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Series in Pure Mathematics Volume I
Total Mean Curvature
and
Submanifolds of Finite Type
Bang-yen Chen
Total Mean Curvature
and
Submanifolds of Finite Type
OTHER BOOKS IN THIS SERIES
Volume 2: A Survey of Trace Forms of AlgebraicNumber Fields
Volume 3: Structures on Manifolds
Volume 4: Goldbach Conjecture
P E Conner & R Perils
K Yano & M Kon
Wang Yuan (editor)
Total Mean Curvature
and
Submanifolds of Finite Type
Bang-yen Chen
Professor of MathematicsMichigan State University
1rWorld Scientific
Published by
World Scientific Publishing Co Pte Ltd.P 0 Box 128Farrer RoadSingapore 9128
Copyright © 1984 by World Scientific Publishing Co Pte Ltd.All rights reserved. This book, or parts thereof, may not be reproducedin any form or by any means, electronic or mechanical, including photo-copying, recording or any information storage and retneval system nowknown or to be invented, without written permission from the Publisher.
ISBN 9971-966-02-69971-966-03-4 pbk
Printed in Singapore by Singapore National Printers (Pte) Ltd.
Dedicated to
Professors T. Nagano and T. Otsuki
PREFACE
These notes are a slightly expanded version of the author's
lectures at Michigan State University during the academic year
1982-1983. These lectures provided a detailed account of
results on total mean curvature and submanifolds of finite types
which have developed over the last fifteen years.
The theory of total mean curvature is the study of the
integral of the n-th power of the mean curvature of a compact
n-dimensional submanifold in a Euclidean m-space and its appli-
cations to other branches of mathematics. Motivated from these
studies, the author introduced the notion of the order of a
submanifold several years ago. He used this idea to introduce
and study submanifolds of finite type.
In Chapter 1, we give a brief survey of differentiable
manifolds, Morse's inequalities, fibre bundles and the deRham
theorem.
In Chapter 2, we review connections, Riemannian manifolds,
Kaehler manifolds and submersions.
Chapter 3 contains a brief survey of Hodge theory, elliptic
differential operators and spectral geometry.
In Chapter 4 we give some fundamental results on submani-
folds. The materials given in the first four chapters can be
regarded as the preliminaries for the next two chapters.
viii Preface
In Chapter 5, results on total mean curvature and its
relations to topology, geometry and the calculus of variations
are discussed in detail.
In the last chapter, the submanifolds of finite type are
introduced and studied in detail. Some applications of the
order of submanifolds to spectral geometry and total mean
curvature are also given.
In concluding the preface, the author would like to thank
his colleagues, Professors D.E. Blair and G.D. Ludden for their
help, which resulted in many improvements of the presentation.
He also wishes to express his sincere gratitude to Professor
Hsiung, who suggested that the author includes this book in the
"World Scientific Series in Pure Mathematics". Finally, the
author wishes to thank Cathy Friess, Tammy Hatfield, Kathleen
Higley, and Cindy Lou Smith, for their excellent work in typing
the manuscript and their patience.
Bang-yen Chen
Autumn, 1983
C O N T E N T S
Preface vii
Chapter 1. DIFFERENTIABLE MANIFOLDS
61_ Tensors 1
§2. Tensor Algebras 5
§3. Exterior Algebras 7
§4. Differentiable manifolds 11
§5. Vector Fields and Differential Forms 15
§6. Sard's Theorem and Morse's Inequalities 20
§7. Fibre Bundles 23
§8. Integration of Differential Forms 28
§9. Homology, Cohomology and deRham's Theorem 37
§10. Frobenius' Theorem 42
Chapter 2. RIEMANNIAN MANIFOLDS
§1 Affine Connections 46
§2. Pseudo-Riemannian Manifolds 53
§3_ Riemannian Manifolds 56
§4. Exponential Map and Normal Coordinates 62
B. Weyl Conformal Curvature Tensor 64
§6. Kaehler Manifolds and QuaternionicKaehler Manifolds 67
§7. Submersions and Projective Spaces 71
Chapter 3. HODGE THEORY AND SPECTRAL GEOMETRY
§1. Operators *, d and A 78
§2. Elliptic Differential Operators 85
§3. Hodge-deRham Decomposition 91
§4. Heat Equation and its Fundamental Solution 95
§5. Spectra of Some Important RiemannianManifolds 100
x Contents
Chapter 4. SUBMANIFOLDS
§.1 Induced Connections and Second FundamentalForm 109
52_. Fundamental Equations and FundamentalTheorems 116
§. Submanifoldc with Flat Normal Connection --1-2-4
§4_. Totally Umbilical Submanifolds 128
I5 Minimal Submanifoldc 135
§6- The First Standard Imbeddings ofProjective Spaces 141
§7. Total Absolute Curvature of Chern andLashof 167
§$ Riemannian Submersions 167
§4. Submanifolds of Kaehler Manifolds 171
Chapter 5. TOTAL MEAN CURVATURE
§1. Some Results Concerning Surfaces in R3 182
12_. Total Mean Curvature 187Conformal Invariants 203
§A. A Variational Problem Concerning TotalMean Curvature 213
B. Surfaces in Rm which are ConformallyEquivalent to a Flat Surface 226
§6. Surfaces in R4 236§7. Surfaces in Real-Space-Forms 244
Chapter 6. SUBMANIFOLDS OF FINITE TYPE
§1. Order of Submanifolds 249
§2. Submanifolds of Finite Type 255
§3. Examples of 2-type Submanifolds 260
§4. Characterizations of 2-type Submanifolds 269
§5. Closed Curves of Finite Type 283
§6. Order and Total Mean Curvature 293
§7. Some Related Inequalities 300
§8. Some Applications to Spectral Geometry 303
Contents xi
§9. Spectra of Submanifolds of Rank-oneSymmetric Spaces 307
§10. Mass-symmetric Submanifolds 320
Bibliography 325
Author Index 341
Subject Index 347
Chapter 1. DIFFERENTIABLE MANIFOLDS
$1. Tensors
Let U,V and W be vector spaces over a field IF of
dimensions m, n and r, respectively. A map f of UxV
into W is called bilinear if it is linear in each variable
separately, i.e., if
f (alu1 + a2u2, blv1 +b 2v 2) = alblf (ul, v1) +a lb2f (ul, v2)
(1.1)+ a2b1f (u2, v1) +a
2b
2f(u2, v2)
for all vectors ul,u2 in U, vl,v2 in V, and scalars
a1,a2,b1, and b2 in IF.
We denote by Hom(V,W) the space of linear maps from V
to W. Then Hom(V,W) is a vector space over IF of dimen-
sion nr. Let V* denote the space of all linear functions
on V, i.e., V* = Hom (V, V* is called the dual spaceof V. We define the map cp : V x W .Hom(V*,W) as follows:To each (v,w) in V x W, we assign a linear map cp(v,w) by
(1-2) cp (v, w) v* = v* (v) w.
for v* E V*. It is easy to see that cp : V X W 4 Hom (V*, W) isbilinear. Moreover, if form a basis of V and
wl, wr form a basis of W, then cp(vi,wi =
1, ,r, form a basis of Hom(V*,W).
2 1. Differentiable Manifolds
Let f be a bilinear map of U X V into W. We define
a linear map
of : Hom(U*,V) aW
by af((P(ui,vi )) = f(ui,vi ), where u1'...,um is a basis of
U and is a basis of V. Then (1.3) defines the
maps a f on the basis (cp(ui,vi ))i,j of Hom(U*,V). we
then extend af
to all of Hom(U*,V) so as to be linear.
One may verify that the linear map af
is in fact independent
of the choice of basis u1," um and V1*...,vn. Consequently,
to each bilinear map f from U x V into W, we have asso-
ciated a linear map af
of Hom(U*,V) into W so that
f = of o cp.
More generally, let be k finite dimensional
vector spaces over a field F. A map f, of V1 X V2 x . . . x Vk
into a vector space W is called multilinear if it is linear
in each of the variables separately.
Let U be the free vector space whose generators are the
elements of V1 X V2 x... x Vk, i.e., U is the set of all finite
linear combinations of symbols of the form (v1,.. ,vk) where
viEVi. Let N be the vector subspace of U spanned by elements
of the form:
a (vl, ...,v k) - (v1, ... , avi, ... , vk)
(v1....,v i + wi, ...,v k) - (vl, ... vk) - (vl, ... wi, ... , vk) .
§ I Tensors 3
We denote by V1 ®V2 ®... ® V,, the factor space U/N.
This vector space is called the tensor product of V1,V2'...'Vk.
We define a multilinear map, qp, of V1 X V2 x ... x Vk
into V®®V2 ®... ®Vk by sending into its coset
mod N. We write
cp(vl,...,vk) = V1 ®... ®vk.
Let W be a vector space and * : V1 x . x Vk - W a multi-linear map. We say that the pair (W,'r) has the universalfactorization property for V1 x... x Vk if, for every vectorspace U and every multilinear map f :V
1x ... x Vk + U, there
exists a unique linear map h : W+U such that f = h o t .
Proposition 1.1. The Pair (Vl e . ® Vk, (p) has theuniversal factorization property for V1 x... x Vk. If a pair
(W,$) has the universal factorization property for V1 x... X Vk,
then (V1 ® ®Vk,cp) and (W,W) are isomorphic in the sense
that there exists a linear isomorphism 0 :V1
® ® Vk + W
such that * = 0 o cp.
Proposition 1.2. If (eir,r
) is a basis of Vr
, (1 s ir
9
dim Vr), then
( e . 1 . ®ei k'k)
is a basis for V1® ®Vk.
4 1. Differentiable Manifolds
Proposition 1.3. (i) There is a unique isomorphism of
U s V onto V ®U which sends u ®v into v u for alluEU and vEv.
(ii) There is a unique isomorphism of (U ®V) ®W onto
u ® (V ®W) which sends (u (9 v) ®w into u 0 (v sw)
uEU, vEV, and wEW.
for all
(iii) If U1 ®U2 denotes the direct sum of U1 and U2,
then
(U1(DU2) sv = U1®v®U2®V,
Us (V1BV2) = U@VIOuev2.
For the proof of these three propositions, see Kobayashi-
Nomizu, [1, vol. 1].
§ 2. Tensor Algebras 5
§2. Tensor Algebras
Let V be a finite dimensional vector space and V*
the dual space of V, i.e., the space of all linear functions
on V. If v E V and w* E V*, we put
<v,w*> = w* (v) .
Let be a basis of V andn*
)
the corresponding basis for V* so that
* 1 if i = j<ei,e> > = bi =
O if i # j
where bi are the Kronecker deltas.
We want to study those spaces of the form V1 0... ®Vk
where each of V. is either V or V*. If there are ri
copies of V and s copies of V*, then the space is called
a space of type (r,s); r is the contravariant degree and
s the covariant degree. Given two tensor spaces U of type
(r,s) and V of type (p,q), the associative law for tensor
products defines a tensor space of type (r +p, s +q). We
consider the ground field ]F as a tensor space of type (0,0).
The tensor product defines a multiplicative structure on the
weak direct sum of all tensor products of V and V*. We
denote this space by T(V), i.e.,
T(V) = IF +V+V*+V®V+V®V*+V*®v+v*®V*+ .
6 1. Differentiable Manifolds
T(V) with its multiplicative structure is called the tensor
algebra of the vector space V.
We shall give the expressions of tensors with respect to
a basis of V. Let be a basis of V and e1*, ,en*
its dual basis. By Proposition 1.2, .(ei ®... ®ei ) is a1
basis of V (r copies) (we denote this space by Vr).
Every contravariant tensor K of degree r can be expressed
uniquely as a linear combination:
il... irK = E 11.
1*** irK e1 . 1
® ®e1 . ,r
where K11 -1r are the components of K with respect to the
basis e1'...'er of V. Similarly, every covariant tensor L
of degree s can be expressed as a linear combination:
31L = E. L. e ®... ®e ]r
....s l...Js
where L.1"' j
are the components of L.sNow, we define the notion of contraction. Let U = VI ®... ®Vk
with Vi = V and Vj = V*. Let U' be the tensor product of
all the terms of U in the same order omitting Vi and Vj.
The map of V1 X ... X Vk into U' defined by
(vl.... , vk) '4 <vi. vj>vl ®... ®vi ®... ®vj ®... ®Vk
is bilinear. Hence, it defines a map of U into U', which
is called the contraction with respect to i and j.
§ 3. Exterior Algebras 7
V. Exterior Algebras
Let V be an n-dimensional vector space over a field IF.
Denote by Tr the permutation group on r letters. Then
Trr acts on Vr = V s . . . (&V (r copies) as follows:
Given any permutation a E1rr and any tensor in Vr of
the form v 1 . . . s vr, we define
a (v1 0 ... s vr ) = va (1) s ... s va (r)
We extend by linearity to all of Vr. A tensor K in Vr
is called symmetric if 0(K) = K for every permutation a
in lrr. K is called skew-symmetric if a(K) = (sgn (Y)K
for every a in err, where sgn a is either 1 or -1
according to a is even or odd.
For any K in Vr, we introduce the following two opera-
tions:
(3.1)
(3.2)
Sr (K) = r rf a (K)a
A (K) = -L E (sgn a) a (K)r .a
Since Sr a = a . Sr = Sr and C. Ar = Ar a = (sgn a)Ar,
Sr(K) is a symmetric tensor and Ar(K) is a skew-symmetric
tensor. Sr is called the svmmetrization and Ar the altera-
tion.
It is easy to check that the alternation Ar : Vr ..Vr is
linear. Denote by NX the kernel of Ar. We have a natural
isomorphism:
8 1. Differentiable Manifolds
Vr/Nr e. Ar (Vr) .
We denote Vr/Nr by Ar(V). The elements of Ar (V) are
called r-vectors. As before, we define a multiplication on
A(V) = AO(V) +Al(V) +A2(V) +
by OAs = Ar+s (a ®S) for Cr E Ar (V) , and 6 E As (V) . Then
a ASE Ar+s (V). The sign of the permutation on r +a letterswhich moves the first r letters past the last s letters
is (-1)rs. Thus we have
(3.3) aAs = (-1)rs3Aa.
The aAs defined above is called the wedge (or exterior)
product of a and S. It is straight-forward to show that
A(V) with the wedge product is an associative algebra, which
is called the exterior (or Grassmann) algebra of V. If
(el,...en) is a basis of V. The exterior algebra A(V) is
of dimension 2n. Moreover, 1 and the elements
ei A...Aei , 1 s it < ... < it s n, r = 1,...,n.1
form a basis of A(V).
Proposition 3.1. Let be r vectors in V.
Then are linearly dependent if and only if
(3.4) V1A...AVr = 0.
§ 3. Exterior Algebras 9
Proof. If are dependent, then we can express
one of them, say vr, as follows:
r-1yr aivi.
i=1r-1
Thus v1A...AVr = v1A...nvr-1A(E aivi) = O.i=1
On the other hand, if are linearly independent,
we can always find Vr+1' ..,vn such that form a
basis of V. Thus vIA...AVr O.
We also need the following.
(Q. E. D. )
Proposition 3.2. (Cartan's lemma). Assume that
are linearly independent in V and are r vectors
in V. If we have
(3.5)
then
r
E winvi = O,i=1
(3.6) wi = v.0 i =
with aij = aji.
Proof. Let (vi,...,vr, be a basis of V.
Then we can write w. as1
r nw . = E a v. + E b..v..
1 j=l i7 j=r+l 17 3
to 1. Differentiable Manifolds
Thus, by (3.5), we find
E (a -a.)viAV + E bi.vinv. = 0.i<jsr
ij j j isr,j>r 7 3
From this we obtain aij = aji and bij = 0. (Q.E.D.)
From the definition of wedge product we obtain
(3.7) (6' ') (X, Y) =2
(9 (X) W (Y) - w (X) 8 (Y) )
for X,Y in V and e,w in V*.
Let X be a vector in V, we define the interior
product tX with respect to x by
(a) tXa = 0, for every a E no(V) ,
(b) (tXw) (Y1....Yr-1) = for wEnr(V*)
and V, where V* is the dual space of V.
It is easy to see that tx is a skew-derivation of A(V*)
into itself, i.e., tX(wnw') = txwnw'+ (-1)rwntXw', where
wEAr(V*).
§ 4. Differentiable Manifolds 11
§4. Differentiable Manifolds
Let M be a separable topological space. We assume that
M satisfies the Hausdorff separation axiom which states that
any two different points can be separated by disjoint open sets.
By an open chart on M we mean a pair (U,4) where U is an
open subset of M and § is a homeomorphism of U onto an
open subset of Euclidean n-space Iltn
A Hausdorff space M is said to have a differentiable
structure of dimension n if there is a collection of open charts
where a belongs to some indexing set A, such that
the following conditions are satisified:
(Ml). M = U Ua, i.e. [U a) is an open covering of M,aEA
(M2) . For any a, B in , A, the map §13 o §21 is adifferentiable map of 4a (Ua fl Ut3 ) onto @13 (Ua flu ) .
(M3). The collection ((U,§a))aEA is a maximal family
of open charts which satisfy both conditions (Ml)
and (M2) .
By "differentiable" in (M2) we mean differentiable of class
Ca unless mentioned otherwise.
By a differentiable manifold of dimension n we mean a
Hausdorff space with a differentiable structure of dimension n.
For simplicity, we call a differentiable manifold a manifold.
Let (U,§) be an open chart of a manifold M of dimension n.
Denote by x11 ,xn the Euclidean coordinates of ,n. The
12 1. Differentiable Manifolds
systems of functions xl e on U is called a local
coordinate system and U a .coordinate neighborhood.
In the definition, if Rn is replaced by Cn and dif-
ferentiable maps by holomorphic maps, then M is called a com-
plex manifold of n complex dimensions. By a compact manifold
we always mean a compact manifold without boundary unless men-
tioned otherwise.
Given two manifolds M and N, a map f :M-ON is calleddifferentiable if for every chart (Ua,*a) on M and every
chart (Vo,$13
) on N such that f(Ua) c V13, the map
13o f o cp-a1 of cpa(Ua) into $ (V is differentiable. Let
u1*...,un be a local coordinate system on Ua and yl'...'ym
a local coordinate system on If f is a differentiable
map of M into N, then locally f can be expressed by a set
of differentiable functions:
Yl = Yl(ul,...,um),....Ym = ym(ul....,un),
In the following, by a differentiable map of a closed
interval [a,b] into a manifold M, we mean the restriction of
a differentiable map of an open interval I D [a,b) into M.
By a (differentiable) curve in M we mean a differentiable map
of a closed interval into M.
Let .7(p) be the algebra of differentiable functions defined
in a neighborhood of p. Let x(t) be a differentiable curve
such that x(to) = p. The vector tangent to the curve x(t) at
p is a map X : 9(p) -. R defined by
§ 4. Differentiable Manifolds 13
xf = df (x (t) )dt t= t0
In other words, Xf is the derivative of f in the direc-
tion of the curve x(t) at t = t0. The vector X satisfies
the following conditions:
(1) X is a linear map of .7(p) into R ,
(2) X (fg) = (Xf) g (p) + f (p) Xg for f, g E T(p) .
The set of maps X of .7(p) into R satisfying these
two conditions forms a real vector space. Let be
a local coordinate system in a coordinate neighborgood U of
p. For each i,(aa )p
is a map of 7(p) into R satisfyingi
conditions (1) and (2) above. We shall show that the set of
vectors at p is the vector space with basis (aul)p. ,(a )p-
Given any curve x (t) with x (t0) = p, let ui = xi (t)be its equations in terms of Then we have
(df(x(t))) = E(af) (dxl t )dt t0 aui p dt t0
Thus every vector at p is a linear combination of (a -)p, ,1
aunp Conversely, given a linear combination X = E a(aui)p.
we consider the curve defined by
ui = ui (p) +a i t' i = 1,.. . , n.
14 1. Differentiable Manifolds
Then X is the vector tangent to this curve at t = 0. To prove
the linear independence, we assume that E ai( a ) = 0. Thenaui p
au.0 = E ai(a )p = aj, 7 = 1,...,n.
Consequently, we obtain the following.
Proposition 4.1. Let M be an n-dimensional manifold
and p E M. If is a local coordinate system on a
coordinate neighborhood containing p, then the set of vectors
at p tangent to M is an n-dimensional vector space over IR
with basis (aul)p, (-aup,
We denote by Tp(M) the vector space tangent to M at
p. Tp(M) is called the tangent space of M at p. And its
elements are called tangent vectors at p.
§5. Vector Fields and Differential Forms 15
$5. Vector Fields and Differential Forms
A vector field X on a manifold M is an assignment of a
vector Xp to each p in M. If f is a differentiable func-
tion on M, Xf is a function on M such that (Xf)(p) = Xpf.
A vector field X is called differentiable if Xf is differen-
tiable for every differentiable function f on M. In terms
of a local coordinate system a differentiable
vector field X may be expressed by X = EXi ( a), where X1i
are differentiable functions.
Let I(M) be the set of all (differentiable) vector
fields on M. 1(M) is a real vector space under the natural
addition and scalar multiplication. Given two vector fields
X,Y on M, we define the bracket [X,Y] as a map from the
ring of functions on M into itself by
(5.1) [X,YJf = X(Yf) -Y(Xf).
Then [X,YJ is again a vector field on M. In terms of local
coordinate system we write
X =F'Xlau.' Y =E Y7 au.1 7
Then we have
(5. 2) [X,Y} = E (Xk(a7)
-Yk(a7
) }au7
With respect to this bracket operation, I(M) becomes a Lie
algebra over R (of infinite dimensions). In particular, we
have the Jacobi identity:
16 1. Differentiable Manifolds
(5.3) [[X,Y],Z]+[[Y,ZJ,X)+[[Z,XJ,YJJ = 0
for X, Y, Z in I (M) .
We may regard I(M) as a module over the ring 9(M) of
differentiable functions on M as follows: If f f 7(M) and
X E I (M) , then we define fX by (fX) p = f (p) Xp. We have
[fX,gY] = fg[X,YJ-f(Xg)Y-g(Yf)X,
for f, g E 9(M) , and X, Y E I (M) .
For each point p in M, we denote by T*p(M) the dual
space of the tangent space Tp(M). Elements of T*p(M) are
called covectors at p. An assignment of a covector at each
point p in M is called a 1-form.
For each f in T(M), the total differential df of f
is defined by
<(df)pX> = Xf
for X ETp(M). If is a local coordinate system in
M, then the total differentials (du form a basis
of T*p(M). In fact, they form the dual basis of the basis
aul)p, ..., (aun)p of Tp(M).
In a coordinate neighborhood of p, every 1-form w can
be expressed as
w = v widui,
§5. Vector Fields and Differential Forms 17
where wi are functions. The 1-form w is called differentiable
if wi are differentiable. It can be verified that this con-
dition is independent of the choice of local coordinate system.
We shall only consider differentiable 1-forms. We denote by
A1(M) the set of 1-forms on M.
Let AT*p(M) be the exterior algebra over T*p(M). An
r-form w on M is an assignment of an element of degree r
in AT*p(M) to each point p in M. In terms of a local
coordinate system we have
w = Zi1<i2<...<ir wil...irduiln...nduir.
The r-form w is called differentiable if the components
wl . l. . . ir
are differentiable. By an r-form we shall always
mean a differentiable r-form. We denote by nr(M) the space
of r-forms on M, r = 0, 1, ., n. We have n0(M) = T(M).
Each Ar (M) is a vector space over R . We set A (M) = E (M) .
With respect to wedge product, n(M) is an algebra over R .
Let d denote the exterior differentiation. Then d is
characterized as follows:
(1) d is an R -linear map of n(M) into itself such
that d (nr (M)) G nr+l (M)
(2) For each f E n0(M), df is the total differential of f,
(3) If WE Ar(M) and § E AS (M) , then
d (wn$) = dwA§ + (-1) rwndf ,
(4) d = O.2
18 1. Differentiable Manifolds
In terms of a local coordinate system, if w = Ei, <... <ir
w 1 . 1... ir du1.
1then
r
dw = E dw. Adu. A Adu.it<...<ir
11...1r 11
1r
We mention the following result for later use. For its
proof, see Kobayashi-Nomizu [1, p. 361.
Proposition 5.1. If w is an r-form, then
(5.4) dw (X0. X1, .'x r) = r + 1 E (-1) 1Xi (w (X0, ... , Xi, . . . I Xr) )i=O
1 rr i++r+1 "osisjsr(-1)
3w([XitX,s...DXr),
in particular, if w is a 1-form, then we have
(5.5) dw(X,Y) = 2(Xw(Y) -Yw(X) -w([X,Y])).
Given a map f of a manifold M into another manifold N,
the differential of f at p is the linear map (f*)p of Tp(M)
into Tf(p) (N) defined as follows:
For any x E Tp (M), we choose a curve x (t) in M such
that X is the vector tangent to x(t) at p = x(t0). Then
(f*)p(X) is defined as the vector tangent to f(x(t)) at
f(p) = f(x(t0)). It is easy to verify that (f*)p is independent
of the choice of the curve x(t) and if g is a function in
a neighborhood of f (p) , then ((f*) pX) (g) = X (g o f) . The
transpose of (f*)p is a linear map of T*f(p)(N) into
§5. Vector Fields and Differential Forms 19
T*p(M). For any r-form w' on N. f*w' is an r-form
on M defined by
(5.4) (f*w') (X1, ...'Xr ) = w' (f*X1, ... f*Xr) ,
for X10 ,Xr E Tp(M), p E M. The exterior differentiation d
commutes with f*, i.e.,
(5.5) dof* = f* od.
Definition 5.1. A submanifold of a manifold M is a pair
(N,f) where f is a differentiable map of a manifold N into
M so that for each point p EN, (f*)p is injective. In this
case, f is called an immersion. If, furthermore, f is also
injective, (N,f) is called an imbedded submanifold of M and
f an imbedding.
Definition 5.2. A map f : N-.M is called proper if f_1 (K)
is compact for any compact subset K of M. If f :N-#M is
a proper imbedding, (N,f) is called a closed submanifold of M.
It is known that a submanifold N of M is closed if and
only if there exist a covering of M by coordinate neighborhoods
(Ua) such that N n Ua is defined by yi(p) _ = ya(p) = 0
where k = dim M -dim N and are local coordinates
on U
20 1. Differentiable Manifolds
§6. Sard's Theorem and Morse's Inequalities
Given a map f of a manifold M of dimension in into
another manifold N of dimension n, the maximum rank that
f can have is the minimum of m and n. If in < n, the
image of M is a lower dimensional object in N. In fact, no
matter what values m and n are, "in general" a point of N
is not an image of a point of N when the rank of f is less
than n. The "generality" is in the sense of measure zero.
We give the following.
Definition 6.1. Let f be a (differentiable) map of
M into N. The points p of M where rank (f*)p < n = dim N
are called the critical points of f. All other points of M
are called regular. A point q E N such that f-1(q) contains
at least one critical point is called a critical value. All
other points of M are called regular values.
It is clear that if dim M < dim N. all points of M
are critical. Moreover, if q EM does not lie in f(M),
it is a regular value.
We state Sard's theorem.
Theorem 6.1. Let M and N be manifolds of dimension in
and n respectively. Let f :M-4N be a (differentiable) map.
Then the critical values of f form a set of measure zero.
For the proof of Sard's theorem, see Stenberg {l].
§ 6. Sard's Theorem and Morse's Inequalities 21
Let M be a compact n-dimensional manifold. Let f be
a differentiable real-valued function on M. A point p in
M is a critical point of f if and only if (f*)p = 0. if
we choose a local coordinate system in M,
this means that of/6yi = 0 at p for i = If
p is a critical point of f, then the matrix
32
)
6yi ayi p
represents a symmetric bilinear map f*,k on the tangent space
Tp(M). A critical point p of f is called non-degenerate
if f** is non-degenerate at p. In this case, the dimension
of a maximum dimensional subspace of Tp(M) on which f** is
negative-definite is called the index of the critical point p.
A function f on M is called non-degenerate if all of its
critical points are non-degenerate. A non-degenerate function
is also called a Morse function. According to Sard's theorem
almost all functions on M are non-degenerate (except a set
of measure zero).
We introduce the following notations:
4(M) = the set of non-degenerate functions on M;
k(f) = the number of the critical points of index k
of f, f E (M) ;nf3
(f) = k Bkf
22 I. Differentiable Manifolds
For any field IF we denote by Hk(M;]) the k-th homology
group of M with coefficients in F. We put
bk(M;I) = dim Hk(M;IF),
nb(M;ff) = bk(M;If)
k=0
b(M) = max (b(M;IF) 1 IF field ).
We mention the following results for later use.
Theorem 6.2 (Weak Morse Inequalities). Let M be a
compact n-dimensional manifold. Then for any field IF and
any function f E I (M) we have
(f) Lt bk(M;IF), (-1)k6k(f) _ (-1)kb k(M;3F) = X(M)
where X(M) is the Euler characteristic of M.
Theorem 6.3 (Reeb). Let M be a compact n-dimensional
manifold. If there exists a differentiable function f on M
with only two non-degenerate critical points, then M is homeo-
morphic to an n-sphere.
Remark 6.3. Theorem 6.3 remains true even if the critical
points are degenerate. It is not true that M must be diffeo-
morphic to Sn with its usual differentiable structure. In
fact, Milnor (2] had constructed a 7-sphere with non-standard
differentiable structure which admits a function on it with
two non-degenerate critical points.
§ 7. Fibre Bundles 23
§7. Fibre Bundles
A Lie group G is a group which is at the same time a dif-
ferentiable manifold such that the group operation (a,b) E G xG +
ab-1EG is a differentiable map into G.
We say that a Lie group G is a Lie transformation group
on a manifold M or that G acts on M if the following con-
ditions are satisfied:
(a) Every element a of G induces a transformation
of M, denoted by x w xa for x EM;
(b) (a, x) E G x M-. xa E M is a differentiable map;
(c) (xa)b = x(ab) for all a,b E G and x E M.
We say that G acts effecitvely (resp., freely) on M
if xa = x for all x E M (resp. , for some x E M) implies
that a = e, where e is the identity element of G.
Definition 7.1. Let M be a manifold and G a Lie group.
A fibre bundle over M with the structure group G and
(typical) fibre F consists of a manifold P and an effective
action of G on P which satisfies the following conditions:
(1) There exists an open covering (Ui) of M and
diffeomorphisms hi : Ui x F + Tr-1 (Ui) which map the fibre7r-1 (x) onto (x) x F, where Tr : P 4 M is the projection;
(2) Define 'Pi, x : F -.'r1 (x) , x E Ui, by cpi, x (u) = hi (x, u) ,-1then gji(x) = cpj x'qi xEG for xEUj RUi;
24 I. Differentiable Manifolds
(3) g j i : Uj fl Ui -G is differentiable.
The family of maps gji are called the transition functions of
the fibre bundle P.
Definition 7.2. Let E,M be two manifolds and 7 : E +M
a map. Then E is called an n-dimensional (real) vector
bundle over M if the following two conditions are satisfied:
(1) 7r-1 W, is a (real) vector space for each x E M;
(2) There exist an open covering (ui) of M and dif-
feomorphisms hi : Ui X Rn - 7r-1 (Ui) such that cpi, x : Rn - 7r (x)
are isomorphisms of vector spaces. In this case, gji(x) == 4 Rn (x E Uj flUi) is an element of the generallinear group GL (n;R).
Similarly, we may define complex vector bundles over M.
Let *r :E-4M be the projection of a vector bundle overM. A map s : M -. E is called a cross-section if 7r o s = id.
A similar definition applies to fibre bundles. We denote by
r(E) the space of all cross-sections of E.
In the following, we mention some important vector bundles
and fibre bundles.
Let M be a manifold of dimension n. The set of all
tangent vectors to M, T(M) = U T (M), has a differentiablepEM P
structure defined as follows:
Let U be a coordinate neighborhood in M with local
coordinates Let V be a real n-dimensional vector
space with a fixed basis We define a map
§ 7. Fibre Bundles 25
0U:UxV-+ UT (M)pEU p
by the condition that V ( U (p, Y) = E Y1 (au ) p, where y = E y'ei ci
It is clear that f6U is one-to-one. We put 91U,p(Y) = OU(p,y).
If V is another coordinate neighborhood with local co-
ordinate system vlf...'vn suppose u n v 3d 0. We define OV
andQ(V,p
to be the maps associated with V. We put
(7.1) -1gUV (p) = 0U, p o 0V, p
for p E U fl V. Then gUV (p) : V -+ V is one-to-one. In terms oflocal coordinate systems, we have the following:
Let y = E ylei. We put
Y- = g5V (p) (y) .
oU,p(Y) = E y-k(aI )p.
aOV,p(Y) = E E Yl(a p(
auk)p
Thus, (7.1), (7.2), (7.3), and (7.4) imply
(7.5) auky'k = E yl().p
These equations define gUV(p) as a linear automorphism of V.
We give to V the topology and differentiable structure of the
26 1. Differentiable Manifolds
Euclidean n-space. Denote by GL(V) the general linear group
of V. Then (7.5) shows that gW (p) defines a map
gW : U f1 V +GL(V) which is differentiable.
We take a covering of M by the coordinate neighborhoods
U, V, W. ets. Since the map gUV is differentiable, it
follows that T(M) is a fibre bundle over M with (71U as
coordinate functions. We call T(M) the tangent bundle of M.
This defines meanwhile a topology of T(M) characterized by
the condition that 91U maps open set of Uf1V into open sets
in T(M). This topology on T(M) is in fact Hausdorff. A
similar argument yields that all tensors of type (r,s) over
a manifold M form a fibre bundle over M. called the tensor
bundle of type (r,s). Similarly, T*(M) = UT*p(M) is a fibre
bundle over M, which is called the cotangent bundle of M.
Denote by it : T (M) -4 M the projection map, i.e., the mapwhich maps every element in Tp(M) onto p. A map s of M
into T(M) is called a cross-section if iro s = identity map.
Similar definitions apply to other fibre bundles over M.
Tangent bundle, cotangent bundle and tensor bundles are special
kinds of vector bundles over M.
In the following, by a linear frame u at a point p in M
we mean an ordered basis u = of the tangent space
Tp(M). Let LF(M) be the set of all linear frames u at all
points of M and let Tr be the map of LF(M) onto M which
maps a linear frame u at p onto p. The general linear group
GL(n;R) acts on LF(M) on the right as follows: Let
§ 7. Fibre Bundles 27
a = (a EGL(n;R) and u= a linear frame at p.
Then, by definition, ua is the linear frame (X;' ...'Xn')at
p with Xi = a?Xj. It is clear that GL(n;R) acts freely
on LF(M) and it (u) = it (v) if and only if v = ua for some a
in GL(n;R). In order to introduce a differentiable structure
on LF(M), let be a local coordinate system on
a coordinate neighborhood U, every linear frame v at p C U
can be expressed uniquely in the form v = with
Y. Y5a
au m.where (Yi) is a non-singular matrix. This
J
shows that tr-1(U) is one-to-one correspondent to UxGL(n;R).
We can make LF(M) into a differentiable manifold by taking
(ui) and (Yi) as a local coordinate system on Tr-1(U).
From these we may verify that LF(M) is a fibre bundle over
M with the structure group GL(n;R). We call LF(M) the
linear frame bundle of M.
If M admits a Riemannian metric, then one may consider
the space F(M) of all orthonormal frames on M. By similar
consideration, F(M) becomes a manifold which is a fibre
bundle over M with the structure group O(n). If one con-
siders the set of all unit tangent vectors to M, then one
obtains the unit tangent bundle over M.
28 1. Differentiable Manifolds
§8. Integration of Differential Forms
In this section, we shall develop the theory of integration
of differential forms on a manifold. We follow closely that
given by Chern [1).
Definition 8.1. A manifold M of dimension n is called
orientable if there is a differential n-form which is nowhere
zero. Two such forms define the same orientation if they differ
from each other by a factor which is positive.
An orientable manifold has exactly two possible orientations.
Let w and w' be two n-forms which determine an orientation
of M. Then w' = fw and f is either positive or negative
everywhere. Thus the only possible orientations are given by
w and by -w. The manifold is called oriented if such a n-form
w is given.
Definition 8.2. The support of a real function f on M
is the closure of the set of points of M at which f is not
equal to zero. More generally, the support of a form w is
the closure of the set of points of M where w is not equal
to zero.
An open covering of M is called locally finite if any
compact subset of M meets only a finite number of its elements.
Theorem 8.1. Let B be a family of open sets of a manifold
M which form a base for the topology of M. Then there is a
locally finite open covering of M whose elements are in B.
§ 8. Integration of Differential Forms 29
Proof. Since M is separable, there is a countable open
covering (Ci) of M such that each Ci has compact closure.
We put
D. = U Ci.lsi-j
Then (Dj) form a countable covering of M by compact sets
Dj with Dj CDj+l'
We now construct compact sets Ej such
that
D. c Ej, E. C Interior of Ej+l'
We use the method of induction. Suppose that E1,...,Ej are
constructed. Because Ej U Dj+l is compact, it has a finite
covering by open sets with compact closures. We put Ej+l to
be the union of these closures.
Let Sj = Interior of E. and let T. = E. n (M -SI I I j-1
where we assume that sets with negative indices are empty set.
Since Ej_1 Sj_1 and Tj C :M - Sj_1, we have
Ti fl Ej-2 = 0.
For p in T. there is a set of B containing p, contained
in Sj+l, and not meeting E7_2. These sets, for all p E Tj,
form a covering of Ti. On the other hand, Ti is a closed
subset of a compact set Ej, it is compact. Therefore, the
above covering has a finite subcovering, which we call Kj. We
denote by K' the family of the sets of Kj for all j. The
30 1. Differentiable Manifolds
sets of K' form a covering of K. Indeed, if p eM, there
is a positive k such that p EEj, p % Ej-1. Hence p belongs
to Tk and is covered by a set of K'. Moreover, if j ? k +2,
Ek meets no set of Kj. Since every compact set of M is con-
tained in a certain Ek, it follows that the covering K' is
locally finite. (Q.E.D.)
We need the following.
Theorem 8.3 (Partition of unity). Let (Ui) be an open
covering of a manifold M. Then there are functions (ga)
satisfying the following conditions;
(1) Eachga is differentiable and 0 a ga
s 1,
(2) The support of each ga is compact and contained in
one of the Ui,
(3) Every Point of M has a neighborhood which meets
only a finite number of the supports of g a,
(4) E ga = 1.
For the proof, see Kobayashi-Nomizu [1, p. 272].
Theorem 8.3. Let M be an oriented manifold of dimension
n. Then there is one and only one functional which assigns to
a differeintial n-form t with compact support, a real number
called the integral of § over M, denoted by f 0, such
that
(1) f Il+12 = f §l+ f 2'
§ 8. Integration of Differential Forms 31
(2) If the support of § is contained in a coordinate
neighborhood U with coordinates such that
defines the orientation and _
du1A...ndun, then
r g = f 4 (u1, ... , un) dulA... ndund U
where the right-hand side is a Riemannian integral.
Proof. Let ' be a differential n-form with compact
support S. We choose an open covering (Ui) of M such
that each U i is a coordinate neighborhood. Let gi be
a corresponding partition of unity. Then every point p
in S has a neighborhood Vp which meets only a finite
number of the supports of gi. These P for all p in S
form a covering of S. Because S is compact, it has a finite
sub-covering. Therefore, there are only a finite number of
gi1 which are not identically zero.
We define
where the right-hand side is a finite sum. Because the differ-
ential n-form in each summand has a support lying in a coordinate
neighborhood Ui, we can evaluate it according to the formula
in condition (2).
We now prove that the above definition is independent of
the various choices made. These are: 1) choice of the neigh-
32 1. Differentiable Manifolds
borhood Ui which contains the support of ga§ and 2) choice
of the covering (Ui) and the corresponding partition of unity
ga.
Suppose the support of gaI be in two coordinate neigh-
borhoods U.V with the local coordinates and
vle...,vn. respectively. We can take an open set W containing
the support in which both coordinate systems are valid. In W
we have
gaff = 1 ( u1 ' ... ' un) dulA... ndun
i (vl, ... , ,fin) dvn... ndun.
where
(va(ul'...,un)
I ... , Vn) = $ ( ul (v l , ... , vn) , ... un ( V l ... , vn)) a (V 1'...,v n)
We may assume that the Jacobian determinant is positive through-
out W. The equation
a(ul,....un)r = r (ul, ... , un) a (vl' ... N vn) dvln... ndun
is then exactly the formula for the transformation of multiple
integrals. It follows that our definition is independent of the
choice of the particular choice of the neighborhood Ui in the
evaluation of the summands.
Next consider a second covering (V7) of M by coordinate
neighborhoods and let g' be a corresponding partition of unity.
§8. Integration of Differential Forms 33
Then (Ua fl VS) will be a covering of M with the functions
gag as a corresponding partition of unity. It follows that
EJ ga§ = EJ gags
a a.
and
ga, f=Ea 1gags§
This proves the independence of the integral of the choice of
covering and the corresponding partition of unity. The uniqueness
is clear. (Q.E.D.)
Let M be an oriented manifold of dimension n. A differ-
ential n-form w is said to be > 0 or < 0 according to w
or -w defines the orientation.
Definition 8.3. A domain D with regular boundary is a
subset of M such that if pE D, either (1) p has a neighbor-
hood belonging entirely to D, or (2) there is a coordinate
neighborhood U of p with coordinates
u fl D is the set of all points q E D with un (q) a un (p)
Points with property (1) are called interior points of D
and points with property (2) are called its boundary points.
The set of all boundary points of D is called the boundary of
D, we will denote it by M. It is known that the boundary of
a domain with regular boundary is a closed submanifold which is
regularly imbedded. If M is orientable, so is the boundary BD.
34 1. Differentiable Manifolds
We now consider a domain D with regular boundary aD
and suppose it is compact. Define the characteristic function
h(p), pEM by
10,
1, pED,h (p)
-pEM-D.
We define the integral over D of an n-form I on M by
JDI = f hi.
We give the following well-known Stokes theorem.
Theorem 8.4. 1 M be an oriented manifold of dimen-
sion n and w be an (n -1)-form on M with compact support.
Then for any domain D with regular boundary in M, we have
(8. 1) fD= w.
D aD
Proof. Let (Ui) be an open covering of M by coordinate
neighborhoods such that for each Ui either Ui 020 = (( or
Ui has property (2) of Definition 8.3. Let the functions (ga)
be a corresponding partition of unity. Since aD and D are
both compact, each of them meets only a finite number of the
support of ga. Thus we have
j W =Ef g w,aD a
BD
and
§ 8. Integration of Differential Forms
dw aJ
d (g w) .D D
Therefore, it suffices to prove (8.1) for each summand of
the above sums, i.e., under the assumption that the support
of w lies in a coordinate neighborhood Ui.
Let be the local coordinate system in Ui
such that 0. Let
w = E
Then we have
as.dw = a
35
We first consider the case when Ui n 6D = 0. Then the
right-hand side of (8.1) is zero. The set Ui either belongs
to M -D or to'the interior of D. If the first possibility
holds, h = 0, so (8.1) holds. Now suppose that Ui lies
in the interior of D, we have h = 1, and the integral in
the left-hand side of (8.1) is equal to
as .(dw = (
au1) duIA... ndun.D
where C is a cube in the space of the coordinates uj which
contains the support of w in its interior. we may choose m
sufficiently large so that C is defined by the inequalities
luj1 a M. This integral is now just an iterated integral in
classical analysis. Thus we have
36 1. Differentiable Manifolds
raa.
Ja dul...dun = ±
J
C F.
-aj(u1,...,uj-i' m'uj+l'...,un))dul...duj-iduj+l...dun'
where Fj is the union of the appropriate faces of the cube.
Because the support of w lies inside C, the above integral
is equal to zero. Thus (5.1) holds.
Now, we consider the case when Ui has property (2) of
Definition 8.3. We assume that aD is contained in the sub-
set defined by un = 0. When un(p) : 0, h(p) = 1. In the
space of the coordinates uj we take a cube C defined by
JukI s in. k 1,...,n-1; 0 s un g m
such that the support of w is contained in the union of its
interior and the side un = 0. As in the above, we have
C
Also we have
raaan
du ...du = (-1)n
C
a (u O)du duJ au
n 1 n J n 1,un-1, 1... n-1
aDD
But the right-hand side of the last equation is equal to
askk
auk0 for k = -1.
Thus we have (8. 1) . (Q . E . D. )
§9. Homology, Cohomology and deRahm's Theorem
§9. Homology, Cohomology, and deRham's Theorem
The purpose of this section is to introduce homology
groups, cohomology groups on a manifold M. And use Stokes'
theorem to prove deRham's theorem.
First we give the following.
Definition 9.1. We will denote by Ap the simplex inP
RP defined by 0 s xi s 1, xi 3 1.i=1
37
Thus AO is a point, Al is the unit interval [0,1]
and A2 is a triangle, etc. On the simplex Ap we introduce
the so-called barycentric coordinates defined by
pYO = 1 - 7- xi, Yj = xj, j = 1,...,p,
i=1
So that +yn = 1 and 0 s y7 . 9 1.
The map bp, i (i = O, " . , p + 1) from Ap into Ap+1 isdefined by
1'k if k < iyk (bP, i
(yO, ... yp) 0 if k = i
Yk-1 if j > 1
where yk are the barycentric coordinates of Ap+1'
Lemma 9.1. We have
(9.1) 6p+l, i o 6P, j 6P+l, j c 6p, i+l' if j a i.
38 1. Differentiable Manifolds
By comparing both sides of (9.1) acting on a p-simplex,
we obtain the lemma.
Definition 9.2. A (differentiable) singular p-simplex on a
manifold M is a map of Ap into M which can be extended
to a differentiable map of a neighborhood of Ap in RP into
M. A (differentiable) p-chain is a finite formal linear combina-
tion with real coefficients of singular p-simplices.
The set of all p-chains on M form a real vector space
in the obvious way, we denote it by Cp(M). If f is a differ-
entiable map of a manifold M into another N, we define a
linear map f :CP
(M) -4 Cp (N) by
(9.2) f (s) = s ° f
for simplices and extend it by linearity.
If s is a p-simplex, s ° by-l,i
We define the boundary of s by
is a (p - l)- simplex.
p(9.3) a (s) _ 7 (-1) is ° b
1=O P-l,i
We extend a by linearity to a map of Cp(M) into Cp-1(M).
Lemma 9.2. We have
(9.4) a°a=0.
