Total Mean Curvature and Submanifolds of Finite Type (Series in Pure Mathematics, Vol 1)

Series in Pure Mathematics Volume I

Total Mean Curvature

and Submanifolds of Finite Type Bang-yen Chen

Total Mean Curvature

and Submanifolds of Finite Type

OTHER BOOKS IN THIS SERIES

Volume 2: A Survey of Trace Forms of Algebraic Number Fields

P E Conner & R Perils

Volume 3: Structures on Manifolds

K Yano & M Kon

Volume 4: Goldbach Conjecture

Wang Yuan (editor)

Total Mean Curvature

and Submanifolds of Finite Type Bang-yen Chen Professor of Mathematics Michigan State University

1r World Scientific

Published by World Scientific Publishing Co Pte Ltd. P 0 Box 128 Farrer Road Singapore 9128

Copyright © 1984 by World Scientific Publishing Co Pte Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retneval system now known or to be invented, without written permission from the Publisher.

ISBN 9971-966-02-6 9971-966-03-4 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 applications 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 submanifolds.

The materials given in the first four chapters can be

regarded as the preliminaries for the next two chapters.

Preface

viii

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

vii

Preface

DIFFERENTIABLE MANIFOLDS Chapter 1. 61_ Tensors

1

§2.

Tensor Algebras

§3.

Exterior Algebras

§4.

Differentiable manifolds Vector Fields and Differential Forms

11

Sard's Theorem and Morse's Inequalities Fibre Bundles

20

§7.

§8.

Integration of Differential Forms

28

§9.

Homology, Cohomology and deRham's Theorem

37

§5. §6.

5 7

15

23

§10. Frobenius' Theorem

42

RIEMANNIAN MANIFOLDS Chapter 2. Affine Connections §1

46

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 Quaternionic Kaehler Manifolds

67

§7.

Submersions and Projective Spaces

71

§2.

Chapter 3.

HODGE THEORY AND SPECTRAL GEOMETRY and

78

§1.

Operators *,

§2.

Elliptic Differential Operators

85

§3.

Hodge-deRham Decomposition Heat Equation and its Fundamental Solution

91

§4.

§5.

d

A

Spectra of Some Important Riemannian Manifolds

95

100

x

Contents

Chapter 4.

§. 1

SUBMANIFOLDS

Induced Connections and Second Fundamental Form

52_.

Fundamental Equations and Fundamental Theorems

§.

Submanifoldc with Flat Normal Connection

§4_.

Totally Umbilical Submanifolds Minimal Submanifoldc

I5

109 116 --1-2-4

128 135

§6-

The First Standard Imbeddings of Projective Spaces

141

§7.

Total Absolute Curvature of Chern and Lashof

167

§$

Riemannian Submersions

167

§4.

Submanifolds of Kaehler Manifolds

171

Chapter 5.

TOTAL MEAN CURVATURE

§1.

Some Results Concerning Surfaces in R3

12_.

Total Mean Curvature

187

Conformal Invariants

203

§A.

A Variational Problem Concerning Total Mean Curvature

213

B.

Surfaces in Rm which are Conformally Equivalent to a Flat Surface

226

§6.

Surfaces in R4

236

§7.

Surfaces in Real-Space-Forms

244

182

Chapter 6. §1.

SUBMANIFOLDS OF FINITE TYPE Order of Submanifolds

249 255

§3.

Submanifolds of Finite Type Examples of 2-type Submanifolds

§4.

Characterizations of 2-type Submanifolds

269

§5.

283

§6.

Closed Curves of Finite Type Order and Total Mean Curvature

§7.

Some Related Inequalities

300

§8.

Some Applications to Spectral Geometry

303

§2.

260

293

xi

Contents §9.

Spectra of Submanifolds of Rank-one Symmetric Spaces

§10. Mass-symmetric Submanifolds

307 320

Bibliography

325

Author Index

341

Subject Index

347

Chapter 1.

DIFFERENTIABLE MANIFOLDS

Tensors

$1.

m, n

dimensions

W

into

and W be vector spaces over a field

U,V

Let

and r, respectively.

A map

IF

of

of UxV

f

is called bilinear if it is linear in each variable

separately, i.e., if

(1.1)

f (alu1 + a2u2, blv1 +b 2v 2) = alblf (ul, v1) +a lb 2 f (ul, v2) + a2b1f (u2, v1) +a 2b 2 f(u2, v2)

for all vectors

ul,u2

a1,a2,b1, and b2 in

in

U,

vl,v2

in

V,

and scalars

IF.

V

We denote by Hom(V,W) the space of linear maps from to

W.

sion

Then nr.

Hom(V,W)

Let

V*

is a vector space over

V,

i.e., V* = Hom (V,

of

V.

We define the map

(1-2)

for

of dimen-

denote the space of all linear functions

on

To each (v,w) in

IF

V x W,

V*

is called the dual space

cp : V x W .Hom(V*,W) we assign a linear map

as follows: cp(v,w)

by

cp (v, w) v* = v* (v) w. v* E V*.

bilinear.

It is easy to see that

Moreover, if

wl, wr form a basis of 1,

,r,

cp : V X W 4 Hom (V*, W)

form a basis of W,

form a basis of

then cp(vi,wi = Hom(V*,W).

V and

is

1. Differentiable Manifolds

2

Let

f

U X V

be a bilinear map of

into

We define

W.

a linear map

of : Hom(U*,V) aW

U

where

af((P(ui,vi )) = f(ui,vi ),

by

is a basis of

and

maps

u1'...,um

Then (1.3) defines the

V.

on the basis (cp(ui,vi ))i,j of

af

then extend

a

f

to all of

Hom(U*,V)

One may verify that the linear map

u1,"

of the choice of basis to each bilinear map ciated a linear map

f

a

f

um

from

of

Hom(U*,V).

we

so as to be linear.

a

f

and

U x V

is a basis of

is in fact independent V1*...,vn.

into

Hom(U*,V)

Consequently,

we have asso-

W,

into W

so that

f = of o cp. be

More generally, let

vector spaces over a field into a vector space

W

F.

A map

finite dimensional

k

f, of

V1 X V2 x

. . .

x Vk

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

viEVi.

Let

N be the vector subspace of

(v1,.. ,vk)

U

where

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

We denote by

Tensors

3

the factor space

V1 ®V2 ®... ® V,,

V1,V2'...'Vk.

This vector space is called the tensor product of

We define a multilinear map,

qp,

U/N.

of V1 X V2 x ... x Vk into its coset

into V®®V2 ®... ®Vk by sending We write

mod N.

cp(vl,...,vk) = V1 ®... ®vk. Let W be a vector space and

linear map. We say that the pair

*

a multihas the universal

: V1 x .

(W,'r)

x Vk - W

factorization property for V1 x... x Vk if, for every vector and every multilinear map f :V 1 x ... x Vk + U, exists a unique linear map h : W+U such that f = h o t space

Proposition 1.1.

The Pair

(Vl e .

universal factorization property for (W,$)

then

there

U

has the

® Vk, (p)

If a pair

V1 x... x Vk.

has the universal factorization property for

(V1 ® ®Vk,cp)

and

® Vk + W

* = 0 o cp.

Proposition 1.2. dim Vr),

V1 x... X Vk,

are isomorphic in the sense

(W,W)

that there exists a linear isomorphism 0 :V 1 ® such that

.

If

(ei

r,r

)

is a basis of

then

(e.

1

is a basis for V1® ®Vk.

. ®ei k'k)

V

r

,

(1 s i

r

9

1. Differentiable Manifolds

4

Proposition 1.3.

UsV

(i) There is a unique isomorphism of

onto V ®U which sends

u ®v

into

v

u

for all

uEU and vEv. (ii) There is a unique isomorphism of

u ® (V ®W)

uEU, vEV,

which sends

(u (9 v) ®w

(U ®V) ®W onto

into u 0 (v sw)

for all

and wEW.

(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 KobayashiNomizu, [1, vol. 1].

§ 2. Tensor Algebras

Tensor Algebras

§2.

V be a finite dimensional vector space and

Let

the dual space of

on

5

If

V.

V, i.e., the space of all linear functions

and

vEV

V*

w* E V*,

we put

= w* (v) . be a basis of

Let

the corresponding basis for

)

1 if i = j O if i # j

*

bi

n*

and

so that

V*

<ei,e> > = bi =

where

V

are the Kronecker deltas.

We want to study those spaces of the form where each of copies of

V

V. i

and

is either s

r

the covariant degree.

(r,s)

and

V

or

V*.

r

is the contravariant degree and Given two tensor spaces

of type (p,q),

]F

U

of type

the associative law for tensor

products defines a tensor space of type consider the ground field

If there are

V*, then the space is called

copies of

a space of type (r,s); s

V

V1 0... ®Vk

We

(r +p, s +q).

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

denote this space by

V

and

V*.

We

T(V), i.e.,

T(V) = IF +V+V*+V®V+V®V*+V*®v+v*®V*+ .

1. Differentiable Manifolds

6

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

its dual basis.

V

be a basis of

Let

V.

By Proposition 1.2,

and

e1*,

.(ei ®... ®ei )

is a

,en*

1

V

basis of

(we denote this space by

copies)

(r

Every contravariant tensor

K

of degree

Vr).

can be expressed

r

uniquely as a linear combination: il... ir

K = E 11.

1*** where

K11 -1r

basis

e1'...'er

of degree

of

V.

®e1 .

®

r

K with respect to the

Similarly, every covariant tensor

s 'j

.

... s

e 31 L. l...Js

Vi = V

are the components of

Vj = V*.

and

all the terms of

U

L.

Let

U'

U = VI ®... ®Vk

Let

be the tensor product of

in the same order omitting

The map of V1 X ... X Vk

and

Vi

Vj.

into U' defined by

(vl.... , vk) '4 vl ®... ®vi ®... ®vj ®... ®Vk is bilinear.

L

®... ®e ]r

Now, we define the notion of contraction.

with

,

can be expressed as a linear combination:

s

L.1"

e1 . 1

are the components of

L = E.

where

ir

K

Hence, it defines a map of

is called the contraction with respect to

into

U i

and

U', j.

which

§ 3. Exterior Algebras

V.

Exterior Algebras V

Let

Denote by Trr

7

be an n-dimensional vector space over a field the permutation group on

Tr

acts on Vr = V s . . . (&V

(r

Given any permutation

a E1rr

the form

v1

. .

. s vr,

r

Then

letters.

as follows:

copies)

and any tensor in

Vr

r ) = va (1) s ... s va (r)

We extend by linearity to all of

Vr.

is called symmetric if

for every permutation

K

lrr.

for every

0(K) = K

A tensor

is called skew-symmetric if a

in

where

err,

of

we define

a (v1 0 ... s v

in

IF.

sgn a

according to

a

is even or odd.

For any

K

in

Vr,

K

in

Vr a

a(K) = (sgn (Y)K

is either

1

or

-1

we introduce the following two opera-

tions:

(3.1)

Sr (K) = r rf a (K)

(3.2)

Ar(K) = -L E (sgn a) a (K) a

a

Since

Sr

a = a . Sr = Sr and C. Ar = Ar a = (sgn a)Ar,

is a symmetric tensor and

Sr(K)

tensor.

Sr

.

Ar(K)

is a skew-symmetric

is called the svmmetrization and

Ar

the altera-

tion.

It is easy to check that the alternation linear.

Denote by

isomorphism:

NX

the kernel of

Ar.

Ar : Vr ..Vr

is

We have a natural

1. Differentiable Manifolds

8

Vr/Nr e. Ar (Vr) We denote

by

Vr/Nr

Ar(V).

.

The elements of

are

Ar (V)

As before, we define a multiplication on

called r-vectors.

A(V) = AO(V) +Al(V) +A2(V) +

for

by

OAs = Ar+s (a ®S) a ASE Ar+s (V). The

(-1)rs.

letters past the last

r

letters

s

aAs = (-1)rs3Aa.

defined above is called the wedge (or exterior)

aAs

product of A(V)

Then

6 E As (V) .

Thus we have

(3.3)

The

and

sign of the permutation on r +a letters

which moves the first is

Cr E Ar (V) ,

a

and

S.

It is straight-forward to show that

with the wedge product is an associative algebra, which

is called the exterior (or Grassmann) algebra of (el,...en)

of dimension

is a basis of

Moreover,

2n.

ei A...Aei ,

The exterior algebra

V. 1

If

V.

A(V)

is

and the elements

r = 1,...,n.

1 s it < ... < it s n,

1

form a basis of

A(V).

Proposition 3.1. Then

(3.4)

Let

be

r

vectors in

are linearly dependent if and only if

V1A...AVr = 0.

V.

§ 3. Exterior Algebras Proof.

9

are dependent, then we can express

If

one of them, say

vr,

as follows: r-1 yr

aivi.

i=1 r-1

v1A...AVr = v1A...nvr-1A(E aivi) = O.

Thus

i=1

On the other hand, if

are linearly independent,

we can always find

Vr+1' ..,vn

basis of

vIA...AVr

V.

Thus

such that

form a

(Q. E. D. )

O.

We also need the following. Proposition 3.2.

(Cartan's lemma).

are linearly independent in in

V.

V

Assume that

and

are

r

vectors

If we have r

(3.5)

E winvi = O, i=1

then

wi =

(3.6)

with

v.0

i =

aij = aji.

Proof.

Let

Then we can write

be a basis of

(vi,...,vr, w.

1

as

r

w. 1

n

E b..v.. = E a i7 v. + j=r+l 17 3 j=l

V.

1. Differentiable Manifolds

to Thus, by (3.5), we find

E

i<jsr

(a ij

-a.)viAV + j j

From this we obtain

aij = aji

E

isr,j>r and

bi.vinv. = 0. 7 3

bij = 0.

(Q.E.D.)

From the definition of wedge product we obtain

(6' ') (X, Y) =

(3.7) for

V

X,Y

in

Let

X be a vector in

product

tX

and

e,w

(9 (X) W (Y) - w (X) 8 (Y) )

2 in

V*. V,

x by

with respect to

for every

(a)

tXa = 0,

(b)

(tXw) (Y1....Yr-1) = and

It is easy to see that

into itself, i.e., wEAr(V*).

we define the interior

a E no(V) ,

for wEnr(V*)

V,

where

tx

is a skew-derivation of

V*

is the dual space of V.

tX(wnw') = txwnw'+ (-1)rwntXw', where

A(V*)

§ 4. Differentiable Manifolds §4.

Differentiable Manifolds Let

M

11

M

be a separable topological space.

We assume that

satisfies the Hausdorff separation axiom which states that

any two different points can be separated by disjoint open sets. By an open chart on open subset of

M

M and

we mean a pair §

M

structure of dimension where

a

where

is a homeomorphism of

open subset of Euclidean n-space A Hausdorff space

(U,4)

U

U

is an

onto an

Iltn

is said to have a differentiable

n if there is a collection of open charts

belongs to some indexing set

A,

such that

the following conditions are satisified: (Ml).

M =

(M2) .

For any a, B in , A,

U Ua, aEA

i.e.

[U a)

differentiable map of (M3).

The collection

is an open covering of

the map

§13 o

4a (Ua fl Ut3 )

((U,§a))aEA

§21

onto

is a @13 (Ua flu ) .

is a maximal family

of open charts which satisfy both conditions

and

(Ml)

(M2) .

By "differentiable" in

Ca

M,

(M2)

we mean differentiable of class

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,§)

Denote by

be an open chart of a manifold x11

,xn

M of dimension

the Euclidean coordinates of

,n.

The

n.

1. Differentiable Manifolds

12

systems of functions

e on

xl

coordinate system and

U

U

a .coordinate neighborhood.

In the definition, if

Rn

is replaced by

M

ferentiable maps by holomorphic maps, then plex manifold of

complex dimensions.

n

is called a local

and dif-

Cn

is called a com-

By a compact manifold

we always mean a compact manifold without boundary unless mentioned otherwise.

Given two manifolds

M

and

differentiable if for every chart chart

(Vo,$

)

N

on

13

ofo a cp-

13

u1*...,un

1

of

a map

N,

(Ua,*a)

such that

a local coordinate system on

If

then locally

N,

and every

the map

is differentiable. Let

be a local coordinate system on

map of M into

on M

f(Ua) c V13,

into $ (V

cpa(Ua)

is called

f :M-ON

f

f

Ua

and

yl'...'ym

is a differentiable

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 By a (differentiable) curve in

I D [a,b)

into

M.

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 such that

p

x(to) = p.

p.

Let

x(t)

be a differentiable curve

The vector tangent to the curve

is a map X : 9(p) -. R

defined by

x(t)

at

§ 4. Differentiable Manifolds

xf = df (xdt(t) ) In other words,

tion of the curve

t= t0

is the derivative of

Xf

at

x(t)

13

t = t0.

in the direc-

f

The vector

X

satisfies

the following conditions:

is a linear map of

(1)

X

(2)

X (fg) = (Xf) g (p) + f (p) Xg

The set of maps

of

X

.7(p)

into R ,

for

f, g E T(p) .

into R

.7(p)

satisfying these Let

two conditions forms a real vector space.

be

a local coordinate system in a coordinate neighborgood p.

For

each

i,(aa )

conditions (1) and (2) vectors at

p

is a map of

p

i

7(p)

into R

satisfying

is the vector space with basis (aul)p.

with x (t0) = p,

be its equations in terms of

(df(x(t))) dt

Thus every vector at

p

t0

=

of

We shall show that the set of

above.

Given any curve x (t)

U

,(a

)p-

let ui = xi (t)

Then we have

E(af) (dxl t ) t0 aui p

dt

is a linear combination of

(a -)p,

,

1

aunp

Conversely, given a linear combination

we consider the curve defined by

ui = ui (p) +a i t'

i = 1,.. . , n.

X = E a(aui)p.

1. Differentiable Manifolds

14

Then

is the vector tangent to this curve at

X

To prove

t = 0.

the linear independence, we assume that E ai( aui a ) p = 0.

Then

au.

0 = E ai(a )p = aj,

7 = 1,...,n.

Consequently, we obtain the following.

Proposition 4.1. and

p E M.

Let

M be an n-dimensional manifold is a local coordinate system on a

If

coordinate neighborhood containing at

p

tangent to M

with basis

(aul)p,

We denote by p.

Tp(M)

p, then the set of vectors

is an n-dimensional vector space over (-au

p,

Tp(M)

the vector space tangent to

is called the tangent space of M at

elements are called tangent vectors at

p.

p.

M

at

And its

IR

§5. Vector Fields and Differential Forms

$5.

15

Vector Fields and Differential Forms A vector field Xp

vector tion on

to each

on a manifold M

p

in

If

M.

is a function on

Xf

M,

X

A vector field

is a differentiable func-

f

M

is an assignment of a

such that

is called differentiable if

X

(Xf)(p) = Xpf.

Xf

is differen-

tiable for every differentiable function

f

of a local coordinate system

a differentiable

vector field

may be expressed by

X

on

M.

In terms

X = EXi ( a), where

X1

i

are differentiable functions. Let

be the set of all (differentiable) vector

I(M)

fields on

M.

is a real vector space under the natural

1(M)

addition and scalar multiplication.

X,Y

on

M,

we define the bracket

ring of functions on

into itself

as a map from the

[X,Y]

by

[X,YJf = X(Yf) -Y(Xf).

(5.1) Then

M

Given two vector fields

[X,YJ

is again a vector field on

coordinate system

In terms of local

M.

we write

X =F'Xlau.'

Y =E Y7 au.

1

7

Then we have

(5. 2)

[X,Y} = E (Xk(a

7

-Yk(a7

)

With respect to this bracket operation, algebra over

R

) }au7

I(M)

(of infinite dimensions).

have the Jacobi identity:

becomes a Lie In particular, we

1. Differentiable Manifolds

16

[[X,Y],Z]+[[Y,ZJ,X)+[[Z,XJ,YJJ = 0

(5.3)

for

in I (M) .

X, Y, Z

We may regard

as a module over the ring

I(M)

M

differentiable functions on

X E I (M) ,

then we define

as follows:

by

fX

(fX) p

If

of

9(M)

and

f f 7(M)

= 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

space of the tangent space called covectors at point

p

in

M

For each

p.

M,

we denote by

Tp(M).

T*p(M)

Elements of

the dual

T*p(M)

are

An assignment of a covector at each

is called a 1-form. f

in

T(M),

the total differential

df

of

f

is defined by

= Xf for M,

of

X ET

If is a local coordinate system in (M). p then the total differentials (du form a basis T*p(M).

aul)p,

In fact, they form the dual basis of the basis

..., (aun)p

of Tp(M).

In a coordinate neighborhood of be expressed as

w = v widui,

p,

every 1-form

w

can

§5. Vector Fields and Differential Forms where if

are functions.

wi

is called differentiable

w

It can be verified that this con-

are differentiable.

wi

1-form

The

17

dition is independent of the choice of local coordinate system. We shall only consider differentiable 1-forms. the set of 1-forms on

A1(M)

Let

r-form in

M

on

w

p

. ir

duiln...nduir.

is called differentiable if the components

w

By an r-form we shall always

are differentiable.

We denote by

mean a differentiable r-form.

of r-forms on

Each

Ar (M)

r = 0,

M,

1, ., n.

is a vector space over

With respect to wedge product, Let

d

r

we have

w = Zi1 0

is said to be

n.

U

of

D,

with regular boundary is a either (1) p

has a neighbor-

or (2) there is a coordinate

p with coordinates

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,

M.

we will denote it by

D

is called the boundary of

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.

1. Differentiable Manifolds

34

We now consider a domain

with regular boundary

D

aD

Define the characteristic function

and suppose it is compact.

h(p), pEM by

h (p)

pED,

1,

pEM-D.

10,

We define the integral over

of an n-form

D

on M by

I

JDI = f hi. We give the following well-known Stokes theorem.

Theorem 8.4. sion

n

and

w

M be an oriented manifold of dimen-

1

be an (n -1)-form on M with compact support.

Then for any domain

D

(8. 1)

with regular boundary in

Proof.

Let

(Ui)

aD

be an open covering of

neighborhoods such that for each Ui

we have

w.

=

fD

M,

Ui

either

has property (2) of Definition 8.3.

be a corresponding partition of unity.

M by coordinate Ui 020 = ((

or

Let the functions Since

aD

and

D

(ga)

are

both compact, each of them meets only a finite number of the support of

ga.

Thus we have

jW aD

and

=Ef a

BD

g w,

§ 8. Integration of Differential Forms

aJ

dw

35

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.

be the local coordinate system in

Let

such that

Ui

Let

0.

w = E

Then we have

as.

dw = a We first consider the case when right-hand side of (8.1) is zero. to

M -D or to'the interior of h = 0,

holds,

so (8.1) holds.

in the interior of

D,

we have

Ui n 6D = 0.

