Iernat. J. Mh. & Mah. Sci.
Vol.
(1978) 421-431
421
(n-2)-TIGHTNESS AND CURVATURE OF SUBMANIFOLDS WITH BOUNDARY
WOLFGANG
KIHNEL
Fachbereich Mathematik
Technische Universitt Berlin
StraBe des 17. Juni 135
i000 Rerlin 12 (West)
Germany
(Received August 2, 1978)
ABSTRACT.
The purpose of this note is to establish a connection between the
notion of
(n-2)-tightness in the sense of N.H. Kuiper and T.F. Banchoff and
the total absolute curvature of compact submanifolds-with-boundary of even
dimension in Euclidean space.
The argument used is a certain geometric in-
equality similar to that of S.S. Chern and R.K. Lashof where equality characterizes (n-2)-tightness.
KEY WORDS AND PHRASES.
tight manifolds, total absolute curvature.
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
Primary 53 C 40, 57 D 40
W. KUHNEL
i.
INTRODUCTION.
M
Let
be a compact n-dimensional smooth manifold with or without boundary-
where the boundary is assumed to be smooth
be a smooth immersion of
M
n+k
M
f
and let
E
(n+k)-dimensional euclidean space.
into the
This
leads to the notion of total absolute curvature
where
N
I
i
TA(f)
Cn+k-
IKI
*i
N
denotes the Lipschitz-Killing curvature of
K
the unit normal bundle (with only the
denotes the volume of the unit sphere
’oute
f
in each normal direction,
normals at points of 3M), and
sm Em+l.
c
m
For detailed definitions, in
particular in the case of manifolds with boundary, see[5] or [6].
[6], 2.2)
Let us state the following equation
TA(f)
The famous result of
TA(flM\3 M)
+
1/2TA(fI M)
(i.i)
R.F. Lashof gives a connection between
S.S. Chern and
total absolute curvature and the number of critical points of so-called height
functions
zf
(zf)(p)
defined by
M------+
< z, f(p) >
z
n+k-I
E S
Extending this result to the case of manifolds with boundary we can write
TA(f)
where
9.
1
(zf)
index
i
.(zf)
1
in
+k-
Ei
(i(zf)
+
denotes the number of critical points of
+
M\ M, and
Cn+k- i z(i
3M.
denotes the number of
Here a point
pM
zf
of index
(+)-critical if
i
zf
(+)-critical points of
is called
(1.2)
(zf)) *I
i
p
in
of
is critical
TIGHTNESS AND CURVATURE OF SUBMANIFOLDS
for
zf[Mp
and grad f
is a nonvanishing inner vector on
P
M
423
(for details, see
[2], [4] or [6] ).
The i-th curvature
[7]
introduced by N.H. Kuiper (cf.
r.1
can be expressed
by
1
Y.(f)
i
(cf.
[6],
Cn+k- i
lemma 4.2 or
[9],
z S
+
(i(zf)
nt-
+(zf))
i
*i
lemma 3.1). So we get
.(f).
TA(f)
The Morse-relations give the following connections between the curvatures and
some topological invariants of
M:
.(f)
TA(f) > b(m):
E(-l)i. i(f)
where b (M)
E.
b. (M)
(-i)
i
i(M)
denotes the i-th Betti-number of homology with coefficients in a
suitable field.
f
x(M)
.
(1.3)
(M)
> b
(cf. [7]).
is called k-tight
all real numbers
c
if for all k’
k
and for almost all
the inclusion map
j: (zf) :=
c
induces a monomorphism in the
,(j)
{pM/ (zf)(p)< c}
M
k’-th homology
,((Zf)c
> ,(M)
As usual we write shortly ’tight’ instead of ’n-tight’.
Then the results of N.H. Kuiper show
TA(f)
b(M)
if and only if
f is tight,
z { S
n+k-I
and
424
W.
bk(M)
k(f)
K’UHNEL
(j)
if and only if
_i (j)
and
are monomorphisms
for almost all z, all c (cf. [7] ).
Results on tightness are collected in the survey article [i0] by T.J. Willmore,
for results on k-tightness we refer in addition to the notes [i] by T. Banchoff
and [9] by L. Rodriguez, who has shown that in some sense
(n-2)-tightness is
closely related to convexity.
2.
RESULTS
As mentioned above there is a relation between tightness on one hand and
total absolute curvature and the sum of the Betti-numbers on the other hand.
The following results give certain connections between (n-2)-tightness on one
hand and usual curvature terms and the sum of the Betti-numbers on the other
Note that in case
hand.
to
But in case
8M
#
by duality arguments tightness is equivalent
n
k
k-tightness for
M
if
z
n
is even and for
k
n-I
if
n
is odd.
there are examples of (n-2)-tight immersions which are not
tight (for example: consider the round hemi-sphere).
THEOREM A
f
and
fMM
M--E
n
Let
n+k
M
be an immersion.