§9. Homology, Cohomology and deRahm's Theorem 39
Proof. By linearity, it suffices to prove (9.4) for a
simplex. Let s be a q-simplex. If q s 1, there is nothing
to prove. If q z 2, we have
q q-1a2 (s) = 6 (as) = Z I (-1) (-1) is o 6 0 6
i=O j=0 q-1, i q-2, j
(-1) itjs o 6 0 6Osjsisq q-1, i q-2, j
+ E (-1) i+js o 6q-1, i o 6q-2, j'Osi<jsq
By Lemma 9.1, sobq-l,i o6q-2,j = sobq-l,j o 6q-2,i+l
for j 3 i. Thus we can rewrite the first sum as
E (-1) i+js o b o6q-2,Osjsisq q-l.j i+1
If we put k = i + 1, this will cancel the second sum which
proves (9.4). (Q. E. D.)
Definition 9.3. A p-chain c is called a cycle if ac = 0.
The p-chains of the form c = ad for some (p + 1)-chain are
called p-boundaries.
Lemma 9.2 shows that the space Bp(M) of p-boundaries is
a subspace of the space Zp(M) of cycles.
Definition 9.4. The quotient space Zp(M)/Bp(M) is called
the p-dimensional homology group of M, denoted by Hp(M).
40 1. Differentiable Manifolds
Remark 9.1. If f : M - N is a differentiable map, f
commutes with a, i.e., of = fa. Thus I maps cycles into
cycles and boundaries into boundaries. Hence, it induces a
map of Hp (M) into Hp (N) .
Definition 6.5. A differential form w is called closed
if dw = 0. It is called exact if w = d for some form §.
The quotient of the space of closed p-forms by the space of
the exact p-forms is called the p-dimensional (deRham) cohomologv
group of M, denoted by HH(M). The quotient of the space of
closed p-forms with compact support over the space of exact form
dt where 4 is a (p +l)-form of compact support is denoted by
H p (M) .C
If M is compact, HP (M) = HP(M). If c = E cs is as
p-chain and w is a p-form of compact support, we define
f w = E f w.s
c cs
Then by Stokes' theorem we have
(9.5) f dw = f w.
c (c)
The following results are well-known.
Theorem 6.1. The bilinear map of Cp (M) x np (M) -. R given
by J, induces a bilinear map of Hp (M) x Hp (M) - R .
c
§ 9. Homology, Cohomolo&y and deRahm's Theorem 41
Proof. From (9.5) it follows that the integral of an
exact form over a cycle vanishes and the integral of a closed
form over a boundary vanishes. Thus the integral of a closed
form over a cycle depends only on the cohomology class of the
closed form and the homology class of the cycle. (Q.E.D.)
A fundamental result in algebraic topology asserts that
this bilinear map is non-singular. Thus we have the following
well-known isomorphism:
Theorem 9.2. Hp(M) is the dual space of Hp(M). In
particular, we have the following isomorphism;
(9.6) Hp (M) Hp (M) .
The dimension of Hp(M) is called the p-th betti number
of M. denoted by bp(M).
Let a and be closed p- and q-forms on M. Let
[a] and [0) denote the corresponding cohomology classes re-
presented by a and a, respectively. It is easy to verify
that the wedge product a A6 is a closed (p +q)-form on M.
Let c a and ca denote the corresponding homology classes
in Hp(M) and Hq(M) associated with [a] and [0] through
the natural ismomorphism (9.6). Then the homology class caAO
corresponding to [aA ] is called the cup product of c a and
cW The product so-defined is denoted by ca U c13.
42 1. Differentiable Manifolds
§10. Frobenius' Theorem
Let M be an n-dimensional manifold. An r-dimensional
distribution on M is an assignment b defined on M which
assigns to each point p in M an r-dimensional linear sub-
space by of Tp(M). An r-dimensional distribution b is
called differentiable if there are r differentiable vector
fields on a neighborhood of p which, for each point q in
this neighborhood, form a basis of bq. The set of these q
vector fields is called a local basis of the distribution b.
A vector field X belongs to the distribution 4 if for any
p EM, Xp E &p. in this case, we denote it by X E b. A distri-bution b is called involutive if, for any X. Y Eh, we have
[X,Y) E.B. By a distribution, we shall mean a differentiable
distribution.
An imbedded submanifold N of M is called an integral
submanifold of the distribution b if f,, (Tp (N)) = bf (p) (M),
for all p E N, where f is the inclusion map. An r-dimen-
sional distribution b on M is called completely integrable
if, for each point p E M, there is a coordinate neighborhood
U and local coordinates on U such that all
the submanifolds of U given by yr+1 = const, , yn = const.
are integral submanifolds of b.
The theorem of Frobenius can be stated as follows:
Theorem 10.1. A distribution b gD M is completely
integrable if and only if b is involutive.
§ 10. Frobenius' Theorem 43
Let .b be an r-dimensional distribution on M. We put
(10.1) 0 = (1-forms w on M I w (X) = 0 for x E .&) ,
and let i(0) be the ideal generated by 0 in the ring of
exterior polynomials on M.
Theorem 10.1 of Frobenius can also be rephrased as the
following.
Theorem 10.1'. The distribution b is involutive if
and only if dO is contained in the ideal 1(0), where 0
is the differential system defined by (10.1).
Proof of Theorem 10.1. The necessity is clear. To prove
the sufficiency, we restrict attention to local matters.
If r = 1, this theorem reduces to the existence theorem of
ordinary differential equations. In fact, given a vector field
X, by this existence theorem, we can introduce a local coordinate
system yl, ,yn on M such that X = ay
We prove the theorem b y induction on r. Let p E M and
be r independent vector fields defined on a coordinate
neighborhood V of p. We introduce local coordinates
such that X1 = aY and yl(p) = 0. We put fj = Xjyl on V,1
Y1 = X1 and Yj = xj - fjx1 for j = Then
are also linearly independent. Thus, they span & on V.
Moreover, we have Yjyl = 0 for j = We consider
W = (q E V I yl (q) = 0). Then W is an (r - 1) -dimensional sub-
44 1. Differentiable Manifolds
manifold of V and Y2,...,Yr are tangent to W, i.e.,
Yj(q) are tangent to W for q E W. Let Zj be the restric-
tion of Yj on W. Then i*(aa )q (aa )q for qEW,7 7
where i is the inclusion. If
nY. r Y1-
7 k=2 7 arkn
kthen we have Zj = R2Z azk , where ZJ are the restriction
of Yj to W. Now, we want to claim that span
an involutive '(r -1)-dimensional distribution on w. In fact,
if [Zi,Zu)q did not lie in the space spanned by Z2....,Zr,
then [Yi,Y3)q would not lie in 9q since none of Y2,...,Yr
has any component relative to a-ya . Therefore, by induction,1
we can find an integral submanifold of through
each point p in W. Thus, we can find local coordinates
in some neighborhood of p in W such that the dis-
tribution spanned by Z2,...,Zr has integral submanifolds given
by ur+l = const. Now we define local coordinates
about p as follows: Let vl = yl and vj(y1,...'yn)=
The Jacobian determinant of the v's with
respect to the y's does not vanish at p so that the v's do
av.form a coordinate system about p. Moreover, because
By,= 0
1
for j = we have Y1 = asav .
Since Ylvr+k = 0,yl 1we have
av (Yjvr+k) - [Yj'Y1)vr+k'1
§ 10. Frobenius' Theorem
Now, because span an involutive distribution, we
may find some function Dih so that
r[Yi,YII = DilY1 + E DilY
=2
Thus we have
rav (Y 'vr+k L Dll (Yivr+k)
1 i=2
45
for j = Thus, for each fixed k, we may regard the
functions Yjvr+k as solutions of the homogeneous linear dif-
ferential system of r -1 differential equations with respect
to the independent variable v1. On the other hand, if v1 = 0,
we have Yjvr+k = Zjvr+k = O. Thus, by the uniqueness of the
solutions, we find that Yjvr+k vanish identically. There-
fore, Y1, ,yr can be expressed in terms ofavall
a.
1FjVr
This proves that b is completely integrable. (Q.E.D.)
Chapter 2. RIEMANNIAN MANIFOLDS
§1. Affine Connections
The concept of an affine connection was first defined by
Levi-Civita for Riemannian manifolds, generalizing significantly
the notion of parallelism for Euclidean spaces. There always
exists an affine connection on a manifold. An affine connection
gives rise to two important tensor fields, the curvature tensor
and the torsion tensor which in turn will describe the affine
connection via Cartan's structural equations.
Definition 1.1. An affine connection on a manifold M is
a rule v which assigns to each vector field X EX(M) a linear
map vX of the vector space 1(M) into itself satisfying the
following two conditions:
vf)+gy = fvX + gvY;
(v2) 7X(fY) = fvXY+ (Xf) Y
for f, g E AO (M) and X. Y E I (M) . The operator vX is calledthe covariant differentiation with respect to X.
An affine connection v on M induces an affine connection
vU on an arbitrary open submanifold U of M. In fact, for
any two vector fields X,Y on U, and any p E U, we put
((VU)X(Y))p = (vXY)p. Then VU is a well-defined affine connec-
tion on U. In particular, if U is a coordinate neighborhood
with local coordinate system we write vi instead
of (vU)a/ayl.
§ 1. Affine Connections
On U, we define the functions rid by
vi a ) = E rid syk=1
47
These rid are called the Christoffel symbols or the connection
coefficients of v. If z10...ozn is another coordinate system
valid on U, we obtain another set of functions Virj by
vi () = E rl .k Fjz k
Using the axioms (vI) and (v2) we find easily
k aya ayb azk c a2yaazk
(1.2) r13 azi azi ayc rab +E aziazi aya
Conversely, suppose that there is a covering of M consisting
of coordinate neighborhoods (Ua) and in each coordinate neighbor-
hood a system of functions ( such that (1.1) holds whenever
two of these neighborhoods overlap. Then we can define an affine
connection on M by using equation (1.1).
One may extend the operator vX to arbitrary tensor fields.
For real functions f on M. we define vXf as Xf. For a
tensor field w of type (0,1) (i.e., a 1-form), we put
(1.3) (vXw) (Y) = VX(w(Y)) -w(vXY).
For a tensor field T of type (1,2) (or of type (0,2)). we
put
(1.4) (VXT) (Y,Z) = vX(T(Y,Z)) -T(vXY,Z) -T(Y.VXZ).
48 2. Riemannian Manifolds
In terms of local coordinates, if T has local components
Tai, VT has local componenets
vkTji = y Tai+rkt Tai-r Tti-rki Tit
In general, if T is a tensor field of type (r,s), VT has
local components
(1.5) vkTji.. j9 - ayk Tjl... js +rtk Tjl2.. jsr+ ... +
+Ti1... ir-lt
- rt T'1 - ' r -rt TIl... it
tk jl...is jlk tj2. .js ask j1...3s-lt
Using affine connection v, one may introduce the concept
of parallelism as follows:
Let y(t), tEI be a curve in M over an open interval I.
Then dt is a vector field over I. We put T(t) = ;(). T(t)
is called the velocity vector field of the curve y. Suppose
that Y(t) is a vector tangent to M at y(t) for t E I.
Assume that Y(t) varies differentiably with t. We say that
Y(t) is parallel along Y(t) if vT(t)Y(t) = 0 identically.
In terms of local coordinates system with
T (t) = E Ti (t)y and Y (t) = E Yj (t)yj , the condition of
parallelism of Y(t) along y(t) is
(1.6) dyk +E T i j Yi at = 0, t E I.
§ I. Affine Connections 49
The curve y is called a geodesic if the velocity vector field
T(t) of y(t) is parallel along the curve y. The condition
for y to be a geodesic is
(1.7)d2y dy. dy.
dt 2 +ETig at at = o, t E I.
Since (1.7) is a system of ordinary differential equations, the
existence and uniqueness theorem of ordinary differential equation
implies the following.
Proposition 1.1. Let M be a manifold with an affine con-
nection. Let p be a point in M ,and X, be a vector in Tp(M).
Then there exists a unique maximal geodesic t w y(t) in M
such that y(O) = XP and T(O) =XP.
Similarly, using (1.6), we see that if J = [a,b] is
a closed subinterval of I, then for each vector X E Ty(a)(M),
there is a unique vector field X(t) along I such that X(a) _
XY(a)and X(t) is parallel along y. The map P : Ty(a) (M) 4+
Ty(b)(M) given by P(X(a)) = X(b) is a linear isomorphism
which is called parallel translation along y from y(a) to
y (b) .
Using the affine connection V on M we define two impor-
tant tensor fields R and T by
(1.8) R (X,Y) = VXVY - VYVX - V[X,yl
(1.9) T (X, Y) = VXY - VYX - MY]
50 2. Riemannian Manifolds
for vector fields X,Y tangent to M. It is easy to verify
that R is a tensor field of type (1,3), called the curvature
tensor and T is a tensor field of type (1,2), called the
torsion tensor of V.
Let e1. ,en be a local frame of vector fields defined on
an open subset U of M. Denote by wl,...,wn be the dual
frame. We define n2 connection 1-forms wj on U by
(1.10)n
vX e. = E w) (X)ei,i=1
(linearity of wj follows from axiom (V1).) From (1.10) we
find
w (X) = wl (vXe
We define functions Tjk, on U by
(1.12) T(ej.ek) = E Tjk ei;
(1.13) R (ej. ek) el E R1jk ei.
Using Tjk and Rijk we define the torsion 2-form T1 and
the curvature 2-form 01 by
(1.14)
(1.15)
T1 = 2 E Tjk w3 A wk
nj = 2 E Rjklwk n w 1.
The forms w1,wj,Tl,(2j are related by the following Cartan
structural equations:
§I. Affine Connections 51
Proposition 1.2. We have
(1.16) dwl = - E wl A w3 + Tl (the first structural equation):
(1.17) dw _ - E wk n w +n' (the second structural equation).
Proof. For vector fields X,Y tangent to U, we have
E Tjk w3twkei = VXY - Vyx - [X,Y)
= v,(E wJ(Y)e) -vY (Ew3(X)ei ) -E w3([X,Y))e
= E (X (w] (Y) ) - Y (w3 (X)) - w3 ([X, Y)) )ei
+ E (wi (Y) w (X) - Wi (X) w (Y) )ek,
from which we find
2Ti (X,Y) = (X (wl (Y) ) - Y (wl (X)) - wl ([X,Y)) )
+ E (wJ (X) wi (Y) - wi (Y) w3 (X)) .
Thus, by combining (1.3.7) and (1.5.1) (i.e. equations (3.7)
and (5.5) of chapter 1) and this equation, we obtain, (1.14).
Equation (1.17) can be obtained in a similar way. (Q.E.D.)
Remark 1.1. In general, let E be a vector bundle of rank
r over a manifold M and let T* = T*(M) be the cotangent
bundle of M. Denote by r(E) and r(T* ®E) the spaces of
sections of E and the tensor product T* ®E. A connection on
E is defined as an operator
52 2. Riemannian Manifolds
(1.18) v : r(E) 4 r(T* IRE)
satisfying the following two conditions:
(a) v (s1 +82
) = vs1 + vs2:
(b) V (f s) = df es + fvs,
for functions f on M and sections s,sl,s2 in r(E).
§ 2. Pseudo-Riemannian Manifolds 53
§2. Pseudo-Riemannian Manifolds
Let M be a manifold of dimension n. A pseudo-Riemannian
metric on M is a tensor field g of type (0,2) which satis-
fies
(a) g (X, Y) = g (Y, X) for X, Y E ; (M) ;
(b) for each p E M, gp is a nondegenerate bilinear form
on Tp (M) x Tp (M), i.e., gp (X, Yp) = 0 for all Yp E Tp N'
implies Xp = 0.
A pseudo-Riemannian metric g on M is called a Riemannian
metric if gp is positive-definite for each p in M. It shall
be remarked that every manifold M admits a Riemannian metric.
To prove this, we take a locally finite covering (Uo) of M by
coordinate neighborhoods. For each Uz, let ga be a positive-
definite quadratic differential form on Ua. Let {ha) be a
corresponding partition of unity (Theorem 1.8.1 and 1.8.2) (i.e.
Theorems 8.1 and 8.2 of Chapter 1.) Then we find that E hug'
defines a Riemannian metric on M. A manifold M with a (pseudo-)
Riemannian metric is called (pseudo- Riemannian manifold.
Proposition 2.1. On a pseudo-Riemannian manifold there
exists one and only one affine connection v satisfying the
following two conditions;
(gl) vxY - 7YX = [X,Y), i.e., v has no torsion,
(g2) Zg (X, Y) = g (vZX. Y) + g (X, vZY) , i. e. , the pseudo-
Riemannian metric g is parallel.
54 2. Riemannian Manifolds
Proof. It suffices to show that such connection V exists
and is unique on every coordinate neighborhood U. The unique-
ness implies that V must agree on overlapping domains; hence
V exists and is unique on M.
Let be local coordinates on U. Let
gij = g(ayi , a -)on U. Denote by (gkl) the inverse matrix
j
of (gij). If (gl) and (g2) hold, then we have
a ag ag i(2.1) 29(Vi ayj ayk) - ayi + a'i ayk
because [ay, , ay ] = 0. From (2.1) we findi 7
2rtagki
+agkiii gtk _ayiayi ayk
Consequently, we obtain
(2.2) k 1 tk agti agti - agii)rji 2
g(
ayi+ ayi
syt.
Giving an affine connection V on U is equivalent to
giving functions rj. on U with Vj ay 1a and
1 i k
demanding properties (v1) and (v2). We use (2.2) to define
V on U. Then the explicit expression (2.2) shows v is
unique. A direct computation yields that V is an affine con-
nection satisfying (g1) and (g2). (Q.E.D.)
In the following, we call the unique affine connection satis-
fying (gl) and (g2) on a pseudo-Riemannian manifold, the
§2. Pseudo-Riemannian Manifolds 55
Riemannian connection of v. We consider only the Riemannian
connection on a pseudo-Riemannian manifold M unless mentioned
otherwise.
Proposition 2.2. The curvature tensor of a Pseudo-Riemannian
manifold satisfies the following relations:
(a) R (X. Y) Z + R (Y. X) Z = 0;
(b) R(X.Y)Z+R(Y.Z)X+R(Z.X)Y = 0;
(c) g (R (X, Y) Z. W) + g (R (X. Y) W. Z) = 0;
(d) g (R (X. Y) Z, W) = g (R (Z, W) X, Y) .
Proof. Formula (a) follows immediately from the definition
For (b), we use the Jacobi identity and property (gl). (c)
follows from (a) and (d).
(d) is proved as follows: Let
S (X. Y ; Z. W) = g (R (X, Y) Z, W) + g (R (Y, Z) X. W) + g (R (Z, X) Y, W) .
Then, by direct computation, we have
0 = S(X,Y;Z,W) -S(Y.Z.W.X) -S(Z,W;X,Y) +S(W,X;Y.Z)
= g (R (X, Y) Z, W) - g (R (Y, X) Z, W) - g (R (Z, W) X, Y) + g (R (W, Z) X, Y) .
Thus, by applying (a), we obtain (d). (Q.E.b.)
Formula (b) is called the first Bianchi identity.
56 2. Riemannian Manifolds
§3. Riemannian Manifolds
Let M be an n-dimensional Riemannian manifold with
Riemannian metric g. We sometime denote this by (M,g). Let
be a local coordinate system on M. We have
g = E gij dyi dyj. Denote by (gi3) the inverse matrix of
(gij). Using the metric tensor g. we can define the inner
product <S,T> of two tensor of the same type. For example,
if S, T are tensor fields of type (0,2). we put
(3.1) <S,T> = E gk1g13Sk1Tij.
where S = E Sijdyi ®dyj, T = E Tijdyi 0dyj. The length IITII
of a tensor T is then defined by
1ITII = <T, T>*.
Let 0 :[&,b)-#M be a curve in M, we define the length
of o by
L(0) = jbIIQ. (dt) Ildt.
If a = E aidyi is a 1-form on M, we define a vector
field a# associated with a by
(3.2) °t#=EgljaiBy
a# is called the associated vector field of a. Similarly,
given a vector field X = E Xiayi , we may define a 1-form
X# by X# = E gijXldyj. We call X# the associated 1-form
of X. In fact, if a is the associated 1-form of vector field
§ 3. Riemannian Manifolds 57
X, we have a (Y) = g (X, Y) for Y tangent to M. Similar
operation can be defined for other tensor fields on (M,g).
For each p in M and each 2-dimensional subspace Tr
of the tangent space Tp(M) of M at p, we define the sec-
tional curvature K (Tr) of Tr by
K (Tr) = g (R (X, Y) Y, X) ,
where X, Y are orthonormal vectors in T. It is easy toverify that X(r) is well-defined. Sometime, we denote K(Tr)
by K (Z n W) if Z and W span the 2-plane Tr.
Given two vectors X, Y in Tp(M) and an orthonormal
basis e1....,en of Tp(M), we define the Ricci tensor S
and scalar curvature T by
n(3.3) S (X,Y) = E g(R(ei,X)Y,ei)
i=1
(3.4) _ 1
T - n (n-1) i S (ei ei).
It is easy to verify that both S and T are independent
of the choice of the orthonormal basis. Given a Riemannian
manifold M, if K(w) is constant for all plane section >r
in Tp(M) and for all points p in M, then M is called
a space of constant curvature or a real-space-form.
The following theorem of Schur is well-known.
Theorem 3.1. Let M be a Riemannian manifold of dimen-
sion n > 2. If the sectional curvature K(Tr) depends only
on the point p, then M is a space of constant curvature.
S8 2. Riemannian Manifolds
The proof of this theorem bases on the following
Lemma 3.1. The curvature tensor of a Riemannian manifold
M satisfies the following second Bianchi identity:
(3.5) (VXR) (Y,Z) + (Vt) (Z,X) + (uZR) (X,Y) = 0.
Proof of Theorem 3.1. Let T be the tensor field of
type (0,4) defined by
(3.6) T(Z,U;X,Y) = g(Z,X)g(Y,U) -g(U,X)g(Y,Z).
We define another tensor field, also denoted by R. by
(3.7) R (Z, U; X, Y) = g (R (Z, U) X, Y) .
Since the sectional curvature of M depends only on the point
p in M, we have
(3.8) R = kT
for some function k on M. Because Vg = 0, we have VT = 0.
Thus
(3.9) (V t) (Z,U;X.Y) = (Wk)T(Z,U;X,Y),
from which we obtain
(3.10) ( (VWR) (X. Y) ) Z = (Wk) (g (Z, Y) X - g (Z, X) Y) .
§ 3. Riemannian Manifolds 59
Taking the cyclic sum of (3.10) with respect to W,X,Y and
applying the second Bianchi identity, we find
(Wk) (g (Z, Y) X - g (Z, X) Y) + (Xk) (g (Z, W) Y - g (Z, Y) W )
+ (Yk) (g(Z,X)W-g(Z,W)X) = 0.
If we choose X,Y,Z,W such that Z = W and X,Y,Z are mutually
orthognoal, then we have (Xk)Y = (Yk)X. Therefore, k is a
constant. (Q.E.D.)
From (3.8) we see that if M is of constant curvature,
then the curvature tensor R of M takes the following form;
(3.11) R(X,Y)Z = k(g(Y,Z)X-g(X,Z)Y);
where k is a constant. Moreover, in this case, the scalar
curvature T satisfies
(3.12) T = k.
Let X be a unit vector in Tp(M). Then the Ricci curvature
at X is given by S(X,X), where S denotes the Ricci tensor.
If the Ricci tensor S(X,X) at p is independent of X, then
the Ricci tensor S takes the following form:
(3.13) S = Tg .
If this is the case at every point p in M. then M is
called an Einstein space. If n > 2, then by the second
60 2. Riemannian Manifolds
Bianchi identity, we may conclude that M has constant scalar
curvature.
A Riemannian manifold is called a locally symmetric space
if its curvature tensor R is parallel, i.e., vR = 0.
For any two points p, q in M, we define d(p,q) as
the greatest lower bound of the lengths of all piecewise differen-
tiable curves joining p and q. It can be shown that d
defines a metric on M. If the metric d is complete, i.e.,
all Cauchy sequences converge, we say that the Riemannian mani-
fold M is complete. It is well-known that any two points p
and q in M can be joined by a geodesic arc whose length is
equal to the distance d(p,q). It is also well-known that the
following conditions on M are equivalent:
(a) M is complete;,
(b) all bounded closed sets of M are compact;
(c) Any geodesic arc in M can be extended in both
directions indefinitely with respect to the arc length.
It is clear that a compact Riemannian manifold (without
boundary) is always complete.
Let elf...Ven be an orthonormal local basis on a compact
2m-dimensional manifold M. Let be the curvature 2-forms
with respect to the basis. We define a 2m-form (2m = dim M)
on M by
n =(-1)m r e 0
1 1A...An1
.2m-1 ,
22mwmm, 'l...i2m12 12m
§ 3. Riemannian Manifolds 61
whereci
is zero if do not form a permu-2m
tation of and is equal to 1 or -1 accordingly as
the permutation is even or odd. The cohomology class [C1] CH2m
(M)
is called the Euler class of M. The famous Gauss-Bonnet-Chern
formula is given by the following.
Theorem 3.2. Let M be a compact, 2m-dimensional, oriented
Riemannian manifold and let X(M) be the Euler-Poincare charac-
teristic of M. Then we have
(3.14) X (M) = JM 0.
In particular, if M is 2-dimensional and G(= r) is
the Gauss curvature of M, then we have the following Gauss-
Bonnet formula.
,(3.15) IGdV = 21r) ((M)
M
where dV = wlnw2 denotes the volume element of M.
62 2. Riemannian Manifolds
§4. Exponential Map and Normal Coordinates
Let v be the Riemannian connection of a Riemannian mani-
fold M of dimension n. For each vector X tangent to M
at p, Proposition 1.1 implies that there is a unique geodesic
yX(t) which is defined on a neighborhood of 0 in R with
yX(O) = p and yX(O) = X. For appropriate s in R,
ysX(t) = yx(st) by the nature of the ordinary differential
equations defining the geodesics. This implies that ycx(1)
is defined if yV(c) is defined. Therefore, yX(1) is a
well-defined point in M for X with sufficient small length
IIXII
Definition 4.1. For each X in Tp(M), we define
exppX as the point in M given by yx(1) when yx(1) is
defined. The map expp is called the exponential map at p.
Proposition 4.1. Let M be a Riemannian manifold. Then
for each point p in M. there exists an open neighborhood U
of 0 in Tp(M) and an open neighborhood U of p in M
such that the mapping expp : U-.U is a diffeomorphism of U
onto U.
By applying this proposition we define a normal coordinate
system about a point p E M as follows:
Let be an orthonormal basis of Tp(M) and
1, nw w its dual basis. Let U and U be neighborhoods of
0 in Tp(M) and p in M, respectively, such that expp
§ 4. Exponential Map and Normal Coordinates 63
is a diffeomorphism of U onto U. For each Y in U,
we put
Y = ylel+Y2 e 2
+ ... + ynen.
Then the componenets are called the normal coordi-
nates of the point q = expp(X) in U.
Proposition 4.2. Let be a normal coordinates
system about p in a Riemannian manifold M. Then we have
(a) gij (p) = 6ij;
(b) JO P) = 0.
Proof. From the definition of normal coordinates about p
we have
gij (p) = g ((ayl) p, (ay,) p) = g (ei, ej) = 6ij.
This proves (a).
For (b), we have v ( a ) =37 I` a . Let G (t) be the
ayj K lj ayk
curve with yi oa(t) = ai = constant. Then a is a geodesic
throughkp. Thus, by (1.7), we obtain z I'i j (p) alaj = 0 for
i, jany constants Therefore, we obtain (b). (Q.E.D.)
64 2. Riemannian Manifolds
§5. Weyl Conformal Curvature Tensor
Let M be a n-dimensional Riemannian manifold with
Riemannian metric g and C a positive function on M.
Then
(5.1) g* = C2g
defines a new Riemannian metric on M which preserves the
angle between any two vectors at any point. We call this a
conformal change of the metric. If C is a constant, the con-
formal transformation is called a homothetic transformation.
Let v* and v be the Riemannian connections associated
with g* and g, respectively. Then we have
(5. 2) vXY -vXY = w (X) Y + w (Y) X - g (X, Y) U,
for vector fields X,Y tangent to M, where w is the 1-form
given by w = d log C and U is its associated vector field,
i.e., w = U#. We put
(5.3) t(X,Y) _ (VxW) (Y) -W(X)w(Y) +*W(U)g(X,Y);
(5.4) g(TX,Y) = t(X,Y).
Then the curvature tensor R* of g* satisfies
R* (X, Y) Z = R (X, Y) Z - t (Y, Z) X + t (X, Z) Y
- g (Y, Z) TX + g (X, Z) TY.
§ 5. Weyl Conformal Curvature Tensor 65
From this we obtain
L* = L+t,
where
(5.5) L (X, Y) = - 1 2S (X.Y) + 2(n-2) g (X,Y)
and L* is the corresponding tensor field of type (0.2) asso-
ciated with g*. Thus, by eliminating t, we find
C* = C.
C (X, Y) Z = R (X, Y) Z + L (Y, Z) X - L (X, Z) Y
+ g (Y, Z) NX - g (X, Z) NY,
(5.8) g (NX,Y) = L(X,Y)
for X,Y,Z tangent to M and C* has a similar expression.
From (5.6) we conclude that the tensor field C is invariant
under conformal changes of the metric. Thus C is a confor-
mal invariant. We call it the Weyl conformal curvature tensor,
or simply conformal curvature tensor. This tensor C vanishes
identically when n = 3.
We also have
(5.9) D*(X,Y,Z) = D(X,Y,Z) +e(C(X,Y)Z),
66 2. Riemannian Manifolds
where
(5.10) D(X,Y,Z) _ (vXL) (Y, Z) - (vYL) (X,Z),
and D* is the corresponding tensor field of type (0,3) asso-
ciated with g*.
A Riemannian metric is called flat if its curvature tensor
R vanishes identically. A Riemannian metric g is called con-
formally flat or conformally Euclidean if it is conformally
related to a flat Riemannian metric g*, locally.
The following theorem is a well-known result of H. Weyl.
Theorem 5 1. A necessary and sufficient condition for a
Riemannian manifold to be conformally flat is that
C = 0 for n > 3
and
D = 0 for n = 3.
It should be noted that if n = 2, M is always conformally
flat. Moreover, if n > 3, then D = 0 is a consequence of
C = 0.
§ 6. Kaehler Manifolds and Quaternionic Kaehler Manifolds 67
§6. Kaehler Manifolds and Quaternionic Kaehler Manifolds
Let M be a complex manifold of n complex dimensions
with a system of local complex coordinates defined
on a coordinate neighborhood U. If zi = xi + yi, then
(xl'''*'xno yl'....yn) forms a system of local coordinates on U.
We put xi = a/axi and yi = a/ayi. Then X1,...'XnI Y1,....Yn form a
basis of Tp(M) for each p E U. Let J be the endomorphism
of Tp(M) defined by
JXi = Yi. JYi = -X it 1 , - - - , n .
Then J2 = -I. It is easy to see that J does not depend on
the choice of (zi). J is called the complex structure of M.
A Riemannian metric g on a complex manifold M is
called Hermitian if g and J are compatible, i.e.,
(6. 1) g (JX. JY) = g (X, Y)
for any X,Y in T(M). Let g be a Hermitian metric on M.
We put
(6-2) $ (X, Y) = g (X, JY) .
Then f is a 2-form on M, which is called the fundamental
form of M. A Hermitian manifold is called Kaehlerian if its
fundamental form I is closed, i.e., df = 0. The following
results are well-known.
68 2. Riemannian Manifolds
Proposition 6.1. A Hermitian metric g on a complex manifold
M is Kaehlerian if and only if J is parallel, i.e., vJ = 0.
Proof. From (7.2) we have
d4 (X, Y, Z) =3
(g (X, (VZJ) Y) - g (Y, (vXJ) Z) + g (Z, (vYJ) X)) .
Thus, if vJ = 0. then dfi = 0. Conversely, because
2g ( (vXJ) Y. Z) = & (X, JY, JZ) - d (X, Y. Z) ,
dt = 0 implies VJ = 0. (Q.E.D.)
Proposition 6.2. Let M be a compact Kaehler manifold.
Then H2k(M) 0; 0 S k s n = dimmM.
Proof. Letin
= OA... A6 (n-times). Then, by (6.2), f
is nowhere zero on M. Thus M is orientable. We orient M
in such a way that §n'> 0. Thus we have f
n> 0. Since M
is compact, the Stokes theorem implies that n is not exact.
Thus [§n] in H2n(M) is non-zero. Therefore [0k] EH 2k X
is non-zero for k = Consequently, H2k(M) # 0.
HO(M) 0 is trivial. (Q.E.D.)
Let (M,J,g) be a Kaehler manifold. For each unit vector
X in T(M), the sectional curvature K(X A JX) of the 2-plane
spanned by X and JX is called the holomorphic sectional
curvature of X. We denote it by H(X). A Kaehler manifold is
called a complex-space-form if it has constant holomorphic sec-
§ 6. Kaehler Manifolds and Quaternionic Kaehler Manfolds 69
tional curvature c. The curvature tensor R of a complex-
space-form takes the following form:
(6.3) R(X,Y)Z = 4{g (Y,Z)X-g(X,Z)Y+g(JY,Z)JX
g (JX, Z) JY + 2g (X, JY) JZ) .
It is well-known that two complete, simply-connected
complex-space-forms of the same constant holomorphic sectional
curvature are isometric and biholomorphic.
Let M be a 4k-dimensional Riemannian manifold with
Riemannian metric g. Then M is called a quaternionic Kaehler
manifold if there exist a 3-dimensional vector bundle V con-
sisting of tensors of type (1,1) over M satisfying the following
conditions:
(a) In any coordinate neighborhood U of M, there is
a local basis (J1,J2,J31 of V such that
J1 = J2 = J3 = -I ;(6.4)
1 11 2 = -J2J1 = J3; 21 3 = -J3J2 = Jl; J3Jl = -J1J3 = J2 .
(b) For any local cross-section cp of V. vXcp is also
a local cross-section of V, where X is a vector tangent to
M and v the Riemannian connection of (M,g).
Let X be a unit vector on M. Then X. J1X, J2X, J3X
are orthonromal vectors in M. We denote by Q(X) the 4-plane
spanned by them. Q(X) is called a quaternionic 4-plane. A
2-plane in Q(X) is called a quaternionic 2-plane. A quarter-
TO 2. Riemannian Manifolds
nionic Kaehler manifold is called a guaternion-space-form if
the sectional curvature of quaternionic 2-plane is constant.
The curvature tensor R of a quaternion-space-form
takes the following form:
3R (X, Y) Z =
4f g (Y, Z) X - g (X. Z) Y + F., g (JrY, 2) JrX
r=1
(6. 5) 3 3- L g (JrX, Z) JrY + 2 E g (X. JrY) JrZ) ,
r=1 r=l
where c is a constant.
The quaternion projectic m-space QPm with its standard
metric is the best known example of quaternion-space-form.
§ 7. Submersions and Projective Spaces 71
§7. Submersions and Projective Spaces
A map rr:R--M of a manifold into another is called a
covering map if rr is surjective and 1r* is bijective at
every point. Thus, in particular, M and M are of the same
dimension. A covering map rr:R-#M of a Riemannian manifold
into another is called a Riemannian covering map if r is
locally isometric. By an isometry cp of a Riemannian manifold
(N,g) into another (N',g'), we mean a diffeomorphism
cp : N -. N' such that g' = cp*g.
Let (M,g) be a Riemannian manifold and t a group of
isometries of M. We say that r acts properly discontinuously
and freely on M if for each point p EM, there is an open
neighborhood U of p such that a(U) fl U = 0 for each element
a in r with 0 ,-E e. Let N = M/I' be the quotient space.Then M is a covering manifold of N and t is the group of
the covering transformations of M over N. Let
rr : M -4 N = M/t
be the projection. Then, for each point p in M and for each
element a in t, 'r (a (p) ) = Tr (p) and, moreover, N admits
a canonical Riemannian structure g' such that, with respect
to this metric g', 7r : M - N becomes a Riemannian covering map.
Example 7.1. Let 0 : Sn 4 Sn be the antipodal map of the
standard unit n-sphere Sn onto itself, which sends a point p
72 2. Riemannian Manifolds
in Sn onto its antipodal point. Then a is an isometry of
Sn with a2 = e (i.e., a is involutive). Let r = (e,o)
then r acts properly discontinuously and freely on Sn. The
quotient space Sn/i', denoted by RPn, with its canonical
metric gp, is a Riemannian manifold of constant sectional
curvature 1. we call this Riemannian manifold RPn the real
projective n-space.
Example 7.2. Regard Rn as an n-dimensional vector space.
Let vl. .'vn be a basis of Rn. We put
nA = ( Elmivil mi integers).
Then A is a free abelian group (or a lattice generated by
vie...,vn. Acting on Rn as translation, A acts properly
discontinuously and freely on Rn. The quotient space Rn/A
with the canonical metric is a compact, flat, n-dimensional
Riemannian manifold which is called a flat n-torus. Two flat
n-tori Rn/A and Rn/A' 'are isometric if and only if the
lattices A and A' are related by an isometry of Rn.
Example 7.3. Let a,b be two non-zero real numbers.
Consider the following two isometries of R 2 onto itself:
a 1: (xl, x2) - (xl, x2 + b) ,
a2: (xl, x2) .4 (x1 + 2, - x2) .
Letra,b be the group of isometries of R2 generated by
01,02. Thenra,b
acts on Rn properly discontinuously and
§ 7. Submersions and Projective Spaces 73
freely. The quotient space R2/I'a,b with its canonical
metric is a compact, flat, 2-dimensional unorientable Riemannian
manifold, denoted by Ka,b' which is called a Klein bottle.
Two Klein bottles Ka,b' Ka',b' are isometric if and only if
a = a' and b = b'.
A map Tr : M + B of a manifold M onto another B iscalled a submersion if (*r4,)p is surjective for each point p
in M. (O'Neill [1]) In particular, a covering map of a manifold
onto another is a submersion.
If Tr : M + B is a submersion, then, for each b E B, 7r-1 (b)is a submanifold of M of dimension dim M - dim B. We call
the submanifold 'r-1(b) a fibre. A tangent vector of M is
called vertical if it is tangent to a fibre. If M is a
Riemannian manifold, then a vector of M is called horizontal
if it is orthogonal to a fibre.
A submersion ir: M 4 B of a Riemannian manifold into another
is called a Riemannian submersion if Tr* preserves lengths of
horizontal vectors.
Let M be a Riemannian manifold and G a group of isometries
such that the projection Tr: M 4 B = M/G is a submersion. Then,
by imposing Tr* to preserve lengths of horizontal vectors, one
may induce a Riemannian metric on B. With respect to this
metric on B, Tr:M-#B becomes a Riemannian submersion. We
state this as the following.
Lemma 7.1. 1&t M be a Riemannian manifold and G a group
of isometries such that the projection I: M -OB = M/G is a
74 2. Riemannian Manifolds
submersion. Then B admits a canonical metric such that
rr :9 -. B is a Riemannian submersion.
Example 7.4. RegardCn+l
=12n+2 as a (2n+2)-dimensional
Euclidean space with the usual Euclidean metric. Denote by
S2n+l the standard unit hypersphere of Cn+1. Let
G = (z E C 11zI = 1). Then G is a group of isometrics
acting on 52n+1 by multiplication. Denote by CPn the quotient
space S2n+1/G. Then CPn admits a canonical complex structure
and, moreover, the projection
(7.1) 'rr : S 2n+ 1 4 CPn
is a submersion. By Lemma 7.1, CPn admits a cononical metric
gosuch that v becomes a Riemannian submersion. With respect
to this canonical metric g0, CPn becomes a Kaehler manifold
of constant holomorphic sectional curvature 4. In fact, (6.3)
implies that the sectional curvature K of CPn satisfies
1 K 9 4. CPn is called the complex projective n-space. The
canonical metric g0 on CPn is called the Fubini-Study metric.
Moreover, the submersion (7.1) is also known as the Hopf fibration.
Example 7.5. Regard Qn+l = R4n+4 as a (4n+4)-dimensional
Euclidean space with the usual Euclidean metric. Denote by
S4n+3 the standard unit hypersphere ofQn+l.
Let
G - (z E Q `JzJ = 1). Then G is a group of isometrics acting
on 54n+3 by multiplication. Denote by QPn the quotient
space S4n+3/G. Then QPn admits a canonical quoternionic
structure and the projection rr:54n+3 .QPn is a submersion.
§ 7. Submersions and Projective Spaces 75
Lemma 7.1 implies that QPn admits a canonical metric go
such that 7r:S4n+3 + QPn is a Riemannian submersion. With
respect to this canonical metric, QPn becomes a quaternionic
Kaehler manifold with maximal sectional curvature 4. QPn with
this canonical quaternionic Kaehlerian structure is called the
quaternion projective n-space.
It is known that spheres, real projective spaces, complex
projective spaces, quaternion projective spaces together with
the Cayley plane form the class of compact symmetric spaces
of rank one.
In general, symmetric spaces can be defined as follows:
Let M be a Riemannian manifold. Given a point p E M, an
isometry sp :M + M is called a symmetry at p if sp is
involutive (i.e., sp = id.) and p is an isolated fixed
point of sp. A Riemannian manifold M is called a symmetric
space if, for each point q E M, there exists a symmetric
sq of M at q. A symmetric space is always complete and
locally symmetric. The dimension of maximal flat totally
geodesic submanifold of a symmetric space M is called the
rank of M. A symmetric space is also a homogeneous space.
In fact, if G denotes the closure of the group of isometries
generated by symmetries (sq Iq E M) in the compact-open
topology. Then G is a Lie group which acts transitive on
the symmetric space M; hence the typical isotropy subgroup K,
say at 0, is compact and M = G/K.
76 2. Riemannian Manifolds
Let M be a manifold and G a compact Lie group acting
on M. Let M be a Riemannian manifold with group I(M) of
isometries. An immersion f :M -. M of M into M is called
G-equivariant if there is a homeomorphism C :G -. I(M) such that
(7.3) f(a(p)) = C(a)f(p)
for a E G and P E M. We mention the following result of
Mostow and Palais [1) for later use.
Proposition 7.1. Let G be a compact Lie group, K a
closed subgroup of G and M = G/K. Then there is a G-equivariant
imbedding of M into the standard m-sphere Sm for m
large enough.
Let n :M -. B be a Riemannian submersion of a compact
Riemannian manifold M into another compact Riemannian manifold B.
If dim M = dim B, then, for each function f on M and each
b in B, we define a function f on B by
(7.4) f(b) = E f(p) .
pEir-1(b)
If dim M > dim B, then we define a function f on B by
(7.5) f(b) = J -1 f do ,
(b)
where do denotes the volume element of the fibre wr 1(b). The
following Lemma is well-known.
Lemma 7.2. Let f be a function on M and f the
associated function on B. Then we have
§ 7. Submersions and Projective Spaces
(7.6)SM
f dVM = fB f dVB ,
77
where dVM and dVB denote the volume elements of M and B,
respectively.
Denote by cn the volume of the unit n-sphere. Then we
have
(7.7)
(7.8)
2(27)mc2m (2m-1)(2m-3).. 3.1
c27T
M+ I
2mf1 m
If we choose f = 1, then we have f = cl and c3,
respectively, for the submersions (7.1) and (7.2). Thus, by
Lemma 7.2, we obtain the following.
Lemma 7.3. Let CPn and QPn be the complex and quaternion
projective n-space with canonical metrics of maximal sectional
curvature 4. Then we have
(7.9) vol(CPn) = n:
2n(7.10) vol(QPn) = (2n+1)!
Chapter 3. HODGE THEORY AND SPECTRAL GEOMETRY
¢1. Operators *, b and A
Let M be an n-dimensional, oriented Riemannian manifold.
We choose an orthonormal local basis el,...,en whose
orientation is compactible with that of M. Denote by
l,...,n the dual basis of e1,...,en. Then wl A... Au?w w
is the volume element of M. We define an isomorphism
* :AP(M) 4 An-p(M), called the Hodge star isomorphism, of
p-forms into (n-p)-forms as follows:
Since w11...,wn form a local basis of A1(M), every
p-form a on M can be expressed locally as follows:
it i(1.1) a= E ai w A...AwP
it<...<iP 1 p
We define
(1.2) *a = E ei i j - jj l<...<jn-P 1' P 1 n-P
ai i w71 A ... A wan-p
1 p
where as before ei1..,ip
il...in-P is zero if i-1..1pi1...]n-p
do not form a permutation of 1,...,n, and is equal to 1
or -1 according as the permutation is even or odd. It is
easy to check that the star operator * is well-defined.
§ 1. Operators , S and a 79
The form *a is called the adjoint of the form a. The adjoint
of 1 is just the volume element:
*1 = wln... nwn .
The adjoint of any function is its product with the volume
element. If vl,...,vn-1 are n -1 vectors in Rn
where Rn equips with the usual orientation and the usual
metric, then (*(vi n ... A vn-1)); is called the vector
product of vl,...'vn-l'
Proposition 1.1. The star operator * has the following
properties:
(a) *(a+8) _ *a+*e; *(fa) = f(*a);
(b) *(*a) _ (-1)np}p a;
(c) aA*1 = Hn*a;
(d) a A *a 0 if and only if a = 0, where a and
p are p-forms and f is a O-form.
This Proposition follows from straight-forward compu-
tation. So we omit it.
Let a and S be p-forms given by
it ia= E ai ...i w n ... nw p1 p
and
it iB = E bi ..i w n ... A p
1 p
80 3. Hodge Theory and Spectral Geometry
Then we have
(1.3) aA*s = E a. b * 1 .p ll. .ip
11 'lp
For any two p-forms a and s on M we define a (global)
scalar product of a and s by
(1.4) (a, 0) = JaA*s ,
M
whenever the integral converges.
Proposition 1.2. The scalar product ( , ) has the
following properties:
(a) (a,a) 0 and is equal to zero if and only if a = 0;
(b) (a,s) _ (s,a);
(c) (a,0 1+ s2) = (a.8 ) + (a,82);
(d) (*a,*s) = (a,s), where a,s,s1,82 are p-forms on M.
This proposition follows from Proposition 1.1.
Two p-forms are called orthogonal if (a,s) = 0. Using
the star operator we define the co-differential operator as
follows:
(1.5) ba = (-1)npfn+l *d*a
for p-forms a on M.