The set D.

Ui

Then the

either belongs

If the first possibility

Now suppose that h = 1,

Ui

lies

and the integral in

the left-hand side of (8.1) is equal to (dw =

as . (

au

1) duIA... ndun.

D

where

C

is a cube in the space of the coordinates

contains the support of

w

sufficiently large so that luj1 a M.

in its interior. C

uj

which

we may choose

is defined by the inequalities

This integral is now just an iterated integral in

classical analysis.

m

Thus we have

36

1. Differentiable Manifolds

aa.

r a dul...dun = ± J

J

C

F.

m'uj+l'...,un))dul...duj-iduj+l...dun'

-aj(u1,...,uj-i'

where

is the union of the appropriate faces of the cube.

Fj

Because the support of is equal to zero.

lies inside

w

set defined by

the above integral

Thus (5.1) holds.

Now, we consider the case when Definition 8.3.

C,

We assume that When

un = 0.

space of the coordinates

interior and the side

has property (2) of

is contained in the sub-

un(p) : 0, h(p) = 1.

we take a cube

uj

1,...,n-1; 0 s un

JukI s in. k such that the support of

aD

Ui

C

In the

defined by

gm

is contained in the union of its

w

As in the above, we have

un = 0.

ask

0

auk

for

k =

-1.

C

Also we have

J au

raaan

C

n

du ...du 1

n = (-1)n

J

a

n (u 1

,un-1, O)du 1... dun-1

aD D

But the right-hand side of the last equation is equal to

Thus we have (8. 1) .

(Q . E . D. )

§9. Homology, Cohomology and deRahm's Theorem §9.

37

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'

Ap

the simplex in

theorem to prove deRham's theorem. First we give the following.

We will denote by

Definition 9.1.

P

RP

defined by 0 s xi s 1,

xi 3 1.

i=1 Thus

and

A2

is a point,

AO

Al

is a triangle, etc.

is the unit interval [0,1]

On the simplex

Ap we introduce

the so-called barycentric coordinates defined by p Yj = xj,

YO = 1 - 7- xi, i=1

+yn = 1

So that The map

j = 1,...,p,

and 0 s y7 . 9 1.

bp, i (i = O, " . , p + 1)

from

into

Ap

defined by

yk (bP, i

where

yk

1'k

if k < i

0

if k = i

Yk-1

if

j > 1

are the barycentric coordinates of

Lemma 9.1.

(9.1)

(yO, ... yp)

6p+l, i

Ap+1'

We have o6

P, j

P+l, j c 6p, i+l'

6

if

j a i.

Ap+1

is

1. Differentiable Manifolds

38

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 M.

Ap

in

RP

into

A (differentiable) p-chain is a finite formal linear combina-

tion with real coefficients of singular p-simplices.

M

The set of all p-chains on

linear map f :C

P

is a differ-

Cp(M).

If

f

into another

N,

we define a

in the obvious way, we denote it by

entiable map of a manifold M

form a real vector space

by

(M) -4 Cp (N)

f (s) = s ° f

(9.2)

for simplices and extend it by linearity. If

is a p-simplex,

s

We define the boundary of

s

is a (p - l)- simplex.

by

p

a (s) _ 1=O 7 (-1) is ° b P-l,i

(9.3)

We extend

a

by linearity to a map of

Lemma 9.2.

(9.4)

s ° by-l,i

We have

a°a=0.

Cp(M)

into

Cp-1(M).

§9. Homology, Cohomology and deRahm's Theorem

By linearity, it suffices to prove (9.4) for a

Proof.

simplex.

Let

to prove.

a2 (s)

be a q-simplex.

s

If

= 6 (as)

q

=Z

By Lemma 9.1,

there is nothing

I

(-1) (-1) is o 6q-1, i 0 6q-2, j

(-1) itjs o 6 q-1, i

06

q-2, j

E (-1) i+js o 6q-1, i o 6q-2, j' Osi<jsq

sob q-l,i o6q-2,j = sobq-l,j o 6 q-2,i+l

Thus we can rewrite the first sum as

j 3 i.

E Osjsisq If we put

q-1

i=O j=0

+

q s 1,

If

we have

q z 2,

Osjsisq

for

39

k = i + 1,

(-1) i+js o bq-l.j

o

6q-2, i+1

this will cancel the second sum which

proves (9.4).

Definition 9.3.

(Q. E. D.)

A p-chain

The p-chains of the form

c = ad

is called a cycle if

c

for some

ac = 0.

(p + 1)-chain are

called p-boundaries.

Lemma 9.2 shows that the space a subspace of the space

Definition 9.4.

Zp(M)

Bp(M)

of p-boundaries is

of cycles.

The quotient space

the p-dimensional homology group of

M,

Zp(M)/Bp(M)

denoted by

is called Hp(M).

40

1. Differentiable Manifolds Remark 9.1.

commutes with

If

a,

is a differentiable map,

f : M - N

i.e.,

Thus

of = fa.

cycles and boundaries into boundaries.

map of

into

Hp (M)

Definition 6.5. if

maps cycles into

I

Hence, it induces a

Hp (N) .

A differential form

w

is called closed

w = d for some

It is called exact if

dw = 0.

f

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

denoted by

M,

The quotient of the space of

HH(M).

closed p-forms with compact support over the space of exact form dt

where

is a (p +l)-form of compact support is denoted by

4

H p (M) . C

If M

is compact,

HP (M) = HP(M).

If

c = E cs

is a

s

p-chain and

w

is a p-form of compact support, we define

fw =E

f w.

c

c

s

s

Then by Stokes' theorem we have

(9.5)

f dw =

f

c

(c)

w.

The following results are well-known.

The bilinear map of Cp (M) x np (M) -. R induces a bilinear map of Hp (M) x Hp (M) - R .

Theorem 6.1.

by J, c

given

41

§ 9. Homology, Cohomolo&y and deRahm's Theorem

From (9.5) it follows that the integral of an

Proof.

exact form over a cycle vanishes and the integral of a closed Thus the integral of a closed

form over a boundary vanishes.

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.

is the dual space of

Hp(M)

Hp(M).

In

particular, we have the following isomorphism;

Hp(M) is called the p-th betti number

The dimension of of

[a]

M.

denoted by bp(M).

Let

a

and

[0)

be closed p- and q-forms on

and

presented by

in

c a

Hp(M)

and

a

and

a,

Let

ca

and

corresponding to

It is easy to verify

respectively. a A6

is a closed

(p +q)-form on

M.

denote the corresponding homology classes associated with

Hq(M)

the natural ismomorphism (9.6).

cW

M.

denote the corresponding cohomology classes re-

that the wedge product Let

Hp (M) .

Hp (M)

(9.6)

[aA

]

[a]

and

[0]

through

Then the homology class

is called the cup product of

The product so-defined is denoted by

ca U c13.

c a

caAO

and

42

1. Differentiable Manifolds

§10.

Frobenius' Theorem An r-dimensional

Let M be an n-dimensional manifold.

distribution on M

is an assignment b defined on M which

assigns to each point

by

space

of

p

an r-dimensional linear sub-

M

in

An r-dimensional distribution b

Tp(M).

called differentiable if there are

differentiable vector

r

p which, for each point

fields on a neighborhood of

this neighborhood, form a basis of

is

bq.

q

The set of these

vector fields is called a local basis of the distribution

in

q 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

A distri-

X E b.

bution b

is called involutive if, for any

[X,Y) E.B.

By a distribution, we shall mean a differentiable

X. Y Eh,

we have

distribution.

of M

is called an integral

submanifold of the distribution b if

f,, (Tp (N)) = bf (p) (M),

N

An imbedded submanifold

for all

p E N,

where

f

sional distribution b on if, for each point U

p E M,

is the inclusion map.

M

is called completely integrable

there is a coordinate neighborhood

and local coordinates

the submanifolds of

U

An r-dimen-

on

given by

are integral submanifolds of

U

such that all

yr+1 = const,

, yn = const.

b.

The theorem of Frobenius can be stated as follows: Theorem 10.1.

A distribution b gD M

integrable if and only if b

is involutive.

is completely

§ 10. Frobenius' Theorem

Let

.b

43

be an r-dimensional distribution on

(10.1)

0 = (1-forms

and let

i(0)

for

on M I w (X) = 0

w

be the ideal generated by

exterior polynomials on

We put

M.

x E .&) ,

in the ring of

0

M.

Theorem 10.1 of Frobenius can also be rephrased as the following.

Theorem 10.1'.

and only if

The distribution b

is involutive if

is contained in the ideal

dO

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. r = 1, this theorem reduces to the existence theorem of

If

ordinary differential equations. X,

In fact, given a vector field

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

be neighborhood such that

r

r.

Let

and

p E M

independent vector fields defined on a coordinate

V of

X1 = aY

We introduce local coordinates

p.

and

yl(p) = 0.

We put

fj = Xjyl

on

V,

1

Y1 = X1

and

Yj = xj - fjx1

for

are also linearly independent.

Moreover, we have

Yjyl = 0

W = (q E V I yl (q) = 0).

for

j =

Then

Thus, they span & on j =

V.

We consider

Then W is an (r - 1) -dimensional sub-

1. Differentiable Manifolds

44

manifold of

V

Yj(q) are tangent to W

tion of where

on

Yj

for

i*(aa )q 7

Then

W.

Let

q E W.

is the inclusion.

i

are tangent to

Y2,...,Yr

and

(aa )q for qEW, 7

If

k=2 7 ark

7

n

of

Yj

to

R2Z

Zj =

be the restric-

Zj

n r Y1-

Y. then we have

k ZJ

where

azk ,

are the restriction

Now, we want to claim that

W.

span

an involutive '(r -1)-dimensional distribution on if

then

i.e.,

W,

w.

did not lie in the space spanned by

[Zi,Zu)q

[Yi,Y3)q

would not lie in

has any component relative to

Z2....,Zr,

since none of Y2,...,Yr

9q

a-ya

In fact,

Therefore, by induction,

.

1

through

we can find an integral submanifold of each point

p

in

W.

Thus, we can find local coordinates

in some neighborhood of tribution spanned by by

Z2,...,Zr

about

p

W

in

such that the dis-

has integral submanifolds given Now we define local coordinates

const.

ur+l =

p

Let

as follows:

vl = yl

and

The Jacobian determinant of the respect to the

does not vanish at

y's

p

v's

vj(y1,...'yn)=

with

so that the

v's

do

av.

form a coordinate system about for

j =

we have

p.

Y1 = as yl

Moreover, because av 1

.

we have

av 1

(Yjvr+k) - [Yj'Y1)vr+k'

Since

By,1

= 0

Ylvr+k = 0,

§ 10. Frobenius' Theorem

45

span an involutive distribution, we

Now, because

may find some function

so that

Dih

r

[Yi,YII = DilY1 + E DilY =2

Thus we have

r L Dll (Yivr+k)

av 1 (Y 'vr+k for

i=2

Thus, for each fixed

j =

ferential system of

differential equations with respect

r -1

to the independent variable we have

v1.

Yjvr+k = Zjvr+k = O.

solutions, we find that fore,

Y1,

,yr

we may regard the

as solutions of the homogeneous linear dif-

Yjvr+k

functions

k,

Yjvr+k

On the other hand, if

Thus, by the uniqueness of the vanish identically.

can be expressed in terms of

This proves that b

v1 = 0,

is completely integrable.

all

av

1

There.

a FjVr

(Q.E.D.)

RIEMANNIAN MANIFOLDS

Chapter 2.

Affine Connections

§1.

The concept of an affine connection was first defined by Levi-Civita for Riemannian manifolds, generalizing significantly There always

the notion of parallelism for Euclidean spaces.

An affine connection

exists an affine connection on a manifold.

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. a rule

map

vX

An affine connection on a manifold M

which assigns to each vector field

v

of the vector space

1(M)

X EX(M)

is

a linear

into itself satisfying the

following two conditions:

vf)+gy = fvX + gvY; 7X(fY) = fvXY+ (Xf) Y

(v2)

for f, g E AO (M)

The operator

and X. Y E I (M) .

the covariant differentiation with respect to An affine connection vU

on an arbitrary open submanifold X,Y

((VU)X(Y))p = (vXY)p.

Then

tion on

U.

on

(vU)a/ayl.

U,

X.

U

of

and any

M.

In fact, for

p E U,

we put

is a well-defined affine connec-

VU

In particular, if

with local coordinate system of

is called

on M induces an affine connection

v

any two vector fields

vX

U

is a coordinate neighborhood we write

vi

instead

§ 1. Affine Connections

On

we define the functions

U,

rid

47

by

vi a ) = E rid sy k=1

These

are called the Christoffel symbols or the connection

rid

coefficients of valid on

is another coordinate system

z10...ozn

If

v.

we obtain another set of functions

U,

vi () = E rl . k

Fjz

Virj

by

k

Using the axioms (vI) and (v2) we find easily

(1.2)

azk

aya ayb azk

k

c a2ya azi azi ayc rab +E aziazi aya

r13

Conversely, suppose that there is a covering of M consisting of coordinate neighborhoods (Ua) hood a system of functions

and in each coordinate neighborsuch 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 For real functions

tensor field

(1.3)

w

f

on

M.

of type (0,1)

vX

to arbitrary tensor fields.

we define

vXf

as

Xf.

(i.e., a 1-form), we put

(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)

For a

(VXT) (Y,Z) = vX(T(Y,Z)) -T(vXY,Z) -T(Y.VXZ).

2. Riemannian Manifolds

48

In terms of local coordinates, if has local componenets

VT

Tai,

has local components

T

vkTji = y Tai+rkt Tai-r In general, if

T

Tti-rki Tit

is a tensor field of type

(r,s), VT

has

local components

(1.5)

vkTji.. j9 - ayk Tjl... js +rtk Tjl2.. jsr+ ... +

+

tk

Ti1... ir-lt - rt jl...is

Using affine connection

l... it -rtask TIj1...3s-lt

T'1 - ' r tj2. .js

jlk

one may introduce the concept

v,

of parallelism as follows: Let

Then

y(t), tEI be a curve in M over an open interval I.

T(t) = ;().

dt is a vector field over I. We put

is called the velocity vector field of the curve that

Y(t)

Assume that Y(t)

is a vector tangent to M Y(t)

at

is parallel along

Y(t)

if

t.

vT(t)Y(t) = 0

T (t) = E Ti (t)y and Y (t) = E Yj (t)yj ,

(1.6)

dyk +E T i j

y(t)

Suppose t E I.

We say that identically.

with

In terms of local coordinates system

parallelism of Y(t) along

for

y(t)

varies differentiably with

y.

the condition of

is

Yi at = 0, t E I.

T(t)

49

§ I. Affine Connections The curve of

T(t)

for

y

is called a geodesic if the velocity vector field is parallel along the curve

y(t)

The condition

y.

to be a geodesic is

y

d2y

(1.7)

Tig

dt 2 +E

dy. dy.

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. nection.

Let

M be a manifold with an affine con-

Let

be a point in M

p

,and

X, be a vector in

Then there exists a unique maximal geodesic such that

and

y(O) = XP

T(O) =

there is a unique vector field

XY(a)

Ty(b)(M)

X(t)

is parallel along

and X(t) given by

M

J = [a,b] is

then for each vector

I,

in

XP.

Similarly, using (1.6), we see that if a closed subinterval of

t w y(t)

Tp(M).

P(X(a)) = X(b)

along

y.

X E Ty(a)(M),

such that

I

The map

X(a) _

P : Ty(a) (M) 4+

is a linear isomorphism

which is called parallel translation along

y

from

y(a)

to

y (b) . Using the affine connection tant tensor fields

R

and

T

V

on M we define two impor-

by

(1.8)

R (X,Y) = VXVY - VYVX - V[X,yl

(1.9)

T (X, Y) = VXY - VYX - MY]

50

2. Riemannian Manifolds

tensor and

T

is a tensor field of type (1,2), called the

torsion tensor of

V.

e1. ,en be a local frame of vector fields defined on

Let

wl,...,wn

Denote by

U

of

M.

We define

n2

connection 1-forms

an open subset frame.

It is easy to verify

M.

is a tensor field of type (1,3), called the curvature

R

that

tangent to

X,Y

for vector fields

wj

be the dual on

U

by

n

vX e. = i=1 E w) (X)ei,

(1.10) (linearity of

wj

follows from axiom (V1).)

From (1.10) we

find

w (X) = wl (vXe We define functions

(1.12)

on

U by

T(ej.ek) = E Tjk ei;

R (ej. ek) el

(1.13) Using

Tjk,

Tjk

and

E R1jk

ei.

we define the torsion 2-form

Rijk

01

(1.14)

T1 = 2 E Tjk w3 A wk

(1.15)

nj = 2 E Rjklwk n w 1. w1,wj,Tl,(2j

structural equations:

and

by

the curvature 2-form

The forms

T1

are related by the following Cartan

51

§I. Affine Connections

We have

Proposition 1.2.

(1.16)

dwl =

(1.17)

dw _ - E wk n w +n'

Proof.

- E wl A w3 + Tl

For vector fields

(the first structural equation):

(the second structural equation). 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)

and (5.5) of chapter 1) and this equation,

(i.e. equations (3.7)

we obtain, (1.14).

Equation (1.17) can be obtained in a similar way.

Remark 1.1. r

over a manifold

bundle of sections of E

In general, let

M

and let

Denote by

M.

E

r(E)

E

be a vector bundle of rank

T* = T*(M)

and

be the cotangent

r(T* ®E)

and the tensor product

is defined as an operator

(Q.E.D.)

T* ®E.

the spaces of A connection on

52

2. Riemannian Manifolds

(1.18)

v : r(E) 4 r(T* IRE)

satisfying the following two conditions:

(a)

v (s1 +8 2 ) = 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

§2.

53

Pseudo-Riemannian Manifolds M be a manifold of dimension

Let

metric on

M

is a tensor field

A pseudo-Riemannian

n.

of type

g

which satis-

(0,2)

fies

for each

(b)

on

p E M,

gp

X, Y E ; (M) ;

is a nondegenerate bilinear form

for all

gp (X, Yp) = 0

Tp (M) x Tp (M), i.e.,

implies

for

g (X, Y) = g (Y, X)

(a)

Yp E Tp N'

Xp = 0.

A pseudo-Riemannian metric metric if

gp

on

g

M

is called a Riemannian

is positive-definite for each

be remarked that every manifold

M

p

in

It shall

admits a Riemannian metric.

To prove this, we take a locally finite covering coordinate neighborhoods.

M.

For each

Uz,

definite quadratic differential form on

let Ua.

ga Let

(Uo)

be a positive{ha)

be a

corresponding partition of unity (Theorem 1.8.1 and 1.8.2)

M.

A manifold M with a (pseudo-)

Riemannian metric is called (pseudo-

Proposition 2.1.

Riemannian manifold.

On a pseudo-Riemannian manifold there

exists one and only one affine connection

v

satisfying the

following two conditions;

has no torsion,

(gl)

vxY - 7YX = [X,Y), i.e.,

(g2)

Zg (X, Y) = g (vZX. Y) + g (X, vZY) ,

Riemannian metric

(i.e.

Then we find that E hug'

Theorems 8.1 and 8.2 of Chapter 1.)

defines a Riemannian metric on

M by

of

g

is parallel.

v

i. e. , the pseudo-

2. Riemannian Manifolds

54

It suffices to show that such connection

Proof.

and is unique on every coordinate neighborhood ness implies that V

on

Denote by

U.

(gij). If (gl)

of

M.

be local coordinates on

Let a -) j

and

(g2)

29(Vi ayj

hold, then we have ag

ag

ayk) - ayi + a'i

i

ayk

From (2.1) we find

because [ay, , ay ] = 0. i

Let

U.

the inverse matrix

(gkl)

a

(2.1)

The unique-

U.

must agree on overlapping domains; hence

V

exists and is unique on

gij = g(ayi ,

exists

V

7

agki gtk _ayi

2rtii

+

agki ayi

+

agti ayi

ayk

Consequently, we obtain

k

(2.2)

1

rji

g

2

tk agti (

ayi

Giving an affine connection giving functions demanding V

on

1

properties

V

U with

on (v1)

and

.

syt

on

is equivalent to

U

a

Vj ay i (v2).

1

A direct computation yields that

nection satisfying (g1)

and

k

and

We use (2.2) to define

Then the explicit expression (2.2) shows

U.

unique.

rj.

agii)

V

v

is

is an affine con-

(g2).

(Q.E.D.)

In the following, we call the unique affine connection satisfying

(gl)

and

(g2)

on a pseudo-Riemannian manifold,

the

§2. Pseudo-Riemannian Manifolds

Riemannian connection of

v.

55

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).

Formula (b)

is called the first Bianchi identity.

(Q.E.b.)

2. Riemannian Manifolds

56

Riemannian Manifolds

§3.

M

Let

be an n-dimensional Riemannian manifold with

Riemannian metric

We sometime denote this by

g.

be a local coordinate system on g =

E

gij dyi dyj.

product

<S,T>

We have

M.

(gi3)

the inverse matrix of

g.

we can define the inner

Using the metric tensor

(gij).

if

Denote by

Let

(M,g).

of two tensor of the same type.

For example,

are tensor fields of type (0,2). we put

S, T

<S,T> = E gk1g13Sk1Tij.

(3.1)

S = E Sijdyi ®dyj, T = E Tijdyi 0dyj. The length of a tensor T is then defined by

where

IITII

1ITII = *. 0 :[&,b)-#M be a curve in

Let

of

M,

we define the length

by

o

L(0) = jbIIQ. (dt) Ildt.

field

is a 1-form on

a = E aidyi

If

a#

associated with

(3.2) a#

we define a vector

by

°t#=EgljaiBy

is called the associated vector field of

given a vector field

X = E Xiayi ,

X# by X# = E gijXldyj. of

a

M,

X.

In fact, if

a

We call

a.

Similarly,

we may define a 1-form X#

the associated 1-form

is the associated 1-form of vector field

57

§ 3. Riemannian Manifolds

X,

we have

for

a (Y) = g (X, Y)

tangent to

Y

Similar

M.

operation can be defined for other tensor fields on For each

in

p

M

of the tangent space

tional curvature

and each 2-dimensional subspace

M

of

Tp(M)

of

K (Tr)

Tr

at

X(r)

by

if

K (Z n W)

Z

by

e1....,en

T.

It is easy to

is well-defined. Sometime, we denote and W span the 2-plane Tr.

