ali
hypersurfaces f : Mn - Rn + l , n ;::: 5, with a principal curvature of
multiplicity n - 2, which are not conformally rígido The set of non­
congruent conformai deformations ís a one-parameter family, and the
( n - 2)-dimensional umbilical subspace remains invariant throughout
the deformation.
For the proof of Theorem 9. 1 we will make use of the following
algebraic result due to Dajczer [D3 ].
9.2
Lemma
Suppose that the biJinear form j3: V x V - W is non-zero and flat,
where W is a 4-dimensional space of type (2, 2) . Moreover, assume
dim N (j3) � dim V - 5. Then W admits an orthogonal direct sum
decomposition W = W1 EB W2 such that W1 is a nondegenerate subspace
oftype (q, q ) , q = 1 or 2, and if j31 and f32 are the W1 and W2 components
of j3, respectively, then
(i) j31 is non-zero and null,
(ii) f32 is flat and d im N (f32) ;::: dim V - 2 .
Proo!. It is clear that j3(X, V) is a degenerate subspace of W for
any X E RE(j3) since ker j3(X) t= {O} . Moreover, if j3(X, V) is an
isotropic subspace for ali X E RE(j3), we conclude that j3 is null
from Proposition 6.4. Setting W1 = W and W2 = O, we obtain the
conclusion of the lemma in this case.
It remains to consider the situation in which a regular element
X E V exists so that j3(X, V) is not an isotropic subspace. ln this
154
9.
CONFORMAL RIGIDITY OF HYPERSURFACES
case, the isotropic subspace U(X) = (3(X, V) n (3(X, V) l. satisfies
dim U(X) = 1 . To see this, observe that if dim U(X) = 2 we
would have, from Proposition 6.2, 4 = dim W = dim (3(X, V) +
dim (3(X, V)l. , arid thus (3(X, V) = (3(X, V)l. = U (X) , which is a
contradiction. From dim U(X) = 1, it follows that 2 � dim (3(X, V)
� 3, and hence
dim ker (3(X)
2:
dim V - dim (3(X, V)
2:
dim V
-
3.
(1)
We claim that the subspace S«(3) is orthogonal to U(X). Other­
wise, there would exist Uo, Vo E V such that ( (3( Uo, Vo), {I ) -::f O, where
6 E W is an isotropic vector spanning U(X). For n E ker (3(X) and
Y E V we have, from Proposition 6.5 and flatness, that
iff ( (3(Y, n), (3(Uo, Vo)) = ( (3 ( Uo , n ) , (3(Y, Vo ) ) = O .
(2 )
Consider the linear map L : ker (3(X) -7 U(X) given by
(3(Y, n )
O
=
L(n) = (3(Uo, n).
By (2) ker L
dim N «(3)
C
2:
N «(3), and therefore, using ( 1 )
dim ker L 2: dim ker (3(X) - dim U(X )
2:
dim V - 4,
which is a contradiction, and proves the claim.
Now we complete {I to a pseudo-orthonormal basis 6, . . . , {4 of
W so that ({I, {2) = 1, ( 6 , {2) = O, and ({i, {j ) = O for 1 � i � 2,
3 � j � 4 or i = 3 , j = 4, (cf. Proposition 6. 1), and we
write (3 = L ;=l cpi {j , where each cpi is a real valued bilinear formo
From 6 E S«(3), we obtain </> 1 -::f O, and from the fact that {I is
2
orthogonal to S«(3), we conclude that </> = O. Set W1 = span {{I, {Ú,
W2 = span {6, {4}, (31 = </> 1 6 and f32 = </>36 + ifJ4{4' Then (31 -::f O
is null and thus f32 is flat. I f (32 -::f O, we claim that S«(32) is nonde­
generate. To see this observe that if ( L�=l f32(Xj, Yj), f32(Y, Z)) = O
for alI Y, Z E V , then ( L�=l (3(Xj, Yj ), (3(Y, Z)) = O, and thus
L� =l (3(Xj , Yj) E W1 . This implies L�=l f32(Xj, Yj) = O. On the
other hand, if dim N (f32) < dim V - 2, we conclude, by the sarne
argument as the one used in the beginning of the proof, that for any
X E RE«(3), f32(X, V) is degenerate, and thus isotropic by dimension
9. l . CARTAN'S CONFORMAL RIGIDI TY
155
reasons. This is a contrad iction, and completes the proof of the
lemma . •
Proof of 9.1 . L et g : Mn � R n + l be another conformai immersion.