Proposition 1.3. The co-differential operator
6 :AP(M) 4 Ap-l(M) has the following properties:
§ 1. Operators *, 6 and A
(a) b(a+ 13) = ba+ 613
(b) bba = O ,
(c) *ba = (-1)p d*a, *da = (-1)P+l b*a
where a and p are p-forms on M.
This proposition follows from (1.5), Proposition 1.1 and
the property; d2 = O.
81
A form a is called co-closed if ba = O. If a = bH
for some form p, then a is called co-exact. In contrast
with the differential operator d, the co-differential operator
b involves the metric structure of M.
Using the operators d and A. we define an operator A
by
(1.6) A = db+bd .
Then a maps p-forms into p-forms. The operator A is called
the Hodge-Laplace operator. Sometime we simply call A the
Laplacian of M, particularly, when A applies to functions.
Definition 1.1. A form a on M is called harmonic if
Aa = O.
Proposition 1.4. If M is a compact, oriented Riemannian
manifold and a and S two forms of degree p and p+ 1,
respectively, then we have
(1.7) (da,g) = (a.oP)
82 3. Hodge Theory and Spectral Geometry
i.e., the operator b is the adjoint of d. Consequently, the
Hodge-Laplace operator A is self-adjoint.
Proof. Since M is compact, the Stokes theorem implies
$ d(aA«p) = 0 .M
Therefore, by using the properties of d, we find
J daA*13 = (-l)P-lJ
aAdwp .M M
Thus, by using (1.5), we obtain (1.7). (Q.E.D.)
Corollary 1.1. On a compact, oriented Riemannian manifold
M, we have the following
(a) A p-form on M is closed if and only if it is orthogonal
to all co-exact p-forms;
(b) A p-form on M is co-closed if and only if it is
orthogonal to all exact p-forms.
This Corollary follows immediately from Proposition 1.4.
Another application of Proposition 1.4 is the following
well-known result.
Theorem 1.5. On a compact, oriented Riemannian manifold
M, a form a is harmonic if and only if do = 6a = O.
Proof. Let a be a p-form on M. Then Proposition 1.4
implies
§ 1. Operators , 6 and A 83
(Aa,a) = (d6a,a) + (6da,a) = (da,da) + (6a,6a) .
From this we conclude that a is harmonic if and only if a
is closed and co-closed. (Q.E.D.)
Theorem 1.5 implies immediately the following
Corollary 1.2. A harmonic function on a compact Riemannian
manifold is a constant.
Definition 1.2. Let X be a vector field on M and Xf
its associated 1-form. Then -6X# is called the divergence
of X, denoted by div X.
Proposition 1.6. (Divergence Theorem). Let X be a
vector field on a compact, oriented Riemannian manifold M.
Then
(1.8) $ (div X) *1 = 0 .M
Proof. By Proposition 1.4, we have
J (div X) * 1 = -f (6X#) * 1= -(a#,dl) = 0.M M
(Q.E.D.)
Corollary 1.3. If f is a differentiable function on a
compact, oriented Riemannian manifold M. then we have
(1.9) $(,&f) * 1 = 0 .
M
84 3. Hodge Theory and Spectral Geometry
Proof. Since f is a O-form, Af = bdf. Thus, from
Proposition 1.6, we obtain (1.9). (Q.E.D.)
Corollary 1.4 (Hopf's Lemma). Let M be a compact
Riemannian manifold. if f is a differentiable function on
M such that Af Z 0 everywhere (or Af S 0 everywhere), then
f is a constant function.
Proof. We may assume that M is orientable by taking the
two-fold covering of M if necessary. If Af 2 0 (or Af S 0)
everywhere, then Corollary 1.3 implies Af = 0. Thus, by
applying Corollary 1.2, f is constant.(Q.E.D.)
Remark 1.1. Let (M,g) be a Riemannian manifold and c
a positive constant. Then g = c2g defines a homothetic
change of metric g. From the definition of A, we have
(1.10) Ag = c-2 Ag .
Remark 1.2. For further results concerning * and 6,
see Chern (1], Goldberg (1].
§ 2. Elliptic Differential Operators 85
§2. Elliptic Differential Operators
Let U be an open set in Rn with Euclidean coordinates
x1,...,xn. For each n-tuple t = (t1 ...,tn) of non-negative
integers we put
(2.1)
(2.2)
Itl=tl+...+tn,
Dt = aitItl to
.axn
A linear differential operator D of degree r over U takes
the following form:
(2.3) D = a (x) DtItI<r t
where at(x) :L -4 V is a homomorphism of vector spaces and
depends differentiably on x in U.
For each y = (yl,...,yn) E Rn, we set
(2.4) E at(x)yt
t twhere we use the multi-indices; yt = yll ...ynn. a(D,y) is
called the characteristic polynomial of D and
a(D, - ) : Rn -+ Hom(L,V)
is called the symbol of D. The differential operator D is
called elliptic if the characteristic polynomial a(D,y) of
86 3. Hodge Theory and Spectral Geometry
D has no real zeros except y = 0 at each point x E U.
The notion of elliptic differential operator has a natural
generalization to manifolds defined as follows: Let M be an
n-dimensional manifold and E, F two (complex) vector bundles
over M. A linear differential operator is a linear map
D : r(E) .. r(F)
which, when restricted to each coordinate neighborhood U of
M (over which E and F are trivial), is expressible in the
form (2.3). The differential operator D is again called
elliptic if the characteristic polynomial o(D,y) has no real
zeros except y - 0 for each x E M. It is clear that composites
of elliptic operators of elliptic operators are elliptic.
Remark 2.1. The symbol also has an intrinsic
definition as a bundle map
(2.5) a(D,.) :T*(M) -+Hom(E,F).
which is done as follows: Let p E M. y E Tp(M), s E E We
choose a local section s in E such that s = ap and a
differential function f(p) = 0 with (f*)p = y. Then
(2.6) a(D,y)(s) = r: D(fr s)P
It can be verified that this is well-defined and it coincides
with the former definition in a coordinate neighborhood.
§2. Elliptic Differential Operators 87
Perhaps the most important example of an elliptic operator
is the Hodge-Laplace operator A. Take M = Rn and E = F =
the trivial line bundle. Take
n a2D - - F
i=1 a i
Since o(D,y) E y?,i
elliptic.
it is clear that D is
If M is a compact Riemannian manifold, p E M and
(y1,...,yn)a normal coordinate system about p, then, from
Proposition 2.4.2, we obtain
2A = at p
ayi
Thus A is elliptic.
We give another proof of the ellipticity of A as follows:
Let P E M, W E A(M), and y E Tp(M). Choose a function
f E A0(M) such that f(p) = 0 and (f*)p = y. Then we have
(p(d,y) )wp = d(fw) = y A wpP
by (2.6). This implies
(2.7) a(d,y) = y A
Similarly, we may prove that
(2.8) a(6.y) = -tyf .
88 3. Hodge Theory and Spectral Geometry
where ty is the interior product with respect to y#,
y# E T(M) the associated vector of y on the Riemannian manifold
M. Therefore, by (2.7) and (2.8), we find
(2.9) a(A,Y) = a(db + 6d,y) = -(Y n ty# + ty# y n )
_ -{{Y{{2
where denotes the length of y E Tp(M) induced from the
Riemannian metric g on M. From (2.9), we see that p is
elliptic. Proposition 1.4 shows that p is also self-adjoint, too.
Let E, F be two complex vector bundles over a compact
Riemannian manifold M. Let us equip E and F with Hermitian
metrics. This allows us to define an inner product
on r(E), e.g., if s, s' E r(E),
(s,s') -J <s,s'> * 1 .
M
Let D :r(E) -. T(F) be an elliptic operator. It is
possible to define the adjoint of D; D* :T(F) -+ NE), as a
differential operator characterized by the property; if s E r(E),
u E r(F), then
(2.10) (Ds,u) = (s,D*u) .
*It can be verified that D is also elliptic. It follows from
* *(2.10) that ker(D) (kernel of D) and im(D ) (image of D
are orthogonal with respect to ( , ). Moreover, ker(D) is
§ 2. Elliptic Differential Operators 89
*precisely the complement of im(D ), in fact, if s is
*orthogonal to im(D ), then 0 = (s,D Ds) = (DS,Ds), thus
De = 0. Therefore, we have
*(2.11) r(E) = ker(D) ® im(D
In fact, an elliptic operator D :r(E) + r(F) has many other
important properties. (see, for instance, Palais [1)): For
example, an elliptic operator D is a Fredholm operator, i.e.,
D is a differential operator which has finite-dimensional
kernel and cokernel and closed image. From (2.11), we may
conclude that for any f in the orthogonal complement of ker(D),
there exist a solution of the equation
*(2.12) D u = f
*Now, we consider the special case: E = F and D = D
(i.e., D is self-adjoint.). For each X E 3R, we put
(2.13) r X= (s e r (E) I Ds = Xs I .
Let L2(E) denote the completion of r(E) with respect to
( ). Then it is known that there are only countably many
X such that rX 1 0 and each rl 0 and each rX is
finite-dimensional and
(2.14) L2(E) = ®lrl
where ® is the completion of orthogonal sum.
90 3. Hodge Theory and Spectral Geometry
Because the Hodge-Laplace operator A :Ak(M) 4 Ak(M) is
elliptic and self-adjoint. We have
(2.15) k(M) = ®A iVk,i
where Vkj is the i-th eigenspace of A acting on k-forms.
In fact, the eigenvalues of A : Ak(M) -4 Ak(M) satisfy
0 g Ak,l < 'k.2 < ... T m
Since the kernel of A :Ak(M) 4 Ak(M) is finite-dimensional,
we obtain the following well-known result.
Theorem 2.1. If M is a compact Riemannian manifold of
dimension n, then the spaces of harmonic k-forms, k = 0,1,...,n,
are finite-dimensional.
We simply denote VO.i by Vi and denote Xby by X..
The decomposition (2.15) gives the following.
Theorem 2.2. If M is a compact Riemannian manifold, then
(2.16) C0°(M) = ®iVi-
Denote by Speck(M) the set of all eigenvalues of
A :Ak(M) 4 Ak(M) enumerated with multiplicity. We simply
denote Speck(M) by Spec(M). Speck(M) is called the spectrum
of k-forms. The geometry which studies spectra of M is called
spectral geometry.
§ 3. Decomposition
¢3. Hodge-de Rham Decomposition
Let M be a compact, oriented n-dimensional Riemannian
manifold. For each k; k = 0,1,...,n, denote by na(M)
A6(M), and /1(M), the subspaces of nk(M) consisting of
k-forms on M which are exact, co-exact, and harmonic,
respectively.
Lemma 3.1. The subspaces Ad (M), Ak 6(M), and N(M) are
mutually orthogonal.
Proof. If W E A(M), w is closed. Thus corollary 1.1
implies that w is orthogonal to A (M). Let w = do and
B E nH(M). Then Proposition 1.4 and Theorem 1.5 imply
(w,8) = (da,ft) = (a,bp) = 0 .
This shows that Ak(M) is orthogonal to both na(M) and
91
#H (M). Similar argument applies to the remaining cases. (Q.E.D.)
We give the following well-known Hodge-de Rham decomposition
theorem.
Theorem 3.1. A k-form a on a compact, oriented Riemannian
manifold M may be uniquely decomposed into the orthogonal sum:
(3.1) a = ad+aa+aH ,
where ad E Ak(M), as E na(M) and aH E nk(M).
92 3. Hodge Theory and Spectral Geometry
Proof. Theorem 2.1 implies that AH(M) is finite-
dimensional. Thus, we may choose an orthonormal basis
w w ,I(M). Let a be any k-form on M, we putl,...,h kof nf
haH = (a,w')w'
i=1
Then AaH = 0, i.e., aH is harmonic. Moreover, it is
clear that a - aH is orthogonal to AH(M) because a -aH is
orthogonal to each wl, i = 1,...,h. Since A = A, there is
a solution of the equation pu = a - aH. Thus, we may find a
k-form A such that pR = a -aH*
We put
ad = d6ft , a6 = r, d13 .
Then we obtain a = ad + a6 + aH.
if a = ad '+ a + aH is another decomposition of a, then
(ad - a') + (a6 - as) + (aH - aH) = 0. Thus by Lemma 3.1, we obtain
ad aa, a6 = a6, and aH = aH. This proves the uniqueness.
(Q.E.D.)
Now, we mention some important applications of the
Hodge-de Rham decomposition theorem.
Theorem 3.2. (Hodge-de Rham). Every cohomology class in
Hk(M) is represented uniquely by a harmonic k-form on M.
Proof. Let § E Hk(M). Then I = [a] where a is
a closed k-form. Thus, by Corollary 1.1, we have
a = ad + aH
§ 3. Hodge-deRahm Decomposition 93
where ad E Ak(M) and aH E AH(M). Since ad is exact, §
is represented by the harmonic form aH. If p is any other
k-form which represents §, then a -13 is exact. Thus, by
Theorem 3.1, we obtain aH = SH (Q.E.D.)
Combining Theorem 1.9.2 and Theorem 3.2, we obtain
Theorem 3.3. If M is a compact, oriented Riemannian
manifold, then the k-th betti number of M is equal to the
number of linearly independent harmonic k-forms on M, for
k = 0,1,...,n. In particular, we have b0(M) = bn(M) = 1.
We also have the following Poincarg duality theorem.
Theorem 3.4. If M is a compact, oriented Riemannian
manifold of dimension n, then we have the following natural
isomorphisms; p :Hk(M) y Hn-k(M).
Proof. If I E Hk(M), there is a harmonic k-form
aH representing 4. Since G*aH = *paH = 0, *aH is again
harmonic. We put 04) = [*%] E Hn-k(M). Then it is easy to
see that µ defines an isomorphism from Hk(M) onto Hn-k(M).
(Q.E.D.)
Another easy consequence of Hodge-de Rham decomposition
theorem is the following: If M is a covering manifold of
M which is also compact, then
(3.2) bk(M) S bk(M)
94 3. Hodge Theory and Spectral Geometry
for k = 1,...,n-1. This can be seen as follows: If a is
a nonzero harmonic k-form on M, then there is a periodic
extension a on R given by a = where a is the
projection of the covering map. It is clear that a is also
a non-zero harmonic k-forms on M. Moreover, linearly independent
harmonic forms on M lift to linearly independent harmonic
forms on A. Thus, we have (3.2).
§4. Heat Equation and its Fundamental Solution
¢4. Heat Equation and its Fundamental Solution
Ih this section we will mention some well-known results
95
on the fundamental solution of the real heat equation for later
use. For the detail, see Berger-Gauduchon-Mazet (1).
Let M be a compact Riemannian manifold. A heat operator
on M is the operator
acting on functions defined on M x R+ , which is of class C2
on the first variable and of class C1 on the second. The
heat equation on M works on functions F : M x R+ + R which
satisfy
(4.1) L(F) = 0, F(p,O) = f(p), p E M,
where f : M + R is a given initial condition.
Definition 4.1. A fundamental solution of the heat equation
on M is a function h : M x M x R+ -0 R which satisfies thefollowing conditions:
(h1) : h is continuous on the three variables, of class
C2 on the first two variables and of class Cl on the third,
(h2) : L2h = 0, where L2 = p2+ at, p2 the Laplacian
on the second variable,
(h3) : for each p E M,
limt-#O+ 6y ,
96 3. Hodge Theory and Spectral Geometry
where 6 is the Dirac distribution at p E M, i.e., for a
function f on M with p E supp(f), we have
lim J' h(p,x,t)f(x)dx = f(p) ,
t..o+ M
where dx denotes the volume element of the second M.
The following result is well-known.
Theorem 4.1. A fundamental solution of heat equation on
M exists and is unique.
For each eigenspace Vi of a :C'(M) -. C"(M), we choose
an orthonormal basis i
" 'gymi
of Vi (mi dim Vi). The
set of (maa is called an orthonormal set of eigenfunctions
of A. According to Theorem 2.2, we have, for each function
f:M -. R ,
(4.2) f = (ma,f)cpQ (in the L2-sense)a,iProposition 4.2. If (mQ) is an orthonormal set of
eigenfunctions, then for each (p,x,t) in M xM x R+ , the series
(4.3)
converges and
ealt
CpQ(p)oDQ(x)
(4.4) h(p,x,t) e lit cpi(p)cp (x)i,a
The proof bases on the uniqueness of the fundamental
solution of the heat equation given in Theorem 4.1. Now
§4. Heat Equation and its Fundamental Solution 97
tL2(E a-li ma(P)tpa()
= E e alt ma (P) (ACP ) (x) +E at (e llt)ma(P)GDQ(x)
t t= E Ca (P)coa(x) -E lie-11 apa(p)cpa()
= O .
Moreover, for any f : M -. R , we also have
lim J E ealt
TaTal(x)f(x)dxt..0 M
-xt= lim E ei (ma,f)CPat-+O+
=E(ma,fa=f
These show that (4.3) satisfies conditions (h2) and (h3). (Note
that we omit the proof of convergence.) Moreover, in fact,
(4.3) also satisfies condition (h1). Thus (4.3) is a fundamental
solution of the heat equation. By the uniqueness, it is
h(p,x,t).
By integrating h(x,x,t) over M and using (4.4) we
obtain the following.
Proposition 4.3. For each t > 0, the series E m(ai)ei
converges and
p -xi t
(4.5)J
h(x,x,t)dx = E m(ai)eM i
where m(%i) denotes the multiplicity of ai.
98 3. Hodge Theory and Spectral Geometry
To construct the fundamental solution, we use a successive
approximation method. Although Rn is not compact, there
exists a unique fundamental solution of heat equation on Rn
under the condition to be decreasing at infinity. The solution
is given by
n d(p,x)2-2 - 4th0(p,x,t) = (4rrt) e
where d(p,x) denotes the Euclidean distance between p and x.
Now, using the idea that on a Riemannian manifold M
the fundamental solution of heat equation differs little from
the pull-back of h0 by exp 1, one may arrive at the
following asymptotic expansion of Minakahisundaram-Pleijel after
rather long computation.
Theorem 4.4. For each compact Riemannian manifold M,
there are constants ai's (i = 0,1,2,...) with
n
(4.6) Fi MO. )e3t ,., (4Trt)2 aitij j t-00 i-O
where n = dim M.
The first four coefficients a0, al, a2' a3 have been
computed. In fact, it is well-known that
(4.7)
(4.8)
a0 =J
1 = vol(M) ;
M
a1
6
1 r * 1 (already folklore in 1965)
§4. Heat Equation and its Fundamental Solution 99
a2 has been determined by McKean and Singer [1] in 1967; and
a3 was determined by Sakai [1] in 1971. a2 is given by
1(4.9) a2 = 360 J (2IIRII2 - 2IISII2+5 n2(n_1)2,23 *1M
In particular, (4.7) gives the following.
Proposition 4.5. Let M be a compact Riemannian manifold.
Then the volume of M, vol(M), is a spectral invariant.
By a spectral invariant we mean a global Riemannian
invariant which depends only Spec(M).
Corollary 4.1. Let M and M' be compact Riemannian
manifolds. if Spec(M) = Spec(M'). then vol(M) = vol(M').
This corollary is an immediate consequence of Proposition 4.5.
100 3. Hodge Theory and Spectral Geometry
¢5. Spectra of Some Important Riemannian Manifolds
In this section we consider the Laplacian A acting on
functions. For general information in this direction see
Berger-Gauchuchon-Mazet (1]. In this case, we have a = bd.
In the following we mention various expressions of a:
(1) For a function f on a Riemannian manifold (M,g),
df is a 1-form on M. Denote by (df)t the associated vector
field of df. Since div(df) # is -bdf, we have of = -div(df),.
(2) Since df is a 1-form, the covariant derivative
vdf of df is a 2-form which is called the Hessian of f.
The trace of vdf is -Af.
(3) Let p be a point in M and ul,...,un a normal
coordinate system about p. Then we have
n(5.1) (af)(p) E a?2 (p)
i=l aui
This is equivalent to say that for each point p E M, pick
an orthonormal set of geodesics (yi) parameterized by arc
length and passing through p at s = 0, then
(5.2) (Af)(p) = -.F, 2 (0)
i=1 ds
(4) In terms of local coordinates yl,...,yn of M, of
takes the following general form:
n d2(f o yi)
1 a(g gi3(af/ay.))of = -
9ayl
§5. Spectra of Some Important Riemannian Manifolds 101
where q = det(gij) and gij the components of the metric g
with respect to yl .... 'yn'
In the following, a submanifold N of a Riemannian
manifold M is called totally geodesic if geodesics of N are
carried into geodesics of M by the immersion. Using the
expression (5.2) of A we have the following.
Proposition 5.1. Let ir:(M,g) + (B,g') be a Riemannian
submersion with totally geodesic fibres. Then, for functions
f on B, we have
(5.3) AM(f oTr) = (ABf) or .
Proof. Let p be a point in M. Let u1,.. .,uk,vl " " 'vn-k
be an orthonormal basis of Tp(M) such that u1,...,uk are
horizontal and vl,...,Vn_k are vertical, where k = dim B.
Let 1Yi1i=l,...,n be the corresponding orthonormal set of
geodesics through p. Then we have, from (5.2), that
k 2 n d 2(5.4) am(fotr) _ - E d2 (forroyi)- E 2 (fo rroyj)
i=1 ds j=k+1 ds
Since rr:(M,g) + (B,g') is a Riemannian submersion with
totally geodesic fibres, 7royi form an orthonormal set of
geodesics in B through p and Yk+1' " 'Yn are geodesics
of the fibre n 1(ir(p)) . Thus, we obtain A o 1r) (p) =
(a5f)(7r(p)). Because the Laplacian is well-defined, it is
independent of the choice of local coordinates. Thus we obtain
(5.3). (Q.E.D.)
102 3. Hodge Theory and Spectral Geometry
Sometime, we called an eigenfunction of A :C (M) -4 C'(M)
a proper function of (M.g). Using Proposition 5.1 we may
prove the following.
Proposition 5.2. Let ir: (M,g) -6 (B,g') be a Riemannian
submersion with totally geodesic fibres. Then the proper functions
of (B,g') are those functions f on B such that r*(f) = for
are proper functions of (M,g).
Proof. Let f be a proper function of (B,g') with
eigenvalue X. Then, by (5.3), we have AM(f or) = if or. Thus
f or is a proper function of (M,g) with eigenvalue X.
Conversely, if f or is a proper function of (M,g)
with eigenvalue 7, then AM(for) = 1(f oir) = (ABf) air. Since
f or is constant along fibres, this shows that ABf = Xf. (Q.E.D.)
Although one may use the so-called Freudenthal formula
concerning the eigenvalues of Casimir operators to calculate
the eigenvalues of A for functions, in this section, we shall
use only elementary methods to calculate eigenvalues of a for
spheres and projective spaces. In order to do so, we first
obtain a relation between the Laplacian of Rl and that of S.
Let p be a point in Sn. Then p determines a unit
vector en+l in R 1. Let e1,....en be an orthonormal
basis of Tp(Sn). Then e1,...,en+1 form an orthonormal basis
of T p(Rn+1) . Let y1,...,Yn be the associated orthonormal
set of geodesics of Sn through p. If we regardell ... 'en+l
§ 5. Spectra of Some Important Riemannian Manifolds 103
as n + 1 points in R 1, then the geodesic yi through p
with velocity vector ei at p is given by
yi(s) = (cos s)en+l+ (sin s)ei. i = 1,...,n .
(in fact, this gives a great circle lies in the 2-plane spanned
by ei, en+1) Let f be a function on Rn+1 and
be the Euclidean coordinates associated with e1,...,en+l.
Consider the functions (f oyi)(s) = f(yi(s)). By using the
chain rule, we have
d(f oy.)da 1
= -(sin s) 211 + (cos s) Bn+1 i
2d
(f 2yi)(0) _ -a(P) + af(p)ds n+1 21x1
Therefore, by (5.2), we get
n(5.5) 21(f n(P) L(P) + n ax (p)
S i=1 axi n+l
On the other hand, the Laplacian Z of Rn+ l satisfies
(5.6)n+1 2
(Sf)(P) _ - E 2(P)J=1 axe
Combining (5.5) and (5.6) we obtain
a(f ( n)(p) n(P)+ 0 f (p) + n a f (P)S S 21xn+l n+l
104 3. Hodge Theory and Spectral Geometry
Consequently, if we denote by r the distance function from a
point in Rn+l to the origin, then we obtain
(5.7) (if) I Sn = A(f (Sn) - I Sn-n ar I Sn .
for functions f on Rn+l, where A denotes the Laplacian
of R 1
Consider a homogeneous polynomial P of degree k _> 0
onRn+l
. Let P - P In. Then we have P = rkP. Thus we
find
aP k-1 a?P k-2( 5 . 8 ) ar = kr P ; 2 k(k - 1)r P
ar
Substituting (5.8) into (5.7) we obtain
pP I n = AP-k(n+k-1)P .S
In particular, if P is harmonic, we find
(5.9) AP = k(n+k-1)P
In the following, we denote by Wk the vector space of
harmonic homogeneous polynomials of degree k on Rl, and
by uk the restriction of ik on Sn. It is known that
dim Vk = rnkk1 - (nk_221
where the last term is assumed to be zero for k = 0,1. Because
dim Vk > 0 for each k _> 0. We obtain from (5.9) the following.
§ 5. Spectra of Some Important Riemannian Manifolds 105
Proposition 5.3. The spectrum Spec(S) of the unit
n-sphere Sn is given by
(5.10) kk = k(n+k+l) , k _> 0 ,
with the multiplicity m(ak) of1'k
given by m(%0) = 1,
m(%1) = n+ 1, and
(5.11) m(kk) = (n+ 2
Moreover, the eigenspace Vk is A(k.
Now, consider the Riemannian covering map
it : (Sn,gO) - (R Pn,gO) .
According to Proposition 5.2, proper functions of (R Pn,gO)
are induced from the proper functions of (Sf,gO) which are
invariant under the antipodal map. Thus, proper functions of
(R pn,gO) are obtained from harmonic homogeneous polynomials
of even degree. From this and Proposition 5.3 we obtain the
following.
Proposition 5.4. The spectrum Spec(R Pn) (n > 1) of
the real projective n-space (R Pn,gO) is given by
(5.12) Xk = 2k(n+ 2k - 1) , k 0
and the multiplicities are given by m(O) = 1 and
(n+2k-2 n+2k-3 n+1 n2k:) (n+4k-1), k 2 1
106 3. Hodge Theory and Spectral Geometry
Using Proposition 5.2 and making further studies on the
following two Riemannian submersions:
S1 S2n+l CPn
S3 S4n+3QPn
one may obtain the following.
Proposition 5.5. The spectrum Spec(CPn) of the complex
projective n-space (CPn,go) with maximal sectional curvature 4
is given by
(5.13) ?,k = 4k(n+k), k , 0
and the multiplicities are given by
(n(n+1)...(n+k-1) 12m(Xk) -n(n+ 2k) ` k! J
Proposition 5.6. The spectrum Spec(QPn) of the quaternion
projective n-space (QPn,g0) with maximal sectional curvature 4
is given by
(5.14) Xk - 4k(2n+k+ 1) .
From Remark 1.1, we obtain immediately the following.
Lemma 5.1. Let (M,g) be a compact Riemannian manifold
and g - c2g with c a positive constant. Then we have
(5.15) Ak = c2lk , k - 0,1,...
§ 6. Spectra of Some Important Riemannian Manifolds 107
whereXk
andXk
denote the k-th eigenvalues of Laplacians
of (M,g) and (M,g), respectively.
Now, we determine the spectrum of a flat torus Rn/ A,
where A is a lattice of Rn . Put
A = (u E Rn <u, v> E a for any v E A )
Then it can be verified that A is also a lattice which is
called the dual lattice of A. Moreover, (A*)* = A. In fact,
if vl,...,vn is a basis for it. then its dual basis
vi,.... vn is a basis for A'. For each x E Vt, we define
a function fx on Rn by
f(Y) = e2irix(y)
where i = and y E Rn on the right is regarded as a
vector. It is clear that fx defines a function on Rn / A
which is also denoted by fx. If we denote by xi and yi the
components of x and y with respect to the bases vl,...,vn
and vl,...,vn respectively, then we find
fx (Y) =e2ni E xjyj
.
By taking differentiation with respect to yj we get
dye xx = 2rrixJf (y) ,
a2fx(Y) 2
aY2= _4m'xjj fx(Y)
iThus
108 3. Hodge Theory and Spectral Geometry
pfx = 47r2IIxII2fx .
This shows that ) = 4rr2IIxlI2 is an eigenvalue of A with proper
function fx for each x E A . To each eigenvalue X. the
corresponding eigenspace V). is generated by the fx's with
11x112 = X2 . The multiplicity m(%) of X is equal to the47r
number of x inA*
such that 11XI12 = 2. We summarize this
4Tr
to give the following result of Milnor [4].
Proposition 5.7. Let (Rn/ A, go) be a flat n-torus
and A* the dual lattice of A. Then the spectrum of Rn/A
is given by
(4rr2IIxII2 I x E A*) ,
and the multiplicity of ),. = 4ir2IIxII2 is equal to the number of
u E A* such that IIull = IIxII.
In 1941, Witt [1] discovered two lattices A. A' in R16
not isometric but with the same number of elements of any given
norm. Using these Milnor showed that there exist two 16-dimensional
flat tori which are not isometric, but nevertheless they have
the same spectrum.
Remark 5.1. The spectrum Spec( Ka,b) of a flat Klein
bottle Ka,b is also completely known. In fact, it is given by
m2 m24rr2 a2 +
2 ) with ml, m2 E 2Z, subject to the condition
that if m1 is odd, m2 ¢ 0.
Chapter 4. SUBMANIFOLDS
41. Induced Connections and Second Fundamental Form
Let i :M + M be an immersion of an n-dimensional manifold
M into an m-dimensional Riemannian manifold M with the
Riemannian metric g. Denote by g = the induced metric
on M. Equipped with g, i becomes an isometric immersion.
We shall identify X with its image i,X for any X E TIM).
If X,Y are vector fields tangent to M, we put
pXY = pXY+h(X,Y) ,
where pXY and h(X,Y) are the tangential and the normal components
of CXY, respectively. Formula (1.1) is called the Gauss formula.
Proposition 1.1 V is the Riemannian connection of the
induced metric g = ig on M and h(X,Y) is a normal vector
field over M which is symmetric and bilinear in X and Y.
Proof. Replacing X and Y by aX and AY,
a, a being functions on M, we have
from which we find
(1.2)
Vax (RY) = a{ (XR)Y+ 13VXY)
= (a(X9)Y + aj3VXY)+ al3h(X,Y) ,
Vax (oy) = a(Xp)Y+alsVXY ,
respectively,
(1.3) h(aX,$Y) = aj3h(X,Y) .
110 4. Submanifolds
Equation (1.2) shows that v defines an affine connection on
M and equation (1.3) shows that h is bilinear in X and
Y since additivity is trivial.
Since the Riemannian connection v has no torsion, we
have
O = vXY - vy C - IX,Y]
= vXY+h(X,Y) -vyC-h(Y,X) - IX,Y]
from which, by comparing the tangential and normal parts,
we have
vxY-vyx = Ix,Y]
and
h(X,Y) = h(Y,x) .
These equations show that v has no torsion and h is symmetric.
Since the metric g is parallel, we have
vxg(Y,Z) = Oxg(Y,Z) = g(vXY,Z) + q(Y,OxZ)
= g(PXY,Z)+ g(Y,VxZ)
= g(vxY,Z)+ g(Y,vxZ)
for any vector fields X. Y, Z tangent to M. This shows that
v is the Riemannian connection of the induced metric g on
M. (Q.E.D.)
§ 1. Induced Connections and Second Fundamental Form 111
We call h the second fundamental form of the submanifold
M (or of the immersion i).
Let g be a normal vector field and X a tangent vector
field on M. We decompose vXg as
(1.4) pXg = -ASX+ DXS ,
where -ASX and DXg are the tangential and normal components
ofvXg, respectively. It is easy to check that ASX and
DXS are both differentiable on M. Moreover, (1.3) implies
that h(Xp,Yp) depends only on X. Yp E TpM. not on their
extensions X, Y. Formula (1.4) is called the Weingarten
formula.
Proposition 1.2. (a) A9(X) is bilinear in g and X.
And (b) For each normal vector g of M and tangent vectors
X, Y of M,
(1.5) g(A9X,Y) = g(h(X,Y),g)
Proof. Let a and A be any two functions on M. Then
(1.6) %X(at) = avx(Pg) = a(X(3)S + ftvXgI
= a(X(3)g -aMAX+aj3DXS .
This implies
Aat(aX) = aAASX .
112 4. Submanifolds
Thus Amt is bilinear in g and X, since additivity is trivial.
This proves (a). To prove (b), we notice that for any arbitrary
vector field Y tangent to M. we have
0 = g(VXY,u + q(Y,vXs)
= g(h(X,Y),s) - g(Y,A9X)
This shows (b). (Q.E.D.)
Let T'(M) denote the normal bundle of the immersion
i:M -. M. From (1.6) we find
(1.7) DaX(Og) = a(Xp)g+apDXs .
Moreover, it is easy to verify that
(1.8) DX+Y = DX + DY
Equations (1.7) and (1.8) justified that D is a connection
on the normal bundle T1(M). In fact, we have the following.
Proposition 1.3. D is a metric connection in the normal
bundle T'(M) of M in M with respect to the induced metric
on T1(M).
Proof. For any two normal vector fields g and r on
M, we have
Vx = -ASX+DX9 ; VXn = -AnX+DXf
§ 1. Induced Connections and Second Fundamental Form
Hence, we get
g(DXg,n) + g(g,DXr) = g(v"Xg,n) + g(g,v"Xn)
vXg(g,r,) =
D is a metric connection. (Q.E.D.)
Definition 1.1. A normal vector field g is said to be
parallel if DXg = 0 for any X tangent to M.
Definition 1.2. H =
n
trace h is called the mean curvature
vector of the submanifold M in M. The submanifold M is
called minimal if H = 0 identically. And M is called totally
umbilical if
h(X,Y) = g(X,Y) H
for any X, Y tangent to M.
In §2.5 (i.e., §5 of Chapter 2), we have defined a sub-
manifold M of a Riemannian manifold M to be totally geodesic
if geodesics of M are carried into geodesics of M. In fact,
we have the following.
Proposition 1.4. Let i :M -. M be an isometric immersion
of a Riemannian manifold M into another. Then M is totally
geodesic in M if and only if h = 0 identically.
Proof. Assume that the second fundamental form h of
the submanifold M in M vanishes identically. Then, for any
114 4. Slubmanifolds
vector field X tangent to M, we have
(1.9) vXX = vXX .
If y(s) is a geodesic in M, then we have vT(x)T(s) s O.
where T(s) = y(s) = y,w(d-ds). Thus by (1.9) we find vT(s)T(s) O.
This shows that y(s) is also a geodesic in M. Hence, M is
totally geodesic in M.
Conversely, assume that M is totally geodesic in R.
Let Xp E Tp(M) be any unit vector at p E M. Choose a geodesic
y(s) in M such that y(O) = p and y(O) = T(O) = Xp. Then
we have vT(s)T(s) = VT(s)T(s) = O. Thus we find h(Xp,Xp) = O.
Since h is symmetric and bilinear, h = 0 at p. (Q.E.D.)
The following elementary results shows that fixed point
set of isometries are always totally geodesic.
Proposition 1.5. Let I+1 be a Riemannian manifold and
G a set of isometrics of M. Let F(G,M) = (p E M ja(p) = p
for any a E G) be the fixed point set of G. Then each
connected component of F(G,F1) is a closed totally geodesic
submanifold of M.
Proof. If F(G,M) is empty, then this proposition is
trivial. Assume that F(G,M) is not empty. Let p be a
point in it and let Vp be the subspace of Tp(M) consisting
of vectors fixed by all elements of G. According to
Proposition 2.4.1, there is a neighborhood U of the origin
0 in Tp(M) such that the exponential map expp :U -. M
§ 1. Induced Connections and Second Fundamental Form 115
is an injective diffeomorphism. We put U = expp(U). Assume
that U is convex. Then we have u (1 F(G,M) = expp(U fl Vp).
Thus, we find that the neighborhood u f1 F(G,M) of p in
F(p,M) is a submanifold expp(U f1 Vp). Hence F(G,M) consists
of submanifolds of M. It is clear that F(G,M) is closed.
To prove that each connected component of F(G,M) is totally
geodesic, choose any two points p, q of F(G,M) which are
sufficiently close so that they can be joined by a unique
minimizing geodesic a(s). For each element a E G, (ao a)(s)
is also a geodesic joining p and q. Thus a *a is just a.
Thus every point of this geodesic must be fixed by any element
a E G. Hence, each component of F(G,M) is totally geodesic.
(Q.E.D.)
Although totally geodesic submanifolds are the "simplest"
submanifolds of a Riemannian manifold and it is known for a
long time that totally geodesic submanifolds of Rm and Sm
are linear subspaces and great spheres, respectively. It is
somewhat surprising that totally geodesic submanifolds of rank
one symmetric spaces are not classified until 1963 by Wolf [1].
For totally geodesic submanifolds of other symmetric spaces,
see Chen and Nagano [2) in which the (M ,M_) - method was
introduced.
116 4. Submanifolds
§2. Fundamental Equations and Fundamental Theorems
Let M be an n-dimensional submanifold of an m-dimensional
Riemannian manifold M. Let R denote the curvature tensor
of M. Then, for any vector field X, Y, Z tangent to M,
we have
R(X,Y)Z - VXVYZ - VYVXZ --V [X,YIZ .
Thus, by Gauss' formula (1.1), we find
R(X,Y)Z = X(VYZ+ h(Y,Z)) - VY(VXZ+ h(X,Z) )
- (V[X.YjZ+h([X,YJ,Z))
R(X,Y)Z+h(X,VYZ) -h(Y,VXZ) -h([X,Y),Z)
+ VXh(Y,Z) - VYh(X,Z) ,
where R denotes the curvature tensor of the submanifold M.
By using Weingartan formula (1.2) we obtain
(2.1) R(X,Y)Z = R(X,Y)Z - h(Y,Z)X +--h(X,Z)Y
+ h(X,VYZ) -h(Y,VXZ) -h([X,Y),Z)
+ DXh(Y,Z) -DYh(x,z) .
Thus, for any vector field W tangent to M, we have
(2.2) R(X,Y;Z,W) = R(X,Y;Z,W)+ g(h(X,Z),h(Y,W))
- g(h(X,W),h(Y,Z)) ,
§ 2. Fundamental Equations and Fundamental Theorems 117
where R(X,Y;Z,W) = g(R(X,Y)Z,W). Equation (2.2) is called
the equation of Gauss. Moreover, from (2.1), we see that the
normal component of R(X,Y)Z is given by
(2.3)
where
(2.4)
(R(X,Y)Z)L = (-Oxh)(Y,Z) -(iyh)(X,Z) ,
(OXh)(Y,Z) = DXh(Y,Z) - h(vxY,Z) - h(Y.vxZ)
Equation (2.3) is called the equation of Codazzi.
If g and n be two normal vector fields of M, then
we have
R(X,Y; g,n) = g(ox Yg,T)) - g(iy Xg,T1) - g(V [X,Y] g,Ti)
_ -9(vx(AgY) ,I) + g(vXDyt,r1)
+ g(vY(AgX),n) -g(vYDXg,r1)
- g(D[X'Yjg,T1)
_ -g(h(X,AgY)n)+ g(h(Y,A9X),T>)
+ -g(DYDXS,n) -g(D[X,YJS.n)
Thus, if we denote by RD the curvature tensor of the normal
connection on the normal bundle TL(M), i.e.,
RD(X,Y)g = DXDYt-DYDXg-D[X,Y)S
then we have
118 4. Submanifolds
(2.5) RD(X,Y;S.,n) = R(X,Y;S,1)+ g([AS.A11 )(X),Y)
where
(2.6) (AA,) = ASA'n-A'n AS
Equation (2.5) is called the equation of Ricci.
If the ambient space Si is a space of constant (sectional)
curvature k, then equations (2.2), (2.3) and (2.5) of Gauss,
Codazzi and Ricci reduce to
(2.7) R(X,Y:Z,W) = k(g(X,W)g(Y.Z) - 9(X.Z)g(Y.W))
(2.8)
+ g(h(X,W),h(Y,Z))- g-(h(X,Z),h(Y,W)) ;
(vxh)(Y,Z) = (vYh)(X.Z)
(2.9) g([AS.A11 )(X),Y) ,
where k is a constant.
Sometime we denote (vxh)(Y,Z) by (vh)(X,Y.Z). It
is clear that vh is a normal-bundle-valued tensor field of
type (0.3). For k > 1. we define the k-th covariant derivative
of h with respect to T(M) ® T1(M) by
(2.10) (vkh)(Xl,X2.....Xk+2) = DX1((vk-lh)(X2.....Xk+2)
k+2 k-1- i=2 (v h)(X2,...,VX1Xi.....Xk+2)
where v0h = h. It is clear that vkh is a normal-bundle-
valued tensor field of type (O,k+2). By direct computation
§ 2. Fundamental Equations and Fundamental Theorems 119
we have
(2.11) (vkh)(Xl,X2,X3,...,Xk+2)-(vkh)(X2,Xl.X3,...,Xk+2)
= RD(Xl.X2)((vk-2h)(X3,...,Xk+2))
k+2+ E
(vk-2h)(X3.....R(Xl,X2)Xi.....Xk+2),
i=3
for k > 2.
In the following, we call an r-dimensional vector bundle
over a manifold M a Riemannian r-plane bundle if it is equipped
with a bundle metric and a compatible metric connection. If
E is any vector bundle over a Riemannian manifold M, a
second fundamental form in E is a cross-section A in
Aom(T 0 E,T) satisfying
(2.12) g(A(X,S),Y) = g(X,A(Y,S))
for any vector fields X, Y tangent to M and a section S in
E, where g is the metric of M and T = T(M). If E is
a Riemannian vector bundle with a second fundamental tensor A.
we define the associated second fundamental form h by
(2.13) g(h(X,Y),S) = g(A(X,S),Y) .
where g is the bundle metric tensor of E.
We now state the fundamental theorems of submanifolds
as follows.
120 4. Submanifolds
Existence Theorem. Let M be a simply connected
n-dimensional Riemannian manifold with a Riemannian r-plane
bundle E over M equipped with a second fundamental form h
and associated second fundamental tensor A. If they satisfy
the equation (2.7) of Gauss, the equation (2.8) of Codazzi
and equation (2.9) of Ricci, then M can be isometrically
immersed in a complete, simply-connected Riemannian manifold
of constant curvature k with normal bundle E.
Rigidity Theorem. Let i, i':M -.IP(k) be two isometric
immersions of an n-dimensional Riemannian manifold M into
a complete, simply-connected Riemannian manifold of constant
curvature k. Let E and E' be the associated normal bundles
equipped with their canonical bundle metrics, connections, and
second fundamental forms. Suppose that there is an isometry
f :M -. M such that f can be covered by a bundle map
f :E -. E' which preserves the bundle metrics, the connections,
and the second fundamental forms. Then there is an isometry F
of 14m(k) such that F o i = i' of.
The problem of isometric immersions is almost as old as
differential geometry itself, beginning with the theory of curves
and surfaces. The first general result is the Theorem of
Janet-Cartan which states that a real analytic n-dimensional
Riemannian manifold M can be locally isometrically imbedded
in any real analytic Riemannian manifold of dimension
2
n(n+l).
A global isometric imbedding theorem was obtained by Nash [1):
§ 2. Fundamental Equations and Fundamental 77teorems 121
Nash's Theorem. Every compact n-dimensional Riemannian
manifold of class Ck (3 S k < e) can be Ck-isometrically
imbedded in any small portion of a Euclidean N-space AN
where N =
2
n(3n+ 11). Every non-compact n-dimensional
Riemannian manifold of class Ck (3 < k < -) can be
Ck-isometrically imbedded in any small portion of a Euclidean
N-space RN , where N = 2 n(n+ 1)(3n+ 11).
In particular, Nash's theorem implies that every compact
2-dimensional Riemannian manifold can be isometrically imbedded
in Rl7
In views of Nash's Theorem, we mention the following
result.
Theorem 2.1. Let M be a compact n-dimensional Riemannian
manifold isometrically immersed inRn+r
. If, at every point
p of M, Tp(M) contains a k-dimensional subspace Tp such
that the sectional curvature for any 2-plane in Tp' is non-
positive, then r _, k.
Proof. Let x(p) denote the position vector of p in
Rn+r. We put f(p) = <x(p),x(p)>, where < , > denotes
the Euclidean inner product. Let p0
be a point of M such
that f takes a maximum at p0. For a vector X E Tp (M), we0
have Xf = 2<vXx,x> = 2<X,x> and this is zero at po. Thus
the vector x(p0) is normal to M at p0. Moreover, at po,
we have
122 4. Submanifolds
X2 f = 2<p C,xo>+ 2<X,X> = 2<h(X,X) ,xo>+ 2<X,X> .
Since f has a maximum at po, X2f < 0 at p0. Thus, we
obtain h(X.X) 1 0 for any non-zero vector X E T (M). Considerpo
the restriction of h to T' xT' . By assumption, the sectionalpo po
curvature of M is non-positive for any 2-plane X AY in
T' . Thus, by equation (2.7) of Gauss with k = 0, we have0
g(h(X,X),h(Y,Y)) < g(h(X,Y),h(X,Y)) ,
where X, Y are orthonormal vectors in T' . By linearity,0
this inequality holds for all X, Y in T' . Thus, thep0
theorem follows from the following Lemma of T. Otsuki [1).
Lemma 2.1. Let h : Rk x Rk _, Rr be a symmetric bilinear
map and g a positive-definite inner product in Rr . if
g(h(X,X),h(Y,Y)) < g(h(X,Y),h(X,Y))
for all X, Y in Rk and if h(X,X) ql 0 for all non-zero
X in Rk , then r 2 k.
Proof: We extend h to a symmetric complex bilinear
map of Ck xCk Cr. Consider the equation h(Z,Z) = 0. Since
h is Cr-valued, this equation is equivalent to a system of
r quadratic equations:
h1(Z,Z) = 0,...,hr(Z,Z) = 0 .
§ 2. Fundamental Equations and Fundamental Theorems 123
If r < k, then the above system of equations has a non-zero
solution Z. By assumption, Z is not in Rr. Thus
Z = X + Y, where X, Y in Rr and Y ¢ O. Since
0 = h(Z,Z) = h(X,X) -h(Y,Y)+ 2 h(X,Y) ,
we have h(X,X) = h(Y,Y) I 0 and h(X,Y) = O. This is a
contradiction.