Given two vectors basis

we define the sec-

p,

X, Y are orthonormal vectors in

verify that

Tr

= g (R (X, Y) Y, X) ,

K (Tr)

where

(M,g).

of

X, Y

in

and an orthonormal

Tp(M)

Tp(M), we define the Ricci tensor

and scalar curvature

K(Tr)

S

by

T

n

S (X,Y) = E g(R(ei,X)Y,ei)

(3.3)

i=1 _

(3.4)

i

1

T - n (n-1)

It is easy to verify that both

S (ei ei). and

S

of the choice of the orthonormal basis.

manifold in

M,

if

K(w)

T

are independent Given a Riemannian

is constant for all plane section

Tp(M) and for all points

p

in

M,

then M

>r

is called

a space of constant curvature or a real-space-form. The following theorem of Schur is well-known.

Theorem 3.1. sion

n > 2.

on the point

Let

M be a Riemannian manifold of dimen-

If the sectional curvature p,

then

M

K(Tr)

depends only

is a space of constant curvature.

2. Riemannian Manifolds

S8

The proof of this theorem bases on the following

Lemma 3.1.

M

The curvature tensor of a Riemannian manifold

satisfies the following second Bianchi identity:

(VXR) (Y,Z) + (Vt) (Z,X) + (uZR) (X,Y) = 0.

(3.5)

Let

Proof of Theorem 3.1.

T be the tensor field of

type (0,4) defined by T(Z,U;X,Y) = g(Z,X)g(Y,U) -g(U,X)g(Y,Z).

(3.6)

We define another tensor field, also denoted by

(3.7)

R.

by

R (Z, U; X, Y) = g (R (Z, U) X, Y) .

Since the sectional curvature of M depends only on the p

in

M,

we have

R = kT

(3.8) for some function

k

on

M.

Because

Vg = 0, we have

Thus

(3.9)

(V

t) (Z,U;X.Y) = (Wk)T(Z,U;X,Y),

from which we obtain (3.10)

point

( (VWR) (X. Y) ) Z = (Wk) (g (Z, Y) X - g (Z, X) Y) .

VT = 0.

59

§ 3. Riemannian Manifolds

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

Z = W and X,Y,Z

such that

orthognoal, then we have

(Xk)Y = (Yk)X.

are mutually k

Therefore,

is a

constant.

(Q.E.D.)

From (3.8) we see that if R

then the curvature tensor

k

is a constant.

curvature

takes the following form;

T

satisfies

T = k.

Let

X

M

of

Moreover, in this case, the scalar

(3.12)

at

is of constant curvature,

R(X,Y)Z = k(g(Y,Z)X-g(X,Z)Y);

(3.11) where

M

X

be a unit vector in

Tp(M).

S(X,X), where

is given by

Then the Ricci curvature

denotes the Ricci tensor.

S

If the Ricci tensor

S(X,X)

the Ricci tensor

takes the following form:

S

(3.13)

S = Tg

If this is the case at every point called an Einstein space.

If

is independent of

p

at

X,

.

p

n > 2,

in

M.

then

M

is

then by the second

then

2. Riemannian Manifolds

60

Bianchi identity, we may conclude that M

has constant scalar

curvature.

A Riemannian manifold is called a locally symmetric space if its curvature tensor

is parallel, i.e., vR = 0.

R

For any two points

p, q

in

we define

M,

d(p,q)

as

the greatest lower bound of the lengths of all piecewise differentiable curves joining defines a metric on

p

and

q.

It can be shown that

If the metric

M.

d

is complete, i.e.,

d

all Cauchy sequences converge, we say that the Riemannian mani-

M

fold

and

q

It is well-known that any two points

is complete.

M

in

p

can be joined by a geodesic arc whose length is

equal to the distance

It is also well-known that the

d(p,q).

following conditions on

M

are equivalent:

is complete;,

(a)

M

(b)

all bounded closed sets of

(c)

Any geodesic arc in

M

M

are compact;

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.

with respect to the basis. on

be the curvature 2-forms

Let

We define a 2m-form (2m = dim M)

M by (-1)m n = 22mwmm,

1 r e 0 'l...i2m

12

1A...An1

.2m-1 12m

,

§ 3. Riemannian Manifolds

where

ci

61

do not form a permu-

is zero if 2m

and is equal to

tation of

the permutation

is even or odd.

is called the Euler class of

M.

1

or

-1

accordingly as

The cohomology class

[C1] CH

2m (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 teristic of

M.

X(M)

be the Euler-Poincare charac-

Then we have

(3.14)

X (M) = JM 0.

In particular, if

the Gauss curvature of

M

is 2-dimensional and G(= r)

then we have the following Gauss-

M,

Bonnet formula.

(3.15)

IGdV = 21r) ((M) , M

where

dV = wlnw2

is

denotes the volume element of

M.

2. Riemannian Manifolds

62

Exponential Map and Normal Coordinates

§4.

Let

M

fold at

of dimension

For each vector

n.

tangent to M

X

Proposition 1.1 implies that there is a unique geodesic

p,

yX(t)

be the Riemannian connection of a Riemannian mani-

v

yX(O) = p

and

For appropriate

yX(O) = X.

in R with

0

which is defined on a neighborhood of

R,

in

s

by the nature of the ordinary differential

ysX(t) = yx(st)

equations defining the geodesics.

This implies that

ycx(1)

is defined if

Therefore,

is a

is defined.

yV(c)

M

well-defined point in

yX(1)

with sufficient small length

for X

IIXII

For each

Definition 4.1. exppX

defined.

The map

for each point 0

in

p

Tp(M)

Tp(M), yx(1)

we define

when

is

yx(1)

p.

Then

be a Riemannian manifold.

M. there exists an open neighborhood U

and an open neighborhood

such that the mapping onto

given by

Let M

in

in

is called the exponential map at

expp

Proposition 4.1.

of

M

as the point in

X

expp : U-.U

U

of

p

M

in

is a diffeomorphism of

U

U.

By applying this proposition we define a normal coordinate system about a point p E M

be an orthonormal basis of

Let w 1,

wn

0

Tp(M)

in

as follows:

Let

its dual basis.

and

p

in

M,

U

and

U

Tp(M)

and

be neighborhoods of

respectively, such that

expp

63

§ 4. Exponential Map and Normal Coordinates onto

U

is a diffeomorphism of

U.

For each Y

in

U,

we put

Y = ylel

+Y2 e 2

Then the componenets nates of the point

are called the normal coordiq = expp(X)

Proposition 4.2.

system about

p

+ ... + ynen.

in

U.

be a normal coordinates

Let

in a Riemannian manifold

M.

Then we have

gij (p) = 6ij;

(a)

JO P) = 0.

(b)

From the definition of normal coordinates about

Proof.

p

we have

gij (p) = g ((ayl) p, (ay,) p) = g (ei, ej) = 6ij.

This proves

(a).

For (b), we have curve with through

v

(ayj a)

=

37 I` a K lj ayk

yi oa(t) = ai = constant.

p.

any constants

.

Then a k

Let G (t)

be the

is a geodesic

Thus, by (1.7), we obtain z I'i j (p) alaj = 0 for i, j Therefore, we obtain (b).

(Q.E.D.)

64 §5.

2. Riemannian Manifolds

Weyl Conformal Curvature Tensor Let

M be a n-dimensional Riemannian manifold with

Riemannian metric

g

and

a positive function on

C

M.

Then

(5.1)

g* = C2g

defines a new Riemannian metric on M which preserves the We call this a

angle between any two vectors at any point. conformal change of the metric.

If

C

is a constant, the con-

formal transformation is called a homothetic transformation. Let

with

g*

and

v*

and

g,

be the Riemannian connections associated Then we have

respectively.

vXY -vXY = w (X) Y + w (Y) X - g (X, Y) U,

(5. 2)

for vector fields

given by i.e.,

v

X,Y

w = d log C

w = U#.

tangent to and

U

M,

where

w

is the 1-form

is its associated vector field,

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) and

L (X, Y) = L*

- 1 2S (X.Y)

+

g (X,Y)

2(n-2)

is the corresponding tensor field of type (0.2) asso-

ciated with

Thus, by eliminating

g*.

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,

g (NX,Y) = L(X,Y)

(5.8) 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.

C

is a confor-

mal invariant.

Thus

We call it the Weyl conformal curvature tensor,

or simply conformal curvature tensor. identically when

This tensor

n = 3.

We also have

(5.9)

D*(X,Y,Z) = D(X,Y,Z) +e(C(X,Y)Z),

C

vanishes

2. Riemannian Manifolds

66

where

D(X,Y,Z) _ (vXL) (Y, Z) - (vYL) (X,Z),

(5.10)

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

locally.

g*,

The following theorem is a well-known result of

Theorem 5 1.

H. Weyl.

A necessary and sufficient condition for a

Riemannian manifold to be conformally flat is that

C=0

for n > 3

D = 0

for

and

It should be noted that if flat.

C = 0.

Moreover, if

n > 3,

then

n = 3.

n = 2,

D = 0

M

is always conformally is a consequence of

67

§ 6. Kaehler Manifolds and Quaternionic Kaehler Manifolds

Kaehler Manifolds and Quaternionic Kaehler Manifolds

§6.

M

Let

be a complex manifold of

complex dimensions

n

defined

with a system of local complex coordinates on a coordinate neighborhood

If

U.

then

yi,

zi = xi +

forms a system of local coordinates on U.

(xl'''*'xno yl'....yn)

We put xi = a/axi and yi = a/ayi. Then X1,...'XnI Y1,....Yn form a basis of of

for each

Tp(M)

Tp(M)

Let

J

be the endomorphism

defined by

JXi = Yi.

Then

p E U.

It is easy to see that

J2 = -I.

the choice of

A Riemannian metric called Hermitian if

g

J

-

-,n.

does not depend on

is called the complex structure of

J

(zi).

1,-

JYi = -X it

g

and

on a complex manifold M

M.

is

are compatible, i.e.,

J

g (JX. JY) = g (X, Y)

(6. 1)

for any

X,Y

in

T(M).

Let

g

be a Hermitian metric on

M.

We put

(6-2) Then

$ (X, Y) = g (X, JY) . f

form of

is a 2-form on M.

M,

which is called the fundamental

A Hermitian manifold is called Kaehlerian if its

fundamental form

I

is closed, i.e.,

results are well-known.

df = 0.

The following

2. Riemannian Manifolds

68

A Hermitian metric

Proposition 6.1.

M

is Kaehlerian if and only if

is parallel, i.e.,

J

vJ = 0.

From (7.2) we have

Proof.

d4 (X, Y, Z) = Thus, if

on a complex manifold

g

3

(g (X, (VZJ) Y) - g (Y, (vXJ) Z) + g (Z, (vYJ) X)) .

then

vJ = 0.

dfi = 0.

Conversely, because

2g ( (vXJ) Y. Z) = & (X, JY, JZ) - d (X, Y. Z) ,

dt = 0

implies

Let

Proposition 6.2.

Then

M be a compact Kaehler manifold.

0; 0 S k s n = dimmM.

H2k(M)

in

Let

Proof.

(Q.E.D.)

VJ = 0.

is nowhere zero on

M.

§n

in such a way that

Then, by (6.2),

= OA... A6 (n-times).

Thus '> 0.

M

is orientable.

Thus we have

f

n

is compact, the Stokes theorem implies that Thus

H2n(M)

in

[§n]

is non-zero for 0

HO(M)

Let

X

in

is non-zero.

Therefore

Consequently,

k =

n

spanned by curvature of

X

and

be a Kaehler manifold.

X.

JX

> 0.

Since

M

is not exact. [0k] EH 2k X

(Q.E.D.)

the sectional curvature

T(M),

M

We orient

H2k(M) # 0.

is trivial.

(M,J,g)

f

For each unit vector

K(X A JX)

of the 2-plane

is called the holomorphic sectional

We denote it by

H(X).

A Kaehler manifold is

called a complex-space-form if it has constant holomorphic sec-

69

§ 6. Kaehler Manifolds and Quaternionic Kaehler Manfolds

tional curvature

c.

of a complex-

R

The curvature tensor

space-form takes the following form:

R(X,Y)Z = 4{g (Y,Z)X-g(X,Z)Y+g(JY,Z)JX

(6.3)

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. M

Let

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

sisting of tensors of type (1,1) over M

V

con-

satisfying the following

conditions:

In any coordinate neighborhood

(a)

a local basis

of

(J1,J2,J31

V

of

U

M,

there is

such that

J1 = J2 = J3 = -I ; (6.4)

1 11 2 = -J2J1 = J3;

For any local cross-section

(b)

a local cross-section of

M

and

Let

where

V,

X

be a unit vector on

are orthonromal vectors in

M.

M.

of

cp

V.

vXcp

(M,g).

Then

We denote by

X. J1X, J2X, J3X Q(X)

the 4-plane

is called a quaternionic 4-plane.

spanned by them.

Q(X)

2-plane in

is called a quaternionic 2-plane.

Q(X)

is also

is a vector tangent to

X

the Riemannian connection of

v

J3Jl = -J1J3 = J2 .

21 3 = -J3J2 = Jl;

A

A quarter-

2. Riemannian Manifolds

TO

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: 3

R (X, Y) Z =

(6. 5)

where

4

f g (Y, Z) X - g (X. Z) Y + F., g (JrY, 2) JrX r=1 3

3

r=1

r=l

- L g (JrX, Z) Jr Y + 2 E g (X. JrY) JrZ) , 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

Submersions and Projective Spaces

§7.

of a manifold into another is called a

rr:R--M

A map

covering map if

is surjective and

rr

rr:R-#M

A covering map

dimension.

and M

M

Thus, in particular,

every point.

is bijective at

1r*

are of the same

of a Riemannian manifold

into another is called a Riemannian covering map if By an isometry

locally isometric.

into another

(N,g)

Let

M.

and freely on

M

r with M

we mean a diffeomorphism

(N',g'),

We say that

p

of

0

,-E

e.

such that

rr

a

in

t,

Example 7.1.

M

and

N

over

Then, for each point

Let

standard unit n-sphere

Let

:M-N

p

in

M and for each

and, moreover,

'r (a (p) ) = Tr (p)

7r

N.

is the group of

t

: M -4 N = M/t

a canonical Riemannian structure

to this metric g',

for each element

a(U) fl U = 0

is a covering manifold of

be the projection.

there is an open

p EM,

Let N = M/I' be the quotient space.

the covering transformations of

element

a group of

acts properly discontinuously

r

if for each point

U

neighborhood

Then

of a Riemannian manifold

be a Riemannian manifold and t

(M,g)

isometries of

in

is

such that g' = cp*g.

cp : N -. N'

a

cp

r

g'

N

admits

such that, with respect

becomes a Riemannian covering map.

0 : Sn 4 Sn be the antipodal map of the Sn

onto itself, which sends a point

p

2. Riemannian Manifolds

72

onto its antipodal point.

in

Sn

Sn

with r

then

a2 = e

Let

r = (e,o)

acts properly discontinuously and freely on

quotient space metric

is an isometry of

a

is involutive).

a

(i.e.,

Then

Sn/i',

denoted by

RPn,

The

Sn.

with its canonical

gp, is a Riemannian manifold of constant sectional we call this Riemannian manifold

curvature 1.

RPn

the real

projective n-space.

Example 7.2.

Regard Rn

as an n-dimensional vector space.

.'vn be a basis of Rn.

Let vl.

We put

n A = ( Elmivil mi

Then

A

integers).

is a free abelian group (or a lattice Acting on

vie...,vn.

Rn

as translation,

discontinuously and freely on

Rn.

A

generated by acts properly

The quotient space

Rn/A

with the canonical metric is a compact, flat, n-dimensional Riemannian manifold which is called a flat n-torus. n-tori lattices

Rn/A A

Rn/A' 'are isometric if and only if the

and and

Example 7.3.

Two flat

are related by an isometry of

A'

Let

a,b

be two non-zero real numbers.

Consider the following two isometries of R 2

a

1:

a2:

Rn.

onto itself:

(xl, x2) - (xl, x2 + b) , (xl, x2) .4 (x1 + 2, - x2) .

be the group of isometries of R2 generated by ra,b 01,02. Then acts on Rn properly discontinuously and ra,b Let

73

§ 7. Submersions and Projective Spaces

The quotient space

freely.

R2/I'a,b

with its canonical

metric is a compact, flat, 2-dimensional unorientable Riemannian manifold, denoted by Two Klein bottles

which is called a Klein bottle.

Ka,b'

are isometric if and only if

Ka,b' Ka',b'

a = a' and b = b'. A map

Tr

of a manifold

:M+B

called a submersion if in

M.

(*r4,)p

onto another

M

is

B

is surjective for each point

p

In particular, a covering map of a manifold

(O'Neill [1])

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

the submanifold

M

of dimension a fibre.

'r-1(b)

We call

dim M - dim B.

A tangent vector of

called vertical if it is tangent to a fibre. Riemannian manifold, then a vector of

M

If

M

M

is

is a

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

preserves lengths of

Tr*

horizontal vectors. Let

M

be a Riemannian manifold and

such that the projection by imposing

Tr*

Tr: M 4 B = M/G

a group of isometries

is a submersion.

Then,

to preserve lengths of horizontal vectors, one

may induce a Riemannian metric on metric on

G

B, Tr:M-#B

B.

With respect to this

becomes a Riemannian submersion.

We

state this as the following.

Lemma 7.1.

1&t M

be a Riemannian manifold and

of isometries such that the projection I: M -OB = M/G

G

a group

is a

2. Riemannian Manifolds

74

submersion.

rr :9 -. B

Then

B

admits a canonical metric such that

is a Riemannian submersion.

Example 7.4.

Cn+l

Regard

12n+2

=

as a (2n+2)-dimensional

Euclidean space with the usual Euclidean metric. Denote by S2n+l the standard unit hypersphere of Let Cn+1.

G = (z E C 11zI = 1). acting on space

52n+1

S2n+1/G.

Then

is a group of isometrics

G

Denote by CPn

by multiplication.

Then CPn

the quotient

admits a canonical complex structure

and, moreover, the projection

(7.1)

'rr

is a submersion. go

such that

v

:S

2n+ 1

By Lemma 7.1,

4 CPn admits a cononical metric

CPn

becomes a Riemannian submersion.

to this canonical metric

CPn

g0,

becomes a Kaehler manifold

of constant holomorphic sectional curvature implies that the sectional curvature 1

K 9 4.

CPn

canonical metric

With respect

In fact, (6.3)

4.

K of CPn

satisfies

is called the complex projective n-space. g0

on

CPn

The

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. S4n+3

the standard unit hypersphere of

G - (z E Q `JzJ = 1). on

54n+3

space

Qn+l.

Then

G

by multiplication.

S4n+3/G.

Then

QPn

structure and the projection

Denote by Let

is a group of isometrics acting

Denote by QPn

the quotient

admits a canonical quoternionic rr:54n+3

.

QPn is a submersion.

§ 7. Submersions and Projective Spaces

Lemma 7.1 implies that

QPn

7r:S4n+3 + QPn

such that

75

admits a canonical metric

go

With

is a Riemannian submersion.

respect to this canonical metric,

becomes a quaternionic

QPn

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. sp :M + M

isometry

point of

is called a symmetry at sp = id.) and

involutive (i.e.,

of

M

p

at

q.

q E M,

there exists a symmetric

The dimension of maximal flat totally

G

is called the

denotes the closure of the group of isometries

generated by symmetries

Then G

the symmetric space 0,

is

A symmetric space is also a homogeneous space.

M.

In fact, if

say at

sp

is called a symmetric

geodesic submanifold of a symmetric space M

topology.

if

A symmetric space is always complete and

locally symmetric.

rank of

p

an

p E M,

is an isolated fixed

A Riemannian manifold M

sp.

space if, for each point sq

Given a point

(sq Iq E M)

in the compact-open

is a Lie group which acts transitive on M;

hence the typical isotropy subgroup

is compact and

M = G/K.

K,

2. Riemannian Manifolds

76

M be a manifold and

Let on

a compact Lie group acting

G

Let M be a Riemannian manifold with group

M.

M

f :M -. M of

An immersion

isometries.

G-equivariant if there is a homeomorphism

is called

such that

C :G -. I(M)

f(a(p)) = C(a)f(p)

(7.3)

a E G

for

M

into

of

I(M)

and

We mention the following result of

P E M.

Mostow and Palais [1) for later use. Proposition 7.1.

closed subgroup of imbedding of

M

G

be a compact Lie group, K

Let

G

and

M = G/K.

a

Then there is a G-equivariant

Sm

into the standard m-sphere

m

for

large enough. Let

n :M -. B

Riemannian manifold If

b

dim M = dim B, in

B,

be a Riemannian submersion of a compact

then, for each function

we define a function

(7.4) If

f

E

f(b) =

on

M and each

on

f

by

B

f(p)

.

pEir-1(b)

dim M > dim B,

then we define a function

f(b) = J -1

(7.5)

where

into another compact Riemannian manifold

M

do

f do

f

on

B

by

,

(b)

denotes the volume element of the fibre

wr 1(b).

following Lemma is well-known. Lemma 7.2.

Let

associated function on

f

be a function on M and B.

Then we have

f

the

The

B.

§ 7. Submersions and Projective Spaces

(7.6)

77

f dVM = fB f dVB , SM

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

2(27)m

(7.7)

(2m-1)(2m-3).. 3.1

c2m

(7.8)

c

If we choose

27T

M+ I

2mf1 m then we have

f = 1,

f = cl

and

respectively, for the submersions (7.1) and (7.2).

c3,

Thus, by

Lemma 7.2, we obtain the following. Lemma 7.3.

Let

and

CPn

QPn

be the complex and quaternion

projective n-space with canonical metrics of maximal sectional curvature

(7.9) (7.10)

4.

Then we have

vol(CPn) = n: 2n vol(QPn) = (2n+1)!

HODGE THEORY AND SPECTRAL GEOMETRY

Chapter 3.

¢1.

Operators

and

*, b

A

M be an n-dimensional, oriented Riemannian manifold.

Let

el,...,en whose

We choose an orthonormal local basis

orientation is compactible with that of w l,...,n w

the dual basis of

is the volume element of

M.

M.

e1,...,en.

Denote by

Then

wl A... Au?

We define an isomorphism

called the Hodge star isomorphism, of

* :AP(M) 4 An-p(M),

p-forms into (n-p)-forms as follows: Since

p-form a

form a local basis of

w11...,wn

on M can be expressed locally as follows:

it

ai E a= it * 1

.