By stereographic projection , we can assume that f, g are conformai
immersions in Sf+1 , and consider Sf+ l to be isometricalIy embedded
in the light cone Vn+2 C Ln + 3 , as in Chapter V I I . The immersion
f, considered as a submanifold of Vn+2 C Ln+3, with the induced
metric, will be denoted by F. Let � be the inner unit normal vector
to S1 + 1 in the affine hyperplane of L"+3 which contains S.+ 1 . The
second fundamental form aI of F is given by
(3)
where AN is the shape operator of f .
Now, since g : Mn � S1+1 is an immersion conformai to t, there
exists a positive smooth function rp : M � R such tha t
for alI X, Y
Vn+2
C
E
TxM . Define an isometric immersion G :
Ln+3 by
G(x )
=
1
g (x).
rp (x )
Mn
-+
To conclude the proof, it is sufficient to show that there exists an
isometry T of Ln+ 3 such that G = T o F. The n , usi n g Proposition 7 .4,
it is easy to see that the conformai transfor mation T : S.+ 1 -+ S1+ 1 ,
induced by T , is such that g = T o f . l n order to produce such a
T, we need to obtain a bundle map between the normal bundles of
the immersions F and G , as submanifolds of L n + 3 , preserving the
metric, the second fundamental form and the normal connection.
Notice the following facts. If IV is a unit normal vector field to
g , then
-
1
1
(G * (x) X , N ) = ( X ( - )g (x) + -g * ( x)( X ) , N )
rp
rp
that is, N
E
=
0,
Tc (x)M � Also G(x) E Tc(x)M � . It follows th at
.
Tc(x) M � =
span
-
2
{N } e L ,
1 56
9. CON FORMAL RIGI D ITY OF HYPERSURFACES
where L2 is a Lorentzian plane, and the direct sum is orthogonal.
Since G E L2 is isotropic, there exists a basis �, 1/ for L 2 such that
G = � + 1/,
(1/, 1/)
=
1,
( 1/, O =
(�, Ü
0,
=
-1.
Thus the second fundamental forrn 02 of G can be written as
Furtherrnore, since
A G(x)( X ) = -G .. (x )X = - X,
we conclude that A � + 7] =
-
I It
follows that
=
(A � + 7] X, Y)
.
( 02 ( X, Y) , � + 1/)
=
- (X, Y) .
Hence
02 = «( 02 , 1/ )
+
( , ) )�
+
(02, 1/) 1/ + (AN,
)N.
( 4)
Now let
be given the natural rnetric (( , )) of type (2, 2). Define f3 : TxM
TxM --+ W by
We clairn that f3 is fiat.
A
x
calculation shows that
((f3 (X, Y), f3 ( Z, W) )) - ((f3 ( X, W ) , f3 (Z, Y ) )) = (X, Y) ( 02 (Z, W ) , 1/)
+ (Z, W ) (02(X, Y ) , 1/) - (X, W ) (02( Z, Y), 1/ ) - (Z, Y) ( 02( X, W), 1/ )
- (ANX, Y) (ANZ, W) + (ANX, W) (ANZ, Y) + (X, Y) (Z, W)
- ( X, W) ( Z, Y) - (A N X, W) (ANZ, Y) + (AN X, Y) (AN Z, W)
+ (X, Y) ( Z, W) - (X, W) (Z, Y) .
9. 1 .