From Theorem 2.1 we obtain immediately the following.
Theorem 2.2. Every compact n-dimensional Riemannian manifold
of non-positive sectional curvature cannot be isometrically
immersed into R2n-1
Remark 2.1. Lemma 2.1 was conjectured by Chern and
Kuiper (1]. They showed that it implied Theorem 2.1. The lemma
was then proved by Otsuki. The proof above is due to
T.A. Springer. (See, Kobayashi and Nomizu [2].)
124 4. Submanifolds
03. Submanifolds with Flat Normal Connection
Let M be an n-dimensional submanifold of an m-dimensional
Riemannian manifold M. If the normal connection D is flat,
we have
RD(X,Y) = DX Y - DYDX - D(X,Y) - 0
for any vector fields X, Y tangent to M.
Proposition 3.1. Let M be an n-dimensional submanifold
of an (n+ p)-dimensional Riemannian manifold M. Then the
normal connection D is flat if and only if there exist locally
p orthonormal parallel normal vector fields.
Proof. If there exist r orthonormal parallel normal vector
fields p, locally. Then we have Dgl = ... = Dgf = 0.
Hence RD(X,Y)gr = 0. Since RD is tensorial, this implies
RD a 0. Thus the normal connection D is flat.
Conversely, if the normal connection D is flat, we have
(3.2) DXDYgr - DY XSr -D (X,Y) 9r = 0 ,
for any p orthonormal normal vector fields gl,...,g P.
put
(3.3) DXgr = £ er()g5 , r = 1,...,Ps=1
We
where er are local 1-forms on M. For simplicity, we express
(3.3) in matrix form. In fact, let
§ 3. Submanifolds with Flat Normal Connection 125
(3.4) t9 = tS1,...,Sp) ,
Then (3.3) can be written as
(3.5) DS = ®S
The matrix 9 completely determines the connection D.
In terms of 0, (3.2) is given by
(3.6) dO=82
Moreover, since ?,1,....Sp are orthonormal, we also have
(3.7)
We need the following lemma.
Lemma 3.1. Let 8 = (8r) be a (p xp)-matrix of
1-forms defined in a neighborhood of 0 in Rn . If 8 satisfies
(3.6) and (3.7), then there exist a unique (p xp)-matrix A
of functions in a neighborhood of 0 such that
(3.8) A = -A-1(dA); AO = I; tA = A-' .
where I is the identity matrix.
Proof. (Uniqueness). Assume that A and B are two
solutions. Then 8 = -A-1dA = -B 1dB and AO = BO = I. Thus
126 4. Submanifolds
d(AB) _ (dA)B-1 -A(B-1(dB)B 1)
-A®B 1+ AB 1BOB- 1 = 0
Thus AB1 is constant. Hence, by AO = B0 = I, we obtain
A = B.
(Existence). We pass to (n+ p2)-dimensional space2
Rn+p with coordinates x1,...,xn,zr (r,s = 1,...,p) and
introduce the p2 1-forms which are coefficients of the matrix
A = dZ + Z® , Z = (zr)
Then we have
dA = dZA®+Zd® _ (A - ZC) A ®+Z®2
= AA0 .
Thus, by Frobenius' theorem, A is completely integrable and
hence, there is a matrix A of functions with AD = I such
that A = Z gives an integral manifold of the system A = 0.
From this we obtain dA = -A0. Now, because 0 is skew-symmetric,
if we put C = to-1, then
dC = -C(dtA)C = Ct®tAC = Ct0
Thus, by the uniqueness, we obtain C = A, i.e., A is
orthogonal. This proves the lemma.
§ 3. Submanifolds with Fiat Normal Connection 127
Applying the lemma to the normal connection D, we have
a matrix A defined locally on M such that dA = -A®. Let
A = (as). Then
(3.9) dar = -E atwt
Put gr = E argt. Then gl,...,gP are orthonormal and
(3.10) Dgr = E (dar + arwt)gs .
Substituting (3.9) into (3.10) we find that Dg{ = = Dgr = 0.
(Q.E.D.)
If the ambient space M is of constant curvature, then
we have the following result of Cartan [1).
Proposition 3.2. Let M be an n-dimensional submanifold
of a Riemannian manifold M of constant curvature. Then the
normal connection is flat if and only if all the second fundamental
tensors A are simultaneously diagonalizable.r
This proposition follows immediately from equation (2.9)
of Ricci.
128 4. Submanifolds
§4. Totally Umbilical Submanifolds
Let Rn be the Euclidean n-space with natural coordinates
xl,...,xn. Then the Euclidean metric on Rn is given by
go = (dxl)2 + ...+ (dxn)2
It is well-known that (Rf ,g0) is a complete, simply-
connected Riemannian manifold of curvature zero.
We put
(4.1) Rn(k) = ((x1,...,xn+ 1) E Rn+I
I.JT ((x1)2+ --- + (xn)2+ (sgn k)(xn+l)2 )
-2xn+1= 0, xn+1 2 0) ,
where sgn(k) = 1 or -1 according as k 2 0 or k < 0.
The Riemannian connection induced by
go = (dxl)2+ ...+ (dxn)2+sgn(k)(dxn+1)2
onRn+1
is the ordinary Euclidean connection for each value
of k. In each case the metric tensor induced on Rn(k) is
complete and of constant curvature k. Moreover, each
Rn(k) is simply-connected.
A Riemannian manifold of constant curvature is called
elliptic, hyperbolic or flat according as the sectional curvature
is positive, negative or zero. These spaces are real-space-forms.
Two complete, simply-connected real-space-forms of the same
constant sectional curvature are isometric.
§4. Totally Umbilical Submanifolds 129
The hyperspheres in Rn(k) are those hypersurfaces
given by quadratic equations of the form;
fxl-al)2+ + (xn-an)2+sgn(k)(xn+1 an+l)2 = constants ,
n+ 1where a = (al,...,an+1) is an arbitrary fixed point in IR
In Rn(0) , these are just the usual hyperspheres. Among
these hyperspheres the great hyperspheres are those sections
of hyperplanes which pass through the center
(0,...,O,sgn(k)/'qk{) of Rn(k) in R'1 , k y( 0. For
k = 0, we consider the point at infinite on the xn+
-axis
as the center in Rn+1. The intersection of a hyperplane
through the center in Rn+l is just a hyperplane in Rn (0).
Great hyperspheres in Rn(k) are totally geodesic hypersurfaces
of Rn(k). All other hyperspheres in Rn(k) are called
small hyperspheres. Small hyperspheres of Rn(0) are called
ordinary hyperspheres or simply hyperspheres if there is
no confusion.
Proposition 4.1. An n-dimensional totally umbilical
submanifold M in the real-space-form Rm(k) is either
totally geodesic in Rm(k) or contained in a small
hypersphere of an (n+ 1)-dimensional totally geodesic submanifold
of Rm(k) .
Proof. If M is a totally umbilical submanifold of
Rm(k), then the second fundamental form h satisfies
(4.2) h(X,Y) = g(X,Y)H
130 4. Submanifolds
for X, Y tangent to M. Substituting this into equation
(2.8) of Codazzi, we find
g(Y,Z)DXH = g(X,Z)DI
By choosing Y = Z 1 X, we obtain DXH = 0. Let a = JHJ be
the mean curvature and g a unit normal vector field such
that H = at. Then we have (Xa)g+ aDXg = 0. Since t and
DXg are orthogonal, we see that the mean curvature a is
constant. If a = 0, (4.2) implies that h = 0. Thus M
is totally geodesic. Assume that a ¢ 0. Then we may choose
m- n orthonormal normal vector fields!l,. . 'gym-n
locally
on M such that
(4.3)
From (4.1) we find
(4.4)
(4.5)
2 Am-n = 0
Dtl = 0 .
Using (4.4), (4.5) and Weingarten's formula, we get
(4.6) VX(g2 n ... A tm-n) = 0 ,
where v is the Riemannian connection of the ambient space
Rm(k) .
§ 4. Totally Umbilical Submanifolds 131
Case (i). k = 0. In this case, Rm(0) is the
Euclidean m-space Rm. Equation (4.6) shows that the normal
subspace spanned by g2,...,gm-n is parallel in Rm. Hence,
the linear subspaces of Rm spanned by the tangent space
Tp(M) and the mean curvature vector H is a fixed (n+ 1) -
dimensional linear subspace of Rm, say Rn+1.
Let
x = (xl,...,xn) be the position vector of Rm. Then, by (4.1)
and (4.5), we find
Y(x+ al) = v x-a-lAl(Y)+Dy(a lgl)
= Y-Y = 0 ,
for Y tangent to M. Thus x + a 1gl is a constant vector,
say c. This shows that M is contained in a hypersphere of
Rn+l with radius a-1 and center c.
Case (ii). k = 1 (resp., case (iii) k = -1). For
simplicity, we consider the position vector x relative to the
center
(0,...,0,1) (resp., (0,...,0,-1))
of Rm(1) (resp., Rm(-1)) in Rm+1 . For each point p
in Rm(1) (resp., Rm(-1)), r = x is a unit normal vector
to Rm(1) (resp., Rm(-1)) in Rm+l. It is easy to verify
that VWr1 = W for any vector W tangent to Rm(1) (resp.,
m(-1) where v' is the Riemannian connection on Rm+lR
Moreover, we have
132
(4.7)
4. Submanifolds
VUV = VUV -g (U,V),,
for any vector fields U, V tangent to Rm(1) (resp., ,m(-1)) .
In particular, we have
7x r = °X'r r = 1,...,m-n ,
for any X tangent to M. Thus, the submanifold M is also
totally umbilical inRm*1.
Hence, we may conclude that M is
contained in the intersection of an (n+ 1)-dimensional linear
subspace of 3k m+1 and Rm(1) (resp., 3Rm(-1)) . From this,
we see that M is contained in a small hypersphere of an
(n + 1)-dimensional totally geodesic submanifold. (Q.E.D.)
Remark 4.1. Totally umbilical submanifolds in complex-
space-forma and in quaternion-space-forms are classified in
Chen-Ogiue (2] and Chen (14], respectively. For a systematic
study of totally umbilical submanifolds in locally symmetric
spaces or in Kaehler manifolds, see Chen (17, Chapter VII].
Let M be a submanifold of a Riemannian manifold M.
If the second fundamental form h and the mean curvature vector
H of M in M satisfy
(4.7) g(h(X,Y),H) = fg(X,Y)
for some function f on M, then M is called pseudo-
umbilical. As a generalization of Proposition 4.1 we have
the following (Yano and Chen (1)).
§4. Totally Umbilical Submanifolds 133
Proposition 4.2. Let M be a pseudo-umbilical submanifold
of the real-space-form Rm(k) . If M has parallel mean
curvature vector, then either M is a minimal submanifold of Rm(k)=
F1, or M is a minimal submanifold of a small hypersphere of
Rm(k) .
Proof. Let M be a pseudo-umbilical submanifold of
Rm(k) with parallel mean curvature. Then the mean curvature
a = (HJ is constant. If a = 0, M is minimal in M. Assume
that a is non-zero. Then the unit vector in the direction
of H is parallel,i.e., DP = 0. If Rm(k) = Rm , we
consider the vector field
(4.8) y(p) = x(p) + Sap
where x is the position vector of M in Rm. Let X be
any tangent vector on M. We have
Xy = v x+1a XS = X--a AgX
Since M is pseudo-umbilical, we find AS = al. Thus y is
constant. This shows that M lies in the hypersphere S of
Rm centered at y = c and with radius a-1. Now, because
the mean curvature vector H of M in Rm is parallel to
and g is parallel to the radius vector x _c, we find that
H is always perpendicular to S. Thus, M is minimal in
the hypersphere S.
134 4. Submanifolds
If k ¢ 0. we just regard Rm(k) as the hypersurface
ofRm+1
defined by (4.1). Then a similar argument yields
the result. (Q.E.D.)
§ 5. Minirnal Submanifolds 135
§5. Minimal Submanifolds
Let x : M - Rm be an isometric immersion of an
n-dimensional Riemannian manifold M into Rm. Let ell...len
be an orthonormal local frame on M such that ve ei = 0 at
a fixed point p in M. Let x denote the position vector
of M in Rm . Then we f ind
n(Ax) (e) (ex) (v e)
P i=1 1 P 1 ei 1 Pn
h(ei,ei)p = -nHpi=1
Hence, we have the following well-known results.
Lemma 5.1. Let x : M -+ Rm be an isometric immersion.
Then
(5.1) Ax = -nH .
Corollary 5.1. x : M -4 Rm is a minimal immersion if
and only if each coordinate function xA of x = (x1,...,xm)
is harmonic.
This corollary follows immediately from Lemma 5.1.
Since every harmonic function on a compact Riemannian manifold
is constant (Corollary 2.1.2), Corollary 5.1 implies
Corollary 5.2. There are no compact minimal submanifolds
of Rm .
136 4. Subnwnifolds
Proposition 5.1. (Takahashi [1)). Let x :M + Rm
be an isometric immersion. If Ax = Xx, X ¢ O, then
(1) ).>0,
(2) x(M) c So-1(r), where So 1(r) is a hypersphere
of Rm centered at the origin 0 and with radius r
(3) x :M 4 So-1(r) is minimal.
Furthermore, if x :M + So-1(r) is minimal, then
Ax = (n/r2)x.
Proof. If Ax = Xx, . 31 0, then by Lemma 5.1 we have
H= -()./n)x. Let X be a vector field tangent to M, we have
(5.2) <x,X> = 0 .
Thus X<x,x> = 2<x,vxx> = 2<x,X> = O. Therefore <x,x> is
constant on M. This proves that IxI is constant. Thus, M
is immersed into a hypersphere So 1(r) of Rm centered at
the origin. Let h, h' and Ti be the second fundamental
forms of M in Fm , M in So 1(r), and So-1(r) in Rm
respectively. Then we have h(X,Y) = h'(X,Y)+I(X,Y). Thus,
the mean curvature vectors H, H' of M in Rm and So1(r)n
satisfies H = H'+ H, where H n E 1i(e.,e.), andi=1
e1,....en an orthonormal frame of M. Since x(p) is
perpendicular to So-1(r) at p and Hp is parallel to x(p),
this implies that H' = O. Thus M is minimal in So-1(r).
Because X is an eigenvalue of a on M, X > O. Now,
§ S. Minimal Submanifolds
n
nHp = E <' ei'x-rPZ>(xZ) = - 2 E <ei.ei>x(p)ri=1 e i r
137
Thus ), = 2 . This proves (1) and (2). The last statementr
is clear. (Q.E.D.)
Proposition 5.1 shows that minimal submanifolds of
spheres are given by proper functions associated with a nonzero
eigenvalue a of A. For a compact symmetry space M and
a nonzero eigenvalue a of a on M, we may indeed construct
such a minimal immersion as follows:
Let M = G/K be a compact symmetric space where G is
a compact connected subgroup of i(M) and K a closed subgroup
of G. Assume that M is orientable and the isotropy action of
K is irreducible. Let < , > be a G-invariant Riemannian
metric on M (such < > is unique up to scalar multiple
and thus naturally reductive). Let a be the Laplacian of
(M. < > ). For each ) we denote by m.& the multiplicity
of X. Let be an orthonormal basis of the
eigen-space V), (with respect to ( )). We define a map
(5.3)
by
(5.4)
xX : M 4 RR\
x(p) = 2 ($1(p).....4 (p))
mx
138 4. Submanifolds
Then xIL defines an isometric immersion of (M,c< , >) into
1
S0 (1) for some c > 0. (Indeed E d4i is an invariant
nonzero bilinear form on T(M); thus X d4i 0 d4i = c< , >
for some c > 0). Now, applying Takahashi's result, we conclude
that xIL is a minimal immersion and c = n. We summarize
these as the following well-known result. (Takahashi (1),
Wallach [1)).
Theorem 5.1. Let M = G/K be an irreducible compact
symmetric space equipped with a G-invariant Riemannian metric
< >. Then for any nonzero eigenvalue A of & on
(M,< there is an isometric minimal immersion of M1 = r
into a hypersphere S' (r) of R where r /
If ). i is the i-th nonzero eigenvalue of A. then
x4l of M = G/K is sometime called the i-th standard immersion
of M.
Example 5.1. Let S2(r) = ((x,y,z) E R3 Ix2+y2+z2 = r2}.
Then, according to Proposition 2.5.3. we know that the
eigen-space Vk (associated with the k-th nonzero eigen-value
kk of p) is given by 11k, the space of harmonic homogeneous
polynomials of degree k on R3 restricted to S2(k).
From this, we see that the standard immersion of S2(1) in
R 3 is the first standard imbedding of S2(1).
We consider the following homogeneous polynomials of
degree 2;
§5. Minimal Submanifolds
(5.5)
u1=yz, u2=xz, u3=xy,
U4 = z(x2-y2) , u5 = 6 (x2+y2-2z2)
It can be verified that u1,...,u5 are harmonic on R3 and
their restrictions to S2(1) form an orthonormal basis of
V2 = V2. Thus, the map x2 of S2(1) into 3R5 defined by
(5.5) gives a minimal isometric immersion of S2(1) into
S4( ). It is the second standard immersion of S2(1) and3
it also gives the first standard imbedding of R P2 into R5
Similarly, the following homogeneous polynomials of
degree 3;
ul = 12
z(-3x2 - 3y2+ 2z2)
(5.6)
u = 15 z(x2-Y 23 12
5 = 24 Y(-x2 - y2 + 4z2 )
u7 = 24 y(3x2-Y2)
u2 = 24 x(-x2 -y2+4z2) ,
u4 = 24 x(x2 - 3y2)
u6 = 116 xyz ,
139
are harmonic and their restriction to S2(1) form an orthonormal
basis of A(3. The map x3 of S2(1) into S6(1) c R7 is
a minimal isometric imbedding. It is the third standard imbedding
of S2(1).
The k-th standard immersion xk of a rank one symmetric
space M is an imbedding if M is different from a sphere
140 4. Submanifolds
or k is odd. In the case of the k-th standard immersion
of Sn with even k, the immersion is a two-sheet covering
map of R Pn.
§ 6. The First Standard Imbeddings of Projective Spaces 141
06. The First Standard Imbeddings of Projective Spaces
In this section we will construct the first standard
imbedding of a compact symmetric space of rank one. Such
imbedding had been considered in various places. (cf. Tai [11,
Little [2), Sakamoto (1), Ros (1), Chen [24)).
Throughout this section, F will denote the field R
of real numbers, the field C of complex numbers or the field
Q of quaternions. In a natural way, R c C c Q. For each
element z of F , we define the conjugate of z as follows:
If
z = z0+ z1i+ z2j+ z3k E Q ,
with z0,z1,z2,z3 E R , then
z = z0-z1i-z3j-z3k
If z is in C, z coincides with the ordinary complex conjugate
of z. If z is in R , z = z.
It is convenient to define
1 if F = R ,d = d(F) = 2 if F = C
4 if F = Q
For a matrix A over F , denote by At and A the
transpose of A and the conjugate of A, respectively.
142 4. Submanifolds
Let z = (zi) E Fm+1 be a column vector. A matrix
A = (aij), 0 < i, j S m; operates on z by the rule:
/a00. . aOm z0
(6.1) Az =
\aMO
We will use the following notations:
M(m+ l;F) = the space of all (m+ 1) x (m+ 1)
matrices over F ,
H(m+ 1;F) = (A E M(m+ 1;F) A* = A) _
the space of all (m+ 1) x (m+ 1)
Hermitian matrices over F ,
U(m+ 1;F) = (A E M(m+ 1;F) I A*A = I)
where A* = A and I is the identity matrix. If
A E H(m+ 1;3R) then A is a symmetric matrix. Moreover,
U(m+ 1;R) = 0(m+ 1), U(m+ 1;(C) = U(m+ 1), and U(m+ 1;Q) _
Sp(m+ 1).
Fm+l can be considered as an (m +1)d-dimensional vector
space over R with the usual Euclidean inner product:
(6.2) <z,w> = Re(z*w) .
And M(m+ 1;F) can be considered as an (m+ 1)2d-dimensional
Euclidean space with the inner product given by
§ 6. The First Standard Imbeddings of Projective Spaces 143
(6.3) <A,B> =
2
Re tr(AB*)
If A, B belong to H(m+ 1;F) , we have
(6.4) <A,B> =
2
tr(AB)
Let F Pm denote the projective space over F. F Pm is
considered as the quotient space of the unit hypersphere
S(mtl)d-1 = (z EFm+l
1z*z = 1] obtained by identifying
z with zX, where z is a column vector and ). E F such
that lx = 1. The canonical metric go on F Pm is the
invariant metric such that the fibering n-S(mFl)d-1
. F Pm
is a Riemannian submersion. Thus, the sectional curvature of
R Pm is 1, the holomorphic sectional curvature of QPm is 4,
and the quaternion sectional curvature of QPm is 4.
Using (6.1), we have an action of U(m + 1;F) acting on
S(m+l)d-1 Such an action induces an action of U(m+ 1;F) on
F Pm. Denote by 0 the point in F Pm with the homogeneous
coordinates (zi) with zp = 1, zl = .. = zm = 0. Then
the isotropy subgroup at 0 is U (1: F) x U (m; F) . Thus we
have the following well-known isometry:
(6.5) µ : F Pm y U(m+ 1;F)/t)(1;F) xU(m;F)
The metric on the right is U(m+ 1;F)-invariant.
Define a mappingm
: S (m+l)d-1 -s H(m+ 1;F) as follows
144 4. Submanifolds
2IZ0I Z0Z1 . . . Z0Zm
m(Z) = zz* . . . . . . . . . . . . . .
2zmz0
zm21 . . . Izml
for z = (zi) E S(m+l)d-1 Then it is easy to verify that
induces a mapping of F Pm into H(m+ 1;F) :
*(6.7) m(7r(z)) = cp(z) = zz
We simply denote cp('rr(z)) by V(z) if there is no confusion.Define a hyperplane H1(m+ 1;F) by H1(m+ 1;F) = (A E H(m+ 1;F) I
tr A = 1). Then we have dim Hl(m+1;F) = m+m(m+1)d/2.
From (6.6), we can prove that the image of F Pm under cp is
given by
(6.8) cp (F m) = (A E H(m+ 1;F) I A2 = A and tr A = 1)
Let U(m+ 1; F) act on M(m+ 1;]F) by
(6.9) P(A) = PAP 1
for P E U(m+ 1;F) and A E M(m+ 1;F) . Then we have
(6.10) <P(A).P(B)> = <A,B> .
Hence, the action of U(m+ 1;F) preserves the inner product
of M(m+ 1;F) . Moreover, we also have
(6.11) cp(Pz) = P(cp (z) ) E cp(F Pm)
§6. The First Standard Imbeddings of Projective Spaces 145
for z E F Pm and P E U (m + 1;F ). Thus, we have the following.
Lemma 1. (Tai [1)) The imbedding cp of F Pm into
H(m + 1;F) given by (6.7) is equivariant with respect to and
invariant under the action of U(m+ 1;F) .
Now, we want to show that the imbedding tp is the first
standard imbedding of F Pm. Let A be a point in cp(F Pm).
Consider a curve A(t) in M with A(O) = A and
A'(O) = X E TA(F Pm). From A2(t) = A(t), we find XA + AX = X.
Because the dimension of the space of all X in H(m+ 1;F)
such that XA + AX = X is md, we obtain
(6.12) TA(F Pm) = (X E H(m+ 1;F) I XA+AX = X) .
There is another expression of TA(F Pm) given as
follows:
M+lFor u, v EIF
, we define a(u,v) = u v. Let z be
a point in S(m+l)d-1 and v a vector in Tz(S(m'l)d-1). We
identify v and its image in T7 r (
under 7r*. Let
a(t) be a curve in S(m+l)d-1 with a(O) = z and a'(O) = v.
Then A(t) = a(t)a(t)* is a curve in tp(F Pm) through A = zz*.
From this we find
* *M*(v) = vz +zv .
Therefore, we have
* *(6.13) TA(F Pm) = (vz + zv I v E Fm} and a(z,v) = 0) ,
146 4. Submonifolds
where A = zzz E S(mfl)d-1
A vector g in H(m+ I; F ) is normal to F Pm at A
if and only if <X,g> = 0 for all X in TA(F Pm). Thus,
is in TA(F Pm) if and only if tr(Xg) = 0 for all x in
TA(F Pm). Therefore, by (6.12), we obtain
(6.14) TA (F Pm) = (g E H(m+ 1;F) 1 Ag = gA)
For each A in tp(F Pm) we have
<A- 1 I, A-m+l I> =2
tr(A-m1 1)2
Therefore, F Pm is imbedded in a hypersphere S(r) of
H(m +1;F ) centered at mIland with radius r = (m/2(m +
l))1/2.
Let X be a vector in TA(F Pm) and Y a vector field
tangent to F Pm. Consider a curve A(t) in cp(F Pm) so
that A(O) = A and A'(O) = X. Denote by Y(t) the restriction
of Y to A(t). Because Y(t) E TA(t)(F Pm), (6.12)
A(t)Y(t)+ Y(t)A(t) = Y(t) ,
from which we find
(6.16) vXY = Y'(O) = A(VXY)+ (pXY)A+XY+YX ,
where v denotes the Riemannian connection of the Euclidean space
H(m+1; F) . Using (6.12) we have
§6. The First Standard Imbeddings of Projective Spaces 147
(6.17) AXY = XYA
Thus we find
A (XY + YX) (I - 2A) = -XYA-YXA = (XY + YX) (I - 2A) A .
Hence, by (6.14), we obtain
(6.18) (XY+YX)(I-2A) E TA(FPm)
On the other hand, by multiplying A to (6.15) from the
right, we get
(6.19) (XY+YX)A+A(vxY)A = 0 .
Therefore, from (6.8), (6.17) and (6.18), we obtain
(6.20) 2(XY+YX)A+A(VXY) + (VXY)A E TA(F Pm)
Combining (6.16), (6.18) and (6.20), we find
(6.21) h(X,Y) = (XY+YX)(I-2A)
(6.22) VXY = 2(XY+YX)A+A(VXY) + (-vXY)A
where S is the second fundamental form of F Pm in Hfm+ 1;3r)
at A and v the induced connection on F Pm. From (6.21) we
find that the mean curvature vector ft of F Pm in
H(m+ 1;F) at A is given by
148 4. Submanifolds
(6.23) H = m (I - (m+ 1)A)
which is parallel to the radius vector A - m11 I. Thus,
F PM is imbedded in the hypersphere S( ) as a minimal
submanifold. Using the result of Takahashi (Proposition 5.1),
cp is a standard imbedding of F Pm associated with an
eigenvalue X of A. From Theorem 5.1 we obtain ). = 2(m+ 1)d.
Since 2(m+ 1)d is exactly the first non-zero eigenvalue
X1 of p on F Pm, we conclude that cp is the first standard
imbedding. We summarize these results as the following
well-known theorem.
Theorem 6.1. The isometric imbedding cp :F PM . H(m+ 1;F)
defined by (6.7) is the first standard imbedding of Pm into
H(m + 1;F) . Moreover, the second fundamental form h and the
mean curvature vector i of F Pm in H(m+ 1;F ) are given
by (6.21) and (6.23), respectively. And F Pm lies in a
hypersphere S(r) of H(m+ 1;F) centered at (Il)I and
with radius r = [2(m +l))1/2
Let A -zzw
be a point in cp(F Pm). For each vector
X in TA(F Pm), there is a vector v in ]Fm+1 such that
a(z,v) = 0 and X = vz*+ zv*. If F = a, we put
(6.24) ix = viz* - ziv* .
Then J defines the complex structure of cp(CPm). Similarly,
we may define the quaternionic structure (J1,J2,J3) on
ep(QPm) in a similar way.
§ 6. The First Standard Imbeddings of Projective Spaces 149
Let X = uz* + zu* and Y = vz* + zv* be two vectors
in TA(CPm), where A = zz*, a(v,z) = a(u,z) = 0, and
2(m}1)-1z E
SThen we have
ix = uiz* - ziu* , JY = viz* - ziv*
Thus, we find
(6.25) (JX)(JY) =uv*+ zu*vz*
= XY
Consequently, by using (6.21), we obtain
(6.26) h(JX,JY) = h(X,Y) for X,Y E TA(CPm) .
A similar formula holds for QPm in H(m+ 1;Q).
In the remaining part of this section, we shall study the
second fundamental form fi of F Pm in H(m + 1;3F) in more
details.
Let z0 = (110,...10)t and AO = zOzO. Then (6.13)
implies
O b*
(6.27) TA (F Pm) = X = b E Fm0 b 0
Using (6.4), we see that a vector X E TA (F Pm) is a unit0
vector if and only if x takes the following form:
150 4. Submanifolds
(6.28) X =
O b*
b b = 1
b O
Therefore, by using (6.4), (6.21), and (6.28), we obtain
f-1 0(6.29) fi(X,X) = 2
0 bb*
Therefore, we get
(6.30) llh(X,X)JI = 2 ,
for unit vectors X in TA (F Pm). Since F Pm is imbedded0
equivariantly in H(m+ 1;3F), we obtain
(6.31) flh(X,X)jj = 2, for unit vectors X E TA(F Pm) ,
where A E m(F Pm) .
We need the following.
Lemma 6.2. Let F Pm be imbedded by 4V into H (m + 1; F)
Then the sectional curvature K of F Pm satisfies
(6.32) <h(X,X),h(Y,Y)> = 3(4 + 2f((X,Y))
for orthonormal vectors X, Y tangent to F Pm in H i m + 1;F )
Proof. Since X, Y are orthonormal, (6.31) implies
§6. The First Standard Imbeddings of Projective Spaces 151
(6.33) 32 = <fi(X+Y,X+Y),fi(X+Y,X+Y)>
+ <h(X-Y,X-Y),1i(X-Y,X-Y)>
= 16+ 4<h(X,X),fi(Y,Y)> + 8<h(X,Y),h(X,Y)>
On the other hand, from the equation of Gauss, we find
(6.34) K(X,Y) _ <h(X,X),h(Y,Y)> - <h(X,Y),h(X,Y)>
Combining (6.33) and (6.34), we obtain (6.32). (Q.E.D.)
By using (6.31), (6.32) and Lemma 6.2, we obtain the
following.
Lemma 6.3. Let M be an n-dimensional submanifold of
F Pm which is imbedded by cp into H (m + 1; F) . Then the
mean curvature vectors H and H' of M in H(m+ 1;F) and
in F Pm satisfy
(6.35) IHI2- (H'I2 = 4(n+2) + 2 X IC(ei,e.)3n 3n2 i i 7
where e1,...,en form an orthonormal basis of T(M).
Let X and Y be two orthonormal vectors in T(F Pm).
Then X A Y is called totally real if X 1 JY when F = C
and if X i JaY, a = 1,2,3, when F = Q. A submanifold M
of F Pm is called totally real if every plane section in T(M)
is totally real. A submanifold M in CPm is called a
complex submanifold if J(T(M)) = T(M) and a submanifold M
152 4. Submanifolds
of QPm is called a quaternionic submanifold if JJ(T(M)) - T(M)
for a - 1,2,3. Complex submanifolds and quaternionic submanifolds
are also called invariant submanifolds. A submanifold M of
R Pm is regarded as a totally real submanifold and as an invariant
submanifold of R Pm in a trivial way.
From Lemma 6.3, we have the following.
Lemma 6.4. Let M be an n-dimensional submanifold of
F Pm which is imbedded by cp into H(m+ 1; F) . Then we have
(6.36) IHI2 2IH'I2+2 nl
equality holding if and only if M is totally real in F Pm.
Proof. It is known that the sectional curvature K(X,Y)
of F Pm is 2 1, equality holding if and only if X A Y is
totally real (cf. Chen and Ogiue (1).) Thus, by Lemma 6.3, we
obtain Lemma 6.4. (Q.E.D.)
Lemma 6.5. Let M be an n-dimensional (n > d) minimal
submanifold of F Pm which is imbedded into H(m + 1;F)
q. Then we have
(6.37) IHI2 <2(n+d)
n
equality holding if and only if n s 0 (mod d) and M is an
minvariant submanifold of F P.
§ 6. The First Standard Imbeddings of Projective Spaces 153
Proof. If F = R , this lemma follows immediately
from the fact that K(ei,ej) = 1 for any i, j = 1,...,n.
If F = C, then, from formula (2.6.3), we have
(6.38) K(ei,ej) = 1+ 3<ei,Jej>2
Thus, by Lemma 6.3, we find
n2 = 2 n+l) + 2
F, <eiJej>2(6.39) IHI
nn2 i,j=l
Denote by P the endomorphism of T(M) defined by
<PX,Y> = <JX,Y> for X, Y E T(M). Then, by (6.39), we find
(6.40)IH12=2nn1
+ 2IIP112n
Since P is nothing but the tangential component of JITM,
have IIPII2 S n, with the equality holding if and only if
n is even and T(M) is invariant under J, i.e., M is
a complex submanifold of QPm Thus, we find that
1H12 2(n+2), with equality holding if and only if M isn
a complex submanifold of CPm
If F = Q, the curvature tensor R takes the form
given by (2.6.5). Thus, for any orthonormal vectors X, Y
tangent to M, we have
(6.41)3
K(X,Y) = 1+ 3 E <X,JrY>2r=l
we
Thus, by combining (6.34) and (6.41), we find
154 4. Submanifolds
(6.42) IH:1
= 2(n+1) +
2
-L3
F,
nE <ei,J
r j>e.2
n r=1 i,]=1
Define the endomorphism Pr of Tp(M) by <PrX,Y> = <JrX,Y>
for X, Y E T(M), we have
(6.43)IH12
=2 nl + 2 (11P1112+ 11P2112+ 11P3112)
n
Since IIPr112 < n, (6.43) implies IH12 < 2(n+4)/n, equality
holding if and only if n is a multiple of 4 and M is a
quaternionic submanifold of QPm. (Q.E.D. )
Let A E cp(CPm) and X,Y,Z E TA(CPm). Then the second
fundamental form h of CPm in H(m + 1;C) satisfies
h(JX,JY) = h(X,Y). Thus, we find
(vXfi) (JY,JZ) = DXh(Y,Z) - Fi(vxY,Z) - Fi(Y,vxZ)
(vXFi) (Y,Z)
from which we find (v,I)(Y,JY) = 0. Applying Codazzi equation,
we obtain (vYh)(X,JY) = 0. In particular, this implies
(V c)(Y,Y) = (vlri)(JY,JY) = 0. Since the ambient space
H(m+ 1;C) is Euclidean, this implies that vh = 0, that
is, h is parallel.
Remark 6.1. It was proved in Little (2) and Sakamoto [1]
that the second fundamental form Fi of each F Pm in
H(m+ 1;F) , F = R , C or Q, under cp is parallel, that is,
(6.44) vh = 0 .
§6. The Fast Standard Imbeddings of Projective Spaces 155
Remark 6.2. One may obtain the first standard imbedding
of the Cayley plane OP2 in a similar way as follows: Let
Cay denote the Cayley algebra over I R. Let (e0 = l,e1,...,e7)
be the usual basis for Cay. For a z = z0 +Z1e
1+ . + z7 e7
in Cay, the conjugate of z is defined by
z = z0-zle1-... -z7e7 .
The usual inner product in Cay = R8 is given by
<x,y> = Re(xy) = E xiyi for x xiei, Y = E Yiei
and the norm of x is defined as jxI = <x,x>1"2. Let
H(3;Cay) be the space of 3 x3 Hermitian matrices over Cay.
Then H(3;Cay) is a Jordan algebra under the multiplication:
A *B = Z(AB+ BA), for A,B E H(3;Cay)
Define an inner product in H(3;Cay) = R 27 by
<A,B> =2
tr(AB) for A,B E H(3;Cay)
Let H1(3;Cay) = (A E H(3;Cay) Itr A = 1). Then the Cayley
projective plane OP2 is defined as
OP2 = (A E H1(3;Cay) IA2
= A) .
The OP2 with its induced metric becomes a compact rank-one
symmetric space with maximal sectional curvature 4. Moreover,
it is known that OP2 is a minimal 16-dimensional submanifold
156 4. Submanifolds
of a hypersphere S23(--) with radius-L
in H1(3;Cay).
(see, Tai (1), Little [2), and Sakamoto [11). From this, we
see that the mean curvature IHl of OP2 in H1(3;Cay) satisfies
(6.45) (Hl = .
Moreover, if M is a minimal submanifold of OP2. then the
mean curvature vector H of M in H1(3;Cay) satisfies
(6.46) JHI S 2 ,
equality holding if and only if n 8, and each tangent space
TA(M) at A E M is a subspace of a Cayley 8-plane of
TA(OP2).
Furthermore, one may prove that the first nonzero eigenvalue
k1of Laplacian and the volume of OP
2are given respectively by
(6.47) 11 = 48 ,
(6.48)
where w is given by
7vol(OP2)
=7T, w
(6.49) w =r fr sin8(y -X) I sin 2(y-x) I7 dy dx
20 x
§7. Total Absolute Curvature of Chern and Lashof
{7. Total Absolute Curvature of Chern and Lashof
157
Let C be a closed oriented curve in the 2-plane 3R2
As a point moves along C, the line through a fixed point
0 and parallel to the tangent line of C rotates through
an angle 2nrr or rotate n times about 0. This integer
n is called the rotation index of C. If C is a simple curve,
n = *1.
Two closed curves are called regularly homotopic if one
can be deformed to the other through a family of closed smooth
curves. Because the rotation index is an integer and it
varies continuously through the deformation, it must keep
constant. Therefore, two closed smooth curves have the same
rotation index if they are regularly homotopic. A theorem of
Graustein and Whitney says that the converse of this is also
true. Thus, the only invariant of a regular homotopy class
is the rotation index.
Let (x(s),y(s)) be the Euclidean coordinates of the
closed curve C in R2 which is parameterized by its arc
length s. Then we have
(7.1) x"(s) = -x(s)Y'(s) ,Y"(s)
- x(s)x'(s)
where x(s) denotes the curvature of C. Let 8(s) be the
angle between the tangent line and the x-axis. We have
x1# x m !
de =xxy2+ y
ds = x ds2
Y
158 4. Submanifolds
From this we obtain the following formula:
(7.2)J
x ds = 2nrr, n = the rotation index .
C
Using (7.2) we may conclude that the total absolute curvature,
f InIds, of C satisfies
(7.3) Ix Ids > 2nC
The equality holds if and only if C is a convex plane curve.
This result was generalized by W. Fenchel [1] in 1929 to
closed curves in R3 and by K. Borsuk [1] in 1947 to closed
curves in Rm, m > 3. In 1949 - 1950, Fary [1] and Milnor [1]
obtained the following improvement to knotted curves.
Theorem 7.1. If C is a knotted closed curve in Rm ,
then
(7.4)S
Ix Ids > 4rr .C
The Fenchel-Borsuk result was extended by S.S. Chern
and R.K. Lashof (1] in 1957 to arbitrary compact submanifolds
in Rm which we will discuss as follows.
Let x : M - Rm be an isometric immersion of an
n-dimensional closed manifold M into a Euclidean m-space.
The normal bundle T1(M) of M in Rm is an (m - n) -
dimensional vector bundle over M whose bundle space is the
subspace of M X Rm, consisting of all points (p,g) so
§ 7. Total Absolute Curvature of Chern and Lashof 159
that p E M and g is a normal vector of M at p. With
respect to the induced metric from Rm the normal bundle
rI(M) is a Riemannian (m-n) - plane bundle over M. Let B1
denote the subbundle of the normal bundle whose bundle space
consists of all points (p,g) in T'(M) such that p E M
and C is a unit normal vector at p. Then B1 is a bundle
of (m-n-1) -spheres over M and is a Riemannian manifold of
dimension m -1 endowed with the induced metric. Let dV denote
the (Riemannian) volume element of M. Then there is a differential
form do of degree m -n- 1 on B1 such that its restriction
to a fiber Sp is the volume element of the sphere SP of
unit normal vectors at p. Then dV A do is the volume element
of B1. We denote it by dVB . In fact, this can be seen as1
follows:
Suppose that Rm is oriented. By a frame p, el,...,em
in Rm we mean a point p in Rm and an ordered set of
orthonormal vectors e1,....em whose orientation consistent
with that of Rm. Denote by F(Rm) the space of all frames
in Rm M. Then F(Rm) is a manifold of dimension2
m(m+ 1).
Moreover, F(Rm) is a fibre bundle over Rm with the
structure group SO(m). In what follows it is convenient to
agree to the following range of indices:
1 S i,j,k S n; n+ 1 S r,s,t S m; 1 S A,B,C S m.
Let wA denote the dual 1-forms of eA. Let W_A be the
connection 1-forms defined by
160 4. Submanifolds
(7.5) V eB = E WAB eA
Then -A, -A satisfy the following structural equations of
Cartan:
(7.6) dwA= - E wB n wB
( 7 .7 ) 4A = - E wA A wB , WA + W-BB B C B A= 0
Throughout this section, we shall consider WA, WB as
forms defined on F(Rm) in a natural way.
Let x : M .4 Rm be an isometric immersion of an
n-dimensional Riemannian manifold into Rm. We identify a
tangent vector t with its image under x,,. Let B denote
the bundle whose bundle space is the set of M xF( Rm)
consisting of (p,x(p),el;...,en'en+l" ..,em) such that el,...,en
are tangent to M and en+l,...,em are normal to M. The
projection B a M is denoted by i. We define the map
B , by
(7.8) $l(P,x(P),el,...,em) _ (P,em)
Consider the maps
(7.9) B l-- M x F(Rm) -2-> F(Rm)
where i is the inclusion and k is the projection onto the
second factor. Put
§7. Total Absolute Curvature of Chern and Lashof 161
(7.10) WA =(),i)*wA
WAB = (li)*w8
*Since d and n commute with (Xi) , (7.6) and (7.7) imply
(7.11) dwA = - E W n W
From the definition of B it follows that wr = 0 and
wl...,Wn are linearly independent. If we restrict these
n 1-forms to M, then the volume element dV of M is given by
(7.13) 1 ndV = w n ... nw
Moreover, the volume element of B1 is
(7.14) dVndo = wln ...nu,nnwn+ln... nwm-1
do being equal to the product w n+1 n n wm-1'the (m - n - 1) -form on B1 which we are looking for.
Since wr = 0, (7.11) gives
o=dwr= wf Awi
This is
Hence by Cartan's Lemma (Proposition 1.3.2), we may write
(7.15) wi = E hij W
M in Rm , we have
162 4. Submanifolds
(7.16) hid = <h(ei,ej),er> ,
where < , > denotes the Euclidean metric of Rm .
Consider the map
v : B1 -+ Sm-1
of BI into the unit sphere Sm-1 of Rm defined by
v(p,e) = e. Denote by dj the volume element of Sm-l. Since
e = em is the position vector ofSm-1
in Rm, (7.5) implies
(7.17) = win ... A wm_1
Therefore, by (7.15) and (7.17) we find
v c E = G(p,em) W 1 A ... A wn A Wn+ A... A W m_1
G(p,em) = det(h'.)
is called the Lipschitz-Killing curvature at (p,em).
The total absolute curvature TA(x) of the immersion
x :M -o Rm, in the sense of Chern and Lashof, is then defined by
(7.20) TA(X) - 1 f Iv*dZI = 1 f G*(p)dVcm-1 B1 cm-1 M
where cm_1 is the volume of unit (m-1) -sphere and
(7.21) G*(p) = J IG(p,em)Ida
p
§7. Total Absolute Curvature of Chern and Lashof 163
The famous Chern-Lashof inequality is given by the
following.
Theorem 7 .1 (Chern-Lashof (1, 21) . Let x : M - Mm be an
immersion of an n-dimensional compact manifold M into 1M
R
Then the total absolute curvature of x satisfies the following
inequality:
(7.22) TA(x) 2 b(M) .
Proof. For each unit vector a in Sm-1 we define the
height function ha in the direction a by
ha(p) = <a,x(p)>, p E M .
If g is a unit normal vector at p, i.e., (p,g) E Bl, then
dhS(p) = <t,dx(P)> = 0 .
Hence, p is a critical point of the height function hg.
Conversely, if p is a critical point of the height function
ha, then
dha(P) = <a.dx(p)> = 0 .
Thus, a is a unit vector normal to M at p, i.e.,
(p,a) E B1. Consequently, we see that the number of all critical
points of ha which is denoted by $(ha) is equal to the
number of points in M with a as its normal vector. Hence,
we obtain
164 4. Submanifolds
JBIv* I
=SaESr-1
13 (ha)c .
1
Since for each a in Sm-l, ha has degenerate critical
point if and only if a is a critical value of the map
v :B1 -4 Sm-l. By Theorem 1.6.1 of Sard the image of the set of
critical points of v has measure zero in Sm-1 Thus, for
almot all a in Sm-1, ha is a non-degenerate function.
Therefore, g(ha) is well-defined and is finite for almost
all a in Sm-1. By applying Theorem 1.6.2 of Morse we
obtain (7.22). (Q.E.D.)
Theorem 7.2 (Chern-Lashof [1)). Under the hypothesis of
Theorem 7.1, if
(7.23) TA(x) < 3
then M is homeomorphic to an n-sphere.
Proof. Suppose that (7.23) holds. Then there exists a
set of positive measure on Sm-1 such that if a is a unit
vector in this set, the height function ha has exactly two
critical points. Since, by Sard's theorem, the image of the
set of critical points under v is of measure zero, there is
a unit vector a such that ha has exactly two non-degenerate
critical points. Applying Theorem 1.6.3 of Reeb, M is
homeomorphic to an n-sphere. (Q.E.D.)
§7. Total Absolute Curvature of (here and Lashof 165
For a hypersurface M in Rn+1 if for each point
p E M, the tangent plane Tp(M) at p does not separate
M into two parts, then M is called a convex hypersurface
of Rn+l . For an immersion of M with total absolute curvature
2 we have the following.
Theorem 7.3 (Chern-Lashof (11). Under the hypothesis
of Theorem 7.1, if TA(x) = 2, then M belongs to a linear
subspace Rn+l of Rm and is imbedded as a convex hypersurface
in Rn+l The converse of this is also true.
For the proof of this theorem, see Chern and Lashof [1).
If dim M = 1, Chern-Lashof's results reduce to the famous
result of Fenchel-Borsuk. The Chern-Lashof results also gave
birth to the important notion of tight immersion which serves
as a natural generalization of convexity.