M

be an elliptic operator.

possible to define the adjoint of

It is

D* :T(F) -+ NE),

D;

differential operator characterized by the property; if

as a s E r(E),

u E r(F), then (Ds,u) = (s,D*u)

(2.10)

.

*

It can be verified that (2.10) that

ker(D)

D

is also elliptic.

D) and

(kernel of

are orthogonal with respect to

(

,

).

It follows from * im(D ) (image of D *

Moreover,

ker(D)

is

89

§ 2. Elliptic Differential Operators * im(D ),

precisely the complement of orthogonal to

*

im(D ),

is

s

0 = (s,D Ds) = (DS,Ds),

then

thus

Therefore, we have

De = 0.

r(E) = ker(D) ® im(D

(2.11)

In fact, an elliptic operator

*

has many other

D :r(E) + r(F)

(see, for instance, Palais [1)):

important properties.

example, an elliptic operator

D

in fact, if

For

is a Fredholm operator, i.e.,

D

is a differential operator which has finite-dimensional

From (2.11), we may

kernel and cokernel and closed image. conclude that for any

f

in the orthogonal complement of

ker(D),

there exist a solution of the equation *

D u = f

(2.12)

Now, we consider the special case: D

(i.e.,

is self-adjoint.).

X

denote the completion of

L2(E) ).

(

X E 3R,

r X= (s e r (E) I Ds = Xs I

(2.13)

Let

For each

r(E)

we put

.

with respect to

rX

1 0

and each

rl

0

and each

finite-dimensional and

where ®

*

D = D

Then it is known that there are only countably many

such that

(2.14)

and

E = F

L2(E) = ®lrl

is the completion of orthogonal sum.

rX

is

90

3. Hodge Theory and Spectral Geometry Because the Hodge-Laplace operator

A k(M)

where Vkj

= ®iVk,i

is the i-th eigenspace of

In fact, the eigenvalues of

A : Ak(M)

acting on k-forms.

A -4

0 g Ak,l < 'k.2 < ...

Since the kernel of

is

We have

elliptic and self-adjoint.

(2.15)

A :Ak(M) 4 Ak(M)

Ak(M)

T

A :Ak(M) 4 Ak(M)

satisfy

m

is finite-dimensional,

we obtain the following well-known result. Theorem 2.1.

dimension

M

If

is a compact Riemannian manifold of

then the spaces of harmonic k-forms,

n,

k = 0,1,...,n,

are finite-dimensional.

We simply denote

VO.i

by Vi

and denote

X

by by

X..

The decomposition (2.15) gives the following. Theorem 2.2.

If

M

is a compact Riemannian manifold, then

C0°(M) = ®iVi

(2.16)

-

Denote by

Speck(M)

the set of all eigenvalues of

A :Ak(M) 4 Ak(M)

enumerated with multiplicity.

denote

by

Speck(M)

of k-forms.

Spec(M).

Speck(M)

We simply

is called the spectru m

The geometry which studies spectra of

spectral geometry.

M

is called

Decomposition

§ 3.

¢3.

91

Hodge-de Rham Decomposition

Let M be a compact, oriented n-dimensional Riemannian manifold.

For each

and

A6(M),

k-forms on

k;

/1(M),

k = 0,1,...,n,

the subspaces of

denote by

na(M)

consisting of

nk(M)

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

implies that B E nH(M).

W E A (M),

w

w

is closed.

is orthogonal to

A (M).

Thus corollary 1.1 and

w = do

Let

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

Similar argument applies to the remaining cases.

#H (M).

(Q.E.D.)

We give the following well-known Hodge-de Rham decomposition theorem.

Theorem 3.1.

manifold

a

on a compact, oriented Riemannian

M may be uniquely decomposed into the orthogonal sum:

a = ad+aa+aH ,

(3.1)

where

A k-form

ad E Ak(M),

as E na(M)

and

aH E nk(M).

3. Hodge Theory and Spectral Geometry

92

Theorem 2.1 implies that

Proof.

Thus, we may choose an orthonormal basis

dimensional. w l,...,h w

is finite-

AH(M)

of

Let

nf k,I(M).

be any k-form on

a

M,

we put

h

aH =

Then

AaH = 0,

clear that

aH

i.e.,

a - aH

is harmonic.

is orthogonal to

a solution of the equation such that

A

Moreover, it is

Since

pu = a - aH.

pR = a -a

A

a -aH

is

there is

= A,

Thus, we may find a

We put

H*

ad = d6ft ,

because

AH(M)

wl, i = 1,...,h.

orthogonal to each

k-form

(a,w')w'

i=1

a6 = r, d13

.

Then we obtain a = ad + a6 + aH. if

a = ad '+ a + aH

is another decomposition of

(ad - a') + (a6 - as) + (aH - aH) = 0. ad

aa, a6 = a6,

and

aH = aH.

a,

then

Thus by Lemma 3.1, we obtain 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 Proof.

Let

a closed k-form.

§ E Hk(M).

Then

I = [a]

where

Thus, by Corollary 1.1, we have

a = ad + aH

a

M.

is

93

§ 3. Hodge-deRahm Decomposition

where

ad E Ak(M)

and

is represented by the harmonic form k-form which represents

Theorem 3.1, we obtain

then

§,

ad

Since

aH E AH(M).

aH.

is exact, p

If

is any other

is exact.

a -13

§

Thus, by

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

M

manifold, then the k-th betti number of

is equal to the

number of linearly independent harmonic k-forms on In particular, we have

k = 0,1,...,n.

M,

for

b0(M) = bn(M) = 1.

We also have the following Poincarg duality theorem. Theorem 3.4.

If

manifold of dimension

aH

If

representing

harmonic.

see that

is a compact, oriented Riemannian

n,

then we have the following natural

p :Hk(M) y Hn-k(M).

isomorphisms; Proof.

M

there is a harmonic k-form

I E Hk(M), 4.

Since

G*aH = *paH = 0,

We put 04) = [*%] E Hn-k(M). µ

defines an isomorphism from

*aH

is again

Then it is easy to Hk(M)

onto

Hn-k(M). (Q.E.D.)

Another easy consequence of Hodge-de Rham decomposition theorem is the following:

If

M

is a covering manifold of

M which is also compact, then (3.2)

bk(M) S bk(M)

94 for

3. Hodge Theory and Spectral Geometry This can be seen as follows:

k = 1,...,n-1.

a nonzero harmonic k-form on

extension a

on R given by a =

a non-zero harmonic k-forms on harmonic forms on forms on

A.

M

a

is

then there is a periodic

M,

projection of the covering map.

If

where

It is clear that M.

a

is the

is also

Moreover, linearly independent

lift to linearly

Thus, we have (3.2).

a

independent harmonic

§4. Heat Equation and its Fundamental Solution ¢4.

95

Heat Equation and its Fundamental Solution Ih this section we will mention some well-known results

on the fundamental solution of the real heat equation for later For the detail, see Berger-Gauduchon-Mazet (1).

use.

Let

on M

M be a compact Riemannian manifold.

A heat operator

is the operator

acting on functions defined on

which is of class

M x R+ ,

on the first variable and of class

on the second.

C1

heat equation on M works on functions

F : M x R+ + R

C2

The

which

satisfy L(F) = 0,

(4.1)

where

f : M + R

M

p E M,

is a given initial condition.

A fundamental solution of the heat equation

Definition 4.1.

on

F(p,O) = f(p),

is a function

h : M x M x R+

-0 R which satisfies the

following conditions: (h1) :

C2

h

is continuous on the three variables, of class

on the first two variables and of class (h2)

:

L2h = 0, where

L2 = p2+ at

p2

,

on the second variable, (h3) :

for each

p E M,

limt-#O+

6y

Cl

,

on the third,

the Laplacian

96

3. Hodge Theory and Spectral Geometry

where

is the Dirac distribution at

6

function

f

on M with lim

where

dx

we have

p E supp(f),

h(p,x,t)f(x)dx = f(p)

J'

t..o+

i.e., for a

p E M,

,

M

denotes the volume element of the second

M.

The following result is well-known. Theorem 4.1.

M

A fundamental solution of heat equation on

exists and is unique.

For each eigenspace

Vi

of

i " 'gym

an orthonormal basis

a :C'(M) -. C"(M),

of

Vi (mi

we choose

dim Vi).

The

i

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 , f =

(4.2)

a,i

Proposition 4.2.

(in the L2-sense)

(ma,f)cpQ

If

eigenfunctions, then for each

(4.3)

e

is an orthonormal set of

(mQ)

(p,x,t)

in

M xM x R+ ,

the series

alt CpQ(p)oDQ(x)

converges and

(4.4)

h(p,x,t)

i,a

e

lit cpi(p)cp (x)

The proof bases on the uniqueness of the fundamental solution of the heat equation given in Theorem 4.1.

Now

§4. Heat Equation and its Fundamental Solution

L2(E

97

a-li t ma(P)tpa()

=Ee =E

alt ma (P) (ACP ) (x) +E at (e llt)ma(P)GDQ(x) t t Ca (P)coa(x) -E

= O

lie-11 apa(p)cpa()

.

Moreover, for any f : M -. R , we also have lim t..0

J

Ee

alt

TaTal(x)f(x)dx

M

-xt

= lim E ei (ma,f)CPa t-+O+

=E(ma,fa=f These show that (4.3) satisfies conditions (h2) and (h3). Moreover, in fact,

that we omit the proof of convergence.)

Thus (4.3) is a fundamental

(4.3) also satisfies condition (h1).

solution of the heat equation.

(Note

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

the series E m(ai)e

t > 0,

i

converges and -xi t

p

(4.5)

h(x,x,t)dx = E m(ai)e

J

M

where

m(%i)

i

denotes the multiplicity of

ai.

3. Hodge Theory and Spectral Geometry

98

To construct the fundamental solution, we use a successive approximation method.

Rn

Although

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

d(p,x)2 n -2 4t h0(p,x,t) = (4rrt) e where

d(p,x)

denotes the Euclidean distance between

and

p

Now, using the idea that on a Riemannian manifold

x.

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

there are constants

(4.6)

Fi MO. j

where

(i = 0,1,2,...)

ai's

)e3t j

,.,

n (4Trt)2

aiti

n = dim M.

computed.

(4.8)

with

i-O

t-00

The first four coefficients

(4.7)

M,

a0, al, a2' a3

have been

In fact, it is well-known that

a0 =

J

1 = vol(M)

;

M

a1

1 r * 1

6

(already folklore in 1965)

§4. Heat Equation and its Fundamental Solution a2

has been determined by McKean and Singer [1] in 1967;

a3

was determined by Sakai [1] in 1971.

(4.9)

1 a2 = 360 JM

a2

99

and

is given by

- 2IISII2+5 n2(n_1)2,23 *1

(2IIRII2

In particular, (4.7) gives the following. Proposition 4.5.

Then the volume of

M

Let

M,

be a compact Riemannian manifold.

vol(M),

is a spectral invariant.

By a spectral invariant we mean a global Riemannian invariant which depends only Corollary 4.1. manifolds.

if

Let

M

Spec(M). and

M'

Spec(M) = Spec(M').

be compact Riemannian then

vol(M) = vol(M').

This corollary is an immediate consequence of Proposition 4.5.

100

3. Hodge Theory and Spectral Geometry Spectra of Some Important Riemannian Manifolds

¢5.

In this section we consider the Laplacian

A

For general information in this direction see

functions.

In this case, we have

Berger-Gauchuchon-Mazet (1].

In the following we mention various expressions of (1)

df

For a function

is a 1-form on

field of (2)

vdf

of

Since df

Let

f

div(df) #

a:

the associated vector

(df)t

we have

-bdf,

is

of = -div(df),.

is a 2-form which is called the Hessian of vdf p

(M,g),

is a 1-form, the covariant derivative

df

f.

-Af.

is

M

be a point in

coordinate system about

p.

ul,...,un

and

a normal

Then we have

n E a?2 (p)

(af)(p)

(5.1)

a = bd.

on a Riemannian manifold

Denote by

M.

Since

df.

The trace of (3)

acting on

i=l aui

This is equivalent to say that for each point an orthonormal set of geodesics length and passing through

p

n

(yi)

at

then

d2(f o yi)

i=1

(4)

2

(0)

ds

In terms of local coordinates

yl,...,yn

takes the following general form: 1

of =

9

pick

parameterized by arc

s = 0,

(Af)(p) = -.F,

(5.2)

p E M,

a(g gi3(af/ay.)) ayl

of

M,

of

§5. Spectra of Some Important Riemannian Manifolds

where

with respect to

gij

and

q = det(gij)

the components of the metric

M

N

of a Riemannian

is called totally geodesic if geodesics of M

carried into geodesics of expression (5.2) of

B,

be a Riemannian

ir:(M,g) + (B,g')

Let

Then, for functions

we have

AM(f oTr) = (ABf) or .

(5.3) Proof.

Let

be a point in

p

be an orthonormal basis of

1Yi1i=l,...,n

geodesics through

(5.4) Since

Tp(M)

vl,...,Vn_k

horizontal and Let

are

Using the

by the immersion.

submersion with totally geodesic fibres. on

N

we have the following.

A

Proposition 5.1.

f

g

yl .... 'yn'

In the following, a submanifold manifold

101

such that

k = dim B.

k

n

2

2

E dsd 2 (fo rroyj) - i=1 E dsd2 (forroyi)-j=k+1 is a Riemannian submersion with 7royi

geodesics in

p

through

and

form an orthonormal set of Yk+1' " 'Yn

of the fibre n 1(ir(p)) . Thus, we obtain

A

are geodesics

o 1r) (p) =

Because the Laplacian is well-defined, it is

independent of the choice of local coordinates. (5.3).

are

u1,...,uk

are vertical, where

totally geodesic fibres,

(a5f)(7r(p)).

u1,.. .,uk,vl " " 'vn-k

Then we have, from (5.2), that

rr:(M,g) + (B,g')

B

Let

be the corresponding orthonormal set of p.

am(fotr) _

M.

Thus we obtain (Q.E.D.)

3. Hodge Theory and Spectral Geometry

102

Sometime, we called an eigenfunction of a proper function of

A :C (M) -4 C'(M)

Using Proposition 5.1 we may

(M.g).

prove the following. Let

Proposition 5.2.

Then the proper functions

submersion with totally geodesic fibres. of

are those functions

(B,g')

are proper functions of Let

Proof.

eigenvalue

B

such that

r *(f)

with

(B,g')

Then, by (5.3), we have AM(f or) = if or.

X.

Conversely, if

= for

(M,g).

is a proper function of

f or

on

f

be a proper function of

f

be a Riemannian

ir: (M,g) -6 (B,g')

(M,g)

with eigenvalue

is a proper function of

f or

Thus

X.

(M,g)

with eigenvalue 7, then AM(for) = 1(f oir) = (ABf) air. Since is constant along fibres, this shows that

f or

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

for functions, in this section, we shall

A

use only elementary methods to calculate eigenvalues of spheres and projective spaces.

a

for

In order to do so, we first

obtain a relation between the Laplacian of Rl and that of S. Let

vector basis of of

p

be a point in

en+l

R 1.

in

Tp(Sn).

T p(Rn+1) .

Then

Let

set of geodesics of

p

determines a unit

e1,....en

be an orthonormal

Then

Sn.

Let

e1,...,en+1

y1,...,Yn Sn

through

form an orthonormal basis

be the associated orthonormal p.

If we regard

ell ... 'en+l

§ 5. Spectra of Some Important Riemannian Manifolds as

n + 1

R

points in

with velocity vector

then the geodesic

1,

p

at

ei

103

yi

through

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

be the Euclidean coordinates associated with Consider the functions

and

e1,...,en+l.

(f oyi)(s) = f(yi(s)).

By using the

chain rule, we have

d(f oy.) da

d

1

= -(sin s) 211 + (cos s)

n+1

Bi

2

(f 2yi)(0) ds

_ -a(P) + n+1

af(p) 21x1

Therefore, by (5.2), we get

n (5.5)

L(P) + n ax (p) n+l

n(P)

21(f

i=1 axi

S

On the other hand, the Laplacian

Z

n+1 (Sf)(P) _

(5.6)

- E J=1

of Rn+ l

satisfies

2

2(P) axe

Combining (5.5) and (5.6) we obtain

a(f (

n(P)+ 0 f (p) + n a f (P)

n)(p) S

S

21xn+l

n+l

3. Hodge Theory and Spectral Geometry

104

Consequently, if we denote by

Rn+l

point in

to the origin, then we obtain

(if) Sn = A(f (Sn) -

(5.7)

I

for functions

of

the distance function from a

r

R

f

on

Rn+l,

where

ar I Sn .

denotes the Laplacian

A

1

Consider a homogeneous polynomial on

Sn-n

I

Rn+l .

Let

P - P In.

of degree

P

Then we have

P = rkP.

k _> 0

Thus we

find

aP ar

(5.8)

= kr k-1 P ;

k(k - 1)rk-2 P

a?P2

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

AP = k(n+k-1)P

(5.9)

In the following, we denote by Wk

the vector space of

harmonic homogeneous polynomials of degree

by uk

the restriction of ik

on

dim Vk = rnkk1

Sn.

-

k on Rl,

It is known that

(nk_221

where the last term is assumed to be zero for dim Vk > 0

for each

k _> 0.

and

k = 0,1.

Because

We obtain from (5.9) the following.

§ 5. Spectra of Some Important Riemannian Manifolds

The spectrum Spec(S)

Proposition 5.3.

Sn

n-sphere

of the unit

is given by

kk = k(n+k+l) ,

(5.10)

with the multiplicity

m(%1) = n+ 1, (5.11)

105

m(ak)

of

k _> 0 ,

given by

1'k

m(%0) = 1,

and

m(kk) =

(n+

Moreover, the eigenspace

Vk

is

2

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

which are

invariant under the antipodal map. (R pn,gO)

(Sf,gO)

Thus, proper functions of

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)

the real projective n-space

(5.12)

(R Pn,gO)

Xk = 2k(n+ 2k - 1)

and the multiplicities are given by (n+2k-2

n+2k-3

2k:)

is given by

k

,

m(O) = 1

n+1 n

of

0 and

(n+4k-1),

k21

3. Hodge Theory and Spectral Geometry

106

Using Proposition 5.2 and making further studies on the following two Riemannian submersions: S1

S2n+l

S4n+3

S3

CPn

QPn

one may obtain the following. Proposition 5.5.

The spectrum Spec(CPn)

projective n-space (CPn,go)

of the complex

with maximal sectional curvature 4

is given by

?,k = 4k(n+k),

(5.13)

k,0

and the multiplicities are given by

m(Xk) -n(n+ 2k) `(n(n+1)...(n+k-1) J12 k! Proposition 5.6.

projective n-space

The spectrum Spec(QPn)

(QPn,g0)

of the quaternion

with maximal sectional curvature 4

is given by

Xk - 4k(2n+k+ 1)

(5.14)

From Remark 1.1, Lemma 5.1. and

Let

g - c2g with

(5.15)

we obtain immediately the following. (M,g)

c

.

be a compact Riemannian manifold

a positive constant.

Ak = c2lk ,

k - 0,1,...

Then we have

§ 6. Spectra of Some Important Riemannian Manifolds

where of

and

Xk

denote the k-th eigenvalues of Laplacians

Xk

and

(M,g)

107

respectively.

(M,g),

Now, we determine the spectrum of a flat torus

where

Rn

is a lattice of

A

= (u E Rn

A

called the dual lattice of if

a function

Moreover,

A.

is a basis for

vi,.... vn

on

fx

Rn

it.

It is clear that

vector.

which is also denoted by components of and

vl,...,vn

x

then its dual basis

x

fx fx.

e2irix(y)

on the right is regarded as a

defines a function on Rn / A

If we denote by xi

(Y) =

Thus

and

e2ni E xjyj .

By taking differentiation with respect to

x

we define

respectively, then we find

f

dye

x E Vt,

y with respect to the bases

and

In fact,

by

and y E Rn

i =

(A*)* = A.

For each

A'.

f(Y) =

where

for any v E A )

is also a lattice which is

A

is a basis for

vl,...,vn

Put

.

E a

Then it can be verified that

Rn/ A,

= 2rrixJfx (y) ,

a2fx(Y) aY2i

yj

we get

= _4m'xjj2fx(Y)

yi

the

vl,...,vn

3. Hodge Theory and Spectral Geometry

108

pfx = 47r2IIxII2fx

.

This shows that ) = 4rr2IIxlI2 is an eigenvalue of function

fx

x E A

for each

corresponding eigenspace 11x112

X2 .

=

To each eigenvalue

.

is generated by the

V).

The multiplicity

A

m(%)

of

with proper X.

fx's

the

with

is equal to the

X

47r A*

number of

x

in

11XI12

such that

2

=

.

We summarize this

4Tr

to give the following result of Milnor [4]. Let

Proposition 5.7. and

(Rn/ A, go)

the dual lattice of

A*

be a flat n-torus

Then the spectrum of

A.

Rn/A

is given by

(4rr2IIxII2 and the multiplicity of

I x E A*)

),. = 4ir2IIxII2

,

is equal to the number of

such that IIull = IIxII.

u E A*

In 1941, Witt [1] discovered two lattices

A. A'

in

R16

not isometric but with the same number of elements of any given Using these Milnor showed that there exist two 16-dimensional

norm.

flat tori which are not isometric, but nevertheless they have the same spectrum. Remark 5.1.

bottle

Ka,b

m2

4rr2

a2 +

that if

m2 2

m1

The spectrum Spec( Ka,b)

is also completely known.

) with ml, m2 E 2Z, is odd,

m2 ¢ 0.

of a flat Klein

In fact, it is given by

subject to the condition

SUBMANIFOLDS

Chapter 4.

Induced Connections and Second Fundamental Form

41.

M

M with the

into an m-dimensional Riemannian manifold

Riemannian metric on

be an immersion of an n-dimensional manifold

i :M + M

Let

Denote by

g.

Equipped with

M.

g,

the induced metric

g =

becomes an isometric immersion.

i

We shall identify X with its image X,Y

If

are vector fields tangent to

pXY = pXY+h(X,Y) where of

and

pXY

CXY,

Proposition 1.1 induced metric

Proof.

a, a

M,

X E TIM).

we put

,

are the tangential and the normal components

h(X,Y)

Formula (1.1) is called the Gauss formula.

respectively.

field over

for any

i,X

is the Riemannian connection of the

V

g = ig on

M and

is a normal vector

h(X,Y)

M which is symmetric and bilinear in X X

Replacing

being functions on

and M,

Y

by

and

aX

we have

Vax (RY) = a{ (XR)Y+ 13VXY)

from which we find (1.2)

Vax (oy) = a(Xp)Y+alsVXY ,

(1.3)

h(aX,$Y) = aj3h(X,Y)

.