CARTAN'S CONFORMAL RIGIDITY
157
Observe that the first eight su mmands of the above expression m ay
be written, up to sign, as
(a2(X, Y), a2 ( Z, W) ) - ( a2 (X, W), a2 ( Z, Y ) ) .
B y the Gauss equations for F and G, the last four summands give
the curvature of the metric induced by F, and the first eight the
curvature of the metric induced by G . By observing the signs, we
easily conclude that f3 is ftat, as was claimed.
Now, we notice that by definition of f3, f3(X, X) =I O for ali X =I
O. Therefore, N (f3 ) = O. Since n 2': 5, Lemma 9.3 implies that we
can decompose f3 as f3 = f3t EB f3z, where
is nonzero and null, and
is either zero or ftat with dim N (f3z) 2': n 2 .
We claim that f3z = o. If f32 =I O, it follows that both Wt , W2
are Lorentzian planes, and f3t = fjJ6 , where 6 E Wt is an isotropic
vector, and fjJ is a real valued symmetric bilinear formo Thus, there
exist at , a2, a3 , a4 E R not ali zero, such that ai + a� = a� + a�,
and 6 = a t''l + a2N + a3{ + a4N . By definition, n E N (f3z ) iff
f3( n, Y) = f3t ( n , Y) = fjJ ( n , Y)6 for ali Y E TxM . Therefore,
-
(a2 ( n , Y), 1/) + ! ( n , Y) = a t fjJ (n, Y)
( a2(n, Y), 1/ ) + � ( n, Y) = a3fjJ ( n , Y)
( AJÍln , Y) = a2fjJ ( n , Y )
(ANn, Y ) = a4fjJ( n , Y)
(5)
(6 )
(7)
(8)
for ali n E N (f3z) and Y E TxM . From ( 5 ) and (6 ) , we obtain
( n, Y) = ( a3 - a t ) fjJ( n , Y),
(9)
which implies a3 - a t =I O; while from (8) and (9), we get
( 1 0)
158
9.
CONFORMAL RIGI DITY OF HYPERSURFACES
for alI n E N (/32 ) and Y E TxM . Therefore AN has an eigenvalue
with multiplicity equal to dim N (/32) � n - 2 . This contradicts the
hypothesis on f, and proves the claim.
From
/32 = 0, we have that f3 is nulI.
fi =
fi =
Let f3
=
fi EB fi, where
«(a2 , 1]) + ! ( , ))1] + (AN , ) N
«(a2 , 1]) + � ( , ) )� + (A N , ) N.
Since f3 is nu lI, we have for alI X, Y, Z, W that
( fi( x, Y), fi( W, Z )) = ( fi( X, Y ) , fi( W, Z ) ) .
It folIows that there exists a linear orthogonal map
T : span { �, N}
such that
such that
fi = T fi.
o
�
span { 1], N}
So, we can assume that there exists
() E [0, 211")
T (ü =
eos () 1] + sin () N
T (N) = sin ()1] - eos ON.
Therefore, the equation fi = T o fi can be written as
«(a2, 1]) + ! C ))1] + (AN, ) N = « ( a2 , 1]) + i C ) )T(Ü + (A N , ) T(N)
= « ( a2, 1]) + i ( , ))(eos 01] + sin ON ) + (AN , ) (sin 01] - eos ON ) .
It folIows that
(a2 , 1]) + ! C ) = eos O«(a2, 1]) + i C )) + sin O (A N ,
(AN, ) = sin O«(a2, 1]) + i ( , ) ) - eos O (AN, ) .
From ( 1 1 ) we have
(a2, 1]) =
1 -
eos O f. 1 , and
�OS O [ � (3 eos o - 1) ( , ) + sin O (A N, )] .
(1 1)
( 1 2)
( 1 3)
9. 1 .
CARTAN'S CON FORMAL RIGIDITY
159
From ( 1 2) and ( 1 3), we obtain
(A N- , )
=
(A N , ) +
sin O ( , ) .
cos O
1-
( 1 4)
Finally (4), ( 1 1 ) and ( 1 3 ) yield
1
a2 = 2( 1 _ COS O) [( 1 + cos O)� + (3 cos O - 1 )1J + 2 sin ON] ( , )
sin O
+
1 cos O (� + 1J) + N (AN , ) .