If x : M -. R3 is an immersion of a compact surface
M into R3. then the Lipschitz-Killing curvature of M
in R3 reduces to the Gauss curvature G, i.e.,
(7.24) G(p,e3) G(p)
Thus, by (7.22), we obtain the following inequality of Chern
and Lashof:
(7.25) J jGjdV Z 2tr(4 - )((M)) ,M
where X(M) denotes the Euler characteristic of M.
166 4.
Analogous to Fary-Milnor's result on knotted curves,
R. Langevin and H. Rosenburg [1) obtained in 1976 the following
result on knotted tori:
Theorem 7.4. Let T be a knotted torus in Iii . Then
(7.26) f IGIdV > 16aT
Recently, N.H. Kuiper and W.H. Meeks [1) improved (7.26)
to the following.
Theorem 7.5. Let T be a knotted torus in 3t3 . Then
(7.27) f IGIdV > 161rT
For the proof of this theorem, see Kuiper and Meeks [1).
§8. Riemannian Submersions 167
§8. Riemannian Submersions
In this section, we will study Riemannian submersions in
more detail. The fundamental geometry of submersions has been
discussed by B. O'Neill (1].
Let 7r :M - B be a Riemannian submersion. For a tangent
vector X of M, X can be decomposed as VX +}1X, where
YX is vertical and kX is horizontal. Let v and v be
the Riemannian connections of M and B, respectively. To
each tangent vector field X on B there corresponds a unique
horizontal vector field X on M such that 7r*X = X. If
X and Y are any two tangent vector fields of B, we have
(8.1) K(vXY) = vXY .
This can be seen as follows: Let Z be any tangent vector
field of B, we have
2<vXY,Z> + Y<Z,X>
- <t,[Y,ZI> + <Y,[Z,XI> + <Z,[X,Y] >
= X<Y,Z> o 7r + Y<Z,X> o 7r - Z<X,Y> o 7r
- <X, [Y,ZI> o 7r + <Y, [Z,X]> o it + <Z, [X,Y]> o 7r
= 2<vxY , Z> o 7
This shows (8.1). Furthermore, if V is a vertical vector
field on M, then
168 4. Submanifolds
7,t [%,V] = [n*]C,ir V] = 0 .
%,(VXV) = W(VvX) .
We define on M a tensor A of type (1,2) called the fundamental
tensor of the submersion as follows: Let X, Y be tangent
vector fields on M, we put
(8.4) AXY = YV CY + *(VUXYY .
This definition shows that AX is a skew-symmetric linear
operator on the tangent space of M and it reverses the
horizontal and vertical subspaces. Moreover, if X is vertical,
AXY = 0. If X and Y are horizontal fields, then
A X Y = -A X2
Y[X,Y].
Suppose that N is an n-dimensional submanifold of M
which respects the submersion ir:M a B. That is, suppose that
there is a submersion 7r -.N -. N' where N' is a submanifold
of B such that the diagram
N f ) M
-1 1 n
N B
commutes and the immersion f is a diffeomorphism on the
fibers.
§8. Riemannian Submersions 169
We shall now relate the second fundamental forms of the
submanifolds N and N'. The discussion will be local, and
so for convenience we shall consider N and N' imbedded in
M and M', respectively with the usual identification of tangent
vectors. If X is a tangent vector of M at a point p E N,
we denote by XT and XN respectively the projections of
X on the tangent and normal spaces of N at p. Note that
the normal space is always horizontal. Let X. Y be tangent
vector fields of N (or N'). Then the Riemannian connection
and second fundamental form of N (or N') are given respectively
by
vXY = (vXY)T , (or vXY = (vxY)T)
h(X,Y) _ (vxY)N , (or h'(X,Y) = (vxY)N) .
We give the following lemma for later use.
Lemma 8.1 (Lawson [1)) N is a minimal submanifold of M
if and only if N' is a minimal submanifold of B.
Proof. For a point p in N, we choose an orthonormal
local vertical fields E1....,Ed about p. Let Fl,...,Fn
be local, orthonormal tangent fields on N' about ir(p).
Denote by Fl,...,Fn the horizontal lifts of F1,....Fn.
Then N is minimal in M if and only if
170 4. Submanifolds
0 = E (O E ) + E (Vi F.)Nk=1 Ek k j=1 Fj
1 (VF Fj)N N= Fi (VF )j j j=1 j
This is equivalent to tr h' = 0 . (Q.E.D.)
§9. Submanifolds of Kaehler Manifolds 171
§9. Submanifolds of Kaehler Manifolds
Let M be a Kaehler manifold with complex structure
J and Kaehler metric g. Let M be a submanifold of M.
For each point p E M, denote by Xp the maximal holomorphic
subspace of the tangent space Tp(M), i.e.,
(9.1) VP = Tp(M) (1 J(Tp(M)) .
if the dimension of !!p is constant along M and afp defines
a differentiable distribution X over M, then M is called
a generic submanifold of M. The distribution of is called
the holomorphic distribution of the generic submanifold M.
For each point p E M, we denote by Xp the orthogonal
complementary subspace of flp in Tp(M). If M is a generic
submanifold, then A(P, p E M, define a differentiable distribution
1l1 over M, called the purely real distribution. For the
general theory of generic submanifolds, see Chen [17,18). It
is easy to see that every submanifold of M is the closure
of the union of some open generic submanifolds of M.
Let M be a generic submanifold of R. For a vector
X tangent to M, we put
(9.2) JX = PX+FX ,
where PX and FX are the tangential and normal components
of JX, respectively. Then P is an endomorphism of T(M)
and F is a normal-bundle-valued 1-form on T(M).
172 4. Submanifolds
We put
(9.3) a = dim W
Then we have dim M = 2 a + .
dim RV
A generic submanifold M of M is called a CR-submanifold
if its purely real distribution W'L is totally real, i.e.,
JV II C Tp1(M), P E M. (Bejancu [1], Chen [17,20], Blair-Chen [11).
Since
(9.4) (P1u)2
= -id.
we have the following inequality:
(9.5) P 2 , 2a
with equality holding if and only if M is a CR-submanifold.
We mention some fundamental properties of CR-submanifolds
as follows:
Theorem 9.1 (Chen [20]). The totally real distribution
WL of a CR-submanifold M of a Kaehler manifold M is
completely integrable.
Proof. Let X be a vector field in I( and Z and W
vector fields in V. Then $(X,Z) = g(X,JZ) = 0, where
denotes the fundamental form of M. Since M is Kaehlerian,
di = 0. Thus we have
§ 9. Submanifolds of Kaehler Manifolds
0 = d§(X,Z,W)
= X§(Z,W) -Z§(X,W)+W§(X,Z)
- §([X,z] ,W) - §(IW,X] ,Z) - 4([Z,WI ,X)
= -g([Z,WI,JX) .
173
Because JX is arbitrary in V and [Z,W] is tangent to
M, [Z,W] must lie in V1. This shows that the totally real
distribution u1 is involutive. Hence, Frobenius' theorem
implies that u1 is completely integrable. (Q.E.D.)
Remark 9.1. The proof of Theorem 9.1 given above is
simpler than the original proof of the present author done
in early 1978. This simplified proof is essentially given in
Blair and Chen [1], in which the following generalization of
Theorem 9.1 was obtained.
Theorem 9.2 (Blair and Chen [1]). Let M be a Hermitian
manifold with d§ = §n w for some 1-form w. Then in order
that M is a CR-submanifold it is necessary that u1 is
completely integrable.
Let . be a differentiable distribution on a Riemannian
manifold M. We put
0
(9.6) h(X,Y) _ xY)1
for any vector fields X, Y in 9, where (vXY)1 denotes
the component of vXY in the orthogonal complementary
174 4. Submanlfolds
distribution .D in T(M). Let el,...,er be an orthonormal
basis of .8, r = dimR. . If we put
(9.7)r o
H = z E h(ei,ei)i=1
Then H is a well-defined f-valued vector field on m
(up to sign), called the mean-curvature vector of the distribution0
B. A distribution 9 on M is called minimal if H = 0
identically.
For the holomorphic distribution X of a CR-submanifold,
we have the following general result:
Theorem 9.3 (Chen (20)). The holomorphic distribution a!
of a CR-submanifold M of a Kaehler manifold M is a
minimal distribution.
Proof. Let X and Z be vector fields in U and !!j,
respectively. Then we have
(9.8) g(Z,v.X) = g(Z,vXX) = g(JZ,VXJX)
= -g(vXJZ,JX) = g(AJZX.JX)
where v and v denote the Riemannian connections of M
and M, respectively. Thus, we find
(9.9) g(Z,vix JX) = -g(A2JX,X) = -g(AJZX,JX)
Combining (9.8) and (9.9) we get g(g X + vjx3X.z) = 0,
from which we conclude that the holomorphic distribution J!
is always a minimal distribution. (Q.E.D.)
§9. Submanifolds of Kaehler Manifolds 175
Theorem 9.4 (Blair and Chen [11). Let M be a generic
submanifold of a Kaehler manifold M. Then the holomorphic
distribution is completely integrable if and only if
(9.10) h(X,JY) = h(JX,Y)
for X, Y in W.
Proof. If Al is integrable, let N be an integral
submanifold. The second fundamental form h' of N in
M satisfies h'(X,Y) = h(X,Y)+ a(X,Y), where a is the
second fundamental form of N in M. Since V is holomorphic,
N is a Kaehler submanifold of M. Thus
(9.11) h(X,JY) - h(JX,Y) = h(JX,Y) - h(X,JY)
But the left-hand side is normal to M and the right-hand side
is tangent to M. Thus both sides of equation (9.11) vanish
which gives the desired condition.
Conversely, since J is parallel with respect to v,
0 = h (X , JY) - h (JX , Y ) = JvXY - vXJY - JvY{ + v1,JX
= J[X,Y] -vXJY+v1,JX .
Therefore J applied to the tangent vector field [X,Y) is
tangent to M and hence [X,Y] belongs to the holomorphic
distribution. Thus the result follows from Frobenius' theorem.
(Q.E.D.)
176 4. Submanifolds
Theorem 9.4 implies the following.
Corollary 9.1 (Bejancu [1]). Let M be a CR-submanifold
of a Kaehler manifold M. Then the holomorphic distribution
W is completely integrable if and only if h(X,JY) = h(JX,Y)
for X, Y in V.
In contrast with the integrability of W!'` and the
minimality of Al for CR-submanifold, we have the following
theorem.
Theorem 9.5 (Chen [21]). Let M be a compact CR-submanifold
of a Kaehler manifold M. If H2k(N) = 0 for some k S dim V ,
then either A! is not integrable or W is not minimal.
Proof. Let M be a compact CR-submanifold of a Kaehler mani-
fold M. Choose an orthonormal local frame el,...,ea,Jel,...,Jea
of W. Denote by w1 ,...,w2a the 2a 1-forms on M satisfying
(9.12) w?(Z) = 0; wI(ei) = bi; i,j = 1,...,2a
for Z in 0, where ea+j = Jej. Then
( 9 . 1 3 ) W=W ln
Aw2a
is a well-defined global 2a-form on M. From (9.13), we have
(9.14)2a
dw = 7, (-1)1 wl n ... ndwl n ... nw2ai=1
From a straightforward computation, we can prove that dw = 0
if and only if
§ 9. Submanifolds of Kaehler Manifolds 177
(9.15) dw(Z1,Z2,X1,...,X2a-1) = 0 ,
(9.16) dw(Z1,X1,....x2a) = 0
for any vectors Z1, Z2 in Wl and Xl,...,x2,-l in V. But
(9.15) holds when and only when Wl is integrable and (9.16) holds
when and only when W is a minimal distribution. But for a
CR-submanifold of a Kaehler manifold these two conditions hold
automatically (Theorems 9.1 and 9.3), therefore,the 2a-form
w is closed. Consequently, we obtain the following.
Lemma 9.1. For any compact CR-submanifold M of a
Kaehler manifold M, the 2a-form w defines a deRham cohomology
class given by
(9.17) c(M) _ [w) E H2a(M)
where a = dimcl.
We need the following.
Lemma 9.2. The cohomology class c(M) E H2a(M) of a
compact CR-submanifold M of a Kaehler manifold M is a
non-trivial class if Al is integrable and All is minimal.
We choose an orthonormal local frame e1,...,eaJel,...I
Jea,e2a+11....e2a+g
such that el,...,ea,Jel,...,Jea are in
X and e2a+1" .. 'e2a+iB
are in W. Denote by w1,...,w2a+H
the dual form of el,...,ea,ea+l " . . .e2a'e2a+1 " . 'e2a+13'
178 4. Submanifolds
where ea+i = Jei. Then w = wl A ... n w2a. Applying the
Hodge star operator * to w, we obtain
(9.18) *w = 2a+1n n
2a+$w ...w
Sincew2a+1(X)
= ... = w2a+o(X) = 0 for all X in Al,
(9.18) implies that d*w = 0 if and only if
(9.19) d*w(X1,X2,Z1,....Z8 1) = 0 ,
(9.20) d*w(Xl,Z1,...,Z13 ) = 0
for all vectors X1, X2 in Af and Z1....,Z8 in ifl. Because
(9.19) holds if and only if K is integrable; and (9.20) holds
if and only if All is minimal. We see that if V is integrable
and VI is minimal, then the 0-form *w is closed. Thus,
bw =(-1)2an+n+1*d*w
= 0, i.e., w is also co-closed. Therefore,
w is a harmonic 2a-form on M. Because w is non-zero,
Theorem 3.3.2 of Hodge-deRham implies that c(M) _ [w) represents
a non-trivial class in H2a(M). This proves Lemma 9.2.
we choose an orthonormal local frame
e1.
'ea'ea+l'...,eo+8.ea+p+l' ...,em,Je1,...,Jem
in M in such a way that, restricted to M, e1....,ea,Jel,....Jea
are in V and ea+ 1,...,er a are in All. we denote by
1 mU) ,...,w ,W ,...,Um the dual frame of ell .. ,em,Jel, ..,Jem.
We put
§9. Submanifolds of Kaehler Manifolds 179
**
A9A = wA +'r 1
wA , 8A = w - W'r 1
. A = 1,...,m .
*
Restrict 0A.s and8A
s to M, we have
8i = 41 = wl, for i = a+ 1,...,a+ is(9.21)
8r= er=0 for r=a+13+l,...,m
The fundamental 2-form I of M is a closed 2-form on M
given by
= E8AA8AA
*Let § = i% be the 2-form on M induced from } via the
immersion i of M into M. Then (9.21) gives
(9.22) 2 j=1
It is clear that 4 is a closed 2-form on M and it defines
a cohomology class [0) in H2a(M). Equations (9.13) and
(9.22) implies that the class c(M) and the class [1) satisfy
(9.23) [,)a = (-1)a(a!)c(M)
If K is integrable and V 1 is minimal, then Lemma 9.2 and
(9.13) imply that H2k(M) ¢ 0 for k = 1,2,...,c. This
proves Theorem 9.5. (Q.E.D.)
We state the following lemma for later use.
180 4. Submanifolds
Lemma 9.3. Let M be an n-dimensional generic submanifold
of CPm which is imbedded in H(m+ 1;Q) cp given in (6.7).
Then the mean curvature vectors H and H' of M in
H(m + 1;C) and CPm satisfy
(9.24) IHI2 ', IH'I2+ 2 (n2+n+2a], a = dim !n
equality holding if and only if m is a CR-submanifold of CPm.
This lemma follows immediately from Lemma 6.2, (2.6.3)
and (9.5).
Remark 9.2. A. Ros [1] proved that if m is a CR-submanifold
of CPm, then IHI2 = IH' I2+ n (n2+ n+ 2a).
Remark 9.3. If M is a submanifold of a quaternionic
Kaehler manifold M with quaternionic structure
we put
ill J2' J3'
9p = Tp(M) fl J1(Tp(M)) fl J2(Tp(M)) fl J3(Tp(M))
for p E M, then -Ap is the maximal quaternionic subspace
of T(M). If 9 : p -..6p is a differentiable distribution and
its orthogonal complementary distribution .8 in T(M) is
totally real, then M is called a quaternionic CR-submanifold.
For the general theory of quaternionic CR-submanifolds, see
Barros-Chen-Urbano [1]. By using a similar argument as we
give in Lemma 9.3, we also have the following
§9. Submanifolds of Kaehler Manifolds 181
Lemma 9.4. Let M be an n-dimensional guaternionic
CR-submanifold of QPm which is imbedded in H(m+ 1;Q) by
cp. Then the mean curvature vectors H and H' of M in
H(m+ 1;Q) and QPm satisfy
(9.25) IHI2 = 1H,12+ 2(n2+n+ 12a),
na = dimQ.B
Remark 9.4. For quaternionic version of Theorem 9.5,
see Barros and Urbano [1,2].
Chapter 5. TOTAL MEAN CURVATURE
fl. Some Results Concerning Surfaces in R 3
For surfaces in R3 , the two most important geometric
invariants are the Gauss curvature G and the mean curvature.
According to Gauss' Theorema Egregium, the Gauss curvature is
intrinsic. The integral of the Gauss curvature gives the famous
Gauss-Bonnet formula:
(1.1)J
G dV = 2r X(M) ,
M
for a compact Riemannian surface M. Moreover, the integral
of the absolute value of the Gauss curvature satisfies the following
Chern-Lashof inequality:
(1.2) fM JGJdV k 2ir(4 - X(M)) .
The idea of integrating the square of the mean curvature
instead of the Gauss curvature was discussed at meetings at
Oberwolfach in 1%0 (cf. Willmore [4, p.145).) The first published
result of this subject appeared in Willmore [1,2) which states
as follows;
Theorem 1.1. Let M be a compact surface in R3. Then
we have
(1.3) f IHI2dV -. 4, .
M
§ 1. Some Results Concerning Surfaces in IR3 183
The equality of (1.3) holds if and only if M is an ordinary
sphere in R3 .
Proof. Clearly, a2-G = 4(xl-x2)2 O, where
a = IHI and xl and x2 are the eigenvalues of the Weingarten
map Ae = (hid). Thus if we divide the surface M into3
regions for which G > 0 and G S 0, we have
(1.4)J
,H,2dV 2$ IHj2dV 2 f G dV 2 4rM G>O G>O
where the last inequality is obtained by combining (1.1) and
(1.2). This shows (1.3). Moreover, equality of (1.3) holds
if and only if xl = x2, i.e., M is totally umbilical in
R3 . By Proposition 4.4.1, M is an ordinary sphere in R3
(Q.E.D.)
Analogous to Fary-Milnor's results on knotted curves, the
present author obtained in 1971 the following unpublished result
on knotted tori by investigating its Gauss map (cf. Willmore
[5l.)
Theorem 1.2 (Chen 1971). Let T be a knotted torus in
R3 . Then
(1.5)J
1HI2dV > 8ir
M
By using a very recent result of Kuiper and Meeks
(Theorem 4.7.5), Willmore improves inequality (1.5) in 1982
by replacing the sign by strict inequality. Willmore's argument
184 5. Total Mean Curvature
goes as follows: If T is a knotted torus in R3 , Kuiper
and Meeks' result impliesf
IGIdV > 16rr. Combining thisT
with the Gauss-Bonnet formula, one obtains$ G dV > 8ir.
This implies $ JH J 2dV 8v.T
G>O
For tubes in R3 , we have the following result of
K. Shiohama and R. Takagi (1) and Willmore [3):
Theorem 1.3. Let M be a torus imbedded in R3 such
that the imbedded surface is the surface generated by carrying
a small circle around a closed curve so that the center moves
along the curve and the plane of the circle is in the normal
plane to the curve at each point, then we have
(1.6) I I H12dV 2 2,, 2
.
M
The equality sign holds if and only if the imbedded surface
is congruent to the anchor ring in R3 with the Euclidean
coordinates given by
where a
xl = (, a+ a cos u)cos v ,
x2 = (,F2 a+ a cos u) sin v ,
x3 = a sin u ,
is a positive constant.
Proof. Let C be the closed curve mentioned in the
theorem. Let x = x(s) be the position vector field of C
parameterized by the arc length. Denote by x and 7 the
§ 1. Some Results Concerning Surfaces In IR' 185
curvature and torsion of C. Let y denote the position
vector of M in R3 . Then
(1.7) y(s,v) = x(s) + c Cos v N+ c sin v B ,
where N and B are the principal normal and binormal of C.
By a direct computation, we find that the principal curvature
of M in R3 are given by
_ 1 _ x Cos vkl -c' k2-xccos v-1
Thus the mean curvature vector satisfies
12 1- 2xc cos v 2HI12c 1 - xc cos v I
Thus
P IH I2 dV = J_ PpL 2v 1- 2xc cos v 2 dv doJM O JO 12c(1-xc cos v)
it2c J it - x2c2)-1/2 doO
where l is the length of C. Therefore,
p
(1.8)J
IHj2dV = 2it Ixl do
M 0 'xcj 1-x2 c2
A?irI InIds>4ir,O
by virtue of the fact that, for any real variable x. (1- x )
takes its maximum value2
at x 1.
42
186 5. Total Mean Curvature
If the equality sign of (1.6) holds, inequalities in (1.8)
become equalities. Thus, by Fenchel's result, C is a convex
planar curve. Moreover, x =(2c2)-1/2.
Thus, C is a circle
of radius 2 c. This shows that M is imbedded as an anchor
ring of the type given in the theorem. The converse is trivial.
(Q.E.D.)
Willmore conjectured that inequality (1.6) holds for all3
torus in R- . Theorems 1.2 and 1.3 shows that Willmore's
conjecture valids either M is knotted, or M is a tube in
§ 2. Total Mean Curvature 187
{2. Total Mean Curvature
According to Nash's Theorem, every n-dimensional compact
Riemannian manifold can be isometrically imbedded in RN with
N =
-n(3n + 11). On the other hand, "most" compact Riemannian
manifolds cannot be isometrically imbedded in Rn+l as a
hypersurfaces. For example, every compact surface with non-
positive Gauss curvature cannot be isometrically imbedded in
R3 . Furthermore, there are many minimal submanifolds of a
hypersurface of Rm which are not hypersurfaces of Rn+1
Hence, the theory of submanifolds of arbitrary codimensions
is far richer than the theory of hypersurfaces, in particular,
than the theory of surfaces in R3 . Especially, we will see
that this is the case when one wants to study the theory of
total mean curvature and its applications.
The first general result on total mean curvature is
given in the following.
Theorem 2.1 (Chen [2]). Let M be a compact n-dimensional
submanifold of Rm. Then we have
(2.1)J
IHind V 2 cn .
M
The equality holds if and only if M is imbedded as an
ordinary n-sphere in a linear (n+ 1)-subspace Rn+l when
n > 1 and as a convex plane curve when n = 1.
Proof. Let x : M Rm be an isometric immersion
of a compact n-dimensional submanifold M into R. Let Bm
188 5. Total Mean GLrvature
be the bundle space consisting of all frames (p,x(p),el,...I
en'en+l" .. ,em) such that p E M. e1,...,en are orthonormal
vectors tangent to M at p and en+1,...,em are orthonormal
vectors normal to M at p. Choose the frame (p,x(p),
el,...,en,en+l,...,em) in B such that em is parallel
to the mean curvature vector H at p. Then we can easily
find that the mean curvature IHI is given by
(2.2) IHI = n (hll+ ...+ nn
and
(2.3) r = n+l,...,m-1 ,
where 1j= On the other hand, for each (p,em)
in Bi, we can write
(2.4)m
em = E cos 8 ess=n+l s s
where 8s denotes the angle between em and es. For each
(pre) E B1, we put
(2.5) K1(p,e) =
n
trace Ae
From (2.2), (2.3), (2.4) and (2.5) we find
m(2.6) K1(p,em) = E cos 8s K1(P,es) = cos 8mIH(p)I
s=n+ 1
Hence we obtain
§ 2. Total Mean Curvature
(2.7) S 1 K1(p,em)I n dV n do = f IHInIcosn AmI dV n do
B1 B1
= (2cm-1/cn) J I HI ndV .
M
189
Let e be a unit vector in the unit sphere Sm-1. Consider
the height function he = <e,x(p)> on M. It is clear that
he
is a differentiable function on M. For any vector
fields X, Y tangent to M, we have Xhe = <e,X>. Hence
(2.8) YXhe = <e,V YX + h(Y,X)> .
Since h is continuous on M, h has at least one maximume e
and one minimum, say at q and q', respectively. At q
and q', e is normal to M. Thus, we obtain from (2.8) that
(2.9) YXhe = <AeX,Y> .
Since q and q' give the maximum and minimum of he, (2.9)
implies that the Weingarten map Ae is either non-positive
definite or non-negative definite at (q.e) and (q'.e). Let
U denote the set of all elements (p,e) in B1 such that the
eigenvalues k1(p,e),...,kn(p,e) of Ae have the same sign.
Then from the above discussion we see that the unit sphere
Sm-1 is covered by U at least twice under the map v : B1 -0 Sm-1
which is defined by v(p,e) = e. This shows that
(2.10)J
v* dE 2 2cM-1U
190 S. Total Mean Curvature
Since, on U, k1(p,e),...,kn(p,e) have the same sign, we
find
(2.11) IK1(p,e)In = 1n (k1(p,e)+ ...+ kn(p,e) in
? Jkl(p,e) ...kn(p,e)l = IG(p,e)I
Hence, by using (2.7), (2.10), (2.11) and 14.7.18) we obtain
p
(2.12)J
IHIndV2cCn
)I IK1(p,e)in dV Ado
M M-1 B1
c2 (2cn ) S v*d1?cn
M-1 U
This proves (2.1). Now, assume that the equality sign of (2.1)
holds. We want to prove that M is imbedded as an ordinary
hypersphere in a linear subspaceRn+l
of Rm when n > 1.
This can be proved as follows:
We consider the map
(2.13) y : B1 . Rm; (p,e) -# x(p) +ce
where c is a sufficient small positive number which gives
an immersion of B1 into Rm . In this way, we may regard
B1 as a hypersurface in Rm. Moreover, because
<e,dy> _ <e,dx> + c<e,de> = 0, e is in fact a unit normal vector
of B1 in 3Rm at (p,e). Thus el,...,em-1 form an
orthonormal basis of T(p,e)BI' Let wl ,w -1 be the
dual basis of el,...,em-1. Then by direct computation, we have
§2. Total Mean Curvature
n(2.14) wl = wl+ c Z hi. w3
j=1
(2.15) Wr = cwr , r = n+ 1,...,m- 1
191
Let kA(p,e), A be the eigenvalue of the Weingarten
Ae of B1 in Rm at (p,e). Then, by using (2.14) and (2.15),
we may obtain
(2.16)
ki(p,e) _
kr(p,e)
ki(p,e)
1+cki(p,e) '
i = 1,2, ,n ,
r = n+l,...m-1 .
Let el,...lem-1 be the principal directions of Ae. We have
(2.17) wB = kA(p,.e)WA , A
where vem = wm eA. Put veA = WA eB. Taking the exterior
differentiation of both sides of (2.17) we find
(2.18) C BnWA = C kA;C wCnwA + kA wBnwB
where we put dkA = kA;C wC. Let
(2.19) WBI'ABC WC
Then (2.17), (2.18) and (2.19) imply
(2.20) kA.B(p,e) = (k8(p,e) - kA(p,e))i-AA
192 5. Total Mean Curvature
Let U = {(p,e) E BI lkI(p.e) = = kn(p,e) 30'0] and
V = B1 U. Then (2.20) gives
(2.21) kA;B(p,e) = 0 for (p,e) E U .
If we put
dki(p,e) = E ki:A(p,e)wA
then we have
(2.22) ki'j(p,e) = 0 for (p,e) E U .
Now, by the assumption, the equality of (2.1) holds. Thus,
all of the inequalities in (2.11) and (2.12) become equalities.
Hence, we have K1(p,e) = 0 identically on V = B1- U. By
(2.1), we see that U is a non-empty open subset of M.
Let U' be a connected component of U. Then, by (2.22), we
know that w(p) = maxJKI(p,e)l, e runs over (p,e) in U',
is a positive constant function on where a :Bl + M
is the projection. If rr(U') / M, then by the continuity
of K1(p,e) on B1 and the fact K1(p,e) = 0 on V, we
see that for each point p in the boundary of rr(U'), there
exists a point (p,e') in U' such that w(p) = JK1(p,e')I.
Hence there is an open neighborhood of (p,e') in B1 which is
contained in U'. This is a contradiction. Thus ir(U) = M.
From This, we find that, for each point p in M, there
is a non-empty open subset W of the fibre Sm-n-1 of B1
over p such that the principal curvatures k1(p,e),...,kn(p,e)
§ 2. Total Mean Curvature 193
are equal for all (p,e) E W. From this we may conclude that
k1(p,e) _ = kn(p,e) for all e in Sp n-1. Since this
is true for all p in M, M is totally umbilical in Rm .
Consequently, by Proposition 4.4.1, M is imbedded as an
ordinary hypersphere in a linear subspace Rn+1 when n > 1.
If n = 1, M is imbedded as a convex plane curve by the result
of Fenchel-Borsuk. The converse of this is trivial. (Q.E.D.)
Remark 2.1. An alternative proof of inequality (2.1) was
given in Heintze and Karcher [1). However, their method does
not yield the equality case.
Some easy consequences of Theorem 2.1 are the following.
Corollary 2.1 (Chen [5)) Let M be a compact n-dimensional
minimal submanifold of a unit m-sphere Sm. Then the volume of
M satisfies
(2.23) vol(M) _> cn = vol(Sn) .
The equality holds if and only if M is a great n-sphere in Sm.
Proof. Regard Sm as the standard unit hypersphere
ofRm+l.
Since M is minimal in Sm, the mean curvature
of M inRm+l
is equal to one. Thus, (2.1) implies
vol(M) =J
HlndV cnM
This proves (2.23). The remaining part follows easily from
Theorem 2.1, too. (Q.E.D.)
194 5. Total Mean Curvature
Corollary 2.2 (Chen (24]) Let M be a compact n-dimensional
minimal submanifold of a real projective m-space R Pm of
constant sectional curvature 1. Then
c(2.24) vol(M) 2
The equality holds if and only if M is a R Pn imbedded in
R Pm as a totally geodesic submanifold.
Proof. Let M be a compact n-dimensional minimal submanifold
of a real projective m-space R Pm. Consider the two-fold
covering map r : Sm -o R Pm. Then it 1(M) is a minimal
submanifold of Sm with vol(n 1(M)) 2 vol(M). Applying
Corollary 2.1 to 7-1( M ) , we obtain (2.24). If the equality
of (2.24) holds, then vol(n 1(M)) = 2 vol(M) = cn. Thus, by
Corollary 2.1, 7-1 (M) is a great n-sphere in Sm. Thus M
is a R Pn imbedded in RPm as a totally geodesic submanifold.
The converse is trivial. (Q.E.D.)
Corollary 2.3. (Chen (24]) Let M be a compact
n-dimensional (n > 1) minimal submanifold of CPn with
constant holomorphic sectional curvature 4. Then
(2.25) vol(M) Ncn+l
21r
The equality holds if and only if n - 2k is even and m is
a CPk which is isometrically imbedded in CPm as a totally
geodesic complex submanifold.
§ 2. Total Mean Curvature 195
Proof. Let M be a compact n-dimensional minimal
submanifold of CPm. Consider the Hopf fibration rr : S2m+1 -4 CPm
Denote the r 1(M) by M. Then rr:Fl M is a Riemannian
submersion with totally geodesic fibres S1. We consider the
following commutative diagram:
i S2m+1
Since M is minimal in CPm, Lemma 4.8.1 implies that M is
minimal in S21. Thus, by applying 2.1 to M. we obtain
(2.26) vol(M) cml ,
with equality holding if and only if M is a great (n+ 1)-sphere
inS2mf1. On the other hand, because tr:M + M is a Riemannian
submersion with fiber S1, Lemma 2.7.2 gives
(2.27) vol(M) = 2w vol(M) .
Combining (2.26) and (2.27), we obtain (2.25). If the equality
sign of (2.25) holds, then M is a great (n+ 1)-sphere Sn+l
of S2m+ 1. Since n :51 4 M is a Riemannian submersion
with fiber S1, n = 2k is even (Adem (1]). Thus, R is
a great (2k+ 1)-sphere of S2r1. From this we conclude that
M is a CPk which imbedded in CPm as a totally geodesic
complex submanifold. The converse of this is trivial. (Q.E.D.)
196 5. Total Mean Quvature
Remark 2.2. Recently, Roo also obtained a lower bound
of the volume of a compact minimal submanifold of CPm by
applying our Theorem 2.1. However, his estimate is not sharp.
Corollary 2.4. (Chen [241) Let QPm be a quaternion
projective m-space with maximal sectional curvature 4. If
M is a compact minimal submanifold of QPm, then
(2.28) vol(M) c22n
The equality holds if and only if M is a QPk, n - 4k; and
QPk is imbedded as a totally geodesic submanifold in QPm.
Proof. Let it-(M) with it :S4n*3 QPm. Since Mis minimal in QPm, M is minimal in S43. Applying Theorem 2.1
we obtain vol(S) - c3 vol(M) = 2ir2 . vol(M) by Lemma 2.7.2.
Thus, we find (2.28). The equality case can be obtained in
the similar way as Corollary 2.3. (Q.E.D.)
Corollary 2.5. (Chen [24]) Let OP2 be a Cayley plane
with maximal sectional curvature 4 and M an n-dimensional
minimal submanifold of OP2. Then we have
(2.29) vol(M) k cn/2n
Proof. Regard the Cayley plane OP2 as a submanifold in
H(3;O) as mentioned in ¢4.6. Since M is minimal in OP2,
the mean curvature vector of M in H(3;O) satisfies IHl2 S 2.
Then by using Theorem 2.1, we obtain (2.29). (Q.E.D.)
§ 2. Total Mean Curvature 197
Remark 2.3. The estimate of vol(M) given in Corollary 2.5
is sharp if n g 8.
Corollary 2.5 (Chern and Hsiung [1)) There exist no
compact minimal submanifolds in Rm.
This Corollary follows immediately from Theorem 2.1.
It follows from Theorem 2.1 that the total mean curvature
of a compact n-dimensional submanifold in Rm is always
bounded below by cn - vol(Sn). On the other hand, according
to Theorem 4.7.1 of Chern and Lashof, the total absolute
curvature is bounded below by the topological invariant b(M).
Thus, it is natural to ask whether if b(M) is large, the
total mean curvature of M in Rm is also "proportionally
large"? The answer to this is no. This can be seen by using
Lawson's examples of compact minimal surfaces in S3. In
Lawson [2), he had constructed a compact imbedded minimal
surface Mg of genus g (for an arbitrary g 0) in S3
with area less than 8,r. Thus, if we regard Lawson's examples
as surfaces in R4 , they have total mean curvature less
than 87r. However, b(Mg) 2+ 2g which tends to infinity
as g tends to infinity.
Let 11h112 denote the square of the length of the second
fundamental form h of M in Rm. Then by the Gauss
equation, we have
(2.30) n(n - 1).r : n2IHI2 _ 11h112 ,
198 5. Total Mean Curvature
(2.31) (n-l)IIh1I2-n(n-1)T = E (n(hr )2-hiihjj)r,i,j
n E E (hr)2+E E (hr-hr)2 0r i(j ii r i<j 11 ji
Thus we obtain
(2.32) -(n(n 1) )IIhII2 S T S (')IIhII2
In the following, a submanifold M in Rm is called
6-pinched in Rm if we have
(n(nbl )IIhII2 - T - (')IIhII2
for some 6 _> -1. In view of Lawson's examples mentioned above,
we give the following.
Proposition 2.1 (Chen (12)) Let M be a compact n-dimensional
submanifold in Rm. If M is 6-pinched in Rm, then the
total mean curvature of M satisfies
(2.33)
n
fM IHIndV (1 1+6)2cn]b(M)
where b(M) is defined in §1.6. The equality sign of (2.33)
holds when and only when M is (n -1)-pinched in Rm.
Proof. Let M be a compact n-dimensional submanifold
in Rm . Let en+l,... ,em be local orthonormal normal vector
fields of M in Rm . If e is a unit normal vector field
of M, then e = E cos 8 rer. Thus,
§ 2. Total Mean Curvature 199
Ae = E cos grAr Ar = Aer r
Hence, we have
(2.34) IIAeII2 = E cos or cos 8s trace (ArAs)r,s
The right-hand side of (2.34) is a quadratic form on
cos ,cos gm. Thus, we may choose a local orthonormal
normal frame en+l,...,em such that with respect to this
frame field, we have
(2.35) IIAeII2 = E (r Cos2Or Cn+l ...Cm
(2.36) C r= I I e 112r
Let B1 be the bundle of unit normal vectors of M in
Rm . We define a function f on B1 by
(2.37) f(p,e) = IIAe112 ,
for (p,e) E B1. Since all ofCr
are non-negative and
f.,Cos2Or = 1, an inequality of Minkowski implies
(2.38)/p n 2
``2/n /n
iJf2 dv1 = { (E Crcos2gr)2 da)
Sp // ` Sp //
2( `
Icosn0rlda)S E SCr1SSp /
where Sp is the (m -n -1) -sphere of unit normal vectors at
p. On the other hand, we have the following identity,
200 5. Total Mean Curvature
(2.39).
ICosnglda = 2cm-1/CnSp
Thus, by (2.36), (2.38) and (2.39) we find
np(2.40) 11h,12 3
c2cn
d f2 dam-1 Sp
Let G(p,e) denote the determinant of Ae. Then, by using
a relation between elementary symmetric functions, we have
IIAeIIn _ nn IG(p,e)I. Thus, by using (2.40), we find
(2.41)p c
J IIhIIndV nn2cM-1 1 fB IG(p,e) IdV n doM
1
Combining this with Chern-Lashof's inequality, we get
(2.42) J IIhIIndV 2 n b(M)M
Now, by the hypothesis, M is 6-pinched in 3tm . Thus
(2.43) T ? n(nsl)- IIhII2
On the other hand, (2.30) gives n(n -1).r = n2IHI2 - IIhII2. Thus,
(2.43) implies
(2.44) IHI2 2 (1)IIhII2n
Combining (2.42) and (2.44) we obtain (2.33). If the equality
sign of (2.33) holds, then the equality sign of (2.39) holds.
From this we may conclude that m is imbedded as a hypersurface
of a linear subspacegn+l with nIHI2 = IIhII2. Thus, by
(2.30), M is (n- 1)-pinched in Rm . (Q.E.D.)
nn c
§ 2. Total Mean Curvature
If M is a minimal submanifold of a unit hypersphere
Sm-1 of Rm , then M is b-pinched in Rm if and only if
the scalar curvature of M satisfies
(2.45) T-
(
n-1)(1+6
.
n
201
In particular, M is O-pinched in Rm if and only if M
has zero scalar curvature. In this case Proposition 2.1 implies
the following.
Corollary 2.7. Let M be a compact n-dimensional
minimal submanifold of Sm. If M has non-negative scalar
curvature, then we have
n2 c
(2.46) vol(M) _, (n)) 2 b(M)
In the 1973 Symposium on Differential Geometry held
at Stanford University, the present author proposed the
following two problems (See Chen [11)).
Problem 2.1. Let (M,g) be a compact Riemannian manifold
and x : M 4 Rm an isometric immersion of M into Rm . What
can we say about the total mean curvature of x and the
Riemannian structure of (M,g)?
Problem 2.2. Let M be a compact manifold and x :M -4 Rm
an immersion of M into Rm . What can we say about the
total mean curvature of x and the topological structure
of M (or of x(M))?
202 5. Total Mean Curvature
In this book, we shall give a detailed account of the
results on these two problems up to date.
Remark 2.4. Let :(M,g) .. (N,g) be a smooth map between
Riemannian manifolds. We define its energy by the formula
(2.47) 2 fM
jI4 Il2 dVM
where dVM denotes the volume element of (M,g); and t* is
the differential of 4. The Euler-Lagrange operator associated
with E, denoted by T(O), is called the tensor field of
4 (cf., Eells and Sampson (1) and Eells and Lemaire (1]). If
4 is an isometric immersion, the energy E(t) gives the
volume of M. In this case, the tension field .r(4) is
nothing but n times the mean curvature vector of M in N,
where n = dim M. If one defines the total tension of I by
n(2.48) f
i 1J dVMM
then the total mean curvature j IHIdV is exactly the totalM
tension of 4, whenever 0 is isometric.
§ 3. Conformal Invariants 203
3. Conformal Invariants
Let (M,g) be an m-dimensional Riemannian manifold and
P a positive function on M. We put
-* 2g = P 9 .
Then g is a conformal change of metric. Denote by v and
v* the Riemannian connections of g and g*, respectively.
Then equation (2.5.2) gives
(3.2) v*i'-vXY =(X log p)Y+ (Y log P)X-g(X,Y)U
where U = (dp)$.
Let M be an n-dimensional submanifold of A. Denote by
g and g* the metrics on M induced from g and g*,
respectively. For any normal vector field g of M in M
we have
(3.3) v"XS-vXS = (X log P)S-(Slogp)X
for X tangent to M. Thus, by using Weingarten's formula,
we find
(3.4) DXS -D Xt = (X log P)S
*where D and D denote the normal connections of M in
(M,g) and (M,g ), respectively. Thus we have
(3.5) DX = DX + (X log p)I
204 5. Total Mean Curvature
Consequently, the normal curvature tensors RD and RD
satisfy
(3.6) RD (X,Y) = RD(X,Y)+ DX((Y log p)I)
+ (X log p)DY -DYM log p)I)
- (Y log p)DX - ([X.Y]log p)I
Therefore, by using the definition of Lie bracket, we obtain
(3.7) RD (X,Y) = RD(X,Y)
This implies the following (Chen (10]).
Proposition 3.1. Let M be a submanifold of a
Riemannian manifold M. Then the normal curvature tensor RD
is a conformal invariant.
Let h and h denote the second fundamental forms of
M in (M,g) and (A,-9*), respectively. Then, from (3.2),
we find
(3.8) h*(X,Y) = h(X,Y)+ g(X,Y) UN
where UN denotes the normal component of U restricted to M.
Hence, for any normal vector S of M in M, we get
(3.9) 9(A9X,Y) = 9(ASX,Y)+ 9(X,Y)9(U,S)
Let el,...,en be principal normal directions of AS with
respect to g. Then
§ 3. Conformal Invariants
-1 -P e1,...,P en
205
form an orthonormal frame of M with respect to g . Moreover,
they are the principal directions of A If we denote by
k1(g),...,kn(g) the principal curvatures of AS and by
kn(g) that of A, then (3.9) implies
(3.10) ki(S) = ki(S)+ lg , Ag = g(U.g)
Since Ag = pAt* , g* = p-lg is a unit normal vector with
respect tog*,
(3.10) gives
(3.11) P(ki(g*)- k*(S*)) = ki(S) -ki(9)
Now, let g ,l,...,Sm be an orthonormal normal frame
with respect to g. Then the mean curvature vector H is given
by
(3.12) H = n E (E k.(gr))grr i
We put
(3.13)Te n(n--1) E E
r i<7
It is easy to see that Te is well-defined. We call Ge
the extrinsic scalar curvature with respect g. If (M,g) is
of constant sectional curvature k, the equation of Gauss
implies
206 5. Total Mean Curvature
(3.14) Te = T-k ,
where T is the scalar curvature of (M,g). If M is
2-dimensional, the extrinsic scalar curvature relates to the
Gauss curvature G by
(3.15) T = G-R'e
where R' denotes the sectional curvature of (M,g) with
respect to Tp(M). By using (3.11), (3.12), (3.13) we obtain
the following results (Chen [10)).
Proposition 3.2. Let M be an n-dimensional submanifold
of a Riemannian manifold (M,g). Then
(3.16) (IH12-Te)g
is invariant under any conformal change of metric.
In particular, if M is compact, Proposition 3.2 implies
immediately the following (Chen [10]).
Proposition 3.3. Let M be an n-dimensional compact
submanifold of a Riemannian manifold (M,g). Then
n
(3.17)J
(IHI2-Te)IdV
M
is a conformal invariant.
If M is 2-dimensional, Proposition 3.3 reduces to the
following
§ 3. Conformal Invariants
Proposition 3.4. Let M be a compact surface in a
Riemannian manifold (M,g). Then
(3.18) J(IHI2+R')dV
is a conformal invariant.
Remark 3.1. For a problem related to Proposition 3.4,
see Ejiri (1].
By assuming the ambient space to be a real-space-form,
Proposition 3.4 gives the following Corollaries.
207
Corollary 3.1. Let M be a compact surface in Itm and
a diffeomorphism of Itm which induced a conformal change of
metric on 12m . Then we have
(3.19) $
MJH12dV = f
O(M)JH012dV .
For m = 3, this Corollary is due to Blaschke [1],
White (1].
Equation (3.19) says that the total mean curvature of a
compact surface M in IRm is a conformal invariant.
Corollary 3.2. Let M be a compact surface in a complete,
simply connected, real-space-form Hm(-1) of constant sectional
curvature -1. Then we have
(3.20) Hj 2dV Z 47r+ vol(M)M
208 5. Total Mean Curvature
The equality sign holds if and only if M is totally umbilical
in Hm(-1).
Proof. Since Hm(-1) can be obtained from Rm by
a conformal change of the metric on Rm, Proposition 3.4
implies
f(IHI2
- 1)dV = $MIHI2dV
,
M
where M is the surface in Rm with the induced metric.
Thus, by Theorem 2.1, we obtain (3.20). If the equality of
(3.20) holds, then, $IH,2dv
= 47r. Thus, M is an ordinary
2-sphere in Rm. Therefore, M is totally umbilical in Rm.
Since totally umbilicity is a conformal invariant, M is
totally umbilical in Hm(-l). The converse is easy to see.
(Q.E.D.)
Remark 3.2. Maeda [1) also obtain Corollary 3.2 by
using a quite different method.
For the standard m-sphere Sm, one may consider the group
G(Sm) of conformal diffeomorphisms of Sm, i.e., the group
of diffeomorphisms on Sm which induce conformal changes of
metric on Sm. In this case, Proposition 3.4 reduces to
$M(IH12+1)dV = J
(M)(IH+I2+1)dv, E G(Sm) .
Thus, if M is minimal in Sm, then we have
§ 3. Conformal Invariants
vol(4>(M)) SJ
1)dV = vol(ts), p E G(Sm) .