Y.

respectively,

AY,

= (a(X9)Y + aj3VXY)+ al3h(X,Y)

and

,

4. Submanifolds

110

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

has no torsion, we

v

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 Since the metric

g

v

.

has no torsion and

h

is symmetric.

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 v

M.

X. Y, Z

tangent to

M.

This shows that

is the Riemannian connection of the induced metric

g

on

(Q.E.D.)

§ 1. Induced Connections and Second Fundamental Form

We call

M

the second fundamental form of the submanifold

h

(or of the immersion Let

field on

that

a tangent vector

as

pXg = -ASX+ DXS , and

-ASX

ofvXg, DXS

vXg

We decompose

M.

(1.4)

where

i).

be a normal vector field and X

g

111

are the tangential and normal components

DXg

It is easy to check that A X

respectively.

and

S

are both differentiable on

Moreover, (1.3) implies

depends only on X. Yp E TpM.

h(Xp,Yp)

extensions

M.

X, Y.

not on their

Formula (1.4) is called the Weingarten

formula.

Proposition 1.2.

And

(b) For each normal vector

X, Y of

is bilinear in

A9(X)

(a)

g

g

and

X.

M and tangent vectors

of

M, g(A9X,Y) = g(h(X,Y),g)

(1.5)

Proof. (1.6)

Let

a

and

A

be any two functions on

%X(at) = avx(Pg) = a(X(3)S + ftvXgI

= a(X(3)g -aMAX+aj3DXS . This implies

Aat(aX) = aAASX

.

M.

Then

4. Submanifolds

112

Amt

Thus

is bilinear in

This proves (a).

vector field

Y

g

and

X,

since additivity is trivial.

To prove (b), we notice that for any arbitrary 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.)

denote the normal bundle of the immersion

T'(M)

Let i:M -. M.

From (1.6) we find

DaX(Og) = a(Xp)g+apDXs

(1.7)

.

Moreover, it is easy to verify that

(1.8)

DX+Y

= DX + DY

Equations (1.7) and (1.8) justified that on the normal bundle Proposition 1.3. bundle on

T'(M)

M

T1(M).

D in

is a connection

In fact, we have the following.

is a metric connection in the normal

M with respect to the induced metric

T1(M). Proof.

M,

of

D

For any two normal vector fields

g

we have

Vx

= -ASX+DX9 ;

VXn = -AnX+DXf

and

r

on

§ 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,) =

A normal vector field

Definition 1.1.

parallel if

X

for any

DXg = 0

vector of the submanifold nM H = 0

tangent to

trace h

H =

Definition 1.2.

called minimal if

(Q.E.D.)

is a metric connection.

D

in

g

is said to be

M.

is called the mean curvature

M.

identically.

The submanifold M And

M

is

is called totally

umbilical if

h(X,Y) = g(X,Y) for any

X, Y

tangent to

H

M.

In §2.5 (i.e., §5 of Chapter 2), we have defined a submanifold

M

of a Riemannian manifold

if geodesics of

M

M

to be totally geodesic

are carried into geodesics of

In fact,

M.

we have the following. Proposition 1.4.

M

of a Riemannian manifold geodesic in Proof.

M

i :M -. M be an isometric immersion

Let

into another.

if and only if

h = 0

Then M

identically.

Assume that the second fundamental form

the submanifold

M

in

M

is totally

vanishes identically.

h

of

Then, for any

4. Slubmanifolds

114

vector field X

tangent to

vXX = vXX

(1.9)

If

we have

M,

y(s)

where

is a geodesic in

Thus by (1.9) we find

in

y(s)

Since

h

Hence,

M.

is totally geodesic in

be any unit vector at

M such that

and

y(O) = p

vT(s)T(s) = VT(s)T(s) = O.

we have

vT(s)T(s)

M

is

M.

Conversely, assume that M Let Xp E Tp(M)

vT(x)T(s) s O.

is also a geodesic in

y(s)

totally geodesic in

then we have

M,

T(s) = y(s) = y,w(d-ds).

This shows that

.

is symmetric and bilinear,

R.

Choose a geodesic

p E M.

y(O) = T(O) = Xp.

Thus we find h = 0

at

Then

h(Xp,Xp) = O. (Q.E.D.)

p.

The following elementary results shows that fixed point set of isometries are always totally geodesic. Proposition 1.5. G

Let

a set of isometrics of

for any

a E G)

Proof.

trivial.

M.

be a Riemannian manifold and Let

F(G,M) = (p E M ja(p) = p

be the fixed point set of

connected component of submanifold of

I+1

F(G,F1)

is a closed totally geodesic

If

F(G,M)

Assume that

is empty, then this proposition is F(G,M)

Vp

be the subspace of G.

Tp(M)

be a

p

consisting

According to

Proposition 2.4.1, there is a neighborhood Tp(M)

Let

is not empty.

of vectors fixed by all elements of

in

Then each

M.

point in it and let

0

G.

such that the exponential map

U

of the origin expp :U -. M

O.

§ 1. Induced Connections and Second Fundamental Form

We put

is an injective diffeomorphism. U

that

is convex.

Then we have

is a submanifold

of submanifolds of

M.

u (1 F(G,M) = expp(U fl Vp).

Hence

expp(U f1 Vp).

of

To prove that each connected component of p, q

of

p

F(G,M)

F(G,M)

It is clear that

geodesic, choose any two points

Assume

U = expp(U).

Thus, we find that the neighborhood u f1 F(G,M) F(p,M)

115

in

consists

is closed.

F(G,M)

is totally

F(G,M)

which are

sufficiently close so that they can be joined by a unique minimizing geodesic

a(s).

is also a geodesic joining

For each element p

and

Thus

q.

(ao a)(s)

a E G,

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

are linear subspaces and great spheres, respectively.

and Sm 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 introduced.

(M ,M_) - method was

4. Submanifolds

116

Fundamental Equations and Fundamental Theorems

§2.

Let

M

be an n-dimensional submanifold of an m-dimensional

Riemannian manifold of

M.

Let

R

denote the curvature tensor

Then, for any vector field

M.

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

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

(2.2)

.

tangent to

M,

we have

R(X,Y;Z,W) = R(X,Y;Z,W)+ g(h(X,Z),h(Y,W)) - g(h(X,W),h(Y,Z))

,

M.

§ 2. Fundamental Equations and Fundamental Theorems

where

R(X,Y;Z,W) = g(R(X,Y)Z,W).

(2.3)

Equation (2.2) is called

Moreover, from (2.1), we see that the

the equation of Gauss.

normal component of

117

R(X,Y)Z

is given by

(R(X,Y)Z)L = (-Oxh)(Y,Z) -(iyh)(X,Z)

,

where

(2.4)

(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>) +

Thus, if we denote by

-g(DYDXS,n) -g(D[X,YJS.n)

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

4. Submanifolds

118

RD(X,Y;S.,n) = R(X,Y;S,1)+ g([AS.A11 )(X),Y)

(2.5)

where

(2.6)

(AA,) = ASA'n-A'n AS

Equation (2.5) is called the equation of Ricci. If the ambient space curvature

Si

is a space of constant (sectional)

then equations (2.2), (2.3) and (2.5) of Gauss,

k,

Codazzi and Ricci reduce to

R(X,Y:Z,W) = k(g(X,W)g(Y.Z) - 9(X.Z)g(Y.W))

(2.7)

+ g(h(X,W),h(Y,Z))- g-(h(X,Z),h(Y,W))

(vxh)(Y,Z) = (vYh)(X.Z)

(2.8)

g([AS.A11 )(X),Y)

(2.9)

k

where

is clear that (0.3).

type

vh

For

by

(vxh)(Y,Z)

(vh)(X,Y.Z).

It

is a normal-bundle-valued tensor field of k > 1.

h with respect to

(2.10)

,

is a constant.

Sometime we denote

of

;

we define the k-th covariant derivative

T(M) ® T1(M)

by

(vkh)(Xl,X2.....Xk+2) = DX ((vk-lh)(X2.....Xk+2) 1

k+2

k-1 h)(X2,...,VX1Xi.....Xk+2) - i=2 (v

where

v0h = h.

It is clear that

valued tensor field of type

vkh

(O,k+2).

is a normal-bundle-

By direct computation

§ 2. Fundamental Equations and Fundamental Theorems

119

we have

(vkh)(Xl,X2,X3,...,Xk+2)-(vkh)(X2,Xl.X3,...,Xk+2)

(2.11)

= RD(Xl.X2)((vk-2h)(X3,...,Xk+2))

k+2 (vk-2h)(X3.....R(Xl,X2)Xi.....Xk+2)

+ E

,

i=3

k > 2.

for

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. E

is any vector bundle over a Riemannian manifold

second fundamental form in Aom(T 0 E,T)

(2.12)

a

M,

A

in

satisfying

g(A(X,S),Y) = g(X,A(Y,S))

for any vector fields E,

is a cross-section

E

If

where

g

X, Y

tangent to

is the metric of

M

and

M and a section T = T(M).

E

If

S

is

a Riemannian vector bundle with a second fundamental tensor we define the associated second fundamental form

g(h(X,Y),S) = g(A(X,S),Y)

(2.13)

where

g

is the bundle metric tensor of

h

by

.

E.

We now state the fundamental theorems of submanifolds as follows.

in

A.

4. Submanifolds

120

Existence Theorem.

M be a simply connected

Let

n-dimensional Riemannian manifold with a Riemannian r-plane bundle

E

over

M equipped with a second fundamental form

and associated second fundamental tensor

A.

h

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

with normal bundle

k

Rigidity Theorem.

Let

i, i':M -.IP(k)

E.

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.

of

14m(k)

Then there is an isometry

F

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

n(n+l).

2 A global isometric imbedding theorem was obtained by Nash [1):

121

§ 2. Fundamental Equations and Fundamental 77teorems Nash's Theorem.

manifold of class

Every compact n-dimensional Riemannian Ck-isometrically

can be

Ck (3 S k < e)

AN

imbedded in any small portion of a Euclidean N-space

where

N =

n(3n+ 11).

Every non-compact n-dimensional

2

Riemannian manifold of class

Ck (3 < k < -)

can be

Ck-isometrically imbedded in any small portion of a Euclidean

N-space

where

RN ,

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.

M be a compact n-dimensional Riemannian Rn+r If, at every point manifold isometrically immersed in Theorem 2.1.

Let

.

p

of

contains a k-dimensional subspace

Tp(M)

M,

that the sectional curvature for any 2-plane in

positive, then Proof.

Rn+r

f

is non-

Tp'

x(p) denote the position vector of

where

f(p) = <x(p),x(p)>,

the Euclidean inner product. that

such

r _, k.

Let

We put

.

Tp

takes a maximum at

Let p0.

p

0




,

p

M

such

we

X E Tp (M), 0

have

Xf = 2 = 2<X,x>

the vector

we have

x(p0)

and this is zero at

is normal to M

at

p0.

po.

Thus

Moreover, at

po,

4. Submanifolds

122

X

Since

2

f = 2

+ 2<X,X> = 2+ 2<X,X>

has a maximum at

f

X2f < 0

at

Thus, we

p0.

for any non-zero vector X E T

h(X.X) 1 0

obtain

po,

.

Consider

(M).

po

the restriction of

M

curvature of T'

to

h

T'

po

xT'

po

.

By assumption, the sectional

k = 0,

Thus, by equation (2.7) of Gauss with

.

X AY

is non-positive for any 2-plane

in

we have

0

g(h(X,X),h(Y,Y)) < g(h(X,Y),h(X,Y))

X, Y

where

are orthonormal vectors in

T'

,

By linearity,

.

0

this inequality holds for all

X, Y

in

T' p0

Thus, the

.

theorem follows from the following Lemma of T. Otsuki [1). Let

Lemma 2.1.

map and

h : Rk x Rk _, Rr

be a symmetric bilinear

a positive-definite inner product in

g

Rr

.

if

g(h(X,X),h(Y,Y)) < g(h(X,Y),h(X,Y))

for all X

X, Y

in Rk

,

in

then

Rk

and if

Ck xCk Cr-valued,

for all non-zero

r 2 k.

Proof: We extend map of

h(X,X) ql 0

Cr.

h

to a symmetric complex bilinear

Consider the equation

h(Z,Z) = 0.

Since

this equation is equivalent to a system of

h

is

r

quadratic equations:

h1(Z,Z) = 0,...,hr(Z,Z) = 0

.

123

§ 2. Fundamental Equations and Fundamental Theorems If

then the above system of equations has a non-zero

r < k,

solution

By assumption,

Z.

Z = X +

where

Y,

is not in

Z

X, Y

in

Rr

and

0 = h(Z,Z) = h(X,X) -h(Y,Y)+ 2

we have

h(X,X) = h(Y,Y) I 0

and

Rr.

Y ¢ O.

Thus

Since

h(X,Y)

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 Remark 2.1. Kuiper (1].

R2n-1

Lemma 2.1 was conjectured by Chern and

They showed that it implied Theorem 2.1.

was then proved by Otsuki. T.A. Springer.

The proof above is due to

(See, Kobayashi and Nomizu [2].)

The lemma

4. Submanifolds

124

Submanifolds with Flat Normal Connection

03.

M be an n-dimensional submanifold of an m-dimensional

Let

Riemannian manifold

If the normal connection

M.

D

is flat,

we have

RD(X,Y) = DX Y - DYDX - D(X,Y) - 0 for any vector fields

X, Y

Proposition 3.1.

Let

tangent to

M

M.

be an n-dimensional submanifold

of an (n+ p)-dimensional Riemannian manifold normal connection p

M.

Then the

is flat if and only if there exist locally

D

orthonormal parallel normal vector fields. Proof.

If there exist p,

fields

locally.

Since

RD(X,Y)gr = 0.

Hence RD a 0.

orthonormal parallel normal vector

r

Then we have RD

is tensorial, this implies

Thus the normal connection

D

is flat.

Conversely, if the normal connection

(3.2) for any

Dgl = ... = Dgf = 0.

D

is flat, we have

DXDYgr - DY XSr -D (X,Y) 9r = 0 , p

orthonormal normal vector fields

gl,...,g

P.

We

put

£ er()g5

DXgr =

(3.3)

r = 1,...,P

,

s=1

where

er

are local 1-forms on

(3.3) in matrix form.

M.

In fact, let

For simplicity, we express

§ 3. Submanifolds with Flat Normal Connection

125

t9 = tS1,...,Sp) ,

(3.4)

Then (3.3) can be written as

DS = ®S

(3.5)

The matrix In terms of

0,

9

completely determines the connection

D.

(3.2) is given by

dO=82

(3.6)

Moreover, since

?,1,....Sp

are orthonormal, we also have

(3.7)

We need the following lemma. Lemma 3.1.

Let

8 = (8r)

be a

1-forms defined in a neighborhood of

(p xp)-matrix of 0

in

(3.6) and (3.7), then there exist a unique

of functions in a neighborhood of

A = -A-1(dA);

(3.8)

where

I

If

.

8

satisfies

(p xp)-matrix A

such that

AO = I;

tA = A-'

.

is the identity matrix.

Proof.

solutions.

0

Rn

(Uniqueness).

Then

Assume that

8 = -A-1dA = -B 1dB

A

and

B

are two

and AO = BO = I.

Thus

4. Submanifolds

126

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 space

2

Rn+p

with coordinates

introduce the

and

x1,...,xn,zr (r,s = 1,...,p)

p2 1-forms which are coefficients of the matrix

Z = (zr)

A = dZ + Z® , Then we have

dA = dZA®+Zd® _ (A - ZC) A ®+Z®2

= AA0 . Thus, by Frobenius' theorem, hence, there is a matrix

that A = Z

is completely integrable and

of functions with AD = I

gives an integral manifold of the system

From this we obtain if we put

A

A

C = to-1,

dA = -A0.

Now, because

0

such A = 0.

is skew-symmetric,

then t

dC = -C(dtA)C = Ct®tAC = C 0

Thus, by the uniqueness, we obtain orthogonal.

This proves the lemma.

C = A,

i.e.,

A

is

§ 3. Submanifolds with Fiat Normal Connection

Applying the lemma to the normal connection a matrix

A

defined locally on

D,

we have

dA = -A®.

such that

Let

Then

A = (as).

dar = -E atwt

(3.9)

Put

M

127

gr = E argt.

Then

gl,...,gP

are orthonormal and

Dgr = E (dar + arwt)gs

(3.10)

Substituting (3.9) into (3.10) we find that

.

Dg{ =

= Dgr = 0. (Q.E.D.)

If the ambient space

M

is of constant curvature, then

we have the following result of Cartan [1). Proposition 3.2.

Let

of a Riemannian manifold M

M

be an n-dimensional submanifold of constant curvature.

Then the

normal connection is flat if and only if all the second fundamental tensors

are simultaneously diagonalizable.

A r

This proposition follows immediately from equation (2.9) of Ricci.

4. Submanifolds

128

Totally Umbilical Submanifolds

§4.

Rn

Let

xl,...,xn.

be the Euclidean n-space with natural coordinates

Then the Euclidean metric on Rn

is given by

go = (dxl)2 + ...+ (dxn)2 It is well-known that

is a complete, simply-

(Rf ,g0)

connected Riemannian manifold of curvature zero. We put

Rn(k) = ((x1,...,xn+ 1) E Rn+I

(4.1)

I.JT

((x1)2+ --- + (xn)2+ (sgn k)(xn+l)2

-2xn+1

where

sgn(k) = 1

or

-1

= 0,

according as

xn+1 2 0) k 2 0

)

,

or

k < 0.

The Riemannian connection induced by

go = (dxl)2+ ...+ (dxn)2+sgn(k)(dxn+1)2 on

of

Rn+1 k.

is the ordinary Euclidean connection for each value

In each case the metric tensor induced on

complete and of constant curvature Rn(k)

k.

Rn(k)

is

Moreover, each

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.

129

§4. Totally Umbilical Submanifolds

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 a = (al,...,an+1)

where In

Rn(0)

,

an+l)2 = constants

is an arbitrary fixed point in

these are just the usual hyperspheres.

IR

,

n+ 1

Among

these hyperspheres the great hyperspheres are those sections of hyperplanes which pass through the center (0,...,O,sgn(k)/'qk{) k = 0,

Rn(k)

of

we consider the point at infinite on the

as the center in

Rn+1.

Rn+l

Great hyperspheres in

Rn(k)

Rn(k).

n+

-axis

is just a hyperplane in

Rn (0).

are totally geodesic hypersurfaces

All other hyperspheres in

Rn(k)

are called

Rn(0)

Small hyperspheres of

small hyperspheres.

x

For

The intersection of a hyperplane

through the center in

of

in R'1 , k y( 0.

are called

ordinary hyperspheres or simply hyperspheres if there is no confusion. Proposition 4.1.

submanifold

M

in the real-space-form

totally geodesic in hypersphere of an

of

Rm(k)

(4.2)

Rm(k)

Rm(k)

is either

or contained in a small

(n+ 1)-dimensional totally geodesic submanifold

.

Proof.

Rm(k),

An n-dimensional totally umbilical

If

M

is a totally umbilical submanifold of

then the second fundamental form

h(X,Y) = g(X,Y)H

h

satisfies

4. Submanifolds

130

for

X, Y

tangent to

Substituting this into equation

M.

(2.8) of Codazzi, we find

g(Y,Z)DXH = g(X,Z)DI

By choosing

the mean curvature and that

H = at.

DXg

g

DXH = 0.

Let

If

a unit normal vector field such

Then we have

a = 0,

is totally geodesic.

(Xa)g+ aDXg = 0.

(4.2)

implies that

Assume that

a ¢ 0.

orthonormal normal vector fields

m- n

be

a = JHJ

Since

are orthogonal, we see that the mean curvature

constant.

on

we obtain

Y = Z 1 X,

h = 0.

t

a

Thus

and is

M

Then we may choose

!l,. . 'gym-n

locally

M such that

(4.3) From (4.1) we find

2

(4.4)

Am-n = 0

Dtl = 0

(4.5)

.

Using (4.4), (4.5) and Weingarten's formula, we get

(4.6)

VX(g2 n ... A

where

Rm(k)

v .

tm-n) = 0

,

is the Riemannian connection of the ambient space

§ 4. Totally Umbilical Submanifolds

Equation (4.6) shows that the normal

Rm.

Euclidean m-space

subspace spanned by

Rm

Hence,

spanned by the tangent space

and the mean curvature vector

H

Rm,

dimensional linear subspace of

is a fixed (n+ 1) -

say

Let

Rn+1.

Rm.

be the position vector of

x = (xl,...,xn)

Rm.

is parallel in

g2,...,gm-n

the linear subspaces of Tp(M)

is the

Rm(0)

In this case,

k = 0.

Case (i).

131

Then, by (4.1)

and (4.5), we find

Y(x+ al) = v x-a-lAl(Y)+Dy(a lgl) = Y-Y = 0 , tangent to

for

Y

say

c.

Rn+l

Thus

M.

This shows that a-1

with radius

is contained in a hypersphere of

and center

c.

(resp., case (iii) k = -1).

k = 1

Case (ii).

M

is a constant vector,

x + a 1gl

simplicity, we consider the position vector

For

relative to the

x

center

(0,...,0,1)

(resp., (0,...,0,-1))

of

Rm(1)

(resp.,

Rm(-1))

in

Rm(1)

(resp.,

Rm(-1)),

to

Rm(1)

(resp.,

Rm(-1))

that

VWr1 = W

Rm(-1)

in

r = x in

for any vector W

where

Moreover, we have

v'

Rm+1

.

For each point

p

is a unit normal vector Rm+l.

It is easy to verify

tangent to

Rm(1)

is the Riemannian connection on

(resp.,

Rm+l

4. Submanifolds

132

= VUV -g (U,V),,

VUV

(4.7)

U, V

for any vector fields

Rm(1)

tangent to

,m(-1))

(resp.,

In particular, we have

r = 1,...,m-n ,

7x r = °X'r

for any X

tangent to

Rm*1.

totally umbilical in

M

Thus, the submanifold

M.

is also

Hence, we may conclude that

M

is

contained in the intersection of an (n+ 1)-dimensional linear subspace of

3k m+1

we see that

M

and

Rm(1)

(resp.,

3Rm(-1))

.

From this,

is contained in a small hypersphere of an

(n + 1)-dimensional totally geodesic submanifold.

Remark 4.1.

(Q.E.D.)

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

If the second fundamental form H

of

M

in

M

and the mean curvature vector

satisfy

g(h(X,Y),H) = fg(X,Y)

(4.7)

for some function umbilical.

h

M.

f

on

M,

then

M

is called pseudo-

As a generalization of Proposition 4.1 we have

the following (Yano and Chen (1)).