[
_
-]
We define vectors 6, �2 in TxM ..l such that
( 1 5)
It is easily shown that (�i , �j} = Óij , 1 ::; i , j ::; 2. ln particular, it
follows that NP , the first normal space of G , is a Riemannian vector
bundle of rank 2. Since f is not an umbilical immersion, there always
exist local orthonormal vector fields X, Y such that (AN X, Y) t- O .
It follows easily that the vector fields �l , �2 are smooth.
We claim that N1G is parallel. ln order to obtain the claim, it
suffices to verify that by the hypothesis of the theorem the s-nullity
of G satisfies v; < n - 2, 1 ::; s ::; 2 , and that the arguments used in
Proposition 4.5 still work in this special Lorentzian case.
Since NP is parallel, we can reduce the codimension of G to 2.
Thus, both F and G have substantial codimension 2, and therefore
both immersions are contained in Riemannian affine hyperplanes
of L n + 3 . It follows from (3) and ( 1 5) that there exists a smooth
bundle map of the normal bundles of F and G, as submanifolds
of a Riemannian hyperplane in L n + 3 , that preserves the metric and
the second fundamental formo By Corollary 6.9, it also preserves the
normal connection. Then there exists a bundle map T between the
normal bundles of F and G, as submanifolds of Ln + 3 , also preserving
the metric, the second fundamental form and the normal connection.
From the uniqueness part of Theorem 1 . 1 , adapted to the Lorentzian
case, we conclude that T extends to an isometry T : L n + 3 -) Ln + 3
with the property that G = T o F. Thus T induces a conformaI map
T : Sr + 1 -) Sr + 1 such that g = T o f, and the theorem is proved. I
1 60
9. CONFORMAL RIGIDITY OF HYPERSURFACES
Remark. Form ula (7) shows that whenever there exists a confor­
maI deformation of the immersion, an ( n 2)-dimensional tangent
subspace remains umbilical, as observed by Cartan.
-
See [MO?] for results on conformaI immersions more in the spirit
of the results in Chapter III.
Exercises
9.1 .
9.2.
Show that the Euclidean sphere s n C Rn+ l i s not conformalIy rigid.
Let f : Mn --t Q�+2 , n � 5, be an isometric immersion. Assume that
for alI x E M a n d all non-zero { E TxM .L , we have v(x) ::; n 5 ,
and rank A { � n 3. Prove that f i s rigid.
-
-
9.3.
Hint: Use Lemma 9.3.
Let f : Mn --t Q�+ l , n � 5, be an isometric immersion which has no
p ri ncip al curvature of multiplicity at least n 2 at any point.
-
(i) Show that the isometric immersion F = i o f : Mn --t Qr2 is
rigid, where i : Q�+l --t Q r2 , é < c, is the standard umbilical
inclusion.
(ii) If g : Mn --t Qr 1 i s a ny isometric immersion, conclude that
é = c and g is congruent to f .