0M)
Corollary 3.3 (Li and Yau [1)). Let M be a compact
minimal surface of Sm given by an isometric immersion
f :M -. Sm. Then we have
(3.21) vol(M) = Vc(m,f) ,
where Vc(m,f) = sup vol(4>(M)), called the m-conformalEG(Sm)
209
volume of f.
Another consequence of Proposition 3.4 is the following.
Corollary 3.4 (Li and Yau (1).) Let M be a compact
surface in Rm . Then
(3.22)J
IH12dV 2 Vc(m,M) ,
where Vc(m,M) = inf Vc(m,4>),4>
runs over all non-degnerate
conformal mappings of M into Sm.
Proof. Using the inverse of stereographic projection,
one forms a conformal immersion4>
of M into Sm.
Compositing with a Mobius transformation, one may assume that
the area of 4>(M) is equal to Vc(m,4>). From Proposition 3.4
we get
(3.23) fJH12dV = J (JH
12+ 1)dV > vol(4>(M)) .
M (M)
210 S. Total Mean Curvature
This implies (3.22). (Q.E.D.)
In the remaining part of this section, we will use a
result of Haantjes (1) to prove the conformal invariance of
JIHI2dV for surfaces in Rm.
M
Assume that M is a surface in Rm . It is obvious
that the quantity (IHI2- G)dV is invariant under similarity
transformations. (i.e., motions and homothetics on Rm.).
On the other hand, according to Haantjes [1], a conformal mapping
on Rm can be decomposed into a product of similarity
transformations and inversions. Hence, it suffices to prove
that (IHI2- G)dV is invariant under inversions. Let $ be
an inversion on Rm such that the center of $ does not lie
on the surface M. We choose the center of i as the origin
of Rm. Denote by x and x the position vectors of the
original surface M and the inverse surface M = #(M),
respectively. Let c be the radius of inversion. Then we have
2
(3.24) x = (°-2)x , r2 = <x,x>r
From these we find
2 2(3.25) dx = (c2)dx - (2c ) (dr)x
r r
(3.26)_ 4
<dx,dx> _(2
4)<dx,dx>r
Hence, the volume element 0 of M satisfies
§ 3. Conformal Invariants 211
(3.27)4
dV = (cT) dVr
Let e3,...,e be orthonormal normal frame of M in Rm
Then
2<x,e >(3.28) er = ( 2r )x-e r = 3,...,mr rare orthonormal normal frame of M. From 13.25), (3.26) and
(3.28), we find
2c2<x,e > 2
(3.29) <dx,der> 4 r )<dx,dx> - (cf) <dx,der>r r
2 2(3.30) ki(er) = -(!)ki(er) - (2 2 )<x,er>
c
where ki(er) and ki(er) are the principal curvaturs of Ae
and Ae , respectively. From (3.30) we getr
(3.31) (kl(er)+ K2(er))` 4 k1(er)k2(er
(c4)((kl(er)+ k2(er ))2 - 4 kl(er)k2(er))
By taking the sum of both sides of (3.31) over r, we obtain
_IH12-G =
(r44)(IHI2+_G)
.c
Combining this with (3.27) we obtain
(3.32) (IHI2- G)dV = (JH12- G2)dV
212 S. Total Mean Qwvature
This shows that (IHI 2- G)dV is invariant under conformal
mappings of Rm . For surfaces in R3 , (3.32) was already
obtained in G. Thomsen [1) in 1923. If M is compact, (3.32)
and Gauss-Bonnet's formula gives the following.
Proposition 3.5. Let M be a compact surface in Rm
and a conformal mapping of Rm. Then we have
(3.33)JM
IH12dV =
Remark 3.3. For surfaces in R3, Proposition 3.5
is due to Blaschke [1] and J.H. White [1). The proof of
Proposition 3.5 above is a slight modification of Blaschke's.
§4. A Variational Problem Concerning Total Mean Curvature 213
¢4. A Variational Problem Concerning Total Mean Curvature
In order to gain more information about the infimum
of total mean curvature, one may apply standard techniques of
calculus of variations. We will deal with this variational
problem in this section.
Let M be a compact n-dimensional submanifold (with or
without boundary) of a Euclidean m-space Rm. Let x denote
the position vector of M in Rm . Then
(4.1) x = x(ul,...,un) ,
where ul,...,u are local coordinates of M. If S is a
unit normal vector field of M in Rm. We put
(4.2) x(ul,...,un,t) = x(ul,...,un) + t4,(ul, .un)g(ul,...,un) ,
where 0 is a differentiable function and t lies in a small
interval (-c,c). If 4 0 on the boundary aM of M,
(4.2) is called a normal variation of M in Rm . We only
consider the normal variations which leave aM strongly fixed
in the sense that both 0 and its gradient vanishes identically
on aM. If M has no boundary, there is no restriction on
the normal variation.
Throughout this section, we put xi = a2 andi
gi7 = <xi,xi >. Then the induced metric tensor on M is given
by
g = E g dui®dui .
214 S. Total Mean Curvature
Let (g13) denote the inverse matrix of (gig). The volume
element dV of M is given by
(4.3)
where
(4.4)
dV = * 1 = W du1 n ... A dun ,
W = det (gig
If t is a unit normal vector field which is in the
direction of the mean curvature vector H, then the variation
(4.2) is called an H-variation of M in Rm. Let b denote
the operator (a/at)Jt=p. A submanifold M of Rm is called
H-stationary if b San dV = 0 for all H-variations of M,
M
where a denotes the mean curvature. And M is called
stationary if 6 f an dV = 0 for all normal variations of M.M
It is clear that stationary submanifolds are H-stationary.
If M is a hypersurface, an H-stationary hypersurface is always
stationary.
Let en+l " " 'em be a local frame of orthonormal normal
vector fields on M such that en+l = g and xi,...,xn,en+l,...,em
define the natural orientation of Rm. Then we have
mfr(4.5) er =
(_1W[xi,...,xn,en+l'...,@r,.. ,em)
where [v1,...,Vm-i) denote the vector product of m -1 vectors
vl,...,vm-1 and n denotes the omitted term. Let
2(4.6) xij = va/aui(°a/auk x) = ax/auiau
§ 4. A Variational Problem Concerning Total Mean Curvature 215
(4.7) eri - a/aui(er) = aer/aui .
Then the formulas of Gauss and Weingarten give
'4.8)
(4.9)
xij = E ijxk + E hijer
eri = -E hi)+ E Fries
where hri , = <h(xi,xJ),er>, hit9tJhij
(4.10)s
Ls = <Da/auieres>
and D is the normal connection. From 14.2) we obtain by direct
computations that
(4.11) 6x = en+l ,
(4.12) bxi = Oien+l - E hi+1 j xJ + E Rn+l i er
(4.13) bgi3 = bgij = 24hn+lij
(4.14) bW = 4 (trace A )W ,en+1
where 4i = and hrijgit
htJ. Moreover, we also
have
(4.15) erij = -E hikxkj + E au (is )es
+ E Ari Rtj es (mod xk)
216 5. Total Mean Curvature
(4.16) bxij o ijen+l + E j+ +jtn+l i
n+lk s a (s-
hihkj + + auj tn+1 i)
r s+ n+1 i trjyes (mod xk) ,
where erij = 62er/auiauj and ij = a2Vauiauj. Hence we
have
(4.17) <en+1'bxij> = 'ij - hi+lkhkjl - tn+l i n+l 1
(4.18) <er.6xi j> _ itn+l j + jtn+1h+l k hkj
+ au (tn+l i ) + tn+l tsj ' r = n+2,...,m
From (4.12) we have
(4.19) <xi,ber> =
if r=n+1;
rn+l i' if r = n+2,. .,m
where we have used (4.5). From (4.5), we also have
(4.20) <e8.b(W er)>
= (-1)m+r [xl....,xneen+I"..,@r,.. ,em.bes)
for r ¢ s, where we use the notation
(4.21) [vie ...,vm) = (-1)m-1 <v1,[v2,...,vm)>
§ 4. A Variational Problem Concerning Total Mean Curvature 217
for m vectors vl,...,vm in 1m. Since <er,ber> = 0,
(4.19) and (4.20) imply
(4.22)
(4.23)
(-1)m+r-1 A
+ W (xl,.. ,er,sir
...em,besle. , r = n+ 2,...,m
where gti4t and Ln+1 = E
gti bn+rl t.
From (4.8), (4.22) and (4.23) we find
(4.24) <X., en+l = - E 4)k rij
m (-l)phij
+ E Ws=n+1
(xl,...,xn,en+2,...,em,bes)
(4.25) <xij,ber> _ -E 4) An+l krij(-l)m+r-lhsj
+W
sir
n(xl,...,xn,en+l,...,er,...,em,bes)
ben+l - - xim m-n
+m
(-1W [x1,...,xn,en+2,...,em.berlerr=n+ 2
ber = -4 , '1n+1 xi
Thus, by using (4.8), (4.16), (4.24) and (4.25), we find
218 5. Total Mean Curvature
(4.26) bhn+l = 0 - hn+1 t hn+1 - Rr frij is j i tj n+l i n+l jm m-n s
+ E (-1) hi.s=n+2
s,
(4.27) bhij - i r+1+ 7 1n+1 i - E hi+1 t hr
+ au .(fir
n+l i )
is r r k+ E 1n+1i1's7 - 4)L Ln+Ik i7
+ (-1)m+r-1 hss'r W ij
n
fxl''xnen+l....,er,.. ,em.bes]
r = n+ 2....'m
where i,j = i7 - E k Ilj are the components of the Hessian,k
vd4), of 4 on M.
From (4.13) and (4.26) we obtain
(4.28) b(trace An+l) -p4)+4)IIAn+1II2-02
m
E (tr As)<en+l'Res>s=n+2
where Ar = Aer, IIAn+1II2 = trace(An+1) and R2 = E en+1 thn 1'
Similarly, we have
§4. A Variational Problem Concerning Total Mean Curvature
(4.29) b(tr Ar) = 4) tr(ArAn+1)+2Eg'3 4i tn+lj
)i r ]i s r+ 4 E g 1n+l i; j+ g to+l i tsj
- E (tr As)<er.bes>s¢r
219
where tsi;j (tsi)k
tsk rij'From (4.28) and (4.29)
we get
(4.30) 6(a2) = 2 ((tr An+1)(-&0+4) (IAn+iII2-412
n
m
+ E (tr Ar) [4'tr(Ar An+1) + 2<d4).wn+l>r=n+2
+ E g
where Des r es r
3to+l i; j + 4) E g1 to+l i tsj)
In particular, if the normal variation (4.2) is an
H-variation, then tr Ar = 0 for r = n + 2,...,m. Thus
(4.30) reduces to
(4.31) ba = n(-l4-4) 12+4)IIAn+1II2)
Therefore, by applying (4.3). (4.14) and (4.31), we have
(4.32) gJ
c dV = $ f-c ac-l p4) - 4) ac-l 12M M
n n
,n+ n Y 11A n+lII2 - n4) ac+l)dV , c 2 0
220 5. Total Mean Curvature
If we integrate by parts to get rid of the derivative of 0,
we find
(4.33) f (ac-1 A4))dV = J$(Aac-l)dV
,
M M
where the boundary terms one would expect after integration by
parts vanishes because of our hypothesis on 4) on aM. Combining
(4.32) and (4.33) we find
6pM ac dV = pM { c Aac-1- c-1 f2J n
nac+1+n ac-1 flAn+1II2)dV
From this we see that 6 t ac dV = 0 for all H-variations if
and only if
(4.34) c Aac-1 - cac-1 12 - n2 ac+l + c ac-1IIAn+1II2 = 0 .
In particular, if c = n, this gives us the following.
Theorem 4.1 (Chen and Houh [1)). Let M be an n-dimensional
compact submanifold of Rm . Then M is H-stationary if and
only if the mean curvature a satisfies
(4.35) Aan-1 - an-1 { l2 + na2 - IIAn+1
112) = 0 ,
where An+l denotes the Weingarten map with respect to the unit
vector in the direction of H.
From Theorem 4.1 we obtain immediately the following.
§ 4. A Variational Problem Concerning Total Mean Curvature 221
Corollary 4.1. If an n-dimensional compact submanifold
M in Rm is stationary, then
Aan-1 - an-1(.t2 + na2 - IIAn+1112 ) = 0 .
Corollary 4.2. (Chen (3)) Let M be an n-dimensional
compact hypersurface in 3R n+1. Then M is stationary if
and only if
(4.36) pan-l+n(n-1)(an+l-an-1 T) = 0
where T denotes the scalar curvature of M.
Remark 4.1. If M is a surface in R3 , Corollary 4.2
is already known to Thomsen [1).
Using Theorem 4.1, we may obtain the following.
Theorem 4.2. (Chen and Houh [1)). Let M be a stationary
(or H-stationary) submanifold of Rm. Then M has parallel
mean curvature vector if and only if M is either a minimal
submanifold of Rm or a minimal submanifold of a hypersphere
of Rm
Proof. If M is stationary or H-stational, then by
Theorem 4.1 we have (4.35). If M has parallel mean curvature,
then 0 = DXH = DX(a en+1) = (Xa)en+1+aDXen+1. Thus, a isconstant. if a = 0, then M is a minimal submanifold of
R = O. Equation (4.35)m. If a ¢ 0, then Den+1: O. Thus l2
then implies
222 5. Total Mean Curvature
0 = no2 = n E (ki k]) 2i< ]
where kl,...,kn are the eigenvalues of An+l. This shows
that M is pseudo-umbilical in Rm. Therefore, by applying
Proposition 4.4.2, we conclude that M is minimal in a
hypersphere of Rm. The converse of this is trivial. (Q.E.D.)
Theorem 4.3. (Chen and Houh [1]). The only compact
pseudo-umbilical stationary (or H-stationary) submanifolds
(without boundary) of Rm are minimal submanifolds of a
hypersphere of Rm .
Proof. Since M is pseudo-umbilical, we have either a = 0
or no2 = IIAn+1112. Thus, Theorem 4.1 implies
pan-1 - an-1 2 2 = O .
(4.38) -Ao2n-2 = 2a2n-2- 2+ Ildan-lilt 0
Thus, by Divergence Theorem, we conclude that a is a constant
which is non-zero. Thus (4.37) implies k2 = 0. And hence
the mean curvature vector H is parallel. consequently, by
Proposition 4.4.2, M is a minimal submanifold of a hypersphere
of Rm .(Q.E.D. )
Theorem 4.4. (Chen [3]). The only odd-dimensional com act
stationary hypersurfaces in Rn+l are h ers heres.
§4. A Variational Problem Concerning Total Mean Curvature 223
Proof. Let M be an odd-dimensional compact stationary
hypersurface inRn+l.
Then Corollary 4.2 implies
(4.39) pan-l+n(n-1)(an+l-an-l r)=0
Let kl,...,kn be the principal curvatures of M inRn+l
Then
1 ! .a (kl + ... + kn 2 k kn) T= n n-1) i<7 i
Therefore, we have a2 -, 2 O. Thus
tan-1= n(n-1)an-1 (T -a
21 < 0 .
By Hopf's lemma, we obtain tan-l= O. This implies that
T = a2, i.e., M is totally umbilical inRn+1.
Consequently,
by Proposition 4.4.1, M is a hypersphere of Rn+1 (Q.E.D.)
Theorem 4.5. (Chen (3).) If n is even, then the only
compact stationary hypersurfaces in Rn+l such that the mean
curvature does not change sign are hyperspheres.
This theorem can be proved in a similar way as Theorem 4.4.
Remark 4.2. Theorem 4.1 shows that minimal submanifolds
of a hypersphere of IRm are H-stationary submanifolds in
R . However, there are many other H-stationary or stationarym
compact submanifolds which are not of this type. For example,
consider the anchor ring in R3 given by
224 5. Total Mean Gbrvature
x1 = fa+bcosu)cosv, x2 = (a+bcosu)sinv, x3 = bsinu
By direct computation, we have
where
2a(a2-G) = a2 r+bcosu4b3r
Moreover, we also have
&a = -atb+acosu)2b2r3
Consequently, we find
(4.40))pa+ 2a(a2 - G) = a(a2 - 2b2
4b3r3
This shows that the anchor ring is stationary if and only if
a - b.
Remark 4.3. Let M be a surface in a 3-dimensional
Riemannian manifold (M,g). Denote by A the Laplacian on
M with the induced metric If * 2-g. q = p g is a conformal
change of metric on M. Denote by g* the induced metric on
M obtained from g*. Then the Laplacian p* of (M,g*)
satisfies
a+ 2b cos u cos ua = 2b (a+b cos u) ' G = br
r = a + b cos u. Then we have
(4.41) A* =P-2
a .
§4. A Variational Problem Concerning Total Mean Ckirvature 225
Define an operator on M by
(4.42) = A+(2a2-11h,12)I .
Then M is stationary in M if and only if a = 0. From
(3.11) and (4.42), we find
(4.43)
*where denotes the corresponding operator on M with respect
*to g
Remark 4.4. Weiner [1) defined a surface M in a
3-dimensional Riemanifold manifold M to be stationary if
5J
(IHI2+ R')dV = 0 for any variation of M in M. Since the
integralJ
(JH12+ R')dV is a conformal invariant, (Proposition
3.4), the equation Aa2 + a (2a2 - ItA3112) = 0 is itself invariantunder conformal transformations. In particular, if a denotes
the stereographic projection from S3 onto R3, then an
immersion f :M 4 S3 is stationary if and only if a of
M + R3 is stationary. Thus, by using Lawson's examples of
compact minimal surfaces in S3, we obtain compact stationary
surfaces in R 3 of arbitrary genus. (cf. Corollary 1 of
Weiner (11.) In fact, the stationary anchor ring given in
Remark 4.2 is one of the examples given by the sterographic
projection in which the minimal surface in S3 is the square
torus (or called the Clifford torus.)
Remark 4.5. For further results in this direction, see
J. Weiner (1), Willmore and Jhaveri [1).
226 5. Total Mean 04rvature
5. Surfaces in Rm which are Conformally Equivalent to
a Flat Surface
The main purpose of this section is to improve inequality
(2.1) on total mean curvature for certain compact surfaces in
an arbitrary Euclidean m-space Rm. According to Nash's
Theorem, every compact Riemannian surface can be isometrically
imbedded in a Rm for large m.
Definition 5.1. A compact surface M in Rm is called
conformally equivalent to a flat surface if it is the image
of a compact flat surface under a conformal map of Rm,
i.e., M is equivalent to a compact flat surface up to
conformal maps or diffeomorphisms of Rm .
For such surfaces we have the following best possible
result.
Theorem 5.1. (Chen [9,191) Let M be a compact surface
in Rm which is conformally equivalent to a flat surface.
Then we have
(5.1) f1H12dV > 2n2 .
The ecuality sign of (5.1) holds if and only if M is a
conformal Clifford torus, i.e., M is conformally equivalent
to a square torus.
Proof. Since the total mean curvature of a compact
surface in Rm is a conformal invariant (Propositions 3.4
and 3.5), it suffices to prove the theorem only for compact
§5. Surfaces in mm which are Conformally Equivalent to a Flat Surface 227
flat surfaces in Itm. For each point p in M, we denote
by A the map;
(5.2) A : Tp`(M) -4 End(Tp(M),Tp(M))
by A(e) = Ae. Let Op denote the kernel of A. Then we
have dim Op 2 m-5. Denote by Np the subspace of Tp(M)
given by
Tp(M) = Np O 0 p , N p 1 Op
Then we have A(e) = 0 for any e in Op. We choose an
orthonormal normal frame e3,...,em at p in such a way that
e ,...,e . Then, for each unit normal vector e atE 0m6 t.,,p
(5.3)m
e = E cos 9 err=3
Thus the Lipschitz-Killing curvature at (p,e) is given by
(5.4) G(p,e) E cos erh11)( E cos 6sh22) -( E cos eth12)2r=3 s=3 t=3
The right-hand side of (5.4) is a quadratic form on cos er.
Hence, by choosing suitable e3, e4, e5 at p, we have
(5.5) G(p,e) = a1(p) cos2e3+x2(p) cos264 + X3(p) cos2e5
al 2 a2 2 a3
Moreover, since M is a flat surface, we find
(5.6) )'1+a2+a3 = 0 1 aA = det(A2+A) .
228 5. Total Mean CLrvature
In particular, we have )`1 2 0 and )'3 S 0. We consider the
cases)`2
0 and X2 < 0, separately.
Case 1: alP l2 2 0. From (5.6) we have
(5.7) G(p,e) = 11(cos2e3 -cos2e5) +'X2(cos204 -cos2e5) .
Hence,
(5.8) ! IG(p,e)Jda =sp
J 111(cos2e3 - cos2e5) + ))2(cos204 - cos2e5) IdoS
P
X1(p)JS
i cos2e3 - cos285Ida
P
+ X2(P) J i cos2e4 - cos2e5Ida .
Sp
On the other hand, by a formula on spherical integration, we
have
(5.9) fS
I Cos20r - Cos2es 1da = 2cm_1/tr2 , r -/ sp
Hence, by (5.8) and (5.9) we find
(5.10)J
IG(p.e)lda2c
m- 1(Xl(p)+ )L2(P))
By the definition of H, we have
§ 5. Surfaces in IRm which are Conforrnally Equivalent to a Flat Surface 229
(5.11) 4IHI2 = h3 +h22)2+ (hll+h22)2+ (hll+h22)2
11
(3 )2+ (h22)2+2x1+2(hi2)211
4(x1+)6 2)
combining (5.10) and (5.11), we obtain
(5.12)
2
IHI2 (p)2cm_1
where G*(p) =J
IG(p,e)Ida.Sp
Case 2: x2, x3 < 0. From (5.6) we find
G(p,e) = x2(cos204 -cos2g3) +x3(cos295 -cos2g3) .
Thus, from (5.6) and (5.9). we obtain
(5.13) G*(p) S -x2JS
,Icos294 - cos203Idap
- X3 J Icos2a5 - cos293Idasp
2 x1cm-1/rr2
On the other hand, we also have
4IHI2 (hi1)2+ (h2?)2+ 2X1+ 2(hi2)2
2 4x1+4(h12)2 4x1
230 S. Total Mean Curvature
Hence, we get
(5.14) JHJ2 22cTF
G* (p)
M-1
Consequently, we obtain (5.12) in general. Thus, by taking
integration of both sides of (5.12), we obtain
(5.15) fM IHI2dV _2
i b(M)
by virtue of Theorem 4.7.1. Now, since M is compact and flat,
M is either diffeomorphic to a 2-torus or diffeomorphic to
a Klein bottle. In both cases, we have b(M) = 4. Thus (5.15)
implies
(5.16) S IH12dV , 2,2 .
If the equality of (5.16) holds, then the inequalities
in (5.8) and (5.13) become equalities. Hence, at least one
of 1 and X2
is zero for the first case and at least one
of2
andX3
is zero for the second case. However, this
implies that the second case cannot occur. Thus, we find
X2 = 0 identically on M. Furthermore, since the equality
signs of (5.11) hold, we have
3 3 3 4h11 = h22 , h12 = h12 = 0
41=
42,
51+
52 = O .
Now, because)`2
= 0, these imply
§5. Surfaces in IRm which are Con formally Equivalent to a Fiat Surface 231
(5.17) h3 = h3 , h3 = 0, h4 = O, h5 + h5 = 011 22' 12
i.
11 22
Consequently, by choosing suitable orthonormal frame
el,e2,e3, ..,em, we have
a O a0(5.8) A = A = , A = ... = A = 0 .
3 0 a 4 0 -a 5 m
In particular, by Proposition 4.3.2, M has flat normal
connection in Rm. Thus, by applying Proposition 4.3.1, we
see that, there exist locally orthonormal normal frame
e3,...,em such that
(5.18) De3 = = Dem = 0 .
We put
(5.19) er = E arses , r = 3,...,m
Then (ars ) is an orthonormal (m - 2) x (m - 2) - matrix.
Since M is two-dimensional and our study is local, we
may assume that M is covered by an isothermal coordinate
system (x,y) such that the metric on M has the form
g = E(dx2+ dy2). Denote by X1 and X2 the coordinate vector
fields a/ax and 6/by, respectively. We put
(5.20) L = h(X1,X1) , M = h(X1,X2) , N = h(X2,X2)
and vX X _ :i Xk. Then we havei
232
(5.21)
5. Total Mean Curvature
r1 = r12 = -r22 = X1E/2E11
Therefore, the Codazzi equation reduces to
DX2
L - DX1
M = (X2E )H ,
(5.22)
DX2
M -Dx1
N = -(X,E)H .
Since X1 and X2 are orthonormal, we may define a function
e = e(X,Y) by
(5.23)
X1 = cos 9 e1+sin 9 e2 ,
X2 = -sine el + cos g e2 .
With respect to the frame field X1, X2, e3,...,em, the second
fundamental tensors are given by
a(arl +a r2 cos 20) -aar2 sin 2e `
Ar = r=3....,m,-aar2 sin 29 a(arl - ar2 cos 29)
Since M is flat, we have E = 1. Thus, by (5.18), equation
(5.22) of Codazzi reduces to
(5.24) ay (a(arl+ ar2 cos 2e)] a (aar2 sin 2e1
(5.25) -ay (aar2 sin 2 e] = a [a(arl- ar2 cos 29))
§5. Surfaces in IRm which are Confornwily Equivalent to a Flat Surface 233
Multiplying arl to (5.24) and summing over r, then, by
using the fact that (ars) E O(m -2), we get
as as(5.26) a In a = E ( ayl)ar2 cos 29+( axl)ar2 sin 29)
Similarly, multiplying arl to (5.25) and summing over r,
we have
(5.27)asI In a = E ( ( ay)ar2 sin 29 -
(asa_Jr')ar2cos 29) )
Multiplying ar2 to (5.24) and summing over r, we find
ar2(aayl) + (a axn a)sin 29 + (a ay a )cos 29
2 ay sin 29 - 2 ax cos 29
By substituting (5.26) and (5.27) into this equation, we get
(5.28) L ar2 aayl = sin 20 " - cos 28 ax
Similarly, by multiplying ar2 to (5.25) and summing over r
and using (5.26) and (5.27), we get
(5.29) E ar2 aaxl = -cos 29y - sin 29ax
Substituting (5.28) and (5.29) into (5.26) and (5.27), we may
find
Ana = 2& a Ana =-beax by ' by ax
234 S. Total Mean Curvature
From this we get
(5.30)2 2
( + a2)(En a) = 06x by
Since E = 1, (5.30) implies
(5.31) A in JHI2 = A in a2 = 0 .
Because IHJ 2 is a non-negative differentiable function on M
and M is compact, (5.31) implies that IHJ is a positive
constant (cf. Yau [1)). We put
e = cos g e3+ sin g e4(5.32)
e4 = sin 9 e3 - cos 8 e4 , e5 = e5""' em = em
With respect to el,e2,e3, ..,em, we have
a O O OC(5.33) A3 = A4 =A5 = ... = Am = 0
O O O 2a
From (5.33) and the structure equations of Cartan, we may
easily find that both the distributions T. = []R ei), i = 1,2,
are parallel. Thus, by the deRham decomposition theorem, we
see that M = C1 xC2, where C. is the maximal integral
manifold of Ti. Moreover, because h(el,e2) = 0, a result
of Moore (1] implies that M is a product submanifold where
C1 is in a linear r-space IRr and C2 is in a linear
. Thus, by a result of Kuiper [ 1) ,(m - r) - space R-r them
§5. Surfaces in IRm which are Conformally Equivalent to a Flat Surface 235
total absolute curvature of M in Rm is the product of
total absolute curvatures of C1 and C2. Because the total
absolute curvature of M in Rm is equal to 4 by (5.14),
the total absolute curvatures of C1 and C2 are both equal
to 2. Hence, by applying Fenchel-Bosuk's result to C1 and
C2, we conclude that both C1 and C2 are planar curves with
curvature .a which are constants. Therefore, C1 and C2
are circles of the same radius. Consequently, M is a square
torus in a linear 4-space R4 . The converse of this is clear.
(Q.E.D.)
Remark 5.1. If M is a flat torus in Rm such that
M is homothetic to the flat torus RX /I', with r generated
by (1,0) and (x,y) with 0 < x2
and y 2 J1- x2 , Li
and Yau (1) obtain very recently the following inequality.
(5.34) f IH12dV ',2(y+y)M
Remark 5.2. From Theorem 5.1, we obtained immediately the
following.
Corollary 5.1. If M is a compact surface in Rm which
is conformally equivalent to a Klein bottle, then we have
(5.35) JHI2dV > 27r2
M
236 5. Total Mean Curvature
*6. Surfaces in R4
The main purpose of this section is to improve inequality
(2.1) on total mean curvature for certain surfaces in R4
Let M be a compact surface in Rm. We choose an
orthonormal normal frame e3,...,em of M in Rm. Let
e = cos grey be a normal vector of M at p. Then we have
Ae = E cos grAr. Thus
(6.1) G(p,e) = det(Ae) = det(E cos grAr) .
Since the right-hand side of (6.1) is a quadratic form of
cos 03,...,cos gm. Thus we may choose a suitable local orthonormal
normal frame e3,...,em such that with respect to er, we have
m2(6.2) E "r_2 cos9r , Al -' 12 ... 'm-2
r=3
We call such a frame as Otsuki frame. We call lA, A = 1,2,...,m-2,
the A-th curvature of the surface M in Rm. If M is a
surface in R4 , we simply denote by ). the first curvature
ll and by p the second curvature X a.
R4
Theorem 6.1. (Chen [9)) Let M be a compact surface in
If M has non-positive Gauss curvature, then we have
(6.3) P JHj2dV
k 2,r2 .M
If H V( 0, then the equality sign holds if and only if M is
a square torus in R4 .
§ 6. Swjaces in IR 237
Proof. Let M be a compact surface in R4 . Denote by
BI the bundle of all unit normal vectors of M in R4 and
by Tr:B1 4 M the projection of BI onto M. Let
W = (p E M IX(p) Z 0). Then by the hypothesis, we have
(6.4) JG(p,e)J = 1% cos2e+ p sin2el
= IX cos 2e+ G sin2el
S XIcos 291 -G sin2a
on it 1fW), where e = cos a e3 + sin a e4 and e3
an Otsuki frame of M in R4 . Thus, we find
e4 form
(6.5) J -l IG(p,e) JdV Ado S 4 J )LdV -ir J GdVir (W) W W
On Bl - rl(W), we have
(6.6)J -1
IG(p,e)IdV Ado = -f1
G(p,e)dV AdoBl-tr (W) Bl-?r (W)
= -r fM-W
G(p)dV .
On the other hand, by definitions, we have on W,
(6.7) 41HI2 = E (tr A )2 = 2G+ E IIAr=3 r r=3 r
2 2(a+µ)+2(). -4) = 4k .
Thus, we get
238
(6.8)
5. Total Mean Curvature
f IHI2dV , f dVM W
4 f _l IG(p,e)IdVnda+4 f G dVn (W) 4 W
+ IG(p,e)IdV n do4 fBl-r-1(W)
+ 4 G dVfM-W
= 4 f IG(p,e)IdVAdo+4 f G dVB M
2
2 2 (b(M) + X(M)) = 2r2
This proves inequality (6.3). If the equality sign of (6.3)
holds, then all the inequalities in (6.4) - (6.8) become
equalities. Assume that IHI > 0, then (6.8) implies that
W = M. Moreover, from (6.7), we see that M is pseudo-umbilical
in R4 . Furthermore, from (6.4) we find
(6.9) I%cos20+Gsin26l = alcos 2aI -G sin2a
for all q. Thus G = 0, i.e., M is flat. Consequently,
from the proof of Theorem 5.1, we conclude that M is a square
torus in R4 . (Q.E.D.)
Combining Corollary 3.1 or Proposition 3.5 with Theorem 6.1,
we obtain immediately the following.
Corollary 6.1. Let M be a compact surface in Rm
4which is conformally equivalent to a compact surface in R
§6. Surfaces in !R' 239
with non-positive Gauss curvature. Then we have
(6.10) 1 JH12dV 2 2r2 .
M
From Chen (9], we also have the following.
Theorem 6.2. Let M be a compact surface in Rm which
is conformally equivalent to a compact surface in R4 with
non-negative Gauss curvature. If we have
(6.11) S JH12dV s (2+Tr)rr ,
then M is homeomorphic to a 2-sphere.
For the proof of this theorem, see Chen (9].
Let f : M - R4 be an immersion of an oriented compact
surface into R4 . By applying regular deformation to f if
necessary, f(M) intersects itself transversally, thus, f(M)
intersects itself at isolated points. At each point p of
self-intersection, we assign +1 if the direct sum orientation
of the two complementary tangent planes equals to the given
orientation on R4 , and we assign -1 otherwise. Then the
self-intersection number is defined as the sum of the local
contributions from all the points of self-intersections. It is
known that the self-intersection number If is an immersion
invariant up to regular homotopy of M into R4 . We mention
the following result of S. Smale (1] for later use.
Theorem 6.3. Two immersions of S2 into R4 are
regularly homotopic if and only if they have the same self-
240 5. Total Mean Q rvature
intersection number.
This theorem says that the self-intersection number is
the only regular homotopic invariant of S2 in R4
For surfaces in R4 , we also have the following.
Theorem 6.4 (Wintgen (2]). Let f : M .. R4 be an immersion
of a compact oriented surface M into R4 . Then we have
(6.12)SM
IHI2dV 4ir(l+ IIf I -g) ,
where g denotes the genus of M.
Proof. We choose an orthonormal local frame el,e2'e3'e4
in R4 such that, restricted to M, el,e2 are tangent to
M and e3,e4 are normal to M. Then the Gauss curvature G
and the normal curvature GD are given respectively by
G - R(el,e2;e2,e1) = E (h11h22 -(h12)2)
GD = RD(el,e2;e4,e3)- h12(h22 - hll)
-h412(h22 - hll
Thus, the mean curvature vector satisfies
IHI2 1 ((h131+h22)2+ (h411 +h22)2)
E
h3 h32
h4 h4 2112 22 + 112
22 + (h12)2+ (h12)2+G
4 3 3 3 4 421 Ih11 - h221 + Ih12I 1hll - h221 + GIh1
IGDI +G .
§ 6. Surfaces in !R
Hence, we have
(6.13)J
IHI2dV > f IGDIdV + f G dV .
M M M
241
It is known that the integral of the Gauss curvature G gives
2rr X(M) and the integral of the normal curvature GD gives
2rr XD(M), where XD(M) denotes the Euler number of the normal
bundle (see, for instance, Little (1]). Thus, (6.13) implies
(6.14) f IHI2dV > 2?r()((M) + IXD(M) I)M
On the other hand, by a result of Lashof and Smale (1), we
have XD(M) = 2 If. Thus, by (6.14), we obtain (6.12). (Q.E.D.)
Combining Theorems 6.3 and 6.4, we have the following.
Theorem 6.5. (Wintgen' [2]) . Let f -S 2 -. R4 be an
immersion of a 2-sphere S2 into R4 . If
(6.15) f IHI2dV < 8tr ,
then f is regularly homotopic to the standard imbedding of
S2 into a linear 3-space R3
If f :M -+ R4 is an imbedding of a compact surface
M into R4 , the fundamental group irl(R4 - f(M)) of
R4 - f(M) is called the knot group of f. The minimal number
of generators of knot group of f is called the knot number
of f. Wintgen obtained the following relation between total
mean curvature and knot number:
242 5. Total Mean Curvature
Theorem 6.6 (Wintgen (1)). Let f : M -* R4 be an
imbedding of a compact surface M into R4 . Then we have
(6.16) IH 12dV 47r pM
where p denotes the knot number of f.
Proof. We need the following simple lemma:
Lemma 6.1. Let ha be a height function of m in
R4 which has only non-degenerate critical points on M.
Then the number (30(k) of local minima satisfies 00/ha) -> p.
Without loss of generality we can assume that ha takes
different values at the critical points pi (i = 0,1,...,t)
written in the order induced from ha. Let c. be real
numbers with
c0 < ha(p0) < c1 < ha(pl) < ... < ha(pt) < at+1
By a result of van Kampen for the fundamental groups
of the spaces Hi = (p E R4 - M I<P.a> < cj), we have
'rl(Hj+l) p ,r1(Hj) + one generator,
if pj is a local minimum;
7T 1(Hj+1) ,rI(H.) + one relation,if pj is a saddle point;
irl(Hj+l) N Tr1(H.), if pj is a local maximum.
§6. Surfaces in JR" 243
The lemma follows from these relations.
We denote by A2(ha) the number of local maxima of ha.
Since A2(ha) = 13 0(h-a), Lemma 6.1 implies p2(ha) Z p. For
each critical point p of ha, a is normal to M at p.
Moreover, Ae is semi-definite if ha is either local maximum
or local minimum at p. Let U denote the set of all elements
(p,e) in B1 such that Al is semi-definite. Then according
to above observation, we see that the unit sphere S3 is
covered by U at least 2p times under the map v : B1 .. S3
Thus, by a similar argument as given in the proof of Theorem 2.1,
we obtain (6.16). (Q.E.D.)
Remark 6.1. For a surface in I23, Theorem 6.6 improves
Theorem 1.2 if knot number is 2 3 and, for a surface in it4
Theorem 6.6 improves Theorem 2.1 if the knot number is 2 2.
Remark 6.2. Lemma 6.1 is essentially due to Sunday [1).
Remark 6.3. Theorems 1.2, 6.2, 6.4, 6.5 and 6.6 can be
regarded as partial solutions to Problem 2.2.
244 5. Total Mean Curvature
*7. Surfaces in Real-Space-Forms
Let f :M -. FP(c) be an isometric immersion of a compact
oriented surface M into a real-space-form of constant curvature
c. By Ricci's equation, the normal curvature tensor RD
satisfies
(7.1) RD(X,Y)g = h(X,A9Y)-h(A9X,Y)
for X, Y tangent to M and g normal to M. Let (X1,X2)
be an orthonormal tangent frame. We put hij = h(Xi,X
i , j = 1 , 2 . We define a A b as the endomorphism
(7.2) (aAb)(c) = <b,c>a-<a,c>b .
Then (7.1) becomes
(7.3) RD(Xl,X2) = (h11-h22) Ah 12 .
The mean curvature vector H and the Gauss curvature G are
given by
(7.4) 4IHI2 = Ih11 +h22I2
, G = <h11,h22> -Ih12I2 + c
For each point p in M. We put
(7.5) Ep = (h(X,X) IX E Tp(M), IXI = 1)
If X = cos 0 X1+ cos 8 X2, then
§ 7. Surfaces in Real- Space- Forms 245
h(X,X) = H+ cos 2e h11-h22 + sin 2e h12
This shows that Ep is an ellipse in the normal space Tp(M)
centered at H. Moreover, as X goes once around the unit
tangent circle, h(X,X) goes twice around the ellipse. We
notice that this ellipse could degenerate into a line segment
on a point. we call this ellipse EP the ellipse of curvature
at p. The ellipse Ep is degenerate if and only if RD = O
at p.
If RD # 0, then h11-h22 and h12 are linearly independent
and we can define a 2-plane subbundle N of the normal bundle
T.L(M). This plane bundle inherits a Riemannian connection
from that of T1'(M). Let (e3.e4) be an orthonormal oriented
frame of N. We define the normal curvature GD of M in
TP(c) by
(7.6) G- = <RD(Xl.X2)e4,e3>
Since M and N are oriented, GD is globally defined. Let
N' be the orthogonal complementary subbundle of N in T1(N).
Then we have the following splitting of the normal bundle;
TA. (N) - N ® Nl. From the definition of N1. we have
(7.7) RD(X1.X2)S = 0 if P, E N1 .
Let a0 = a0(M) denote the bundle of symmetric endomorphism
of the tangent bundle T(M). Define a map 4 :N -. a0 by
246 5. Total Mean Curvature
(7.8)tr A
A - 2F $ E N .
Thus, because RD 0 by assumption, [A
e3.Ae
] ¢ 0. Thus
(7.8) implies that 4 is an isomorphism. We denote by X(N)
the Euler characteristic of the oriented 2-plane bundle N
over M. We mention the following extension of a result of
Little [1], Asperti [1] and Dajczer [i);
Proposition 7.1 (Asperti-Ferus-Rodriguez [1]). For a
compact, oriented Riemannian surface M isometrically immersed
in a real-space-form Mm(c) with nowhere vanishing normal
curvature tensor, we have
(7.9) X(N) = 2%(M) .
Proof. Let a0 = a0(M) be the bundle of symmetric
endomorphism endowed with the orientation induced by that of
N via 4. Then because4'
is an orientation-preserving
isomorphism, we have X(N) = )((a0(M)). For each X E Tp(M),
let B(X) be the element in a0(M) at p given by
B(X)(Y) = 2<X.Y>X - <X,X>Y .
Then B(cos tX + sin tXl) = cos 2t B(X)+sin 2t B(X)l, where
Xl is a vector in Tp(M) such that IXjI = IXI, X 1X1 and
X, X1 give the orientation of M. Therefore, the index
formula for the Euler characteristic applied to a generic
vector field X and to B(X), respectively, yields the
proposition. (Q.E.D.)
§ 7. Surfaces in Real- Space- Forms 247
The following result is a generalization of Theorem 6.4.
Theorem 7.1. (Guadalupe and Rodriquez [11). Let
f :M -o Mmfc) be an isometric immersion of a compact oriented
surface M into an orientable m-dimensional real-space-form
Mm(c). Then we have
(7.10) f IH12dV 2 27 X(M) + I f GD dVI -c vol(M) .
M M
The equality holds if and only if GD does not change sign
and the ellipse of curvature is a circle at every point.
Proof. From (7.1) and (7.6) we have
(7.11)D
G=
Ih11 - h22IIh12I
Thus, (7.4) and (7.11) imply
0 ( Ih11 - h221 - 21h121 )2
Ihll -h2212+41h1212-41h11 -h221 Ih121
= Ih1112+ 1h22I2+21h1212-2G-4IGDI+2c
IlhIl2-2G-4IGDI+2C .
On the other hand,
41HI2Ih11+ h22I2 Ih11I2 + (h22I2+ 2<hillh22>
=1 h 1 1 1 2 + (h221
2+ 21h1212+ 2G- 2c
= 1Ih1I 2 + 2G - 2c .
248 5. Total Mean CLrvature
Hence, we find
(7.12) IHI2+ c _> G+ IGDI
with equality holding if and only if 2 (h11 - h22) = h12'
i.e., the ellipse of curvature is a circle. Integrating (7.12)
over M gives (7.10). Moreover, the equality of (7.10) holds
if and only if GD does not change sign and the ellipse is
always a circle.
Corollary 7.1. (Guadalupe and Rodriguez [1)). Let M
be a compact oriented surface immersed in R4 . If the normal
curvature GD > 0 everywhere, then
(7.13).
IHI2dV 12ir
The equality holds if and only if the ellipse of curvature is
always a circle.
Proof. If GD > 0 everywhere, X(N) = 2nJGD dV > 0.
Thus M is homeomorphic to S2. Hence, X(N) = 2X(M) = 4,
which yields (7.13) by using (7.10). (Q.E.D.)
Remark 7.1. Atiyah and Lawson (1) have shown that an
immersed surface in S4 has the ellipse always a circle if and
only if the canonical lift of the immersion map into the bundle
of almost complex structure of S4 is holomorphic. Holomorphic
curves in this bundle can also be projected down to S4 in order
to obtain examples of surfaces in S4 with the property that the
ellipse is always a circle, hence giving equality in (7.10).
Chapter 6. SUBMANIFOLDS OF FINITE TYPE
§1. Order of Submanifolds
It is well known that an algebraic manifold (or an
algebraic variety) is defined by algebraic equations. Thus,
one may define the notion of the degree of an algebraic
manifold by its algebraic structure (which can also be defined
by using homology). The concept of degree is both important
and fundamental in algebraic geometry. On the other hand, one
cannot talk about the degree of an arbitrary submanifold in
IItm . In this section, we will use the induced Riemannian
structure on a submanifold M of Rm to introduce two well-
defined numbers p and q associated with the submanifold M.
Here p is a positive integer and q is either + . or an
integer S p. We call the pair [p,q] the order of the sub-
manifold M (Chen [151,22,25]). The submanifold M is said
to be of finite type if q is finite. The notion of order
will be used to study submanifolds of finite type in sections 2
through 5. It was used in sections 6 and 7 to study total mean
curvature and some related geometric inequalities. The notion
of order will be also used to estimate the eigenvalues of the
Laplacian of M in the last three sections.
The order of a submanifold is defined as follows. Let
M be a compact Riemannian manifold and A the Laplacian of
M acting on C+(M). Then A is a self-adjoint elliptic
operator and it has an infinite, discrete sequence of eigen-
values (cf. 43.2):
250 6. Submanifolds of Finite Type
(1 .1) 0 = )`0 < al < %2 ... < lk < ... t
Let Vk = (f E C '(M) I Of = lkf} be the eigenspace of a
with eigenvalue Xk. Then Vk is finite-dimensional. We
define as before an inner product ( , ) on C (M) by
(1.2) (f,g) = f fg dVM
Then E 0 Vk is dense in COO(M) (in L2-sense). Denote
by 0 Vk the completion of E Vk, we have (cf. Theorem 3.2.2)
C (M) ='kVk
For each function f E C(M), let ft be the projection
of f onto the suspace Vt (t = 0,1,2,...). Then we have the
following spectral decomposition
(1.4) f = E ft, (in L2-sense)t=O
Because V0 is 1-dimensional, for any non-constant
function f E C *(M), there is a positive integer p z 1
such that fp 1 0 and
(1.5) f - fO= E fttap
where f0 E V0 is a constant. If there are infinite ft's
which are nonzero, we put q = . Otherwise, there is an
integer q, q a p, such that fq V 0 and
(1.6)qf - fo = E ft
t=p
§ 1. Order of Submanifolds 251
If we allow q to be W, we have the decomposition (1.6)
mfor any f E C (M).
For an isometric immersion x :M 4 IRm of a compact
Riemannian manifold M into IRm, we put
(1.7) x = (xl,...,xm) ,
where xA is the A-th Euclidean coordinate function of M
in 1Rm . For each xA, we have
qAA = 1,...,m .(1.8) xA -(xA)
O= tF (xA)At=PA
For each isometric immersion x : M + ]Rm , we put
(1.9) p = p(x) = iAnf(pA}, q = q(x) = sAup(gA)
where A ranges among all A such that xA - (xA) 71 O. It
is easy to see that p is an integer it 1 and q is either
or an integer z p. Moreover, it is easy to see that p
and q are independent of the choice of the Euclidean coor-
dinate system on 1Rm . Thus p and q are well-defined.