.

§4. Totally Umbilical Submanifolds Proposition 4.2.

Let

M be a pseudo-umbilical submanifold

Rm(k)

of the real-space-form

curvature vector, then either

Rm(k)

M has parallel mean

If

.

M

is a minimal submanifold of Rm(k)=

is a minimal submanifold of a small hypersphere of

M

or

F1,

.

Proof.

Let

M

be a pseudo-umbilical submanifold of Then the mean curvature

with parallel mean curvature.

Rm(k)

M

a = 0,

is minimal in

a = (HJ

is constant.

If

a

is non-zero.

Then the unit vector

that

of

is parallel,i.e.,

H

133

in the direction

If Rm(k) = Rm ,

DP = 0.

Assume

M.

we

consider the vector field

y(p) = x(p) +

(4.8)

where

is the position vector of

x

any tangent vector on

p

Sa

M

constant.

Rm

This shows that

centered at

y = c

the mean curvature vector and

H

g

M

y

is

lies in the hypersphere

S

of

and with radius H

is always perpendicular to S.

be

Thus

of

M

in

is parallel to the radius vector

the hypersphere

X

AgX

is pseudo-umbilical, we find AS = al.

M

Let

We have

M.

Xy = v x+1a XS = X-a Since

Rm.

in

S.

Thus,

Now, because

a-1.

Rm

is parallel to

x _c,

M

we find that

is minimal in

4. Submanifolds

134

If

of

k ¢ 0.

Rm+1

the result.

we just regard

defined by (4.1).

Rm(k)

as the hypersurface

Then a similar argument yields (Q.E.D.)

§ 5. Minirnal Submanifolds

Minimal Submanifolds

§5.

Let x : M - Rm

be an isometric immersion of an M

n-dimensional Riemannian manifold

a fixed point M

in

Rm

p

in

M.

Let

Rm.

into

M

be an orthonormal local frame on

of

135

ell...len

ve ei = 0

such that

at

denote the position vector

x

Then we f ind

.

n (Ax)

Let

(e) (ex)

P

i=1

1P

(v

1

ei e) 1P

n

i=1

h(ei,ei)p = -nHp

Hence, we have the following well-known results.

Lemma 5.1.

Let x : M -+ Rm be an isometric immersion.

Then

Ax = -nH

(5.1)

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.

of

Rm .

There are no compact minimal submanifolds

4. Subnwnifolds

136

Proposition 5.1. (Takahashi [1)). be an isometric immersion.

(1)

).>0,

(2)

x(M) c So-1(r),

of Rm

Ax = Xx,

If

x :M 4 So-1(r)

(3)

Furthermore, if

X ¢ O,

where So 1(r)

centered at the origin

0

x :M + Rm

Let

then

is a hypersphere

and with radius

r

is minimal.

x :M + So-1(r)

is minimal, then

Ax = (n/r2)x. Proof.

If

Let

H= -()./n)x.

Ax = Xx,

X

be a vector field tangent to

<x,X> = 0

(5.2)

Thus

then by Lemma 5.1 we have

31 0,

.

.

X<x,x> = 2<x,vxx> = 2<x,X> = O.

constant on

Therefore

This proves that IxI

M.

Let

forms of

in

M

respectively.

and

h, h'

Fm ,

M

Rm

of

is

Thus,

M

centered at

be the second fundamental

Ti

in So 1(r), and So-1(r)

in

h(X,Y) = h'(X,Y)+I(X,Y).

Then we have

the mean curvature vectors

<x,x>

is constant.

is immersed into a hypersphere So 1(r) the origin.

we have

M,

H, H'

M

of

in

Rm

Rm Thus,

and So1(r)

n

where

n E 1i(e.,e.),

satisfies

H = H'+ H,

e1,....en

an orthonormal frame of

perpendicular to this implies that Because

X

So-1(r) H' = O.

at

H

i=1

and

p

M

Thus

is an eigenvalue of

M.

a

Since

Hp

x(p)

M,

is

is parallel to

is minimal in on

and

X > O.

x(p),

So-1(r). Now,

§ S. Minimal Submanifolds

137

n

ei'x-rPZ>(xZ) r

nHp = E 0).

that

is a minimal immersion and

xIL

is an invariant

d4i

c =

these as the following well-known result.

n.

We summarize

(Takahashi (1),

Wallach [1)). Theorem 5.1.

Let

M = G/K be an irreducible compact

symmetric space equipped with a G-invariant Riemannian metric
.

A

of

&

there is an isometric minimal immersion of

(M,
d)

F Pm which is imbedded into

minimal

H(m + 1;F)

Then we have

(6.37)

IHI2
2

(6.38)

Thus, by Lemma 6.3, we find 2 n+l) +

IHI 2 =

(6.39)

Denote by

n

n 2

n2 i,j=l

the endomorphism of

P

= <JX,Y>

for

i

Jej>2

defined by

T(M)

Then, by (6.39), we find

X, Y E T(M).

IH12=2nn1

(6.40)

<e

F,

+ 2IIP112 n

Since have

n

is nothing but the tangential component of

P

with the equality holding if and only if

IIPII2 S n,

is even and

T(M)

is invariant under

a complex submanifold of 2(n+2),

1H12

n

given by (2.6.5). M,

M

is

M

is

CPm R

takes the form

Thus, for any orthonormal vectors

we have 3

(6.41)

i.e.,

Thus, we find that

the curvature tensor

F = Q,

tangent to

QPm

J,

with equality holding if and only if

a complex submanifold of If

JITM,

K(X,Y) = 1+ 3 E <X,JrY>2 r=l

Thus, by combining (6.34) and (6.41), we find

X, Y

we

4. Submanifolds

154

2(n+1)

=

(6.42)

+

n

3

-L

IH:1

E <ei,J e.2

F,

r j>

n 2 r=1 i,]=1

Define the endomorphism for

IH12

Since

nl

2

=

+ n2

holding if and only if

IH12 < 2(n+4)/n, equality

is a multiple of

n

quaternionic submanifold of

A E cp(CPm)

fundamental form

(11P1112+ 11P2112+ 11P3112)

(6.43) implies

IIPr112 < n,

Let

= <JrX,Y>

by

we have

X, Y E T(M),

(6.43)

Tp(M)

of

Pr

h

of

M

and

CPm

H(m + 1;C)

in

is a

(Q.E.D. )

QPm.

X,Y,Z E TA(CPm).

and

4

Then the second satisfies

Thus, we find

h(JX,JY) = h(X,Y).

(vXfi) (JY,JZ) = DXh(Y,Z) - Fi(vxY,Z) - Fi(Y,vxZ) (vXFi) (Y,Z) (v,I)(Y,JY) = 0.

from which we find

we obtain

(vYh)(X,JY) = 0.

In particular, this implies

(V c)(Y,Y) = (vlri)(JY,JY) = 0.

H(m+ 1;C) is,

h

Since the ambient space vh = 0,

is Euclidean, this implies that

that

is parallel.

Remark 6.1.

It was proved in Little (2) and Sakamoto [1]

that the second fundamental form

H(m+ 1;F) , F = R , C or (6.44)

Applying Codazzi equation,

Q,

Fi

under

vh = 0

.

of each cp

F Pm

in

is parallel, that is,

155

§6. The Fast Standard Imbeddings of Projective Spaces

One may obtain the first standard imbedding

Remark 6.2.

of the Cayley plane

OP2

in a similar way as follows:

Cay denote the Cayley algebra over

(e0 = l,e1,...,e7)

Let

I R.

be the usual basis for Cay. For a z = z0 +Z 1e 1 + in Cay, the conjugate of

z

Cay = R8

and the norm of

is defined as

x

H(3;Cay) be the space of Then

x

for

<x,y> = Re(xy) = E xiyi

. + z7 e7

is defined by

z = z0-zle1-... -z7e7 The usual inner product in

Let

.

is given by xiei,

Y = E Yiei

jxI =

<x,x>1"2.

Let

Hermitian matrices over Cay.

3 x3

H(3;Cay) is a Jordan algebra under the multiplication:

A *B = Z(AB+ BA), Define an inner product in

=

tr(AB)

for A,B E H(3;Cay) H(3;Cay) = R 27

for

by

A,B E H(3;Cay)

2 Let

H1(3;Cay) = (A E H(3;Cay) Itr A = 1).

projective plane

OP2

is defined as

OP2 = (A E H1(3;Cay) I

The

OP2

Then the Cayley

A2

= A)

.

with its induced metric becomes a compact rank-one

symmetric space with maximal sectional curvature 4. it is known that

OP2

Moreover,

is a minimal 16-dimensional submanifold

4. Submanifolds

156

S23(--)

of a hypersphere

with radius

in

H1(3;Cay).

-L (see, Tai (1), Little [2), and Sakamoto [11). OP2

see that the mean curvature IHl of

(Hl =

(6.45) Moreover, if

M

H

of M

then the

,

equality holding if and only if n at

OP2.

H1(3;Cay) satisfies

in

JHI S 2

(6.46)

H1(3;Cay) satisfies

.

is a minimal submanifold of

mean curvature vector

TA(M)

in

From this, we

8, and each tangent space

is a subspace of a Cayley 8-plane of

A E M

TA(OP2).

Furthermore, one may prove that the first nonzero eigenvalue k1

OP

of Laplacian and the volume of

vol(OP2)

(6.48)

(6.49)

are given respectively by

11 = 48 ,

(6.47)

where w

2

7

=

7T, w

is given by

w=

r fr sin8(y -X) I sin 2(y-x) I7 2

0

x

dy dx

§7. Total Absolute Curvature of Chern and Lashof Total Absolute Curvature of Chern and Lashof

{7.

Let

be a closed oriented curve in the 2-plane

C

As a point moves along 0

or rotate

2nrr

3R2

the line through a fixed point

C,

and parallel to the tangent line of

an angle n

157

times about

n

is called the rotation index of

C.

rotates through

C

This integer

0.

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 Because the rotation index is an integer and it

curves.

varies continuously through the deformation, it must keep Therefore, two closed smooth curves have the same

constant.

A theorem of

rotation index if they are regularly homotopic.

Graustein and Whitney says that the converse of this is also Thus, the only invariant of a regular homotopy class

true.

is the rotation index. Let

closed curve length

(7.1)

where

be the Euclidean coordinates of the

(x(s),y(s))

s.

C

in

R2

which is parameterized by its arc

Then we have Y"(s)

x"(s) = -x(s)Y'(s) ,

x(s)

denotes the curvature of

- x(s)x'(s)

C.

Let

angle between the tangent line and the x-axis.

de =

xxy2+ x1# x m y !

Y

2

ds = x ds

8(s)

be the

We have

4. Submanifolds

158

From this we obtain the following formula:

(7.2)

J

n = the rotation index

x ds = 2nrr,

.

C

Using (7.2) we may conclude that the total absolute curvature, of

f InIds,

C

satisfies

Ix Ids > 2n

(7.3) C

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

Rm, m > 3.

curves in

In 1949 - 1950, Fary [1] and Milnor [1]

obtained the following improvement to knotted curves. Theorem 7.1.

If

Rm

is a knotted closed curve in

C

,

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 Let

which we will discuss as follows. x : M - Rm

be an isometric immersion of an

n-dimensional closed manifold

The normal bundle

T1(M)

of

M M

into a Euclidean m-space. 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 and

p E M

that

M

is a normal vector of

g

respect to the induced metric from Rm

at

With

p.

the normal bundle

is a Riemannian (m-n) - plane bundle over

rI(M)

159

M.

Let

B1

denote the subbundle of the normal bundle whose bundle space consists of all points and

(p,g)

is a unit normal vector at

C

M

of (m-n-1) -spheres over

of degree

to a fiber

m -n- 1

unit normal vectors at of

B1.

is a bundle

B1

on

Then

p.

We denote it by

dVB

.

denote

Then there is a differential

M.

B1

dV

Let

such that its restriction

is the volume element of the sphere

Sp

p E M

and is a Riemannian manifold of

the (Riemannian) volume element of do

Then

p.

endowed with the induced metric.

dimension m -1

form

such that

T'(M)

in

dV A do

SP

of

is the volume element

In fact, this can be seen as

1

follows:

Rm

Suppose that in

Rm

we mean a point

with that of Rm M.

Moreover,

Rm.

Then

Denote by

structure group

in

Rm

p, el,...,em

and an ordered set of

F(Rm)

the space of all frames

is a manifold of dimension

F(Rm)

F(Rm)

p

By a frame

e1,....em whose orientation consistent

orthonormal vectors

in

is oriented.

is a fibre bundle over SO(m).

Rm

m(m+ 1).

with 2the

In what follows it is convenient to

agree to the following range of indices:

1 S i,j,k S n;

Let

wA

n+ 1 S r,s,t S m;

denote the dual 1-forms of

connection 1-forms defined by

1 S A,B,C S m.

eA.

Let

W_A

be the

4. Submanifolds

160

V eB = E WAB eA

(7.5)

Then

-A, -A

satisfy the following structural equations of

Cartan:

dwA

(7.6)

= - E wB n wB

4A = - E wA A wB , B B C

( 7 .7 )

WA + W-B

B

A= 0

Throughout this section, we shall consider

F(Rm)

forms defined on Let

x : M .4 Rm

be an isometric immersion of an

with its image under

t

Rm.

We identify a Let

x,,.

the bundle whose bundle space is the set of

projection B,

M and

B a M

en+l,...,em

is denoted by i.

denote

B

M xF( Rm)

consisting of (p,x(p),el;...,en'en+l" ..,em) are tangent to

as

in a natural way.

n-dimensional Riemannian manifold into tangent vector

WA, WB

such that

are normal to

M.

el,...,en

The

We define the map

by $l(P,x(P),el,...,em) _ (P,em)

(7.8)

Consider the maps

B l--

(7.9) where

i

M x F(Rm) -2-> F(Rm)

is the inclusion and

second factor.

Put

k

is the projection onto the

§7. Total Absolute Curvature of Chern and Lashof (7.10)

Since

WA =

and

d

n

(),i)*wA

WAB = (li)*w8 *

commute with

(Xi)

dwA = - E W

(7.11)

From the definition of

wl...,Wn

B

M,

it follows that

dV = w 1

wr = 0

and

If we restrict these

then the volume element

(7.13)

dV

of

M

is given by

n ... nwn

Moreover, the volume element of

(7.14)

(7.6) and (7.7) imply

,

n W

are linearly independent.

n 1-forms to

161

B1

dVndo = wln ...

is

nu,nnwn+ln...

nwm-1

being equal to the product w n+1 n This is n wm-1' the (m - n - 1) -form on B1 which we are looking for.

do

Since

wr = 0,

(7.11) gives

o=dwr=

wf Awi

Hence by Cartan's Lemma (Proposition 1.3.2), we may write

(7.15)

M

in

wi = E hij W

Rm ,

we have

4. Submanifolds

162

(7.16) where

hid =




,

,

denotes the Euclidean metric of

Rm

.

Consider the map

v : B1 -+ Sm-1 of

BI

into the unit sphere

Rm

of

defined by

Denote by dj the volume element of

v(p,e) = e.

e = em

Sm-1

Sm-1

is the position vector of

Rm,

in

Sm-l.

Since

(7.5) implies

= win ... A wm_1

(7.17)

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 The total absolute curvature

TA(X) -

1

cm-1

where

(7.21)

of the immersion

in the sense of Chern and Lashof, is then defined by

x :M -o Rm,

(7.20)

TA(x)

(p,em).

cm_1

Iv*dZI =

f

1

cm-1

B1

f

G*(p)dV

M

is the volume of unit (m-1) -sphere and

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.

M

immersion of an n-dimensional compact manifold x

Then the total absolute curvature of

be an

Let x : M - Mm

Theorem 7 .1 (Chern-Lashof (1, 21) .

into

1R

M

satisfies the following

inequality: TA(x) 2 b(M)

(7.22)

Proof.

For each unit vector

height function

a

is a unit normal vector at

g

in

in the direction

ha

a

Sm-1 we define the by

p E M .

ha(p) = ,

If

.

i.e.,

p,

dhS(p) = = 0

Hence,

p

hg.

is a critical point of the height function

then

dha(P) = = 0

Thus,

then

.

is a critical point of the height function

p

Conversely, if ha,

(p,g) E Bl,

a

is a unit vector normal to

(p,a) E B1.

points of

at

p,

i.e.,

Consequently, we see that the number of all critical

ha which is denoted by

number of points in

we obtain

M

.

M with

a

$(ha)

is equal to the

as its normal vector.

Hence,

4. Submanifolds

164

= Iv*

JB

SaESr-1

1

a

Since for each point if and only if

in a

ha

Sm-l,

has degenerate critical

is a critical value of the map

By Theorem 1.6.1 of Sard the image of the set of

v :B1 -4 Sm-l.

critical points of almot all

13 (ha)c .

I

a

in

v

Sm-1,

has measure zero in ha

Thus, for

Sm-1

is a non-degenerate function.

Therefore,

g(ha)

is well-defined and is finite for almost

all

Sm-1.

By applying Theorem 1.6.2 of Morse we

a

in

obtain (7.22).

(Q.E.D.)

Theorem 7.2

(Chern-Lashof [1)).

Under the hypothesis of

Theorem 7.1, if

TA(x) < 3

(7.23)

then M

is homeomorphic to an n-sphere.

Proof.

Suppose that (7.23) holds.

set of positive measure on

Sm-1

such that if

vector in this set, the height function critical points.

a

critical points.

ha

a

is a unit

has exactly two

Since, by Sard's theorem, the image of the

set of critical points under a unit vector

Then there exists a

such that

v

ha

is of measure zero, there is has exactly two non-degenerate

Applying Theorem 1.6.3 of Reeb,

homeomorphic to an n-sphere.

M

is

(Q.E.D.)

§7. Total Absolute Curvature of (here and Lashof For a hypersurface

of 2

into two parts, then

Rn+l

in

Tp(M)

the tangent plane

p E M,

M

M

M

at

p

does not separate

M with total absolute curvature

we have the following. Theorem 7.3 (Chern-Lashof (11).

of Theorem 7.1, if subspace in

if for each point

is called a convex hypersurface

For an immersion of

.

Rn+1

165

Rn+l

Rn+l

of

TA(x) = 2,

Rm

then

Under the hypothesis

M belongs to a linear

and is imbedded as a convex hypersurface

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

M in

into

R3

x : M -. R3

R 3.

is an immersion of a compact surface

then the Lipschitz-Killing curvature of

reduces to the Gauss curvature

G(p,e3)

(7.24)

G,

M

i.e.,

G(p)

Thus, by (7.22), we obtain the following inequality of Chern and Lashof:

(7.25) where

J

X(M)

jGjdV

Z 2tr(4 - )((M)) ,

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.

T

Let

(7.26)

f

be a knotted torus in

Iii

.

Then

IGIdV > 16a

T

Recently, N.H. Kuiper and W.H. Meeks [1) improved (7.26) to the following. Theorem 7.5.

(7.27)

Let

T

f

be a knotted torus in

3t3

.

Then

IGIdV > 161r

T

For the proof of this theorem, see Kuiper and Meeks [1).

§8. Riemannian Submersions

167

Riemannian Submersions

§8.

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

vector YX

be a Riemannian submersion.

7r :M - B

X

of

can be decomposed as

X

M,

is vertical and kX

is horizontal.

the Riemannian connections of X

each tangent vector field horizontal vector field X

and

Y

X

on

and

VX +}1X,

Let B,

v

K(vXY) = vXY

B,

Let

and

v

be

To

there corresponds a unique

B

on M such that

This can be seen as follows:

where

respectively.

7r*X = X.

are any two tangent vector fields of

(8.1)

field of

M

For a tangent

Z

B,

If

we have

.

be any tangent vector

we have

2

+ Y - + +

= X o 7r + Y o 7r - Z<X,Y> o 7r

- <X, [Y,ZI> o 7r + o it + o 7r = 2 o 7 This shows (8.1). field on

M,

then

Furthermore, if

V

is a vertical vector

4. Submanifolds

168

7,t [%,V] = [n*]C,ir V] = 0

%,(VXV) = W(VvX)

We define on M

(8.4)

we put

M,

AXY = YV

This definition shows that

CY + *(VUXYY

AX

horizontal and vertical subspaces.

A X Y = -A X

X

If 2

and

Y

N

there is a submersion

X

is vertical,

is an n-dimensional submanifold of ir:M a B.

7r -.N -. N'

where

-1

N'

is a submanifold

commutes and the immersion

M

1n

N

B

f

M

That is, suppose that

such that the diagram

Nf)

fibers.

Moreover, if

are horizontal fields, then

which respects the submersion

B

M and it reverses the

Y[X,Y].

Suppose that

of

.

is a skew-symmetric linear

operator on the tangent space of

AXY = 0.

X, Y be tangent

Let

tensor of the submersion as follows: vector fields on

.

of type (1,2) called the fundamental

A

a tensor

.

is a diffeomorphism on the

§8. Riemannian Submersions

169

We shall now relate the second fundamental forms of the submanifolds

N

and

The discussion will be local, and

N'.

so for convenience we shall consider

M

and

If

we denote by

X

and

imbedded in

N'

respectively with the usual identification of tangent

M',

vectors.

N

X

is a tangent vector of and

XT

respectively the projections of

XN

on the tangent and normal spaces of

the normal space is always horizontal. N

vector fields of

N').

(or

M at a point p E N,

and second fundamental form of

at

N Let

Note that

p.

X. Y

be tangent

Then the Riemannian connection (or N') are given respectively

N

by

,

(or

vXY = (vxY)T)

h(X,Y) _ (vxY)N ,

(or

h'(X,Y) = (vxY)N)

vXY = (vXY)T

.

We give the following lemma for later use. Lemma 8.1 (Lawson [1)) if and only if Proof.

N'

N

is a minimal submanifold of

is a minimal submanifold of

For a point

p

in

N,

B.

we choose an orthonormal

about

p.

Let Fl,...,Fn

be local, orthonormal tangent fields on

N'

about

local vertical fields

Denote by

Then N

Fl,...,Fn is minimal in

E1....,Ed

the horizontal lifts of

M

M

if and only if

ir(p).

F1,....Fn.

170

4. Submanifolds

0 = E (O k=1

j This is equivalent to

1

E

Ek k

)

+ E (Vi F.)N j=1

(VF Fj)N =

Fi

j

tr h' = 0

j=1 .

Fj

(VF ) N

j

(Q.E.D.)