N,
and denote by
fundamental form of
be an even-dimensional manifold with non-void boundary
=
N
O
Let
N
O
be the unit normal bundle of
the open set of unit normals where the second
is positive or negative definite.
f
Then there holds the following inequality
1/2TA (f
M)
+
where equality characterizes
In case of hypersurfaces
(k
1/2TA(fM
Cn+k_ 1
J IKI
NON N,
(n-2)-tightness of
*i > b(m)
(2.1)
f
I) (2.1) becomes
+
TA(flM M,\M => b(M)
(2.2)
TIGHTNESS AND CURVATURE OF SUBMANIFOLDS
M,
where
425
M \)M with positive or negative definite
denotes the set of points in
second fundamental form.
In case
M,
2
n
is just the set of points with positive Gaussian curvature,
so we get
Assume
COROLLARY A i.
2
k
1-.
II
ds
0-tightness of
f
n
and
Then there holds the following
inequality
2-
K<0
IKI
do
M
where equality characterizes
[8]
Here
(2.3)
2- (M)
11
denotes the usual
For part of this result see
Prop. 9.
Assume
COROLLARY A 2.
that
.> b(M) ->
considered as a space curve.
fM
curvature of
+
flM
b(M)
2 b(M)
Then (n-2)-tightness of
is tight and that the second fundamental form of
f
implies
f
has either non-
maximal rank of is positive or negative definite.
[9]
This is shown in
embedded in
E
n.
Prop. 5.2 under the assumption that
This condition implies
M
Under the additional assumption that
(n-l)-spheres L. Rodriguez has shown that
convexity (cf.
[9]
Theorem 2).
2 b(M)
b(M)
Mn
can be
by Alexander duality.
consists of a certain number of
(n-l)-tightness is equivalent to
This is not true in general, (See Corollary
B 2 below).
THEOREM B.
or tight (if
Let
M
n
be even and
), and let M
f
M\M
n
M
+
En+ 1
be (n-2)-tight
be a compact submanifold of dimension
which is contained in some coordinate neighborhood in
the set of points in
form.
M\M
(if M # )
M
As above
M,
denotes
with positive or negative definite second fundamental
Then there holds the-following inequality
n
426
K’HNEL
W.
(2.4)
where equality characterizes
REMARK.
If
M
contains only points of vanishing curvature or definite
\M,
second fundamental form, then
S.S Chern and R.K. Lashof for
M
additional condition that
case
n
2
M
and
lies inside of some given
COROLLARY B i.
implies
KIU
0
Let
M
ds
\f-l(\)
Note that for
f
n
2
+
I
,KI
be as in Theorem B and assume moreover that there is
E
be an embedded compact submanifold of
E
E
n
tightness of
M
M
n
which
and assume
is tightly embedded in
and tightness of
this can be obtained easily using the equations
b()
n+l
in a hyperplane of
f
(n-2)-tight if and only if
n
(2.5)
do
Mn{K<0}
f(U)
M
is
>
which is embedded by
Let
by changing the scale
Then
For example in
M
being a disk we get
I
U
and (2.4) reduces to the inequality of
otherwise (2.4) is sharper and reflects the
M
an open region
flM\(\8
(n-2)-tightness of
TA()
M
1/2TA()
n
E
are equivalent:
and
1/2b()
Roughly spoken Corollary B 1 says:
(n-2)-tight minus tight gives (n-2)-tight.
In particular we get the following
COROLLARY B 2.
In each even dimension there exist (n-2)-tight hypersurfaces
which are not tight and not convex in the sense of [9]
in particular where
f(;M) is not contained in the boundary of the convex hull of
3.
f(M)
PROOFS.
In all proofs the immersion
f
is fixed and so we may write
TA(M)
instead
427
TIGHTNESS AND CURVATURE OF SUBMANIFOLDS
TA(fIM
of
and so on.
PROOF OF THEOREM A.
From
I .(M)
TA(M)
i
i
and
Z (_l)ii(M)
x(M)
TA(M) + x(M)
we get
in
MkM
zf
of index
i
(M)
On the other hand by definition
critical points of
2Z 2i (M)
is the average of the number of
which are precisely the strict local maxima
n
But a point is a strict local extremum of so height function
if and only if the second fundamental form in the direction of
or negative definite.
z
zf
is positive
Hence we get
2
n
I
(M)
J IKI *
Cn+k- i
N,
leading to
1
TA(M)
Cn+k- 1 N,
2
>
(r0(M)
2 (b
0(M)
+
2(M)
+...+
n-2(M))
x(M)
+
b
2(M)
+...+
bn_ 2(M))
x(M)
b (M)
where we have used the assumption that
b (M)
n
n
is even and
M #
which implies
0.
The case of equality is equivalent to the following equations:
428
W.
o(M)
b
o(M)
But the equality
Hi_ l(j)
(M)
z
(2.6)
bn_2(M)
is equivalent to injectivity of
(Zf)c
j
n-2(M)
2(M)
bi(M)
i(M)
for all inclusions
(n-2)-tightness of
b
KNEL
+
Hi(j) and
so (2.6) is equivalent to
M
f
The assertion of the theorem then follows from the inequality above using
the equation (I. I)
TA (M\ M)
TA (M)
PROOF
of Corollary A 2.