Consequently, for each compact submanifold M in ]Rm (or,
more precisely, for each isometric immersion x : M + ]Rm), we
have a pair [p,q) associated with M. We call the pair
[p,q] the order of the submanifold M.
By using (1.7), (1.8) and (1.9) we have the following
spectral decomposition of x in vector form:
252
(1.10)
6. Submanifolds of Finite Type
qx = x0 + E xt
t=p
Definition 1.1. A compact submanifold M in Mm is
said to be of finite type if q is finite. Otherwise M is
of infinite type (Chen (22,25])".
Definition 1.2. A compact submanifold M inRm
is
said to be of k-type (k = 1,2,3,...) if there are exactly
k nonzero xt's (t t 1) in the decomposition (1.10).
For a submanifold M of order [p,q), we sometime say
that M is of order z p (or of order s q) if q (or p)
is not considered. A submanifold of order [p,q] is also
called a submanifold of order p.
Remark 1.1. Let M be a compact submanifold of Rm .
It is easy to see that M is of k-type in Rm (resp.,
of infinite type in 3Rm) if and only if M is of k-type
in any Rm+m DJRm (reap., of infinite type in any
Rm+mM
Rm)
Lemma 1.1. Let x : M -0 ]Rm be an isometric immersion
of a compact Riemannian manifold M into Rm. Then x0 is
the centroid of M in Rm.
Proof. Consider the decomposition
(1.11) x = E xtt=O
We have Axt = atxt. If t y/ 0., then Hopf lemma implies
§ 1. Order of Submanifolds 253
(1.12) f xt dv - -11 Ax dV = OM t M
t
Since x0 is a constant vector in 3tm, we obtain from
(1.11) and (1.12) that
(1.13) x0 = f x dV / vol (M) .M
This shows that x0 is the centroid of M. (Q.E.D.)
Lemma 1.1 shows that if we choose the centroid of M
(in 3tm) as the origin of 3tm , then we have
(1.14)q
x = E xtt=p
Let v1 and v2 be two Htm-valued functions on M.
We define the inner product of vl and v2 by
(1.15) (vl,v2) = fM
< v1,v2 >dV ,
where <v1 ,v2 > denotes the Euclidean inner product of
v1.v2. We have the following.
Lemma 1.2. Let x :M -6 IRM be an isometric immersion
of a compact Riemannian manifold M into 3tm. Than we have
(1.16) (xt,xs) = 0 for t ¢ s ,
where xt is the t-th component of x with respect to the
spectral decomposition (1.10).
Proof. Since A is self-adjoint, we have
254 6. Submanifolds of Finite Type
at(xt,xs) = (Axt,xs) = (xt,Axs) = Xs(xt,xs)
Because at i as, we obtain (1.16). (Q.E.D.)
§ 2. Submanifolds of Finite Type
42. Submanifolds of Finite Type
First, we rephrase Proposition 4.5.1 of Takahashi in
terms of order of submanifolds as follows:
Proposition 2.1. Let x :M -]m
be an isometric
immersion of a compact Riemannian manifold M into 7Rm.
Then x is of 1-type if and only if M is a minimal sub-
manifold of a hypersphere of ]Rm
From this proposition, we see that if M is a compact
minimal submanifold of a hypersphere SD-1(r) centered at
the origin, then we have
(2.1)
for some constant X X. Because Ax = - nH (Lemma 4.5.1),
(2.1) implies
(2.2) HH = X H, ap E 7R .
255
In views of this, we give the following characterization
of submanifolds of finite type (Chen (221).
Theorem 2 . 1 . Let x : M + 1 m be an isometric immersion
of a compact Riemannian manifold M into ]Rm. Then M is
of finite type if and only if there is a non-trivial polynomial
P such that
(2.3) P(6)H = 0 .
256 6. Submanifolds of Finite Type
In other words, M is of finite type if and only if the mean
curvature vector H satisfies a differential equation of the
form:
(2.4) AkH+c1Ak-1H+ ...+ck-lAH+ckH = O
for some integer k ? 1 and some real numbers c1....,ck.
Proof. Let x : M -0 IRm be an isometric immersion of a
compact Riemannian manifold M into ]Rm . Consider the
following decomposition
(2.5)q
x = x0 + E xt , Axt
t=p xtxt
If M is of finite type, then q < .. From (2.5) we have
(2.6) i - 0x 1 2, ,...t t , ,
t=p
qLet cl E xt
' c2 ltls 1t=pt<s cq-Pt
(-1)q-p+l lp - Xq. Then by direct computation, we find
(2.7) AkH + c1 Ak 1H + ... + ckH - 0 ,
where k - q -p+ 1. Conversely, if H satisfies (2.7) for
k z 0, then, because m is compact, we have k x 1. Consider
the spectral decomposition (2.5). Using (2.6) and (2.7), we
find
1 i+1-nA1H =
(2.8) E It0t+c1xt'1+ ...+ek-llt+ek)xt - 0t=1
§ 2. Submanifolds of FYnite Type 257
For each positive integer s, (2.8) gives
m
(2.9) t1at(Xk+clat-1+ ... +ck) fM 0
Since (x5.xt) =J
<xs'x
t> dV = 0 for t ¢ s (Lemma 1.2),
we obtain
(2.10) (fig+c1ag-1+ ...+ck)11xs1j2 = 0
where
(2.11) Nall2 = (xs,xs)
If xs ¢ 0, then jjxsjj ¢ 0. Thus (2.10) implies
(2.12) )+clas-1+ ...+ck = 0
Since equation (2.12) has at most k real solutions and
equation (2.10) holds for any positive integer s, at most
k of the xt's are nonzero. Thus the decomposition (2.5) is
in fact a finite decomposition. Consequently, M is of finite
type.
From the proof of Theorem 2.1, we also have the following.
Theorem 2.2. Let M be a compact submanifold of Mm.
Then M is of k-type (k = if and only if there
is a polynomial P of degree k such that P(t) has exactly
k distinct positive roots and
(2.13) P(A)H = 0 .
258 6. Submanifolds of Finite Type
By using exactly the same proof as Theorems 2.1 and 2.2,
we may also obtain the following.
Theorem 2.1'. Let x : M -4 ]Rm be an isometric immersion
of a compact Riemannian manifold M into IRm. Then M is
of finite type if and only if there is a non-trivial poly-
nomial P(t) such that
(2.14) P(A) (x -x0) = 0 .
Theorem 2.2'. Let M be a compact submanifold of Rm.
Then M is of k-type (k = 1,2,3,...) if and only if there
is a polynomial P of degree k such that P(t) has exactly
k distinct positive roots and
(2.15) P(A) (x-x0) = 0 .
Remark 2.1. From the proof of Theorem 2.1, we see that
the positive roots of P(t) in Theorems 2.2 and 2.2' are in
fact eigenvalues of the Laplacian of M.
The following corollary is an easy consequence of
Theorem 2.1.
Corollary 2.1. Let M be a compact homogeneous space.
If M is equivariantly, isometrically immersed in iRm,
then M is of k-type with k s m.
Proof. Let u be an arbitrary point of M. Then the
m+ 1 vectors H, i at u are linearly dependent.
§ 2. Submanifolds of Finite Type 259
Thus, there is a polynomial P(t) of degree s m such that
P(6)H = 0 at u. Because M is equivariantly isometrically
immersed in IRm, P(A)H = 0 at every point of M. Thus,
by Theorem 2.1 we see that M is of finite type. Moreover,
because P(t) = 0 has at most k roots, M is of k-type
with k s m. (Q.E.D.)
260 6. Submanifolds of Finite Type
03. Examples of 2-type Submanifolds
According to Proposition 2.1 of Takahashi, minimal
submanifolds of hyperspheres of R1Q are 1-type submanifolds
of Rm. Moreover, Corollary 2.1 shows that there exist many
important finite type submanifolds in Rm . In this section,
we will give many examples of 2-type submanifolds in Rm
(Chen [22,25]) .
Example 3.1, (Product Submanifolds). Let M and M
be two compact submanifolds of Rm and Rm . respectively.
Then the product submanifold M xM is of finite type if and
only if both M and M are of finite type. Moreover, if
both M and M are of 1-type, then the product submanifold
M xM is either of 1-type or of 2-type. For instance, con-
sider the flat torus T2 = R2 /A, where A is the lattice
generated by (2rra,O) and (0,2rrb). Then T2 is isometricto the product of two plane circles; T2 = S1(a) xS1(b).
Consider the isometric imbedding x of T2 in R4 by
(3.1) x = x(r,s) _ (a cos a, a sin a, b cos b, b sin b)
Then, by direct computation, we find
(3.2)
(3.3)
2-(.+ 2-7) , x0 = (0,0,0,0)
ar as
H I(acosa, asina, cos.fibSsinb)
1(3.4) AH = 7(- .cos S, 1 sing, - cos b, - 'sin b)a a b b
§ 3. Examples of 2-type Submanifolds 261
(3.5) a2H = 1(1 cos r, 1 sin r, 1 cos e, 1 sin s)T a a s a b b b 16
From (3.1) and (3.3). we see that T2 is of 1-type in II24
if and only if a = b. Assume that a 9d b. From (3.3),
(3.4) and (3.5), we obtain
(3.6) a2H -1.7+ -1.f) AH + -.T1,sl = Oa b a
Thus, by Theorem 2.2, T2 is of 2-type. Let
(3.7) P(t) - t2 1 + 1)t+ a2;17a bThen P(t) has roots
-7
and -17. Thus, by using Proposi-a b
tion 3.5.7, we can conclude that if a > b > then T2
is of order (1,2) in 1R4 .
Example 3.2. (A flat torus in 1R6 .) Again consider
the flat torus
(3.8) T2 = IIt2 /A ,
with A generated by ((2Tra,O) , (0,2Trb)) . Let x : T2 _. IIt6
be defined by
(3.9) x = x(s.t) = (a sins, bsin ssint, bsinscosS,a cos s, b cos s sins, b cos s cosh)
Assume that
(3.10) a2 +b 2= 1 and a,b > 0 .
262 6. Submanifolds of Finite Type
By a direct computation, we have
(3.11) H = + (O, sins sin b, sins cos., O, cos s sin b ,
cos s cosb
) ,
(3.12) SH = (1 + - )H - a (sin s, O. O, cos s, O, O)b 2b
(3.13) A = (1 +) 2H -(2 +1 ) (sin s, O, O, cos s, O, O ) .b 2b b
Consequently, we have
(3.14) A2H - (2 + ) GH + (1 + 2)H = 0
This shows that T2 is of 2-type in IIt6
Example 3.3. (Diagonal immersions.) Let x :M -4 IRm
and x :M a ]Rm be two isometric immersions of a compact
Riemannian manifold M into ]Rm and ltm, respectively.
Then the normalized diagonal immersion x' :M + IItM+m
defined by x'(p) = 1 (x(p),x(p)) is of finite type if2
and only if both x and x are of finite type. In partic-
ular, if both x and x are of 1-type, then we can show
that x' is either of 1-type or of 2-type.
For example, consider the unit 2-sphere in 1R3 by
(3.15) S2 = ( (x,y,z) E IIt3 1 x2+ y2+ z2 = 1) .
Define an isometric immersion u :S2 + 7R8 by
§ 3. Examples of 2-type Submanifolds 263
ul = u2 =Y
u3 = 2 ,
(3.16) u = YE u4 , 5 v 2 62
(x2 + y2 - 2z2)u7 (x2 - y2) , u8 = 172
Then, by a direct computation, we can see that S2 is of
order [1.2] in Ilt8 . Thus, S 2 is of 2-type in IIt8
Example 3.4. (MM,n in H(2n +2; C)). Let S4n+3
denote the unit hypersphere inC2n+2 = S4n+4 given by
2n+1S4n+3
=((z0,...,z2n+1)t E C2n+2
A O
IzAI2= 1)
In S4n+3 we have the following generalized Clifford torus
2n+1 1 2n+1 1M2n+1, 2n+1 = S ( ) x S ( )
defined by
(3.17) M2n+1,2n+1
n 2n+l{(z0,...,z2n+1)tEC2n+2 ItO
IztI2. ; t=z1 Izt'2 .)
Let GC = (z E C (IzI = 1). Then GC is a group of isometries
acting on S4n+3 and on M2n+1,2n+1 by multiplication.
Denote the quotient space M2n+1,2n+1 /GC by Mn.n. Then
Mn,n admits a canonical Riemannian structure such that
CM2n+1,2n+1 > Mn,n
264 6. Submanifolds of Finite Type
becomes a Riemannian submersion with totally geodesic fibres
S . Moreover, we have the following commutative diagram:1
1 4n+3M2n+1,2n+1 ) S
(3.18)
MT QP2n+1n,n
where i and i' are inclusions. Since M2n+1,2n+1 is
minimal in S4'3, Mn,n is a minimal (real) hypersurface
of Cp2n+1
Let cp :TP2n+1 > H(2n +2; T) denote the first
standard imbedding of cP2n+1 into H(2n +2; C defined
by (cf. *4.6)
(3.19) ip(z) = zz* .
Then :p induces an isometric imbedding of Mean into
H(2n + 2; C). By a direct long computation, we may prove
that, for any point A E cp(Mn,n), the mean curvature vector
H of Mean in H(2n + 2; T) at A is given by
(3.20) H = T (2I - (4n+3)A -At)
Because, AA - -(4n+ 1)H, (3.20) implies
(3.21) AA = 2(4n+3)A.+2At-4I ,
(3.22) AAt = 2(4n+3)At+2A-4I .
§ 3. Examples of 2-type Submanifolds 265
From (3.20), (3.21) and (3.22) we may obtain
(3.23) P(A)H = 0 ,
where P(t) = (t - 4(2n+ 1) ) (t - 4(2n+ 2)) . Consequently, byapplying Theorem 2.2, we obtain the following
Proposition 3.1. Mn,n is a 2-type submanifold ofH(2n+2; Q). Moreover, 4(2n+1), r(2n+2) E Spec (M11
Example 3.5. (MQ,n in H(2n +2; Q)). Consider the
unit hypersphere S8n+7 in Q2n+2 = S8n+8In S8n+7 we
have the generalized Clifford torus M4n+3,4n+3 defined by
M4n+3,4n+3 =
n 2n+1((z0,...,z2n+1)tEb2n+21 E Iz1,2= 1 )z12=)i=O j=n+l
Let GQ = fz E Q Ijzi = 1). The GQ is a group of isometrics
acting on SBn+7 and on M4n+3,4n+3 by multiplication.
Denote the quotient space M / GQ by Then4n+3,4n+3 y n,n
MQn
is a minimal real hypersurface of QP2n+1
Consider the first standard imbedding cp of QP2n+1
into H(2n + 2; Q) given by rp(z) = zz*. Then, by a long
direct computation, we can prove that the mean curvature
vector H of MQ,n in H(2n+2; Q) at A E cp(MnQ,n) isngiven by
(3.24) H = 8n+3 (21 - (8n + 7)A -At)
266 6. Submanifolds of Finite 7)'pe
Since AA = - (8n+3)H, (3.24) implies
(3.25) P(A)H = 0 ,
where P(t) _ (t-4(4n+3))(t-16(t+l)). Consequently, by
applying Theorem 2.2, we have the following.
Proposition 3.2. MQ,n is a 2-type submanifold in
H(2n+2; Q) .
Example 3.6. (MQ,n,n in H(3n +3; Q)). Consider the
following product of three (4n +3)-spheres in Q3n+3
defined in an obvious way;
_ 4n+3 1 4n+3 1 4n+3 1+3-S (-) xS (-) xS (-)M=M4 +3 4 +3 4n nn , ,
V3 13
CS12n+11(1) CQ3n+3
Then GO = (z E Q (Izi = 1) acts on S12n+ll
(1) and on M
by multiplication. Denote the quotient space M / GQ by
MQ non. ThenMQ,n,n
is a minimal submanifold of codimension
2 in Qp3n+2 Consider the first standard imbedding cP of
3n+2QP into H (3n +3; Q) Then cp induces an isometricimbedding of MQ,n,n into H(3n + 3; Q).
By a long computation, we may prove that the mean
curvature vector H of MQ,n,n in H(3n +3; Q) at
A E cP(M4,n,n) is given by
(3.26) H = n+ (32I - 96 (n + 1) A + 21 (A -At) 1
§ 3. Examples of 2-type Submanifolds 267
Because AA = -6(2n +1)H, this implies P(n)H = 0, where
P(t) = (t -24n - T) (t -24n - 24) . Consequently, by Theorem
2.2, we obtain the following.
Proposition 3.3. MQ is a 2-type submanifold inn,,H(3n+3; Q).
Example 3 .7 . (Qn in H (n + 2; V). Let Cpn+l be
the complex projective (n + 1)-space with constant holomor-
phic sectional curvature 4. Let z0....,zn+l be the homo-
geneous coordinates of CPn+1 Then the complex quadric Qn
is defined by
n+IQn = ((z0,...,zn+l) E CPn+1 E Jzi12
=01
i=O
Denote by cp the first standard imbedding of CPn+i into
H(n+ 2; (r). Then, by a direct computation, we may prove
that the mean curvature vector of Qn in H(n + 2; C) at
a point A E cp (Qn) is given by
(3.27) H = n(I - (n+l)A -At)
Thus we have P(A)H = 0, where P(t) = (t-4n)(t-4(n+2)).Therefore, by applying Theorem 2.2, we have the following.
Proposition 3.4 (A. Ros [ 2 ]) .
manifold in H (n + 2; C) .
Qn is a 2-type sub-
Example 3 .8 . (MI n in H (2n + 2, ]R)) . Consider the2n+1
following generalized Clifford torus inS.
268 6. Submanifolds of Finite Type
Mn,n=
Sn(1) xsn( a S2n+1(1) C IR2n+2V2
/2
defined in an obvious way. Denote by G the group of
isometries generated by the antipodal map. Denote by MIR n
the quotient space Mn,n /G. Then Mn n is a minimal
Einstein hypersurface of 1RP?n+l = S2n+1 / G . Denote by cp
the first standard imbedding of ]RP2n+l into H(2n+ 2; ]R)
Then cp induces an isometric imbedding of Mmn n into
H (2n + 2: Ilt) . By a long computation as before, we many prove
that Mn n is a 2-type submanif old in H (2n + 2; ]R) .
Remark 3.1. Although examples given in this section are
spherical, there exist some finite-type submanifolds which
are not spherical. (cf. Remarks 5.3 and 5.4.)
§ 4. CAaracterizations of 2-type Submanifolds 269
44. Characterizations of 2-type Submanifolds
In this section. we will give some characterizations of sub-
manifolds of 2-type. In order to do so, we need to recall the
definition of allied mean curvature vector introduced in Chen
[7) and to compute t H.
Let M be an n-dimensional submanifold of an m-dimensional
Riemannian manifold N. Let en+1" ",em be mutually orthogonal
unit normal vector fields of M in N such that en+l is
parallel to the mean curvature vector H of M in N. We define
a normal vector field a(H) by
m(4. 1) a (H) = E tr (AH Ar) er.
r=n+2
Then a(H) is a well-defined normal vector field (up to sign) of
M in N. We call a(H) the allied mean curvature vector of M
in N. It is clear that a(H) is perpendicular to H.
Definition 4.1. A submanifold M of a Riemannian manifold
N is called an Q-submanifold of N if the allied mean curvature
vector of M in N vanishes identically.
Remark 4.1. For results on a-submanifolds, see for instance,
Chen [7), Houh [1], Rouxel [1), and Gheysens, Verheyen, and
Verstraelen [1,2).
Let M be a compact submanifold of Rm with mean curvature
vector H. For a fixed vector c in Rm we put
(4.2) fc = < H, C >.
270 6. Submanlfolds of Finite Type
Then, for any tangent vector X of M. we have
(4.3) Xfc = - <AHX, c> + <DXH. c>.
Thus, for vector fields X, Y tangent to M, we find
(4.4) YXfc = - <VY (AHX), c > - <h(Y, AHX), c>
<AD HY, c> + <DYDXH, c>.X
Thus, we obtain
n n(4.5) A<H,c> = E (vE Ei) <H,c> - E EiEi<H,c>
i=l i i=1Dn
< 6 H , c > + E <(VE. AH) E. + AD Ei + h(EiAHEi),c>i=1 1 Ei
where E1,...,E is an orthonormal basis of M andAD the
Laplacian of the normal bundle, that is,
n(4.6) ADH = E (DV
EH - DE D H).
i=1 E i i i EiBecause (4.5) holds for any c in Htm, (4.5) implies
(4.7) AH = ADH + E (h(Ei,AHEi) + ADH
E i + (VE AH) Ei).Ei 1
Regard v AH and ADH as (1,2)-tensors in T M 0 T M 0 TM
defined by
(4.8) (v AH) (X, Y) = (VX AH) Y , (ADH) (X, Y) = ADX HY.
We put
(4.9) V AH = V AH + ADH.
§ 4. Characterizations of 2-type Submanyolds
Then we have
n(4.10) tr (v AH) = E (AD H Ei + (vE AH) Ei) .
i=1 Ei i
We notice that if DH = 0, we have v AH = V AH.
271
Let En+l,...,em be an orthonormal normal basis of M in
32m such that en+l is parallel to H. Then we have
(4.11) E h (Ei,AH Ei) = II An+l112 H + a(H) ,
where II An+1 II2 = tr (A2) .
Combining (4. 7) , (4.10), and (4-11),n+l
we obtain
Lemma 4.1. Let M be an n-dimensional submanifold of Htm.
Then we have
(4.12) AH = CDH + IIAn+lII2H + a(H) + tr(vAH).
For the comparison with 2-type submanifolds, we give the
following
mProposition 4.1. Let M be a compact submanifold of Ht
If M has Parallel mean curvature in )Rm, then M is of 1-type
if and only if (1) II An+1 II2 is constant, (2) tr (v AH) = 0, and
(3) M is an C!-submanifold of I.
Proof. Because DH = 0, we have AD H = 0 and v AH = v AH.
If M is of 1-type in H2m, there is a constant b such that
AH = b H. Thus, by Lemma 4.1, we have
(4.13) IIAn+l II2H + a(H) + tr(VAH) = bH.
272 6. Submanlfolds of Finite Type
Since H. a(H), and tr(v AH) are mutually orthogonal, we obtain
(1). (2). and (3) of the proposition. Conversely, if (1), (2), and
(3) hold, then, by setting b = I I An+l II2, we obtain A H = b H.
Thus, by Theorem 2.2, we conclude that M is of 1-type. (Q.E.D.)
Now, we assume that M is an n-dimensional compact sub-
manifold of a hypersphere Sm-1(r) of radius r in Rm centered
at the origin of Rm. Denote by H and H' the mean curvature
vectors of M in Rm and Sm-1(r), respectively. Then we have
(4.14)
where x denotes the position vector of M in Rm. Let P
be the unit vector parallel to H'. Then we have H' = a'
where a' = IH'1. We choose an orthonormal normal basis
en+1' -,em of M in Rm such that
(4.15) en+l = H / a , en+2 = ( + a' x) / r a,
where1
(4.16) a = (HI = (a'2 + ) 2r2
Because Ax = - I, we have
(4.17) tr (AH An+2) = a ' (IIA9112 -n (a') 2) / r a,
(4.18) tr (AH Ar) = tr (AH , Ar) , r = n+3,. .. , m.
From these, we obtain
§ 4. Otaracterizations of 2-type Submanifolds 273
(4.19) a (H) = a' (H') + r a' n (a')2 ) en+2'
where a'(H') denotes the allied mean curvature vector of M
in Sm-1 (r).
For the normal vector field x, we have Dx = 0, i.e.,
x is parallel in the normal bundle of M in Rm. Thus, for
any normal vector Tj of M in Htm with < x, fl > = 0, we have
< DTI,x > = 0. From these, we find that
(4.20) aD H = QD ' H' is perpendicular to x,
where D' denotes the normal connection of M in Sm-1(r).
From (4.14) and (4.15), we also find
(4. 21) a2 IIA !12 = tr (A , + I 2 2 2 2n (a") 2 + nII +
2 r4n+lH r 2)=
(a ) IIAS r
Therefore, by combining'(4.12), (4.14), (4.19), (4.20), and (4.21),
we obtain the following.
Lemma 4.2. Let M be an n-dimensional submanifold of a
hypersphere Sm-1 (r) of radius r in Mm. Then we have
(4.22) A H = AD' H' + a ' (H') n++ tr (v AH) + «' (IIAtII 2 + 2) -
-r2 (x-c0),
where c0 denotes the center of Sm-1(r).
We need the following.
Definition 4.2. Let M be a symmetric space which is iso-
metrically imbedded in Rm by its first standard imbedding. Then
274 6. Submanifolds of Finite Type
a submanifold M of M is called mass-symmetric in M if the
centroid (i.e., the center of mass) of M in Rm is the centroid
of M in Rm.
Lemma 4.3. Let M be a compact minimal submanifold of a
hypersphere Sm(r) of radius r in Rm+1 Then M is mass-
symmetric in Sm(r).
Proof. Because A x = - n H , Hopf's Lemma implies
HdV=0.M
Since M is minimal in Sm(r), we have H =12
(c - x), wherer
c is the center of Sm(r) inRm+1.
Thus we find
c = f x dV / f dV.M M
This shows that c is the centroid of M in Rm+1 (Q. E. D.)
Lemma 4.3 shows that compact minimal submanifolds of hyper-
spheres are special examples of mass-symmetric submanifolds. In
fact, there are many mass-symmetric submanifolds which are not
miminal submanifolds of a hypersphere (Cf. Examples 3.1-3.8).
By using Lemma 4.2. we have the following.
Theorem 4.1. (Chen (251.) Let M be an-n-dimensional, com-
pact, mass-symmetric submanifold of Sm-1(r). If M is of 2-
type in ]Rm, then
(1) the mean curvature cx' of M in Sm-1 (r) is constant
and is given by
§ 4. Characterizations of 2-type Submanifolds 275
(4.23) (a')2 = (n)2 ( 2 - Xp) (aq - 2)r r
(2) tr (v'A H.) = 0, and
(3) ADH' + a'(H') + (jjAj!l2 + 2) H' = (),p + Xq) H',r
where A' denotes the Weingarten map of M in Sm-1(r) and
VA 'H' + AD' 'H1.
Conversely, if (1), (2), and (3) hold, then M is of 2-typein Rm
Proof. Without loss of generality, we may assume that the
center of Sm-1 (r) is the origin of Eim. If M is of 2-type
in Rm, then Theorem 2.2' and Lemma 4.3 imply
2(4.24) AD H' + a'(H') + tr (v AH) + a' (UAgEl2 - n S - n
2r r
+ bH' -2
x -n
x = 0,r
for some constants b and c. Since tr(v A H) is tangent to
M and other terms in (4.24) are normal to M, we have
tr (v A H) = 0. On the other hand, because A H = A . + 2 I andr
DH = D'H', we have tr (v'A H.) = 0. Furthermore, because x isnormal to Sm-1(r) and other terms in (4.24) are tangent toSm-1(r), we obtain from (4.24) that
(4.25) (a')2 + 1 = a2 = - b - cr2 n 2r n
On the other hand, (4.24) gives
(4.26) P(s) (x) = 0.
276 6. Submanifolds of Finite Type
where P(t) = t2 + bt - n. Since M is of 2-type with order
[p,q], (4.26) implies b = - (lip + Xq) and c = Xp Xq. Thus,
by (4.25), we obtain Statement (1). Statement (3) follows
from Statements (1) and (2) and equation (4.24). The converse
of this follows from Theorem 2.2' and Lemma 4.3. (Q.E.D.)
If M is a hypersurface of Sm-1(r), then we have the
following.
Theorem 4.2. (Chen [25].) Let M be a compact, mass-
symmetric hypersurface of Sn+1(r). If M is of 2-type in
Rm+l then
(1) the mean curvature a of M in Iltn+2 is constant
and is given by
(4.27) a2 =
n(Xp + ),q) - (n)2 lp lq o
(2) the scalar curvature T of M is constant and is given
(4.28) T =n
(lp + ) q) - n (nr
-1) ap aq
(3) the length of the second fundamental form h of M in
An+2 is constant and is given by
(4.29) IIh!I = ap + Xq , and
(4) tr(VAH) = 0.
Proof. Let M be a compact mass-symmetric hypersurface of
Sn+l(r). If M is of 2-type, then Theorem 4.1 implies that the
§ 4. Characterizations of 2-type Submanifolds 277
mean curvature a' of M in Sn+l (r) is a non-zero constant.
Since the codimension of M in Sn+l (r) is one, the mean cur-
vature vector H' of M in Sn+1(r) is therefore parallel,
that is, D'H' = 0. Thus DD H' = 0 too. Because, a2 =
(a')2 +
2, equation (4.23) implies (4.27). Since we have
rA 'H, = A
H- 2 I, statement (2) of Theorem 4.1 implies
rtr(v A H) = 0. Now, because M is a hypersurface of Sn+1(r),
we also have a'(H') = 0. Thus, by statement (3) of Theorem 4.1,
we obtain
(4.30) IIAg112 + 2 = lp + aq.r
Because 11h112 = 11Aj112 + 2 , (4.30) implies (4.29). Equationr
(4.28) follows easily from equations (5.2.30), (4.27), and
(4.29). (Q. E. D. )
As a converse to Theorem 4.2, we have the following.
Theorem 4.3. (Chen 125].) Let M be a compact mass-
symmetric hypersurface of Sn+1(r). If M has constant mean
curvature a and has constant scalar curvature 'r, and if
tr(v A H) = 0, then M is either of 1-type or of 2-type.
Proof. without loss of generality, we may assume that the
center of Sn+1 (r) is the origin of R n+2 Assume that a
and T are constants. Then we have DD H' = a'(H') = 0. Thus,
by Lemma 4.2, we find
278 6. Submanifolds of Finite Type
2(4.31) O H = a'( IIA II2 + 2) n 2 x
r r
= .'11h 112 C - nag X.r
Since a and r are constant, a' and 11h112 are also con-
stants. Because H = a'g - x/r2, (4.31) implies
(4.32) A H - 11h 112 H +Z
(na2-IIhII2) x = O.r
Consequently, by applying Theorem 2.2', we see that M is either
of 1-type or of 2-type. (Q.E.D.)
As a special case of Theorem 4.1, we also have the following.
(Chen [25].)
Theorem 4.4. j,g, M be a compact. mass-symmetric submani-
fold of a hypersurface Sm-1 (r) of Rm. If M has non-zero
parallel mean curvature vector H' in Sm-1(r), then M is of
2-type if and only if (1) IIA H II is constant, (2) tr (V A H .) = 0,and (3) M is an Q-submanifold of Sm-1 (r) .
Proof. Let M be a compact mass-symmetric submanifold of
Sm-1(r) such that M is of 2-type. Assume that M has non-
zero parallel mean curvature vector H' in Sm-1(r). Then, by
Theorem 4.1, we have
(4.33) IIASII2 + '2 = ap + lq, a' (H') = 0.r
Because a' is constant, this implies that IIA H.II is constant.
Since a'(H') = 0, M is an Q-submanifold of Sm-1(r). Because
AH
= A'H , + I/r2 and D'H' = 0, statement (2) of Theorem 4.1implies tr(VAH,) = 0.
§ 4. Characterizations of 2-type Submanifolds 279
Conversely, if IIA H, H is constant, tr (V A H ) = 0 and
a' (H') = 0, then, we have tr (v A H) = 0. And moreover, by
Lemma 4.2, we also have
2(4.34) AH = a'(IlAsII2 + 2) 6 - n X.
r rwhere a'(IIAsll2 + ) and a2 = (a')2 +
2are constants.r2 r
Because H = a'C - x/r2, (4.34) implies
(4.35) A H - (IIASII2 + 2 ) H +2
(na2 - CIA;II2 - 2) x = 0.r r r
Since H' # 0, (4.35) and Theorem 2.2' imply that M is of
2-type in Rm. (Q. E. D.)
For surfaces in S3(r), we have the following classification
theorem (Chen [25].)
Theorem 4.5. Let M be a compact, mass-symmetric surface
of S3 (r) in R4 Then M is of 2-type if and only if M is
the product of two plane circles of different radii, that is,
M = S1 (a) X S1 (b) , a 71 b.
Proof. Let M be a compact mass-symmetric surface of a
hypersphere S3(r) in 3R4. without loss of generality, we may
assume that the center of S3(r) is the origin of R4. If M
is the product of two plane circles of different radii, then, by
Example 3.1, we see that M is of 2-type in 3R4.
Conversely, if M is of 2-type in 3R4, then by Theorem 4.2,
M has constant mean curvature and constant scalar curvature.
Moreover, we also have
280
(4.36)
6. Subma :ifolds of Finite Type
tr(7AH') = 0.
by virtue of D'H' = 0. Let El, E2 be the eigenvectors of AHThen we have
(4.37) A H , Ei = .1i Ei, i = 1, 2,
where _l, -2 are the eigenvalues of A H Because, and
T are constants, .-l,-2
are constants.
We pu t
(4.38)
Then we find
2V E1 = J E..
j=1 3
(4.39) 2(vE1 AH,) E1 = (- wl (E1) E2
Similarly, we also have
(4.40) vE AH) E2 2-..1) w2 (E2) E1.2
Because M is of 2-type, -2. Thus, by tr(v A H,) = 0, we
obtain u = 0. From these, we may conclude that M is in fact
the product of two plane circles. Because, M is of 2-type, the
radii of these two plane circles must be different. (Q.E.D.)
Remark 4.2. In general, if M is a submanifold of Sm-1(r)
with A H E i = M E . , i = 1, ... , n, then tr (17 A H ) = 0 if and
only if
(4.41) Eµ= (u-µ) w(Ei) 1.... ,n.ijii i jjiRemark 4.3. Recently, A. Ros [2,3] has applied the concept
4. of 2-t rpe SubutaniJolds 281
of order and the spectral decomposition (1.10) introduced in
Chen [15, 17. 22) to obtain some further results concerning
2-type submanifolds which we shall mention as follows:
Let . : M -4 Sm-1(r) be a minimal isometric immersion
of an n-dimensional, compact. Riemannian manifold M into a
hypersphere Sm-1(r) of IItm centered at 0. Denote by
X1,.... xm the Euclidean coordinates of Sm-1(r) in IItm.
Let x = (xi .Oxm) be the row matrix given by xl,...Oxm.
Define an isometric immersion f of Sm-1 (r) into H (m ; ]R)
by f (x) = xtx . Then f is an order 2 immersion of Sm-1 (r)
into H(m IR). An isometric immersion of M in Sm-1(r) is
called full if M is not contained in any totally geodesic
submanifold of Sm-1(r). The results obtained by A. Ros [2,3)
are the following.
Theorem 4.6. Let :M -4 Sm-1 (r) be an isometric immersion
of a compact Riemannian manifold M into Sm-1(r) such that the
immersion r is full and minimal. Then the immersion fo$ of
M into H(m ;IR) is of 2-type if and only if M is Einsteinian
and tr (A A') = kg(,") for all normal vectors S, of M in
Sm-1(r), where k is a constant and A' is the Weingarten map
of M in Sm-1 (r) .
Theorem 4.7. Let M be a compact, Kaehler submanifold of
CPm such that the immersion is full. Denote by cp the first
standard imbedding of SPm into H(m + 1 ;C) defined in §4.6.
Then M is of 2-type in H (m +1 ;C) if and only if M is
Einsteinian and the Weingarten map of M in CPm satisfies
282 6. Subnwnifolds of Finite Type
tr (A9'S A') = k g (S , rl) for all normal vectors S, -n of M in £Pm,
The idea of the proofs of these two results is to express
G H in terms of the Ricci tensor and the Weingarten map of M.
§ 5. Closed Curves of Finite Type
¢5. Closed Curves of Finite Type
283
In this section we shall study closed curves of finite
type inIm
. In order to do so, we first recall the Fourier
series expansion of a periodic function.
Let f(s) be a periodic continuous function with period
2'rr. Then f(s) has a Fourier series expansion given by
af(s) _ -2+a1 cos(y) +a2 cos(2r) +
+b1 sin(r) +b2 sin(2r ) +
where ak and bk are the Fourier coefficients given by
pTr r
(5.2) ak = TJ
f(s) cos (ki)ds, k = 0,1,2,---r
pTr r
(5.3) bk = J f(s) sin (ks)ds, k = 1,2,----r -Try
In terms of Fourier series expansion, we have the following
(Chen [221)
Theorem 5.1. Let C be a closed smooth curve in ]Rm
Then C is of finite type if and only if the Fourier series
expansion of each coordinate function xA of C has only
finite nonzero terms.
Proof. Assume that C is a closed smooth curve in
IItm such that the length of C is 2Trr. . Denote by s
the arc length of C. We put
284 6. Subinanifolds of Finite Type
(5.4)
Because 0 = - - in this case, we haveds
(5.5) A H = (-1)jx(2j+2), j = 0,1,2,...
If C is of finite type in IRm, then Theorem 2.1
implies that each Euclidean coordinate function xA of C
in IRm satisfies the following homogeneous ordinary differ-
ential equation with constant coefficients:
(5.6)
x(j) =d]x
ds]
x(2k+2) +c x(2k) + ...+c x(2) = 0A 1 A k A
for some integer k z I and some constants cl,...,ck.
Because the solutions of (5.6) are periodic with period 2-r,
each solution xA is a finite linear combination of the
following particular solutions:
n.s m.s(5.7) 1, cos( r ) , sin( r ) , ni,mi E a
Therefore, each xA is of the following forms:
qA(5.8) x = c., + E a_ (t) cos (ts) +b_ (t) sin (ts) 1
t= pA r
for some suitable constants aA(tl, bA(t), cA and some positive
integers PA, qA ; A = 1,...,m. Therefore, each coordinate
function xA has a Fourier series expansion which has only
finite nonzero terms.
d2
§ S. Closed Curves of Finite Type 285
Conversely, if each xA has a Fourier series expansion
which has only finite nonzero terms, then the position vector
x of C in ]Rm takes the following form:
q(5.9) x = c+ E {at cos(ts) +bt sin(tr)
t=p
for some constant vectors a, bt, c in IRm and some2
integers p, q. Since A = - - , (5.9) impliesds
q 2(5.10) Ax = E (-xt,) (at cos(ts) +bt sin(ts) )
t=p
Let xt = at cos (ts) +bt sin(ts) . Then (5.9) and (5.10) show
that x = c+ Etp
xt is in fact the spectral decomposition of
x for c in Itm . Since q is finite, c is of finite
type. (Q .E .D . )
From the proof of Theorem 5.1, we obtain the following.
Corollary 5.1. Let C be a closed curve of length
2Trr in IItm. If C is of finite type, then we have the
following spectral decomposition:
q(5.11) x = x0+ E xt xt = at cos(ts) +bt sin( rs)
t=p
for some vectors at, bt in IRm and some integer p, q 2 1.
Using Corollary 5.1, we have the following (Chen (251)
Proposition 5.1. Let C be a closed curve in ]RA1
If C is of k-type, then C lies in a linear O-subspace
286 6. Submanifolds of Finite Type
IItS of IItm with S s 2k.
Proof. Since C is of k-type, there exist exactly k
of xp,...,xq which are nonzero. Since each xt is con-
tained in Span(at,bt1, C must lie in a linear S-subspace
IIt8 of ]Rm with s 1 2k. (Q.E.D.)
Remark 5.1. For each positive integer k, there is a
closed curve of k-type which lies fully in IR2k .
Proposition 5.2. Let C be a closed curve of length
2rrr. If C is of finite type in IRm. Then we have
(5.12) E t2(IatI2 + IbtI2] = 2r2t. p
(5.13) E tt'(<bt,bt>-<at,at,>)t+t =k
+ 2E
tt'(<at,at,>+<bt,bt.>) = 0t-t =k
(5.14) E tt' <a tbt I>t+t =k
+ E tt'(<at,bt,>-<at.,bt>) = 0t-t =k
for 1 1 k 3 2q, where at,bt; p s t 9 q are vectors in IIt
given by (5.11).
Conversely, if there exist at,bt; p 3 t s q, in IRm
such that (5.12), (5.13) and (5.14) hold for 1 s k s 2q,
then x(s) = Etp
(at cos(ts) +bt sin(tr)) defines a finite
type closed curve in )Rm.
m
§ 5. Closed Curves of Finite 7Ype 287
Proof. From (5.11) we have
q(5.15) x(s) = xo+ E [at cos (tr) +bt sin (tr ) 1 .
t= P
Thus we find from < x'(s),x'(s) > = 1 that
q ,
(5.16) r2 = t,E ( < atat, > sin (r s) sin (trs)=p
+ <b st1bt> cos (i) cos (trs)
- 2 < at.bt, > sin (rS) cos (trs) l
From this we find
t t(5.17) 2r2 = E, ( <at,aI> [ cos (t rt)s -cos (trt)s)
+ <bt,bt,> [ cos (t rt)s+cos (trt,)s)
- 2<at,bt,> [ sin (t rt/)s+sin
Since 1, cos(y), sin(r) ,...,cos().2sin(?-q-) are inde-pendent, (5.17) implies (5.12), (5.13) and (5.14). The con-
verse of this follows from Theorem 5.1. (Q.E.D.)
Using Proposition 5.2, one may classify closed curves of
finite type.
Theorem 5.2. Let C be a closed curve of length 2irr.
If C is of 2-type in IItm , then, up to a Euclidean motion
of IRm , C takes the following form:
(5.18) x (s) = (a cos ( ) , a sin 13 cos (v) , 13 sin (as) , O, ... O)
288 6. Submanifolds of Finite Type
where a and S are nonzero constants such that (pa)2+
(qB)2 = r2.
Proof. If C is of 2-type in 3Rm , then by Proposition
5.2, we have
(5.18) 2r2 = P2(Iap12+ lbp12) +g2(IagI2+ IbgI2) ,
(5.19) IapI = lbpI. IagI = lbgI ,
(5.20) a p,bp,aq,bq are orthogonal .
Thus, by (5.18), (5.19) and (5.20) we obtain the theorem.
(Q.E .D. )
Remark 5.2. From (5.20) we see that the vectors at,
bt, p s t s q are orthogonal if C is of 2-type. However,
if C if of k-type with k ? 3, then at, bt, p s t s q,
are not orthogonal in general. For example, the following
closed curve in iR6 is of 3-type but <a2'
a3> # 0.
1(1cos2s+1cos3s, 1sin2s+3sin3s, cos s-2 cos 2s
5
sin s -2
sin 2s, 1 cos 2s,2
sin 2s)
Remark 5.3. In views of Proposition 5.1 and Theorem
5.2, it is interesting to give the following closed curve
in ]R3 of 3-type:
§ 5. Qosed Curves of Finite Type 289
x(s) _ (-3 sin (6) + cos (3) , -3 cos (6) + sin (s)
f2(cos (3)+sin(5)
Another application of Proposition 5.2 is to give the following.
Theorem 5.3 (Chen [22]). If C is a closed plane curve
of finite type, then C is of 1-type and hence C is a
circle.
Proof. If C is of finite type in ]R2 , then
Proposition 5.2 implies
(5.21) IagI = lbgl i 0, <aqbq> = 0 .
If C is of 1-type, (5.11) and (5.21) imply that C is a
plane circle. If C is of 2-type, Theorem 5.2 implies that
C lies fully in 1R4 . Thus, this is impossible because C
is assumed to be a plane curve.
Now, we assume that C is of k-type with k ? 3.
From (5.21), we see that, with a suitable choice of Euclidean
coordinates of IIt2, we may assume that
(5.22) aq = (a,O) , bq = (O,a) .
On the other hand, by letting k = 2q -1, (5.13) and
(5.14) of Proposition 5.2 give
(5.23) <bq
bq-1 > = < aqaq-1 >, < aq bq -1> = -< aq-1 'bq
> .
290 6 Submanifolds of Finite Type
Thus, by using (5.22) and (5.23), we see that aq-1 and
bq-1 take the following forms:
(5.24) aq-1 = (uq-l'vq-1) , bq_1 = ( -vq_l,uq_1)
From (5.22), (5.23) and (5.24) we obtain
latl = Ibti < at.bt' = 0, t = q -1,q(5.25)
<aq,aq-1> = <bqbq-1"' <aq_1.bq> -<aq,bq-1
Now, we assume that we have
Iatl = lbtl, <atbt-. = 0 ,
(5.26) <at,bl>+<a., bt>=C, <at,a4>=<btbI>, tal
h s t, t s q,
for some h > p. Then, by (5.13), (5.14) and (5.26), we find
< h-1'bgl =(5.27)
< ah-l,bq> + < aq,bh-1> = 0
From (5.22), (5.26) and (5.27), we obtain
(5.28) at = (ut, vt
) , bt = (-vt,ut)
for t = h -l,h,...,q. Consequently, we obtain latl = Ibtl,
<atbt> = 0. 0, and <at,aI> _t 4 t for h-1 = t,f = q. Therefore, by induction, we have
g 3. Uosed Curves 01 unite Type 291
(5.29) latI = lbtI, < at,bt> = 0
<at,ai > = <bt,b1>, <at,b.>+<a2,bt> = 0
for t x 2 and p f, t, 2 5 q. Substituting these into (5.13)
and (5.14) we f ind
(5.30)
(5.31)
E t2 < at, a2 > = 0t-2=k
E tt < at,b{ > = 0t-2=k
In particular, these imply < aq,ap > = < aq,b2 > = 0. Because
C is assumed to be of k-type with k = 3. we find that
ap,aq,bp,bq are nonzero orthogonal vectors in IIt2 . This
is a contradiction. (Q.E.D.)
Corollary 5.2. Let C be a closed curve of length
2-r. If x :C IIt2 is an isometric imbedding of finite
type, then x is a standard imbedding of C onto a plane
circle of radius r.
Proof. Let C be an imbedded finite type curve in
IR2. Then Theorem 5.3 shows that C is of 1-type. Thus
we have
(5.32) x(s) = x0+ apcos bpsin
Moreover, we also have IapI = lbp1. < ap,bp > = 0 and
p2lap12= r2. Thus, by a suitable choice of the Euclidean
2coordinates of It , x(s) takes the following form
292 6. Submanifolds of Finite Type
(5.33) x(s) = P (cos () , sin
Thus, C is an imbedded curve if and only if p = 1. In
this case C is a circle of radius r. (Q.E.D.)