§9. Submanifolds of Kaehler Manifolds

Submanifolds of Kaehler Manifolds

§9.

Let

J

171

M

be a Kaehler manifold with complex structure

and Kaehler metric

For each point

g.

p E M,

M be a submanifold of

Let

denote by Xp

subspace of the tangent space

Tp(M),

the maximal holomorphic i.e.,

VP = Tp(M) (1 J(Tp(M))

(9.1)

if the dimension of

.

M

is constant along

!!p

a differentiable distribution X over a generic submanifold of

M.

and

then

M,

The distribution

of

afp

M

p E M,

we denote by Xp

complementary subspace of submanifold, then 1l1

over

M,

in

flp

defines

is called

is called

the holomorphic distribution of the generic submanifold For each point

M.

M.

the orthogonal

Tp(M).

If

M

is a generic

define a differentiable distribution

A(P, p E 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 Let

X

M

tangent to

be a generic submanifold of M,

JX = PX+FX ,

where

and

For a vector

we put

(9.2)

of

R.

M.

PX and

JX, F

FX

are the tangential and normal components

respectively.

Then

P

is an endomorphism of

is a normal-bundle-valued 1-form on

T(M).

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

M

of

if its purely real distribution

W'L

is called a CR-submanifold is totally real, i.e.,

(Bejancu [1], Chen [17,20], Blair-Chen [11).

JV II C Tp1(M), P E M.

Since (P

(9.4)

1u)2

= -id.

we have the following inequality:

P 2 , 2a

(9.5)

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]).

WL

of a CR-submanifold

M

The totally real distribution

of a Kaehler manifold

M

is

Z

and W

completely integrable. Proof.

Let

X

be a vector field in

vector fields in V. Then

$(X,Z) = g(X,JZ) = 0,

denotes the fundamental form of di = 0.

Thus we have

and

I(

M.

Since

M

where

is Kaehlerian,

§ 9. Submanifolds of Kaehler Manifolds

173

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) Because M, [Z,W]

JX

is arbitrary in V

must lie in

u1

distribution

and

is tangent to

[Z,W]

This shows that the totally real

V1.

is involutive.

u1

implies that

.

Hence, Frobenius' theorem

is completely integrable.

(Q.E.D.)

The proof of Theorem 9.1 given above is

Remark 9.1.

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]).

manifold with that

M

d§ = §n w

M be a Hermitian

Let

for some 1-form

w.

Then in order

is a CR-submanifold it is necessary that u1

is

completely integrable. Let

manifold

.

M.

be a differentiable distribution on a Riemannian We put 0

(9.6)

h(X,Y) _

for any vector fields

X, Y

the component of

vXY

in

xY)1

9,

where

(vXY)1

denotes

in the orthogonal complementary

4. Submanlfolds

174

distribution

Let

T(M).

in

.D

el,...,er

be an orthonormal

basis of .8, r = dimR. . If we put r

(9.7)

o

H = z E h(ei,ei) i=1

is a well-defined f-valued vector field on

H

Then

m

(up to sign), called the mean-curvature vector of the distribution 0

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)).

of a CR-submanifold

M

The holomorphic distribution

of a Kaehler manifold

M

a!

is a

minimal distribution.

Let X

Proof.

and

Z

be vector fields in U

respectively.

Then we have

(9.8)

g(Z,v.X) = g(Z,vXX) = g(JZ,VXJX)

and

!!j,

= -g(vXJZ,JX) = g(AJZX.JX)

where and

(9.9)

M,

v

and

v

denote the Riemannian connections of

respectively.

M

Thus, we find

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 is always a minimal distribution.

J!

(Q.E.D.)

§9. Submanifolds of Kaehler Manifolds

submanifold of a Kaehler manifold

M

Let

Theorem 9.4 (Blair and Chen [11).

175

be a generic

Then the holomorphic

M.

distribution is completely integrable if and only if

h(X,JY) = h(JX,Y)

(9.10)

for X, Y in Proof.

W. If

Al

is integrable, let N be an integral

submanifold.

The second fundamental form

M satisfies

h'(X,Y) = h(X,Y)+ a(X,Y),

second fundamental form of N

N

in

is a Kaehler submanifold of

where

of a

Since V

M.

M.

h'

N

in

is the

is holomorphic,

Thus

h(X,JY) - h(JX,Y) = h(JX,Y) - h(X,JY)

(9.11)

But the left-hand side is normal to is tangent to

M.

M

and the right-hand side

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 tangent to

J M

distribution.

.

applied to the tangent vector field and hence

[X,Y]

[X,Y) is

belongs to the holomorphic

Thus the result follows from Frobenius' theorem. (Q.E.D.)

4. Submanifolds

176

Theorem 9.4 implies the following. Corollary 9.1 (Bejancu [1]). of a Kaehler manifold

W

M.

M be a CR-submanifold

Let

Then the holomorphic distribution

is completely integrable if and only if

h(X,JY) = h(JX,Y)

for X, Y in V. In contrast with the integrability of minimality of

and the

W!'`

for CR-submanifold, we have the following

Al

theorem.

Theorem 9.5 (Chen [21]).

of a Kaehler manifold then either Proof. fold

M.

of W. (9.12)

for

Z

A!

M.

If

Let

M

be a compact CR-submanifold

H2k(N) = 0

is not integrable or W

for some

k S dim V ,

is not minimal.

Let M be a compact CR-submanifold of a Kaehler mani-

Choose an orthonormal local frame

Denote by

w1 ,...,w2a

the 2a 1-forms on

w?(Z) = 0;

wI(ei) = bi;

in 0, where

ea+j = Jej.

W=W

(9.13)

ln

dw =

2a 7,

i=1

(-1)1 wl n

M

i,j = 1,...,2a

Then

...

M.

From (9.13), we have

ndwl n

...

nw2a

From a straightforward computation, we can prove that if and only if

satisfying

Aw2a

is a well-defined global 2a-form on

(9.14)

el,...,ea,Jel,...,Jea

dw = 0

§ 9. Submanifolds of Kaehler Manifolds

dw(Z1,Z2,X1,...,X2a-1) = 0

(9.15)

177

,

dw(Z1,X1,....x2a) = 0

(9.16)

Z1, Z2

for any vectors

in

Wl

and

(9.15) holds when and only when Wl

Xl,...,x

But

is integrable and (9.16) holds

is a minimal distribution.

when and only when W

V.

in

2,-l

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

Consequently, we obtain the following.

is closed.

Lemma 9.1. Kaehler manifold

For any compact CR-submanifold the 2a-form

M,

w

of a

M

defines a deRham cohomology

class given by

c(M) _ [w) E H2a(M)

(9.17)

a = dimcl.

where

We need the following. Lemma 9.2.

The cohomology class

compact CR-submanifold non-trivial class if

M Al

c(M) E H2a(M)

of a Kaehler manifold is integrable and

We choose an orthonormal local frame Jea,e2a+11....

X

and

e2a+1

e2a+g

such that

" .. 'e2a+iB

the dual form of

All

is a

is minimal.

e1,...,eaJel,...I

el,...,ea,Jel,...,Jea

are in W. Denote by

el,...,ea,ea+l

M

of a

are in

w1,...,w2a+H

" . . .e2a'e2a+1 " . 'e2a+13'

4. Submanifolds

178

where

Then

ea+i = Jei.

Hodge star operator

(9.18)

*

*w =

Since

w2a+1(X)

(9.18) implies that

Applying the

w = wl A ... n w2a. to

w

we obtain

w,

2a+1

n

2a+$ ...w n

= ... = w2a+o(X) = 0

for all

X

in

Al,

if and only if

d*w = 0

(9.19)

d*w(X1,X2,Z1,....Z8 1) = 0 ,

(9.20)

d*w(Xl,Z1,...,Z13 ) = 0

in

Af

and

(9.19) holds if and only if

K

is integrable; and (9.20) holds

for all vectors

if and only if

and VI

X1, X2

All

(-1)2an+n+1*d*w

= 0,

i.e.,

is a harmonic 2a-form on

M.

w

*w

H2a(M).

Because

ifl.

is integrable

is closed.

Thus,

is also co-closed.

Because

Theorem 3.3.2 of Hodge-deRham implies that a non-trivial class in

in

We see that if V

is minimal.

is minimal, then the 0-form

bw = w

Z1....,Z8

w

Therefore,

is non-zero, c(M) _ [w)

represents

This proves Lemma 9.2.

we choose an orthonormal local frame e

in

M

are in 1 U)

'ea'ea+l'...,eo+8.ea+p+l' ...,em,Je1,...,Jem 1.

in such a way that, restricted to

V

m ,...,w ,W

We put

and

ea+ 1,...,er a

,...,Um

are in

All.

the dual frame of

M, e1....,ea,Jel,....Jea

we denote by ell ..

,em,Jel, ..,Jem.

179

§9. Submanifolds of Kaehler Manifolds

'r 1

9A = wA +

*

wA

8A = wA -

,

'r 1

*

W

.

A = 1,...,m

.

*

Restrict

8A

and

0A.s

to

s

we have

M,

8i = 41 = wl,

for

i = a+ 1,...,a+ is

8r= er=0

for

r=a+13+l,...,m

(9.21)

The fundamental 2-form

M

of

I

is a closed 2-form on M

given by

= E8AA8A A

*

Let

§ =

immersion

i% i

of

M

into

(9.22)

4

a cohomology class

}

via the

j=1

is a closed 2-form on M and it defines [0)

in

H2a(M).

(9.22) implies that the class

(9.23)

induced from

Then (9.21) gives

M.

2

It is clear that

If K

M

be the 2-form on

c(M)

Equations (9.13) and and the class

satisfy

[1)

[,)a = (-1)a(a!)c(M)

is integrable and V 1

(9.13) imply that

H2k(M) ¢ 0

is minimal, then Lemma 9.2 and for

k = 1,2,...,c.

proves Theorem 9.5.

We state the following lemma for later use.

This (Q.E.D.)

180

4. Submanifolds

Lemma 9.3. of

Let

M

be an n-dimensional generic submanifold

CPm which is imbedded in

Then the mean curvature vectors H(m + 1;C)

and

(9.24)

IHI2

CPm

',

H(m+ 1;Q)

H

and

given in (6.7).

cp

M

of

H'

in

satisfy

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.

of

CPm,

A. Ros [1] proved that if

then

m

is a CR-submanifold

IHI2 = IH' I2+ n (n2+ n+ 2a).

Remark 9.3.

M

If

Kaehler manifold

is a submanifold of a quaternionic

M with quaternionic structure

ill J2' J3'

we put

9p = Tp(M) fl J1(Tp(M)) fl J2(Tp(M)) fl J3(Tp(M)) for of

p E M,

T(M).

then If

-Ap

is the maximal quaternionic subspace

9 : p -..6p

is a differentiable distribution and

its orthogonal complementary distribution totally real, then

M

.8

in

T(M)

is

is called a quaternionic CR-submanifold.

For the general theory of quaternionic CR-submanifolds, see Barros-Chen-Urbano [1].

By using a similar argument as we

give in Lemma 9.3, we also have the following

§9. Submanifolds of Kaehler Manifolds

Lemma 9.4.

Let

181

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

QPm

H(m+ 1;Q)

and

(9.25)

IHI2 = 1H,12+ 2

H

and

H'

of

M

in

satisfy

(n2+n+ 12a),

a = dimQ.B

n

Remark 9.4.

For quaternionic version of Theorem 9.5,

see Barros and Urbano [1,2].

TOTAL MEAN CURVATURE

Chapter 5.

fl.

R3

Some Results Concerning Surfaces in For surfaces in

R3

,

the two most important geometric and the mean curvature.

G

invariants are the Gauss 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

Moreover, the integral

M.

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.

M

Let

be a compact surface in

we have

(1.3)

f

IHI2dV -. 4,

M

.

R3.

Then

183

§ 1. Some Results Concerning Surfaces in IR3

sphere in

R3

and

a = IHI

map Ae

.

Clearly, a2-G

Proof.

is an ordinary

M

The equality of (1.3) holds if and only if

and

xl

= (hid).

=

4(xl-x2)2

where

O,

are the eigenvalues of the Weingarten

x2

Thus if we divide the surface M

into

3

regions for which

,H,2dV

(1.4)

J

M

and

G > 0

2$

G S 0,

we have

G dV 2 4r

IHj2dV 2 f

G>O

G>O

where the last inequality is obtained by combining (1.1) and (1.2).

if and only if

R3

.

Moreover, equality of (1.3) holds

This shows (1.3). xl = x2,

i.e.,

By Proposition 4.4.1,

M

M

is totally umbilical in

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).

R3

.

(1.5)

Let

T

be a knotted torus in

Then

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

5. Total Mean Curvature

184

goes as follows:

is a knotted torus in

T

If

and Meeks' result implies

f

IGIdV > 16rr.

$

Combining this

T

with the Gauss-Bonnet formula, one obtains

This implies

$

8v.

JH J 2dV

R3 , Kuiper

G dV > 8ir.

G>O

T

R3 , we have the following result of

For tubes in

K. Shiohama and R. Takagi (1) and Willmore [3): M

Let

Theorem 1.3.

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

I

(1.6)

IH

12 dV

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

xl = (, a+ a cos u)cos v

,

x2 = (,F2 a+ a cos u) sin v

,

x3 = a sin u where

a

is a positive constant.

Proof.

theorem.

,

Let

Let

C

be the closed curve mentioned in the

x = x(s)

be the position vector field of C

parameterized by the arc length.

Denote by

x

and

7

the

185

§ 1. Some Results Concerning Surfaces In IR'

curvature and torsion of

Let

C.

y denote the position

vector of M

in

(1.7)

y(s,v) = x(s) + c Cos v N+ c sin v B ,

where

N

and

B

R3

Then

.

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

kl_ -c'

_

1

x Cos v

k2-xccos v-1

Thus the mean curvature vector satisfies

1- 2xc cos v 2 1 - xc cos v I

HI 12

12c

Thus

_ PpL 2v P JM IH I2 dV = J O JO

1- 2xc cos v 2 12c(1-xc cos v) dv do

it 2c J it - x2c2)-1/2 do

O

where

l

is the length of p

(1.8)

J

M

IHj2dV = 2

C.

Therefore,

it

Ixl

do

0 'xcj 1-x2 c2

?irI InIds>4ir, A

O

by virtue of the fact that, for any real variable takes its maximum value

2

at

x

1 .

42

x.

(1- x )

186

5. Total Mean Curvature If the equality sign of (1.6) holds, inequalities in (1.8)

become equalities. planar curve. of radius

Thus, by Fenchel's result,

Moreover,

2 c.

(2c2)-1/2.

x =

This shows that

M

ring of the type given in the theorem.

Thus,

C

is a convex C

is a circle

is imbedded as an anchor The converse is trivial.

(Q.E.D.) Willmore conjectured that inequality (1.6) holds for all 3

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

187

§ 2. Total Mean Curvature

Total Mean Curvature

{2.

According to Nash's Theorem, every n-dimensional compact Riemannian manifold can be isometrically imbedded in n(3n + 11).

N =

with

On the other hand, "most" compact Riemannian

-

manifolds cannot be isometrically imbedded in hypersurfaces.

RN

Rn+l

as a

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]). submanifold of

Rm.

Let

M be a compact n-dimensional

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 n > 1

and as a convex plane curve when Proof.

Let

x : M

Rm

Rn+l

when

n = 1.

be an isometric immersion

of a compact n-dimensional submanifold

M

into R. m Let

B

5. Total Mean GLrvature

188

be the bundle space consisting of all frames

(p,x(p),el,...I

such that

are orthonormal

en'en+l" .. ,em)

M

vectors tangent to

at

vectors normal to M

Choose the frame

p.

in

are orthonormal

en+1,...,em

and

p

at

el,...,en,en+l,...,em)

p E M. e1,...,en

such that

B

at

H

to the mean curvature vector find that the mean curvature

IHI

p.

em

(p,x(p),

is parallel

Then we can easily

is given by

IHI = n (hll+ ...+ nn

(2.2) and

r = n+l,...,m-1 ,

(2.3)

where in

1j

On the other hand, for each

=

we can write

Bi,

m

em = s=n+l E

(2.4)

where

8s

(pre) E B1,

cos 8 s e s

denotes the angle between

em

and

es.

For each

we put

trace Ae

K1(p,e) =

(2.5)

n

From (2.2), (2.3), (2.4) and (2.5) we find

m (2.6)

(p,em)

K1(p,em) =

E

s=n+ 1 Hence we obtain

cos 8s K1(P,es) = cos 8mIH(p)I

§ 2. Total Mean Curvature IHInIcosn A

1 K1(p,em)I n dV n do = f

S

(2.7)

189

B1

B1

= (2cm-1/cn) J

m

I dV n do

I HI ndV

.

M Let

be a unit vector in the unit sphere

e

he = <e,x(p)>

the height function

on

is a differentiable function on

h

X, Y

fields

tangent to

Since

h

is continuous on

e

and one minimum, say at q',

e

Since

q

is normal to

Xhe = <e,X>.

h

M,

and M.

q

and

q'

.

At

respectively.

q',

q

Thus, we obtain from (2.8) that

.

give the maximum and minimum of

implies that the Weingarten map

Ae

denote the set of all elements k1(p,e),...,kn(p,e)

he,

(2.9)

is either non-positive

definite or non-negative definite at

eigenvalues

Hence

has at least one maximum

e

YXhe =

(2.9)

U

For any vector

YXhe = <e,V YX + h(Y,X)>

(2.8)

and

we have

M,

Consider

It is clear that

M.

M.

e

Sm-1.

(q.e)

(p,e)

of

Ae

in

and B1

(q'.e).

Let

such that the

have the same sign.

Then from the above discussion we see that the unit sphere Sm-1

is covered by

which is defined by

at least twice under the map

U

v(p,e) = e.

v* dE 2 2c

(2.10) J

This shows that

U

M-1

v : B1 -0

Sm-1

190

S. Total Mean Curvature

Since, on

k1(p,e),...,kn(p,e)

U,

have the same sign, we

find IK1(p,e)In = 1n (k1(p,e)+ ...+ kn(p,e) in

(2.11)

? 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

Cn

IHIndV

2c

M

)

I

M-1

IK1(p,e)in dV Ado B1

c

2 (2cn ) S v*d1?cn M-1 U This proves (2.1).

Now, assume that the equality sign of (2.1)

M

We want to prove that

holds.

is imbedded as an ordinary

Rn+l

hypersphere in a linear subspace

Rm

of

when

n > 1.

This can be proved as follows:

We consider the map

y : B1 . Rm;

(2.13) where

is a sufficient small positive number which gives

c

an immersion of B1

(p,e) -# x(p) +ce

B1

Rm

into

as a hypersurface in

In this way, we may regard

.

Rm.

Moreover, because

<e,dy> _ <e,dx> + c<e,de> = 0, e of

B1

in

3R

m

at

orthonormal basis of dual basis of

(p,e).

Thus

T(p,e)BI'

el,...,em-1.

is in fact a unit normal vector el,...,em-1 Let

wl

form an

,w -1

be the

Then by direct computation, we have

§2. Total Mean Curvature

n wl = wl+ c Z hi. w3

(2.14)

j=1

(2.15) Let

Ae

r = n+ 1,...,m- 1

Wr = cwr ,

B1

be the eigenvalue of the Weingarten

A

kA(p,e), of

191

Rm

in

at

Then, by using (2.14) and (2.15),

(p,e).

we may obtain ki(p,e) ki(p,e) _ 1+cki(p,e)

i = 1,2,

'

,n

,

(2.16)

r = n+l,...m-1 .

kr(p,e)

Let

el,...lem-1

(2.17)

where

be the principal directions of Ae.

wB = kA(p,.e)WA

vem =

wm eA.

Put

A

,

veA =

WA eB.

Taking the exterior

differentiation of both sides of (2.17) we find

(2.18)

C BnWA = C kA;C wCnwA

where we put dkA = (2.19)

kA;C wC.

+

kA wBnwB

Let

WBI'ABC WC

Then (2.17), (2.18) and (2.19) imply

(2.20)

We have

kA.B(p,e) = (k8(p,e) - kA(p,e))i-AA

5. Total Mean Curvature

192

Let

U = {(p,e) E BI lkI(p.e) =

V = B1

= kn(p,e) 30'0]

and

Then (2.20) gives

U.

(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

(2.1), we see that Let

U

identically on

is a non-empty open subset of

be a connected component of

U'

know that

w(p) = maxJKI(p,e)l,

e

of

K1(p,e)

on

B1

If

and the fact

see that for each point exists a point

rr(U') / M,

(p,e')

p in

runs over

U'.

K1(p,e) = 0

From This, we find that, for each point

over

p

U',

a :Bl + M

on

V,

rr(U'),

we there

w(p) = JK1(p,e')I.

such that

(p,e')

This is a contradiction.

is a non-empty open subset W

in

then by the continuity

Hence there is an open neighborhood of contained in

M.

(p,e)

where

in the boundary of U'

By

Then, by (2.22), we

U.

is a positive constant function on is the projection.

V = B1- U.

p

of the fibre

such that the principal curvatures

in

Thus in

M,

Sm-n-1

B1

which is

ir(U) = M.

there of

B1

k1(p,e),...,kn(p,e)

§ 2. Total Mean Curvature

are equal for all

is true for all

From this we may conclude that

(p,e) E W.

= kn(p,e)

k1(p,e) _

p

in

for all M,

193

M

in Sp n-1.

e

is totally umbilical in

M

Consequently, by Proposition 4.4.1,

M

n = 1,

Rm

.

is imbedded as an

Rn+1

ordinary hypersphere in a linear subspace If

Since this

when

n > 1.

is imbedded as a convex plane curve by the result

of Fenchel-Borsuk.

The converse of this is trivial.

(Q.E.D.)

An alternative proof of inequality (2.1) was

Remark 2.1.

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 M

vol(M) _> cn = vol(Sn)

M

The equality holds if and only if Proof.

of

Then the volume of

satisfies

(2.23)

of

Sm.

Rm+l. M

in

Sm

Regard

Since Rm+l

M

is minimal in

vol(M) =

Theorem 2.1, too.

is a great n-sphere in

Sm.

as the standard unit hypersphere

is equal to one.

This proves (2.23).

.

J

Sm,

the mean curvature

Thus, (2.1) implies

HlndV

cn

M

The remaining part follows easily from (Q.E.D.)

5. Total Mean Curvature

194

Corollary 2.2 (Chen (24]) Let M be a compact n-dimensional minimal submanifold of a real projective m-space R Pm constant sectional curvature

of

Then

1.

c

vol(M)

(2.24)

2 is a R Pn

The equality holds if and only if M

imbedded in

as a totally geodesic submanifold.