II
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Index
Allendoerfer's th e ore m, 8 9
Complex
Almost comple x
manifold , 1 3 7
manifold, 1 37
str u cture, 1 37
structure, 1 37
almost, 1 37
Asperti Daj c ze r theorem, 1 1 2
Cone, 2 1
Associated family, 1 46
C onfo rmally fiat
-
Asymptotic vector, 2 1
manifold, 1 0 7
Axiom of
suhmanifold, 1 1 7
con fo rm al l y fiat hypersur­
r-plan es
12
,
Connection
Levi-Civita, 3
faces , 1 34
normal, 4
r-spheres, 1 3
Convex
body, 28
hypersurface, 28
Beez-Killing theorem, 89
B i l i near form
locally, 2 7
C u rva tu re
conformai , 1 07
fiat, 8 7
Gauss- Kronecker, 26
null, 87
Bounded subset, 45
holomorphic, 1 3 8
mean, 8
Bundle
normal, 2
p rincipal, 26
paral l el
Ricci, 9
,
16
sectional, 4
Riemannian , 1 5
Cylinder, 76
subbundle, 1 4
over a curve, 26, 72
vector, 1 4
Cartan's theorem , 1 5 3
Dajczer's theorem , 5 7 , 6 1
Cartan-Schouten theorem , 1 1 8
Dajc z er Gro moll theorem, 74, 1 44,
Dajczer-Rodriguez the o re m , 1 42 ,
inequalities, 43
1 48
theorem , 44
Clifford torus, 20
Codazzi eq ua ti o n
C od i mens io n
re du cti on
,
,
-
1 46
Chern-Kuiper
,
5, 24
D ive rge n ce , 5 1
Do Carmo-Dajczer theorem, 1 25
2
54
s u bsta n tial , 54
Co m p a tible metric , 1 6
Einstein
hypersu rfac e , 32
1 72
IN DEX
Einstein
(continued)
manifold, 32
conformai, 1 1 0
isometric, 2
Envelope of spheres, 1 2 6 , 1 34
making constant angle, 1 50
Eudidean
minimal, 8
space, 6
sphere, 6
Extrinsic sphere, 1 3
ruled, 36
substantial, 54
totally geodesic, 8
umbilical, 1 1
l-regular, 5 4
Focal locus, 1 5 1
Index of
Free direction, 5 1
minimum relative nullity, 67
Fundamental theorem for
nullity, 43
hypersurfaces, 25
submanifolds, 7
Fwe's theorem, 1 4 1
relative nullity , 43
Inner product of type
(p, q), 83
Isoparametric hypersurface, 38
Isotropic
subspace, 83
Gauss
vector, 83
equation, 4, 24
formula, 3
map, 26
jo rge-Koutroufiotis theorem, 45
generalized, 1 22
spherical, 38
Generalized
Kaehler manifold , 1 38
catenoid, 37
helicoid, 20
Gradient, 45
Laplacian, 5 1
Leung-Nomizu theorem, 1 3
Light cone, 1 09
Hadamard
manifold, 45
theorem, 28
Hartman's theorem, 76
Hartman-Nirenberg theorem, 72
Hasanis' theorem, 1 40
Mean curvature vector, 8
Moore's
lemma, 1 1 9
theorem, 1 1 7 , 1 2 1
Height function, 28
Hessian, 46
Holomorphic map, 1 38
Hopf 's theorem, 5 2
Hyperbolic space, 6
Normal space
first, 54
k -th, 62
Nullity
s-nullity, 58
Immersion, 2
circular, 1 42
subspace of, 43
subspace of relative, 43
I N DEX
1 73
Omori's lemma, 45
Sacksteder's theorem, 96
Otsuki's
Schouten's theorem, 1 08
lemma, 40
Second fundamental form, 3
theorem, 42
Section
global, 1 4
local, 1 4
Parallel
Shape operator,
3
hypersurface, 37
Signature, 83
mean curvature vector, 1 1
Spherical ruled minimal sur­
second fundamental form, 22
section, 1 6
subbundle, 1 6
faces, 20
Support function, 64
Principal
curvature, 26
Takahashi's theorem, 52
direction , 26
Projection
normal ,
3
tangential,
Thomas-Fialkow-Ryan theorem , 32
Tompkin's theorem, 1 0 1
Tube around,
3
Pseudo-orthonormal basis, 84
131
Type number, 60
left and right, 60
Regular element, 86
Relative nullity distribution, 68
Ricci
Umbilical space, 1 1 7
curvature, 9
equation, 5
tensor, 9
Riemannian vector bundle, 1 5
Veronese surface, 20, 52
Rigidity
conformai, 1 52
Weingarten formula, 4, 24
isometric, 82
Rotation hypersurface,
37, 1 3 1
Weyl tensor, 1 07
C
010
Ph o n
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