and Submanifolds of Finite Type - 东北大学faculty.neu.edu.cn/liuhl/Books/Total mean curvature...

Series in Pure Mathematics Volume I

Total Mean Curvature

and

Submanifolds of Finite Type

Bang-yen Chen


and


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)


and


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.


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*+ .


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:


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


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.


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

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:


(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,


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


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


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


§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


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


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


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.


§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


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-


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.


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


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.


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).


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.


§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-


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]


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


(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.


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.


§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) .


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


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


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.


§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.)


§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),


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.


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


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


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


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.


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.


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)


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


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


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 .


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.


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).


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)


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 ,


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


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.


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)


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.


¢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.)


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


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.


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


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,...


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


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


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 .


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


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).


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


(6.3) <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


(6.17) 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.


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) .


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


(6.33) 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.


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

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.)


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.


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


**

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


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


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


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)


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.)


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.


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.)


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 ,


(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,


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,


(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))?


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


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


(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)).


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


If M is 2-dimensional, Proposition 3.3 reduces to the

following


Proposition 3.4. Let M be a compact surface in a

Riemannian manifold (M,g). Then

(3.18) J(IHI2+R')dV


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


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


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)


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 .


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)


(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)>


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


(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


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.


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


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


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


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


*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:


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.


*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


(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


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


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 .


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 .


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.)


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 .


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


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


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 .


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


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.


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.


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).


Definition 4.2. Let M be a symmetric space which is iso-

metrically imbedded in Rm by its first standard imbedding. Then


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.


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


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


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.


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


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)


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


(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


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


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


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].)



(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).


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.


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.



(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)


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



Corollary 8.5. For any isometric immersion of QPn

into IItm , we have

(8.9)IQPn

IHI4ndV x n+ .


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


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


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


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.)


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


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


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


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).


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.)


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.


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.

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