>
1/2TA(SM)
f18M
which implies tightness of
KI
over
N
o\N,
hence
PROOF of Theorem B.
By theorem A (n-2)-tightness of
1/2TA(M) +
b(M)
K
+ 1/2TA (M)
0
>
Cn+k- 1
f
implies
NO\ N,
1/2b(SM)
b(M)
and moreover the vanishing of the integral of
on
N\N,.
o
By assumption and by theorem A we have
TA(MM,\M) + 1/2TA(SM)
TA(M)
or
b(M)
b(M)
if 8M
#
(2.7)
if 8M
which last equality is equivalent to
TA(M\M,)
For
\(\)
b(M)
(2.8)
2
theoremA yields
TA(M\\M,\ M\)
where equality characterizes
+ 1/2TA(M) +
1/2TA(8)
(n-2)-tightness of
_>
b(M\)
flM (\)"
Subtracting (2.9) from (2.7) or (2.8) respectively we get
(2.9)
TIGHTNESS AND CURVATURE OF SUBMANIFOLDS
1/2TA(M)
<__
b(M)
b(M\)
TA(M\M,\) 1/2TA()
<
b(M)
b(M\)
TA(M\M,\$M)
429
(2.10)
(2.11)
2
respectively.
Now the assertion follows directly from the following lemma
LEMMA.
M
Let
M
M
be n-dimensional dompact connected manifolds with
MM and assume that M
M
is contained in some coordinate neighborhood of
Then
or
PROOF.
Let
B
b(M\)
b(M)
1/2b()
if
M #
b(M\)
b(M)
1/2b()
2
if
be an open coordinate neighborhood in
topologically a closed
the Betti-numbers of
M
M\M
M
We can assume
n-ball
M
in terms of that of
M
such that
M\M
B
B
and
M
B
is
To compute
we apply the Mayer-
Vitoris sequence to the following three decompostions
Io
M
(M\ B) (J B
m
(M\B)
(\(\)) f3
III.
M\ (M\ M)
M
S
n-I
8M
(MkB) kJ
(B\(M\))
(M\B)
(\(\M))
8B
=-
S
n-I
The first decomposition leads to
b(M)
b(M\B)
1
b(M)
b(M\B)
+ I
if
M #
if M
(2.12)
(2.13)
4 30
KHNEL
W.
the second one to
b(B\M) +
b() b()
b(M\)
b(M\B) +
+
(2.14)
i
the third one to
b(\)
(2.15)
2
At last we have the equation
b()
because by assumption M
2
b()
(2.16)
can be embedded in
in
E
n+l
(k
> 1 arbitrary)
standard-torus in
E
is an open region
U
3)
B
Consider for example an embedding of
M
torus’
of type
sk X sn-k
lies not in the boundary of
k
xS
n-k
TM
B
n-m
n+l.
from
Now define
U
(m
by
i)
By suitable
we started from we can assume that
M
M
U
So we can obtain an example
M
In the examples of corollary B 2 the boundaryM was always tightly
EMARK.
embedded in
S
E
is (n-2)-tight but of course it is not tight.
lies not in the boundary of the convex hull
M
S
as a tight hypersurface of rotation (like the
contained in some hyperplane of
choice of the embedding of
where
(cf. [9] Prop. 5.1).
and change this embedding a little bit such that there
removing a small tight ’solid
By Corollary
n
E
(2.16)
Now the lemma follows directly from (2.12)
PROOF of Corollary B 2
B
E
n+l.
The natural question whether there exist in higher dimensions
(n-2)-tight inersions with non-tight boundary seems to be open.
an example is due to L. Rodriguaz.
For
n
2
TIGHTNESS AND CURVATURE OF SUBMANIFOLDS
431
REFERENCES
i.
Banchoff, T.F. The two-piece-property and tight n-manifolds-with-boundary
E
Trans. Amer. Math. Soc. 161 (1971) 259-267.
in
n,
2.
Braess, D. Morse-Theorie fHr berandete Mannigfaltigkeiten, Math. Ann. 208
(1974) 133-148.
3.
Chern, S.S. and R.K. Lashof On the toal curvature of immersed manifolds,
I, II, Amer. J. Math. 79 (1957) 306-313, Mich. Math. J. 5 (1958) 5-12.
4.
Friedrich, T. m-Funktionen und ihre Anwendung auf die totale
Math. Nachr. 67 (1975) 281-301.
Absolutkr’mmung,
5.
Grossman, N.
607-614.
7 (1972)
6.
K6hnel, W.
Math Soc.
Relative Chern-Lashof theorems, J. Diff. Geom.
Total curvature of manifolds with boundary in
E
n
J
London
(2) 15 (1977) 173-182.
7.
Kuiper, N.H. Morse relations for curvature and tightness, in: Proc.
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(Lecture Notes in Mathematics 209).
8.
Rodriguez,
Geom.
L.L. The two-piece-property for surfaces with boundary, J. Diff.
Ii (1976) 235-250.
Rodriuez,
L. L.
Convexity and tightness of manifolds with boundary, in"
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Mathematics 597).
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