Remark 5.4. Theorems 5.2 and 5.3 show that both 2-type
curves in Iltm and finite type curves in IIt2 are spherical,
that is, they lie in a hypersphere of Iltm. But, in general,
a finite type curve in Iltm is not necessary spherical. For
example, the 3-type curve given in Remark 5.3 is not spherical.
§ 6. Order and Total Mean Curvature
§6. Order and Total Mean Curvature
In this section, we will relate the notion of the order
293
of submanifolds with total mean curvature. In particular,
we will obtain a best possible lower bound and a best possible
upper bound of total mean curvature.
First we give the following formula of Minkowski-Hsiung
(see, Hsiung [2] for n = 2; Chen (4] and Reilly [1] for
general n).
Proposition 6.1. Let x :M -4 JRm be an isometric
immersion of a compact n-dimensional Riemannian manifold
M into ]Rm . Then we have
(6.1) f E1+<x,H>)dV = 0M
Proof. Because Ax = - nH (Lemma 4.5.1), Proposition
3.1.4 gives
n f <x,H>dV = - (x,Cx) (dx.dx) _ -n f dVM M
where we have used the identity < dx,dx > = n. (Q.E.D.)
Recall that for an isometric immersion x :M - IRm of
an n-dimensional compact Riemannian manifold M into IRm
we have the order [p,q] = [p(x),q(x)] of M. Using the
concept of order we have the following best possible lower
bound of total mean curvature (Chen [15]).
Theorem 6.1. Let M be an n-dimensional compact
294 6. Submanifolds of Finite Type
submanifold of IRm . Then we have
k
(6.2) f IHIkdV (-2)7 vol (M) , k = 2,3, ,nM
The equality holds for some k, k = 2.3,..., or n, if and
only if M is of order p.
Proof. Because
qx = x0 + E xt , Axt = atxt
t= p
qn2 f IHI2dV = n2(H,H) _ (AX, AX) = E atf'xt''2
M t= p
On the other hand, (6.1) and (6.3) imply
(6.5)
q2n j dV = -n(x,H) = (x,Ax) = E XtixtIl
M t= p
Thus, by (6.4) and (6.5), we find
qn2f IHI2dv -na f dV = E Xt(at -Xp) 'xt'`2 - 0
M P M t=p+1
Therefore, we obtain
(6.6) IHI2dV (n)vol(M)fM
with equality holding if and only if m is of order P.
Now, by using Holder's inequality, we find
1 1%
(n vol (M) JM IHI28V f ( JMIF1I2rdV)r( pM dV)s
§ 6. Order and Total Mean Curvature 295
withr+s = 1, r,s > 1. Let r = 'k. We obtain inequality
(6.2). The remaining part is clear. (Q.E.D.)
Since we have p it 1 for any compact submanifold M
in Iltm , Theorem 6.1 implies
Theorem 6.2. Let M be an n-dimensional, compact
submanifold of ]Rm . Then we have
k(6.7) IH,kdV ("1)1 vol (M) , k = 2,3,...,n
M
equality holding for some k, k = 2,3,...,
only if M is of order 1.
or n if and
Remark 6.1. Inequality (6.7) is essentially due to
Reilly (2). In fact, he proved inequality (6.7) for k = 2
by applying the minimum principle without using the concept
of order. He also proved that if the equality sign of (6.7)
holds for k = 2, then M is a minimal submanifold of a
hypersphere. According to Theorem 6.1, we can further say
that the equality holds if and only if M is of order 1.
Remark 6.2. Masal'cev (1] obtained in 1976 the following
result.
Let M be a compact orientable hypersurface of ]Rn+1 .
Then
(6.8) f jjhjj2dV z 11 vol (M) ,M
296 6. Submanifolds of Finite Type
where IIhII denotes the length of the second fundamental form
h. The same inequality for any compact submanifold M in a
IRm with arbitrary codimension was obtained independently by
Sleeker and Weiner [1] about the same time. Moreover, Bleeker
and Weiner showed that the equality sign of (6.8) holds if and
only if M is in fact an ordinary hypersphere in a linear
n+l ofII2m(n+1)-subspace 1R
By using the notion of the orders of submanifolds we may
also obtain the following best possible upper bound of total
mean curvature (Chen [22]).
Theorem 6.3. Let M be an n-dimensional, compact
submanifold of 3Rm . Then we have
k(6.9) J IHlkdV n)
7 vol (M) , k = 1,2,3, or 4
equality holding for some k, k = 1,2,3 or 4, if and only
if M is of order q.
Proof. Let M be an n-dimensional compact submanifold
of IItm . From Lemma 4.1, we have
(6.10) 4H ADH+ IIAn+1R2H+a(H) +tr(VAH) ,
where a(H) is the allied mean curvature vector. Since
both a(H) and tr(DAH) are perpendicular to H, (6.10)
implies
(6.11) <6H,H> _ <LDH,H>+IIAHII2 .
§6. Order and Total Mean Curvature 297
Furthermore, from (6.3), (6.4) and (6.5) we also have
q(6.12) n 2 fM IHI2dV = : atllxtli2
t= pq
(6.13) n2 f < H,AH NdV = £, Xtllxtll2M t= P
q(6.14) n f dV = E atllxtlI2
M t=p
Assume that q < - . We put
(6.15) A = n2 f <H,_H dV - n 2 ( X
p+ Xq) f IHI2dV
+ n Xpaq f dV .
Then we have
q-1(6.16) _ Z' (Xt - p) (at - aq) IIxtII2 0
t=p+1
with equality holding if and only if M is either of 1-type
or of 2-type.
Combining (6.11), (6.15) and (6.16), we find
(6.17) n2 f < H,.DH .dV + n2 f IIAHII2dV
-n2(Xp+Xq) f IHI2dV+nXpaq f dV 0
Since M is compact, Hopf's lemma implies
(6.18) f < H,ADH ;dV = f IIDHII2dV .
Let denote the eigenvalues of AH. Then it
298 6. Submanifolds of Finite Type
is easy to verify that
(6.19) IIAH II2 =nIHI4+n (ki -k2i,j
Combining (6.17), (6.18) and (6.19) and Schwartz's inequality,
we get
(6.20) 0 ? n2 f IIDHII2dV+n3 f IHI4dV
+ n 1 f (ki -kj)2dV -n2(X +Xq) f IHI2dV+nXpaq f dV
z n2 f IIDHII2dV + n3 ( f IH2dV) 2 /J dV
+ n3 f IHI4dV -n2(Xp+), q) f IHI2dV+nXpaq f dV
Hence, we obtain
(6.21) 0 g n vol (M) f IIDHII2dV +vol (M) T f (k. -kj) 2dVi<j
+ (n f IHI2dV - ),p vol (M) ) (n f IHI2dV Xq vol (M) )
Combining Theorem 6.1 with (6.21), we obtain
(6.22) f I H 12dv (-g) vol (M)
Substituting (6.22) into the first inequality of (6.20), we
obtain
(6.23) f IHI4dV g2
vol (M)
By using Holder's inequality, we obtain
§ 6. Order and Total Mean Curvature
k 1 kf IHikdV 5 ( f IHI4dV)4 ( vol (M) ) 4
for k < 4. Thus, by applying inequality (6.23), we obtain
inequalities (6.9) .
299
If the equality sign of (6.9) holds for some k, then
all the inequalities in (6.16) through (6.22) become equal-
ities. Thus, we find that H is parallel and M is pseudo-
umbilical. Hence, by applying Proposition 4.4.2 of Yano and
Chen, we conclude that M is of 1-type. The remaining part
is easy to verify. (Q.E.D.)
An immediate consequence of Theorem 6.3 is the following
(Chen [221).
Theorem 6.4. Let M be an n-dimensional compact sub-
manifold of IRm . If IHI is constant, then
(6.4) xp ` nIHI2 - lq
Either equality sign holds if and only if M is of 1-type.
300 6. Suhmanijolds of Finite Type
§7. Some Related Inequalities
In this section, we give some geometric inequalities
which are also related to the notion of the order (Chen [22].)
Proposition 7.1. Let M be an n-dimensional compact
submanifold of ]Rm . Then we have
(7.1) f 'dH 2dV z n(XI + a2) f IH12dV _ 1 2 p dM M n JM
equality holding if and only if M is of order < 2.
Proof. From (6.12), (6.13) and (6.14) we find
(7.2) n2(SH,H) -n2(XI+?,2) (H,H) -)`1X2 f dV
E at(at -xi)(at -a2)IIXt112 a 0t_'3
On the other hand, we also have (AH,H) = (6dH,H) = (dH,dH).
Thus, from (7.2), we obtain (7.1). If the equality of (7.1)
holds, then the equality of (7.2) holds. Thus, M is of
order s 2. The converse of this follows from (7.2)
immediately.
Remark 7.1. Ros also obtained Proposition 7.1 independ-
ently (see Ros [21).
Proposition 7.2. Let M be an n-dimensional compact
submanifold of 7ltm . Then we have
2k+1(7.3) f I6kHII2rdV )rvol (M)
M
§ 7. Some Related Inequalities
k2k+1
(7.4) f lidokHl!2rdv ? (1n )rvol (M)M
301
for r z 1 and k = 0,1,2,..., where 60H = H. The equality
sign of (7.3) or (7.4) holds for some r and k if and only
if M is of order p.
Proof. Because x = -nil, we have
(7.5)11cc
qn2 f n2(6kH,6kH) = E X21+2Ilxtll2
t= pq
n2 f dV = -n(x,H) = E X llxtIl2t=p
n2 J II6kHII2dv -na2pk+1 vol (M)
q(X2k+l x2k+1)
Xtllxtll2 0t=p+l p
This shows (7.3) for r = 1. By applying Holder's inequality,
we may obtain (7.3) for -r > 1.
For (7.4) we consider
(7.8) n2(dAkH,dAkH) = n2(AkH,Ak+1H) =q
2k+3IIxtII2
t= p
By using (7.6) and (7.8), we obtain (7.4) for r = 1. Thus,
by applying Holder's inequality, we obtain (7.4) for r > 1.
The equality cases can be easily verified. (Q.E.D.)
From Proposition 7.2, we obtain immediately the following.
302 6. Submanifolds of Finite Type
Corollary 7.1. Let M be an n-dimensional compact
submanifold of 7Rm . Then we have
2k+1(7.9) f 6kH112rdV ' ( 1n ) r vol (M)
M
(7.10)2k+2
fM jjdAkH112r
dV a (
I
-n) r vol (M)
for r z 1 and k = 0,1,2, . The equality sign of (7.9)
or (7.10) holds for some r and k if and only if M is of
order 1.
Remark 7.1. If k = 0, r = p = 1, then inequality (7.9)
is due to Reilly [2).
Proposition 7.2 also implies the following.
Corollary 7.2. For each k = there is no
compact submanifold M in IRm with AkH = 0.
Proposition 7.3. Let M be an n-dimensional compact
submanifold of ]Rm . Then we have
(7.11)
(7.12)
S 116kH112dV s vol (M)XqM
2k+2f IIdAk 1II2dV ` (-gn ) vol (M)
M
for k = 0,1,2, . The equality sign of (7.11) or (7.12)
holds for some k if and only if M is of order q.
We omit the proof of this proposition.
§ 8. Some Applications to Spectral Geometry
¢8. Some Applications to Spectral Geometry
In this and the next two sections, we shall apply the
concept of order to obtain some best possible estimates of
the eigenvalues of the Laplacian.
First of all, we give the following best possible
estimate of xl of a surface up to its conformal equivalent
class.
303
Theorem 8.1 (Chen [16)). Let M be a compact Riemannian
surface which admits an order 1 isometric imbedding into
]Rm. Then, for any compact surface M in IItm which is
conformally equivalent to M, we have
(8.1) X1 vol (M) a ll vol (M)
equality holding if and only if M is of order 1.
Proof. Let x : M -+ iRm be an order 1 isometric
imbedding of M into ]Rm . Then, by Theorem 6.2, we have
(8.2) IHj2dV = (4) vol (M)M
Because the total mean curvature f JHj2dV is a conformal
invariant, we have
(8.3) f_ IMI2dV = (4) vol (M)M
On the other hand, by using Reilly's inequality, we also
have
304
(8.4)
6. Submanifolds of Finite Type
f _ aM I H12dy ? (-l) vol (M)
Thus, by combining (8.3) and (8.4), we obtain (8.1).
If the equality of (8.1) holds, then the equality of
(8.4) holds. Therefore, by applying Theorem 6.2, M is
also of order 1 in IItm. The converse of this follows
immediately from Theorem 6.2 and Proposition 5.3.5. (Q.E.D.)
Let M be a Clifford torus (or a square torus). Then
we obtain from §3.5 that X1 vol (M) = 472. The standard
imbedding of the Clifford torus in IItm (m ? 4) is known
to be of order 1. A compact surface M in IRm is called
a conformal Clifford torus or a conformal square torus if M
differs from M by conformal mappings of Rm .
From Theorem 8.1 we obtain immediately the following
Corollary 8.1 (Chen [16]). If M is a conformal
Clifford torus, then we have
(8.5) al vol (M) s 4v2
equality holding if and only if M admits an order 1
isometric imbedding.
Let 3RPn denote the real projective n-space with the
standard metric. Then we have al vol (]RP2) = 127- . The
Veronese imbedding of IRP2 into ]R5 is an order 1
isometric imbedding. A compact surface M in 3RTO
is
called a conformal Veronese surface if it is conformally
§ 8. Some Applications to Spectral Geometry 305
equivalent to the Veronese surface. From Theorem 8.1 we
obtain the following (Chen (161).
Corollary 8.2. If M is a conformal Veronese surface,
then we have
(8.6) X1 vol (M) 9 12'r ,
equality holding if and only if M admits an order 1
isometric imbedding.
From Proposition 3.5.4 and Theorem 6.2 we have the
following.
Corollary 8.3. For any isometric immersion of IRPn
into IItm , we have
(8.7)n
r IHIndv 2(nn 1) 1 n
equality holding if and only if the immersion is of order 1.
Similarly, if we denote by QPn and QPn the complex and
quaternion projective n-spaces, respectively, with the
standard metrics, then, by Proposition 3.5.5, 3.5.6 and
Theorem 6.2, we have the following.
Corollary 8.4. For any isometric immersion of CPn
into IRm , we have
(8 .8) IH'2ndV s (2(n+1)7r)nCPn nn n
306 6. Submanifolds of Finite Type
equality holding if and only if the immersion is of order 1.
Corollary 8.5. For any isometric immersion of QPn
into IItm , we have
(8.9)IQPn
IHI4ndV x n+ .
equality holding if and only if the immersion is of order 1.
Corollary 8.6. For any isometric immersion of the Cayley
plane OP2 into Iltm, we have
(8.10) IHI16dv a (727
)w2OP
equality holding if and only if the immersion is of order 1,
where w is defined by (4.6.49).
Remark 8.1. In Chen [16], Theorem 8.1 and Corollaries
8.1 and 8.2 are stated in a slightly different form in
which the volume of M was normalized.
§9. Spectra of Submanifolds of Rank-one Symmetric Spaces 307
§9. Spectra of Submanifolds of Rank-one Symmetric Spaces
In this section, we will again apply the concept of order of
submanifolds to obtain several best possible estimates of xk for
submanifolds in rank one symmetric spaces.
Theorem 9.1. (Chen [22].) Let M be an n-dimensional com-
pact submanifold of a hypersphere Sm(r) of radius r in Rm+1
Then
(1) if M is mass-symmetric in Sm(r), then X 1i xp 9E -2 andr
xp = 2 if and only if M is minimal in Sm(r) and hencer
M is of 1-type.
(2) if M is of finite type, then xq z 2 and Xq = 2 if
and only if M is of 1-type.
Proof. Denote by H and H' the mean curvature vectors
of M in Itm+l and in Sm(r), respectively. Then we have
(9. 1) IHI2 = IH'I2 + r-2.
Hence, by applying Theorem 6.3, we find
xq(9-2) (r ) vol (M) fM I H 12 dV (n) vol (M) .
This shows that Xqz n/r2. If xq = n/r2, then (9.1) and (9.2)
imply H' = 0, that is, M is minimal in Sm. Consequently, by
a result of Takahashi, M is of 1-type. The converse of this is
clear. Thus, Statement (2) is proved.
For Statement (1), we assume that the centroid of M is the
mcenter of S. Without loss of generality, we may assume that Sm
308 6. Submanifolds of Finite Type
is centered at the origin. From Lemma 4.5.1 and Proposition 6.1,
we have
(9.3) n vol (M) n (x , H) = (x , A x)qEp at llxtll2 ? ap ilxll2.
Since M lies in Sm, we find 11x112 = r2 vol (M). Thus, by
(9.3), we obtain
(9.4) 2 - xp.
rIf the equality of (9.4) holds, then the inequality of
(9.3) becomes equality. Thus, M is of 1-type. The converse
of this is clear. (Q.E.D.)
From Theorem 9.1, we obtain immediately the following. (Chen
[221.)
Corollary 9.1. If M is a compact, n-dimensional mass-
symmetric submanifold of a unit hypersphere Sm in Rm+l then
al 9 n, equality holding when and only when M is of order 1.
Corollary 9.2. Let M be an n-dimensional compact Riemannian
manifold with Xq n for some integer q. Then every isometric
imbedding of M into a unit hypersphere Sm 2 Rm+lof order
q is an imbedding of order q.
Now, we need the following.
Lemma 9.1. Let p R Pm 4 H(m+l; R) be the first standard
imbedding of R Pm into H(m+l ; R). Then an n-dimensional minimal
§9. Spectra of Submanifolds of Rank-one Symmetric Spaces 309
submanifold of 12 Pm is of 1-type if and only if M is a totally
geodesic R Pn in IEtPm. If this case occurs, M is of order 1
in H (m+l ; P) .
Proof. Assume that M is of 1-type in H(m+l; H2). Then
by Takahashi's result; M is minimal in a hypersphere S(r) of
H (m+l ; IR) . Denote by C the center of S (r) . Since C is asymmetric matrix, by choosing a suitable coordinates of H(m+1;JR),
we may assume that C is diagonalized. For any A E cp(M), the
mean curvature vector of M in H(m+l ;3R) at A is given by
(9. 5) H = 2 (A-C) .r
Because M is minimal in IR Pm, H E TA (Ht PI) . Thus, we have
AC = CA. We put L = (Z E H(m+1 ;IR) ,ZC = CZ). Then M c L.
Let
C1I O
(9.6) C = CiIi
0 CkIk
where Ii are identity matrices. For each A E cp(1 m), we have
A2 = A and tr A = I. Thus, we find
T (IR Pm) fl L =
Al O
Ai Ai = Ai and tr Ai = 10 Ar
From this, we obtain the following disjoint union:
rcp (H2 Pm) n L = U wi,
i=1where
310 6. Submanifolds of Finite Type
Wi =
O O
Ai I Ai = A1 and tr Ai = 1 ,
O O
It is clear that each W. is a totally geodesic submanifold of
R Pm in H(m+l ; R). Thus, each W. is a real projective spacek.
R P 1in H(ki+l; R). Since M is connected and M c ca (R Pm) n L,
we see that M lies in a Wi for some i, 1 1 i s r. Since M
k.is minimal in R Pm and R P 1
is totally geodesic in R Pm, Mk.
is minimal in R P 1 which lies in H(ki+l; R). Moreover, M
lies in the hypersphere of H(ki+l ;1R) centered at ciIi
with
radius r as a minimal submanifold. Thus, the mean curvature
vector H of M in H (ki+l ; R) is given by H = -2 (A -ar
for A E cp(M). Thus, by Lemma 4.6.4, we have
2(nnl)r4 = <A - aiIi, A- aiIi>
2 - ai - 2 (ki+1) ai.
On the other hand, because mH1 = r, we also have r2 = 1/1H12
n/2 (n+l) . Thus, we find
(n+l) (ki+l) a2 + 2 (n+l) ai - 1 = O.
Since ai is real, the discriminate of this equation is a O.k.
Thus, we find n 3 ki. Since M lies in I RP 1, this implies
that M is R P 1.Thus, M is a totally geodesic R Pn in
R Pm. Therefore, M is of order 1 in H(m+l ; R). The con-
verse of this is clear. (Q.E.D.)
§9. Spectra of Submanifolds of Rank-one Symmetric Spaces 311
For CR-submanifold of CPm, we also have the following
result of Ros [1).
Lemma 9.2. L M be a minimal, n-dimensional CR-submanifold
of CPm, where CPm is imbedded in H(m+1; C) by its first
standard imbedding. Then M is of 1-type in H(m+l ;C) if and
only if one of the following two cases holds:
a) M is a totally geodesic complex submanifold of CPm,
b) M is a totally real submanifold of a complex n-dimen-
sional totally geodesic complex submanifold of CPm.
Proof. We suppose that M is of 1-type in H(m+l; C).
Then M is minimal in a hypersphere S(r) of H(m+l ;C). Let
Q denote the center of S(r), we can suppose that Q is a
diagonal matrix, othersise we can use an isometry of H(m+l ;C)
of the type A '+ PAP-1, with P E U(m+l), to obtain a diagonal
matrix. Since M is minimal in S(r), the mean curvature
vector H of M in H (m+l ; C) satisfies H = (A - Q) /r2, for
A E T (M) . Since H ETA (CPm), we have AQ = QA for A E cp (M) .
Thus, M is contained in the linear subspace L = (Z E H(m+l ; C)
,ZQ = QZ). We put
alll 0
Q = a
O arIr
Then we have the following disjoint union:
rcp (CPm) f1 L = U Wi,
i=1
312 6. Submanifolds of Finite Type
where
Wi =
o o
A. E H (m+l ; C)I
Ai = Ai and tr Ai =1O O
Each of these components is evidently a totally geodesic complex
submanifold of CPm (it is a CPk, k s m) and M is a minimal
submanifold of a component CPk. Consequently, the problem is
reduced to the study of minimal CR-submanifolds of CPk which
are minimal in some hyperaphere of H(k+1;C) whose center is
al, a E ]R, and whose radius is r.
From Lemma 4.9.3, we have
C(9.7) IHI2 = r-2 = n (n2+n+2a), a = dimk
On the other hand, we have H = - 2 (A - a I). Thus, we findr
(9.8) r2 = IHI2r4 = < A - a I , A - a I > = 2 (k+l) a2 - a + 2
Combining (9-7) and (9.8), we find
(n2+n+2a) (k+l) a2 - 2(n2+n+2a) a + (n+2a) = O.
Since the discriminate of this equation must be ? 0, we get
n2 s (n + 2a) k. Because k a n - a, we get a (2a - n) ? O. This
implies that either a = 0, that is, M is totally real or
2a = n, that is, M is a complex submanifold of CPk. Because
n2 (n + 2a) k, we find that if M is totally real, then k = n.
And if M is complex, n = 2k. If the first case occurs, M is
§9. Spectra of Submanifolds of Rank-one Symmetric Spaces 313
a totally real submanifold of a totally geodesic aPn in QPm
If the second case occurs, M is a totally geodesic CPk with
n = 2k.
Conversely, if M is a totally geodesic aPk in CPm
then M is of order 1 in H (k+l C) c H (m+l ; a) . If M is atotally real submanifold of aPn, then for any A E T(M) and
any orthonormal basis E1,...,En of TA M, E1....,En, J El,...,J En
form an orthonormal basis of TA (aPn). Therefore, by Theorem
4.6.1 and (4.6.26), we obtain H = 2(1 - (n+l) A)/n. This impliesthat M is of 1-type in H (n+l ; C) C H (m+l ; C) . (Q. E. D. )
In the following, we give some best possible estimate of al
for compact minimal submanifolds of projective spaces.
Theorem 9.2. (Chen [24].) Let M be a compact, n-dimensional,
minimal submanifold of R Pm, where R Pm is of constant sec-
tional curvature 1. Then the first non-zero eigenvalue 11 of the
Laplacian of M satisfies
(9.9) %1 s 2 (n+l) ,
equality holding if and only if M is a totally geodesic it Pn in
IIt Pm
Proof. Let M be a compact, n-dimensional, minimal submani-
fold of P Pm. Then, by Lemma 4.6.5, we have IH12 = 2(n+l)/n.
Thus, by Theorem 6.2, we obtain (9.9). If the equality of (9.9)
holds, then Theorem 6.2 implies that M is of 1-type. Thus, by
314 6. Submanifolds of Finite Type
applying Lemma 9.1, we conclude that M is a totally geodesic
IIiPn in R Pm. The converse of this is clear. (Q.E.D.)
Theorem 9.3. (Chen [24].) Let M be an n-dimensional
(n a 2), compact, minimal submanifold of CPm, where QPm
of constant holomorphic sectional curvature 4. Then we have
(9.10) %1 s 2(n+2),
equality holding if and only if (1) n is even, (2) M is a
n
CP2 and (3) M is a complex totally geodesic submanifold of
CPm
Proof. Let QPm be isometrically imbedded in H(m+l ;C) by
its first standard imbedding. If M is a compact, n-dimensional,
minimal submanifold of CPm, then, by Lemma 4.6.5, we obtain
(9.11) 2 2 n+2IHIn
equality holding if and only if n is even and M is a complex
submanifold of LPm. By combining (9.11) with inequality (6.7)
of Reilly, we obtain (9.10).
If the equality sign of (9.10) holds, then the equality
sign of (9.11) holds. Thus, n is even and M is a complex
submanifold of CPm. On the other hand, we also have
p k(9.12) J dV = (n) vol (M).
M
Thus, by applying Theorem 6.2, we conclude that M is of order 1
§ 9. Spectra of Submanifolds of Rank-one Symmetric Spaces 315
in H(m+l ;C). Therefore, by applying Lemma 9.2, we concluden
that M is a CP2 which is imbedded in QPm as a totally
geodesic complex submanifold. Conversely, if M is an
CP2, then we have X1 = 2(n+2). (Q. E. D.)
Remark 9.1. Yang and Yau [1] showed that if M is a
holomorphic curve in CPm, then a1 1 8. Moreover, Ejiri [2)
and Ros [2] proved that if M is a compact, n-dimensional
Kaehler submanifold of CPm, then a1 a 2(n+2), equality
holding if and only if M is a totally geodesic complex sub-
manifold of CPm.
Theorem 9.4. (Chen [24].) Let M be a compact, n-dimen-
sional (n ? 4), minimal submanifold of QPm, where QPm is of
constant guaternion sectional curvature 4. Then we have
(9.13) a1 s 2(n+4),
equality holding if and only if (1) n is a multiple of 4, (2) M
n
j QP4, AD-d (3) M is imbedded in QPm as a totally geodesic
guaternionic submanifold.
Proof. Let M be a compact, n-dimensional, minimal sub-
manifold of QPm. Then, Lemma 4.6.6 implies
(9.14) IHI2 s 2(n+4)n
Therefore, by combining (9.14) with Theorem 6.2, we obtain (9.13).
316 6. Submanifolds of Finite Type
Now, if the equality sign of (9.13) holds, then (9.14) becomes
equality. Thus, by Lemma 4.6.5, n is a multiple of 4 and M
is a quaternionic submanifold of QPm. Thus, by a result ofn
Gray [1], we conclude that M is a totally geodesic QP4 in
QPm. The converse of this is clear. (Q.E.D.)
Remark 9.2. Recently, Martinez, Perez, and Santos informed
the author that they can also obtain (9.13) for compact, generic,
minimal submanifolds of QPm.
Similarly, by using (4.6.46) and Theorem 6.2, we may also
obtain the following.
Theorem 9.5. (Chen [24].) Let M be a compact, n-dimen-
sional, minimal submanifold of the Cavlev Plane OP2, where OP2
is of maximal sectional curvature 4. Then we have
(9.15) x1 s 4n.
For CR-submanifolds, we also have the following
Proposition 9.1. (Ejiri [2] and Ros [1].) Let M be a
compact, n-dimensional, minimal. CR-submanifold of CPm. Then
we have
(9.16) x1 s 2 (n2 + n + 2a) /n,
where a is the complex dimension of the holomorphic distribution.
This Proposition follows easily from Lemma 4.9.3 and Theorem
6.2. Similarly, by using 4.9.4 and Theorem 6.2, we have the
following.
§ 9. Spectra of Submanifolds of Rank-one Symmetric Spaces 317
Proposition 9.2. Let M be a compact, n-dimensional,
minimal CR-submanifold of QPm. Then we have
(9.17) ll s 2(n2+n+12a)/n,
where a is the guaternionic dimension of the guaternion dis-
tribution.
Theorem 9.6. Let M be an n-dimensional, compact submanifold
of IR Pm, where I2 Pm is imbedded in H (m+l ; It) by its firststandard imbedding. If M is of finite type in H(m+l ; I2), then
we have
(9.18) X a 2(n+l),q
equality holding if and only if M is a totally geodesic P Pn
in I2Pm. If this case occurs, q = p = I.
Proof. From Theorem 6.3, we have
pJ j
2 q(9.19) H d 3 (n) vol (M).
M
I
Moreover, from Lemma 4.6.4, we also have
(9.20) IHI2 a 2(n+l)n
Combining (9.19) and (9.20), we obtain (9.18).If the equality of (9.18) holds, then (9.19) and (9.20) be-
come equalities. Thus, by Theorem 6.3 and Lemma 4.6.4, we see
that M is minimal in It Pm and M is of 1-type in H (m+l ; ]R) .
Thus, by Lemma 9.1, we obtain the theorem. (Q.E.D.)
318 6. Submanifolds of Finite Type
Theorem 9.7. L M be an n-dimensional, compact submanifold
of CPm, where CPm is imbedded in H(m+l ; C) by its first stan-
dard imbedding. If M is of finite type, then we have
(9.21) Xq x 2 (n+l) .
The equality of (9.21) holds if and only if M is a minimal to-
tally real submanifold of a totally geodesic complex submanifold
CPn of CPm.
Proof. Let M be an n-dimensional, compact submanifold of
CPm. Then, by Theorem 6.3, we see that the mean curvature vector
H of M in H (m+l C) satisfiesa
(9.22) J H 12 dv n ) vol (M) .M
Thus, by combining (9.22) with Lemma 4.6.4, we find
(9.23) Xq z 2(n+l),
equality holding if and only if M is totally real and minimal
in CPm and M is of 1-type in H(m+l ;C). Thus, by using
Lemma 9.2, we obtain the theorem. (Q.E.D.)
Similarly, we also have the following.
Theorem 9.8. L M be a compact. n-dimensional submanifold
of QPm, where QPm is imbedded in H (m+l ; Q) by its first
standard imbedding. If M is of finite type, then we have
(9.24) Xq 6 2 (n+l) ,
99. Spectra of Submanifolds of Rank-one Symmetric Spaces 319
equality holding if and only if M is a minimal totally real
submanifold of a totally geodesic QP in QPn m
320 6. Subnwnifolds of Finite Type
§10. Mass-symmetric Submanifolds
From Theorem 9.1 and Corollary 9.1, we have a best possible
estimate of X 1 for mass-symmetric submanifolds of a hypersphere.
In this section, we shall study Xp for mass-symmetric submani-
folds in projective spaces.
Theorem 10.1. L g t R Pm be isometrically imbedded in
H(m+l ;]R) by its first standard imbedding. If M is a compact,
n-dimensional, mass-symmetric submanifold of R Pm, then
(10.1) 2n m+11 m
equality holding if and only if n = m and M = R Pn
Proof. Since R Pm is isometrically imbedded in H (m+l ; R)
b y its first standard imbedding, Theorem 4.6.1 implies that R Pm
is imbedded as a minimal submanifold in a hypersphere S(r) of
radius r = [m/2(m+1)]1/2. Thus, by Lemma 4.3, we see that the
centroid of R Pm is the center of S(r). Thus, by the hypothesis,
the centroid of M is the center of the hypersphere S(r). There-
fore, by applying Theorem 9.1, we obtain the inequality (10.1).
If the equality sign of (10.1) holds, then, by Theorem 9.1,
M is of 1-type in H(m+l ; R) and M is minimal in S(r).
Therefore, M is also minimal in R Pm. By applying Lemma 9.1,
we conclude that M is a totally geodesic R Pn in R Pm. Hence
x1 = 2(n+l). On the other hand, we have al = 2n(m+l)/m. Thus,
we obtain n = m and M = R Pn. The converse of this is clear.
(Q. E. D. )
§ 10.
M be an n-dimensional. compact, mass-
symmetric submanifold of IF Pm, IF = Q or Q, where IF Pm is
isometrically imbedded in H(m+l;IF) by its first standard
imbedding. Then we have
(10.2) 2n m+11 s ap m
Moreover, %p = 2n(m+l)/m if and only if m = n and M is a
minimal totally real submanifold of IF Pn. Where M is assumed
to be of order [p , q] in H (m+l ; IF) .
Proof. Since IF Pm is isometrically imbedded in H(m+l; IF)
by its first standard imbedding, Theorem 4.6.1 implies that IF Pm
is imbedded as a minimal submanifold in a hypersphere S(r) of
radius r = rm/2(m+1)]1/2. Thus, by Lemma 4.3, the centroid of
IF Pm is the centroid of S (r) in H (m+l ; IF) . Hence, by thehypothesis, the centroid of M is also the centroid of S(r).
Therefore, by applying Theorem 9.1, we obtain inequality (10.2).
if %p = 2n(m+l)/m, then Theorem 9.1 implies that M is
minimal in S (r) and hence M is of 1-type in H (m+l ; IF) . Thus,
M is also minimal in IF Pm. By applying Lemma 9.2 and its quater-
nionic version, we conclude that M is either a totally geodesic
invariant submanifold of IFPm or a totally real minimal sub-
manifold of a totally geodesic IF Pn in IF Pm. If the first
case occurs, then Xl
= 2(n+d), d = 2 or 4. This contradicts
to (10.2). If the second case ocuurs, ap = 2(n+l). Thus, we
find n = m by our assumption.
322 6. Submanifolds of Finite Type
Conversely, if M is a totally real minimal submanifold of
]F Pn, then, by (4.6.26) and the fact that iF Pn is a minimal
submanifold of the hypersphere S(r) with radius r =
[n/2(n+1)]1/2, we may conclude that M has mean curvature
vector H satisfying H = H, where ft is the mean curvature
vector of IF Pn in H (n+l ; iF) . Thus, we obtain ap =
-IL = 2 (n+l) .
r(Q.E.D.)
Similarly, by using Remark 4.6.2 and Theorem 9.1, we have
the following.
Theorem 10.3. Let M be a compact, n-dimensional (n z 2),
mass-symmetric submanifold of OP2, Where OP2 is isometrical-
ly imbedded in H(3 ;Cay) by its first standard imbedding.
Then we have
(10.3) xl a 3n,
equality holding if and only if M is a minimal, totally real
surface of OP2. Here, by a totally real surface of OP2, A&
mean a surface whose tangent planes are totally real with re-
spect to the Cavley structure of OP2.
Theorems 10.1, 10.2, 10.3 together with corollary 9.1 give
the best possible upper bound of 11 for compact mass-symmetric
submanifolds in rank-one symmetric spaces.
From Theorem 10.1, we have the following.
Corollary 10.1. R Pn cannot be isometrically imbedded in a
§ 10. Submanifolds 323
IR Pm as a mass-symmetric submanifold for m' n.
Proof. If IR Pm can be isometrically imbedded in Et Pm
m >n, as a mass-symmetric submanifold, then Theorem 10.1 implies
l 2n(m+l)/m. This contradicts to the fact that X1 = 2(n+l).
(Q.E.D.)
Although, P n can be isometrically imbedded in ]F Pn
as a mass-symmetric submanifold in a natural way, P Pn cannot
be isometrically imbedded in IF Pm, m >n, as a mass-symmetric
submanifold. This result is a special case of the following.
Corollary 10.2. L M be a compact, n-dimensional,
Riemannian manifold with >l ? 2(n+l). Then M cannot be
isometrically imbedded in IF Pm as a mass-symmetric submanifold
unless m = n, al = 2(n+l) and M can be imbedded as a totally
real minimal submanifold in IF Pn.
This Corollary follows immediately from Theorem 10.2.
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AUTHOR INDEX
Abe, K. , 324Adem, J., 195. 324Asperti, A.C., 246, 324Atiyah, M.F., 24B, 324
Banchoff, T.F., 324Barros, M., 180, 181, 324Bejancu, A., 172, 173. 176, 325Berger, M., 25-. 100, 325Besse, A.L., 325Blair, D.E., 172, 173, 175. 325
Blaschke, W., 207, 22, 325
Bleeker, D., 296, 325
Bechner, S., 339
Borsuk, K., 158. 193, 235, 325
Calabi, E., 326
Cartan, E., 9 120, 127, 160. 161. 234, 326
Carter, S., 326
Cecil, T.E., 32EChen, B.-Y., 115, 132, 141, 152, 171-176, 180, 183, 187,
193. 194, 196, 198, 201, 204, 206, 220-223,226, 236, 239. 249. 255. 260. 269. 274. 276-279.
281, 283, 285. 289, 293, 296, 299, 300, 303-305.
307, 308- 313-316, 324-328, 339
Cheng, S.Y., 328
Chern, S.S., 2$. 84, 122, 157, 158. 162-165. 197. 200, 32.8
Dajczer, M., 246, 329. 330
do Carmo, M., 328
342 Author Index
Fells, J. Jr., 202, 328. 324
Ejiri, N., 207, 315. 316, 330
Erbacher, J., 330
Escobales, R .H . ,. 330
Fary, I., 158, 166, 330
Fenchel, W., 158, 186, 193, 235, 330
Ferns, D., 246, 324, 31Q
Gauduchon, P., 95, 100, 325
Gheysens, L., 269, 330
Goldberg, S.I., 84, 331
Gray, A., 316, 331
Guadalupe, IN., 247, 248, 331
Haantjes, J., 210, 3].
Heintze, E., 193, 331
Helgason, S.. 331
Hersch, J., 331
Hopf, H., 84
Houh, C.S., 220. 221, 222, 26,2, 321. 328. 331
Hsiung, C.C., 197. 293, 328, 331
Husemoller, D., 331
Jhaveri, C., 225. 3 8
Karcher, H., 193. 331
Klingenberg, W., 332Kobayashi, S., 4, 30, 123, 328, 332, 332
Kon, M. , 33.9
Kuhnel, W., 332
Kuiper, N.H., 123. 16.5., 183, 234, 32Q. 332
Author Index 343
Langevin, R., 166. 332
Lashof, R.K., 157, 158, 162, 163, 164. 165, 19L 200,
241- 328, 332
Lawson, 14-B- Jr., 169, 197, 198, 225, 248, 324, 333
Lemaire, L., 202, 328
Levi-Civita, T., 4.6
Li, P., 209, 235, 333
Lichnerowicz, A., 333
Little, J., 141, 154, 156, 241, 246, 333
Ludden, G.D., 328
Lue, Ham., 328
Maeda, M., 248, 333
Martinez, A., 316Masol'cev, L.A., 295, 333
Mazet, E., 95, 1QQ, 322.5McKean, H.P. 99 333Meeks, W.H.. 166. 183. 332
Milnor, J.W., 22, 108, 158, 166. 333, 334
Minakshisundaram, S., 98
Minkowski, H., 223
Montiel, S., 328
Moore, J.D., 234, 334
Morse, M., 20, 21 22, 1fi4Morvan, J.M., 334
Mostow, G.D., 334
Nagano, T., 115, 328. 334
Naitoh, H., 334
Nakagawa, H 334Nash, J.F., 120. 1$Z, 334Nomizu, K., 4 32. 12.3_. 332, 335
344 Author Index
Obata, M., 335
Ogiue, K., 132, 152, 328. 335
O'Neill, B., 73, 167, 3-15
Osserman, R., 335
Otsuki, T., 122, 236. 335
Palais, R.S., 76. $9 335
Patodi, V.K., 335
Perez, J.D., 3116Pleijel, A., 98
Pohl, W.F., 333.
Reckziegel, H., 336
Reeb, G., 22, 164
Reilly, R.C., 293. 295. 302, 3031 314, 336
Rodriguez, L., 246, 247, 248, 324, 331Ros, A., 141, 180, 196, 267, 280, 281, 290, 311. 315.
316, 336Rosenberg, H_, 332
Rouxel, B., 269. 3316Ryan, P.J., 326
Sakai, T., 99, 3.316Sakamoto, K., 141, 154. 156. 336
Sampson, H., 220 314
Santos, F.G., 316Sard, A., 20 154
Shiohama, K., 184, 336
Simons, J., 33¢
Singer, I.M., 99, 324, 333
Smale, S . , 239. 241. 332, 331Spivak, M., 337
Springer, T.A., 123
Author Index
Sternberg, S., 24. 331
Sunday, D., 241. 3-32
Tai, S.S., 141. 145, 156, 33.7Takahashi, T., 136, 138, 148, 307. 309. 312
Takagi, R., 184, 336
Takeuchi, M., 334. 337
Tanno, S., 332
Thomsen, G., 212, 221, 332
Urbano, F., 180, 181. 324
Vanhecke, L., 32B
Verheyen, P., 269. 328. 330
Verstraelen, L., 269, 334
Wallach, N.R., 138, 31 331Weiner, J.L., 225. 296. 325 332West, A., 32LWhite, J.H., 207, 212, 338
Willmore, T.J., 182, 113 184, 186, 225, 318
Wintgen, P., 240, 241, 242, 138
Witt, E., 1081 338
Wolf, J.A., 115. 334
Yamaguchi, S., 334
Yang, P.C., 315, 339
Yano, K., 132. 299, 328, 339
Yau, S.T., 209. 234, 235, 315, 33.31 335, 334
345
SUBJECT INDEX
Q - submanifold, 269
action, effective, 2.3
action, free, 23adjoint, 79
affine connection, 46
allied mean curvature vector, 20
associated vector field, 56
associated 1-form, 56asymptotic expansion, 4.8
betti number, 41
Bianchi identity, 55, 5B
CR-submanifold, 172
Cartan's lemma, 9
Cartan's structural equations, 5Q
Casimir operator, 102
Cayley projective plane, 155
chain, 38
Christoffel symbols, 42
closed manifold, n
codifferential operator, $Q
cohomology group, 40
completely integrable distribution, 42
complex-space-form, 6$
conformal change of metric, 64
conformal Clifford torus, 344
conformal curvature tensor, 65
conformal square torus, 344
conformal Veronese surface, 344
conformally flat space, 66
348 Subject Index
connection, 46. 51
contraction, 6
convex hypersurface, 165
covariant differentiation, 46
critical point, 29
critical value, 29
cross-section, 24
cup product, 41
curvature tensor, 5Q
curvature 2-form, 5Q
cycle, 3.4
Dirac distribution, 46.
Einstein space, 54
ellipse of curvature, 245
elliptic operator, 8, a6
energy function, 2Q2
equation of Codazzi, 117
equation of Gauss, 117
equation of Ricci, 11S
equivariant immersion, 26
exact form, 4Q
exotic sphere, 22
exponential map, 62
exterior algebra, @
exterior product, fl
exterior differentiation, 11
extrinsic scalar curvature, 295
fibre bundle, 23
finite type submanifold, 249
flat torus, 72
Subject Index
frame bundle, 22Fredholm's operator, 84
Freudenthal's formula, 142
Fourier series expansion, 283
Forbenius' Theorem, 43
Fubini-Study metric, 24
fundamental 2-form, 677
Gauss-Bonnet-Chern's formula, 61
Gauss' formula, 1_Q9
Gaustein-Whitney's Theorem, 151
generic submanifold, 1.7.1
geodesic, 44
H-stationary submanifold, 214
H-variation, 214
harmonic form, 81
heat equation, 95
heat operator, 95
Hermitian manifold, 61
hessian, 100
Hodge-de Rham Theorem, 91, 9-2
Hodge-Laplace operator, 81
Hodge star isomorphism, 28
holomorphic distribution, 171
homogeneous space, 75
homology group, 221 3.9Hopf fibration, 24
horizontal vector field, 73
index, 21
infinite type submanifold, 252
interior product, 10
349
350 Subject Index
Janet-Cartan's Theorem, 120
k-type submanifold, 252
Kaehlerian manifold, 61
Klein bottle, 73
knot group, 241
knot number, 241
Laplacian, $1
lattice, 22
Lie group, 23
Lie transformation group, 23
linear differential operator, 85
Lipschitz-Killing curvature, 152
locally finite covering, 2H
locally symmetric space, 60
(M+,M-) -method, 115
mass symmetric submanifold, 274
mean curvature vector, 113. 114
minimal distribution, 174
minimal submanifold, 113
Morse's inequality, 22
Morse function, 21
Nash's Theorem, 121
non-degenerate function, 2.1
normal coordinates, 63
order of submanifold, 2
Otsuki frame, 236
Otsuki's lemma, 122
Subject Index
parallel translation, 49
partition of unity, 3Q
Poincare duality Theorem, 93
projective space, 77 74, 25.
pseudo-Riemannian. manifold, 53
pseudo-umbilical submanifold, 132
purely real distribution, 121
quaternionic CR-submanifold, 180
quaterionic Kaehlerian manifold, 54
quaternion-space-form, 70
rank, 2fxReeb Theorem, 22
regular point, 24
Ricci curvature, 54
Ricci tensor, 51
Riemannian connection, 55
Riemannian manifold, 53
Riemannian submersion, 73, 162
rotation index, 151
Sard Theorem, 2.0
scalar curvature, 51
Schur's Theorem, 51
second fundamental form, 111
self-intersection number, 233
simplex, 32
spectrum, 90
standard immersions, 138
Stokes' Theorem, 34
submanifold of finite type, 252
submanifold of infinite type, 252
351
352 Subject Index
submanifold of order [p,q], 2.52
submanifold of order p, 2552
submersion, 21
symbol of elliptic operator, 85, $fi
symmetric space, 15
tension field, 202
tensor, 1
tensor product, 3
tight immersion, 154
torsion tensor, 50
total differential, 1.61
total mean curvature, 1fl1
total tension, 202
totally geodesic submanifold, 101
totally real distribution, 172
totally umbilical submanifold, 113
2-type submanifold, 260
About the Author
Dr Bang-yen Chen is Professor of Mathematics at Michigan StateUniversity. He has held visiting appointments at many universities,including the Catholic University of Louvain. National TsinghuaUniversity of Taiwan, Science University of Tokyo, University of NotreDame, and University of Granada. Dr Chen's research interests focuson differential geometry, global analysis and complex manifolds. Heis the author of numerous articles and two books Dr Chen receivedhis B.S. degree in 1965 from Tamkang University. his M.S. degree in1967 from Tsinghua University and his Ph.D. in 1970 from theUniversity of Notre Dame. He is a member of the American Mathe-matical Society.
9971-966-03-4 pbk