R Pm

Proof.

Let M

be a compact n-dimensional minimal submanifold

of a real projective m-space R Pm.

covering map r : Sm -o R Pm. submanifold of

7-1(

of (2.24) holds, then

R Pn

is a

it 1(M)

Then

Sm with vol(n 1(M))

Corollary 2.1 to

Corollary 2.1,

Consider the two-fold

M )

,

is a minimal

2 vol(M).

we obtain (2.24).

Applying

If the equality

vol(n 1(M)) = 2 vol(M) = cn.

7-1 (M)

is a great n-sphere in

imbedded in RP m

Sm.

Thus, by Thus

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

Then

(2.25)

vol(M) N

21r

n - 2k

CPk which is isometrically imbedded in

geodesic complex submanifold.

4.

cn+l

The equality holds if and only if a

M

is even and m CPm

is

as a totally

§ 2. Total Mean Curvature Proof.

195

M be a compact n-dimensional minimal S2m+1 -4 CPm CPm. Consider the Hopf fibration rr :

Let

submanifold of

Denote the r 1(M)

by

Then

M.

M

rr:Fl

submersion with totally geodesic fibres

is a Riemannian

S1.

We consider the

following commutative diagram:

i

Since M

is minimal in

S2m+1

CPm,

Lemma 4.8.1 implies that M

minimal in S21. Thus, by applying 2.1 to

(2.26)

M.

is

we obtain

cml ,

vol(M)

with equality holding if and only if M is a great (n+ 1)-sphere S2mf1. On the other hand, because tr:M + M is a Riemannian in submersion with fiber

S1,

Lemma 2.7.2 gives

vol(M) = 2w vol(M)

(2.27)

.

Combining (2.26) and (2.27), we obtain (2.25). sign of (2.25) holds, then of

S2m+ 1.

with fiber

Since S1,

n

:51 4 M

n = 2k

a great (2k+ 1)-sphere of

M

is a

M

is a Riemannian submersion

is even (Adem (1]). S2r

CPk which imbedded in

complex submanifold.

If the equality is a great (n+ 1)-sphere Sn+l

1.

Thus,

R

is

From this we conclude that

CPm

as a totally geodesic

The converse of this is trivial.

(Q.E.D.)

5. Total Mean Quvature

196

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.

Let QPm be a quaternion

Corollary 2.4. (Chen [241)

projective m-space with maximal sectional curvature 4.

M is a compact minimal submanifold of

M

The equality holds if and only if

QPk

then

QPm,

c2 2n

vol(M)

(2.28)

is a

QPk,

n - 4k;

is imbedded as a totally geodesic submanifold in

Proof. Let is minimal in

QPm,

it-(M) with it :S 4n*3 M

If

QPm.

and

QPm.

Since M

is minimal in S43. Applying Theorem 2.1

we obtain vol(S) - c3 vol(M) = 2ir2 . vol(M) by Lemma 2.7.2. The equality case can be obtained in

Thus, we find (2.28).

the similar way as Corollary 2.3.

Corollary 2.5.

(Chen [24])

with maximal sectional curvature

(Q.E.D.)

Let 4

OP2

and M an n-dimensional

minimal submanifold of

OP2.

(2.29)

vol(M) k cn/2n

Proof.

H(3;O)

Then we have

Regard the Cayley plane

as mentioned in ¢4.6.

be a Cayley plane

OP2 as a submanifold in

Since M

the mean curvature vector of M

in

is minimal in

H(3;O)

Then by using Theorem 2.1, we obtain (2.29).

satisfies

OP2, IHl2 S 2. (Q.E.D.)

197

§ 2. Total Mean Curvature

is sharp if

vol(M)

The estimate of

Remark 2.3.

given in Corollary 2.5

n g 8.

Corollary 2.5 (Chern and Hsiung [1))

compact minimal submanifolds in

There exist no

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 bounded below by

cn - vol(Sn).

is always

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 Thus, it is natural to ask whether if total mean curvature of

M

in

Rm

The answer to this is no.

large"?

b(M)

b(M).

is large, the

is also "proportionally

This can be seen by using

Lawson's examples of compact minimal surfaces in

In

S3.

Lawson [2), he had constructed a compact imbedded minimal

Mg

surface

of genus

with area less than

R4

as surfaces in than as

87r.

However,

(for an arbitrary

g

0)

in

S3

Thus, if we regard Lawson's examples

8,r.

,

g

they have total mean curvature less b(Mg)

2+ 2g which tends to infinity

tends to infinity.

g

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 ,

5. Total Mean Curvature

198

(n-l)IIh1I2-n(n-1)T

(2.31)

E (n(hr )2-hiihjj) r,i,j

=

n rEi(j E (hr)2+E E (hr-hr)2 ii r i<j 11 ji

0

Thus we obtain

-(n(n

(2.32)

)IIhII2 S T S (')IIhII2

1)

In the following, a submanifold M

Rm

6-pinched in

is called

-T-

(')IIhII2

In view of Lawson's examples mentioned above,

_> -1.

6

Rm

if we have

(n(nbl )IIhII2 for some

in

we give the following. Proposition 2.1 (Chen (12))

Rm.

submanifold in

M

If

total mean curvature of

M

Let

M be a compact n-dimensional

is 6-pinched in

Rm,

then the

satisfies n

(2.33)

where

fM IHIndV

is defined in §1.6.

b(M)

Let

Proof.

Rm

Let

.

fields of of

M,

M

then

M

Rm

Rm.

be a compact n-dimensional submanifold

en+l,... ,em in

The equality sign of (2.33)

is (n -1)-pinched in

M

holds when and only when

in

(1 1+6)2cn]b(M)

.

If

be local orthonormal normal vector e

e = E cos 8 rer.

is a unit normal vector field Thus,

§ 2. Total Mean Curvature

199

Ar = Ae

Ae = E cos grAr

r

r

Hence, we have

IIAeII2 = E cos or cos 8s trace (ArAs)

(2.34)

r,s

The right-hand side of (2.34) is a quadratic form on cos

Thus, we may choose a local orthonormal

,cos gm.

normal frame

en+l,...,em

such that with respect to this

frame field, we have

(2.35)

IIAeII2

= E (r Cos2Or

(2.36)

B1

be the bundle of unit normal vectors of M

(2.37)

B1

f(p,e) = IIAe112 Since all of

(p,e) E B1.

for

Cm

r

We define a function f on

.

...

C r= I I e 112

Let

Rm

Cn+l

f.,Cos2Or = 1,

by ,

are non-negative and

an inequality of Minkowski implies

n

/p

``2/n

f2 dv1

(2.38) iJ

Cr

in

Sp

//

2

n

/

= {

` Sp

(E Crcos2gr)2 da) //

2 (

S E SCr

where p.

Sp

`

1SSp

Icosn0rlda)

/

is the (m -n -1) -sphere of unit normal vectors at

On the other hand, we have the following identity,

5. Total Mean Curvature

200

ICosnglda = 2cm-1/Cn

(2.39) .

Sp

Thus, by (2.36), (2.38) and (2.39) we find

n

c2cn

(2.40) Let

11h,12

G(p,e)

f2 da

p d

3

m-1 Sp

denote the determinant of

Ae.

Then, by using

a relation between elementary symmetric functions, we have IIAeIIn _

Thus, by using (2.40), we find

nn IG(p,e)I.

c

p

(2.41)

JM

nn

IIhIIndV

f B IG(p,e) IdV n do

2cM-1 1

1

Combining this with Chern-Lashof's inequality, we get nn c

(2.42)

J

IIhIIndV

Now, by the hypothesis,

(2.43)

2

M

M

n b(M)

is 6-pinched in

3tm

.

Thus

IIhII2

T ? n(nsl)-

On the other hand, (2.30) gives n(n -1).r = n2IHI2 - IIhII2. (2.43) implies IHI2 2

(2.44)

Thus,

(1)IIhII2 n

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 of a linear subspace (2.30),

M

gn+l

m

with

is imbedded as a hypersurface nIHI2

is (n- 1)-pinched in Rm .

= IIhII2.

Thus, by (Q.E.D.)

§ 2. Total Mean Curvature

M

If

Sm-1

of

201

is a minimal submanifold of a unit hypersphere

Rm

,

then M

is b-pinched in Rm M satisfies

the scalar curvature of

(2.45)

n

T

.

-

-1)(1+6

(

In particular,

M

if and only if

n

Rm

is O-pinched in

if and only if

M

In this case Proposition 2.1 implies

has zero scalar curvature. the following.

Corollary 2.7.

M be a compact n-dimensional

Let

minimal submanifold of

Sm.

M has non-negative scalar

If

curvature, then we have n 2 c vol(M) _,

(2.46)

(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)). Let

Problem 2.1.

and

be a compact Riemannian manifold

(M,g)

an isometric immersion of

x : M 4 Rm

can we say about the total mean curvature of Riemannian structure of Let

Problem 2.2. an immersion of

M

of M

(or of

x(M))?

x

What

and the

(M,g)?

M be a compact manifold and

Rm

into

total mean curvature of

into Rm .

M

x

.

x :M -4 Rm

What can we say about the

and the topological structure

5. Total Mean Curvature

202

In this book, we shall give a detailed account of the results on these two problems up to date. Let

Remark 2.4.

:(M,g) .. (N,g)

be a smooth map between

We define its energy by the formula

Riemannian manifolds.

(2.47)

2 f

jI4

Il2 dVM

M

where

dVM denotes the volume element of

the differential of with

E,

and

t*

T(O),

is called the tensor field of

4

(cf., Eells and Sampson (1) and Eells and Lemaire (1]).

4

is an isometric immersion, the energy

volume of

nothing but

where

E(t)

In this case, the tension field

M.

n

If

gives the is

.r(4)

times the mean curvature vector of

n = dim M.

is

The Euler-Lagrange operator associated

4.

denoted by

(M,g);

M

If one defines the total tension of

in

N,

I

by

n i 1J

(2.48)

f

dVM

M

then the total mean curvature

IHIdV

j

is exactly the total

M tension of

4,

whenever

0

is isometric.

203

§ 3. Conformal Invariants

Conformal Invariants

3.

Let P

be an m-dimensional Riemannian manifold and

(M,g)

a positive function on

M.

-* g

Then v*

We put

=P

2

9 .

is a conformal change of metric.

g

the Riemannian connections of

g

and

Denote by

and

v

respectively.

g*,

Then equation (2.5.2) gives

v*i'-vXY =(X log p)Y+ (Y log P)X-g(X,Y)U

(3.2)

U = (dp)$.

where

M

Let g

and

be an n-dimensional submanifold of the metrics on

g*

respectively.

M

induced from

For any normal vector field

and

g g

Denote by

A.

of

g*,

M

in

M

we have

(3.3) for

v"XS-vXS = (X log P)S-(Slogp)X tangent to

X

M.

Thus, by using Weingarten's formula,

we find (3.4)

where (M,g)

(3.5)

DXS -D Xt = (X log P)S

D

and

and

D

*

(M,g ),

denote the normal connections of respectively.

Thus we have

DX = DX + (X log p)I

M

in

5. Total Mean Curvature

204

Consequently, the normal curvature tensors

and

RD

RD

satisfy

RD (X,Y) = RD(X,Y)+ DX((Y log p)I)

(3.6)

+ (X log p)DY -D M log p)I) Y - (Y log p)DX - ([X.Y]log p)I

Therefore, by using the definition of Lie bracket, we obtain

RD (X,Y) = RD(X,Y)

(3.7)

This implies the following (Chen (10]).

Let M be a submanifold of a

Proposition 3.1.

Riemannian manifold

M.

Then the normal curvature tensor

RD

is a conformal invariant. Let

M

in

and

h

and

(M,g)

denote the second fundamental forms of

h

(A,-9*),

respectively.

Then, from (3.2),

we find h*(X,Y) = h(X,Y)+ g(X,Y) UN

(3.8)

where UN

denotes the normal component of

Hence, for any normal vector

(3.9)

Let

S

of

M

in

U

restricted to

M,

we get

9(A9X,Y) = 9(ASX,Y)+ 9(X,Y)9(U,S)

el,...,en

respect to

g.

be principal normal directions of Then

AS with

M.

§ 3. Conformal Invariants

-1 P

e1,...,P

-

they are the principal directions of

kn(g) that of A,

Ag = pAt*

respect to

(3.11)

Moreover,

.

If we denote by

AS

and by

Ag = g(U.g)

is a unit normal vector with

g* = p-lg

,

g

then (3.9) implies

ki(S) = ki(S)+ lg ,

g*,

A

the principal curvatures of

k1(g),...,kn(g)

Since

en

M with respect to

form an orthonormal frame of

(3.10)

205

(3.10) gives

P(ki(g*)- k*(S*)) = ki(S) -k (9) i

Now, let

with respect to

be an orthonormal normal frame

g ,l,...,Sm g.

Then the mean curvature vector

H

is given

by

H = n E (E k.(gr))gr

(3.12)

r

i

We put

(3.13)

Te

n(n--1)

It is easy to see that

Te

E E r i _ itn+l j + jtn+1 +

au

khkjl -

h+l k hkj

tn+l

(tn+l i ) +

tn+l i n+l 1

tsj ' r = n+2,...,m

From (4.12) we have

if r=n+1; (4.19)

<xi,ber> =

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,

(4.21)

where we use the notation

[vie ...,vm) =

(-1)m-1



217

§ 4. A Variational Problem Concerning Total Mean Curvature for

m vectors

vl,...,vm

1m.

in

Since <er,ber> = 0,

(4.19) and (4.20) imply

(4.22)

ben+l

--

xi m-n

m m

+

(-1W

[x1,...,xn,en+2,...,em.berler

r=n+ 2

(4.23)

ber = -4 , '1n+1 xi A ,er,

(-1)m+r-1

+

(xl,..

W

sir

r = n+ 2,...,m

...em,besle. , gti

where

4t

and

Ln+1 = E

gti bn+r

l t.

From (4.8), (4.22) and (4.23) we find

(4.24)

<X., en+l

= - E 4)k rij m +

(-l)phij

E

W

s=n+1

(xl,...,xn,en+2,...,em,bes)

(4.25)

<xij,ber> _ -E 4) An+l krij (-l)m+r-lhsj

+

sir

W n

(xl,...,xn,en+l,...,er,...,em,bes)

Thus, by using (4.8), (4.16), (4.24) and (4.25), we find

218

5. Total Mean Curvature

hn+1 t hn+1 i tj

bhn+l = 0 ij is j

(4.26)

m

E

+

(-1)

m-n

fr

Rr

n+l i n+l j

s

s,

hi.

s=n+2

bhij

(4.27)

-i

r+1

+ 7 1n+1 i -

E hi+1 t hr

(fir

n+l i )

au .

+

r

r

k

is + E 1n+1i1's7 - 4)L Ln+Ik +

s

'r

i7

(-1)m+r-1 hs

ij

W

n 'xnen+l....,er,..

fxl'

,em.bes]

r = n+ 2....'m where i,j = i7 - E k Ilj

are the components of the Hessian,

k

of

vd4),

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,

Similarly, we have

IIAn+1II2

= trace(An+1)

and

R2 = E en+1 thn 1'

§4. A Variational Problem Concerning Total Mean Curvature

219

b(tr Ar) = 4) tr(ArAn+1)+2Eg'3 4i tn+lj

(4.29)

+4Eg - E

g ]i

)i r

1n+l i; j+

s to+l

r

i tsj

(tr As)<er.bes>

s¢r

where

tsk rij'

(tsi)

tsi;j

From (4.28) and (4.29)

k we get

(4.30)

6(a2) = 2 ((tr An+1)(-&0+4) (IAn+iII2-4

12

n

m

E (tr Ar) [4'tr(Ar An+1) + 2

+

r=n+2 E g

+

sr

where Des

to+l i; j + 4)

E g1 3 to+l i tsj)

er

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

ba =

(4.31)

12+4)

n(-l4-4)

IIAn+1II2)

Therefore, by applying (4.3). (4.14) and (4.31), we have

(4.32)

g

J

c dV = $ M

M

f-cn ac-l p4) +

n

,n Y

n

4) ac-l

12

11A n+lII2 - n4) ac+l)dV ,

c20

5. Total Mean Curvature

220

If we integrate by parts to get rid of the derivative of

0,

we find

(4.33)

$(Aac-l)dV

(ac-1 A4))dV = J

f

,

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

6

pM ac dV = pM { c Aac-1-

c-1 f2

n

J

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

c

(4.34)

Aac-1

- cac-1 12 - n2 ac+l + c

In particular, if

c = n,

Rm

only if the mean curvature

Aan-1

(4.35)

where

An+l

Then M

.

a

IIAn+1II2

=0.

this gives us the following.

Theorem 4.1 (Chen and Houh [1)). compact submanifold of

ac-1

M be an n-dimensional

Let

is H-stationary if and

satisfies

- an-1 { l2 + na2 - IIAn+

112) 1

=0

,

denotes the Weingarten map with respect to the unit

vector in the direction of

H.

From Theorem 4.1 we obtain immediately the following.

221

§ 4. A Variational Problem Concerning Total Mean Curvature

If an n-dimensional compact submanifold

Corollary 4.1.

M

Rm

in

is stationary, then

Aan-1

- an-1(.t2 + na2 - IIAn+1112 ) = 0 (Chen (3))

Corollary 4.2.

compact hypersurface in

3R n+1.

.

M be an n-dimensional

Let

Then M

is stationary if

and only if

(4.36)

pan-l+n(n-1)(an+l-an-1

where

T

T) = 0

denotes the scalar curvature of

Remark 4.1.

M

If

is a surface in

M.

R3

Corollary 4.2

,

is already known to Thomsen [1).

Using Theorem 4.1, we may obtain the following. (Chen and Houh [1)).

Theorem 4.2.

Rm.

(or H-stationary) submanifold of

mean curvature vector if and only if

Rm

submanifold of

of

Let

M be a stationary

Then M has parallel M

is either a minimal

or a minimal submanifold of a hypersphere

Rm

M

If

Proof.

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. constant.

Rm.

If

if

a = 0,

a ¢ 0,

then implies

then

then

M

Thus,

a is

is a minimal submanifold of

Den+1 : O.

Thus

l2 = O.

Equation (4.35)

5. Total Mean Curvature

222

0=

= n E (ki

no2

k]) 2

i< ]

where

are the eigenvalues of

kl,...,kn

is pseudo-umbilical in

M

that

Rm.

Proposition 4.4.2, we conclude that hypersphere of

Rm.

Theorem 4.3.

An+l.

This shows

Therefore, by applying M

is minimal in a

The converse of this is trivial.

(Chen and Houh [1]).

(Q.E.D.)

The only compact

pseudo-umbilical stationary (or H-stationary) submanifolds (without boundary) of

hypersphere of Proof. or

Rm

Since

no2 = IIAn+1112.

Rm

are minimal submanifolds of a

.

M

is pseudo-umbilical, we have either

Thus, Theorem 4.1 implies

pan-1

(4.38)

-Ao2n-2

=

- an-1 2 2 = O

2a2n-2- 2+

.

Thus (4.37) implies

the mean curvature vector Proposition 4.4.2,

of

Rm

M

a

k2 = 0.

is parallel.

H

0

Ildan-lilt

Thus, by Divergence Theorem, we conclude that which is non-zero.

is a constant

And hence

consequently, by

is a minimal submanifold of a hypersphere

(Q.E.D. )

.

Theorem 4.4.

a = 0

(Chen [3]).

stationary hypersurfaces in

The

Rn+l

only are h

odd-dimensional com act ers heres.

223

§4. A Variational Problem Concerning Total Mean Curvature Proof.

Let

be an odd-dimensional compact stationary

M

hypersurface in

Then Corollary 4.2 implies

Rn+l.

(4.39)

pan-l+n(n-1)(an+l-an-l r)=0

be the principal curvatures of M

kl,...,kn

Let

in

Rn+l

Then

a

n1

tan-1

n(n-1)an-1

tan-l

2

(T -a 1 < 0

= O.

by Proposition 4.4.1,

M

Rn+1.

is a hypersphere of

Theorem 4.5. (Chen (3).)

n

If

compact stationary hypersurfaces in

.

This implies that

is totally umbilical in

M

.

Thus

=

By Hopf's lemma, we obtain i.e.,

2 T= n n-1) kk i 0,

Moreover, from (6.7), we see that .

M

is pseudo-umbilical

Furthermore, from (6.4) we find

I%cos20+Gsin26l = alcos 2aI -G sin2a

(6.9) for all

q.

Thus

G = 0,

i.e.,

M

is flat.

from the proof of Theorem 5.1, we conclude that

torus in

R4

Consequently,

M

is a square

(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

which is conformally equivalent to a compact surface in

R4

§6. Surfaces in !R' Then we have

with non-positive Gauss curvature.

(6.10)

239

JH12dV 2 2r2

1

.

M

From Chen (9], we also have the following. Theorem 6.2.

M

Let

R4

is conformally equivalent to a compact surface in non-negative Gauss curvature.

(6.11) then

M

which with

If we have

JH12dV s (2+Tr)rr

S

Rm

be a compact surface in

,

is homeomorphic to a 2-sphere.

For the proof of this theorem, see Chen (9]. Let

f : M - R4

surface into necessary,

R4

be an immersion of an oriented compact By applying regular deformation to

.

intersects itself transversally, thus,

f(M)

At each point

intersects itself at isolated points. self-intersection, we assign

+1

if

f

p

f(M)

of

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.

known that the self-intersection number invariant up to regular homotopy of the following result of Theorem 6.3.

M

If

It is

is an immersion

into

R4

.

We mention

S. Smale (1] for later use.

Two immersions of

S2

into

R4

are

regularly homotopic if and only if they have the same self-

5. Total Mean Q rvature

240

intersection number. This theorem says that the self-intersection number is the only regular homotopic invariant of

R4

For surfaces in Theorem 6.4

denotes the genus of

Proof. in

M

R4 and

f : M .. R4

R4

into

M.

such that, restricted to are normal to

and the normal curvature

GD

be an immersion

,

We choose an orthonormal local frame

e3,e4

R4

Then we have

.

4ir(l+ IIf I -g)

IHI2dV SM

g

Let

M

of a compact oriented surface

where

in

we also have the following.

,

(Wintgen (2]).

(6.12)

S2

el,e2

M,

el,e2'e3'e4

are tangent to

Then the Gauss curvature

M.

are given respectively by

G - R(el,e2;e2,e1) = E (h11h22 -(h12)2) GD = RD(el,e2;e4,e3) - h12(h22 - hll)

-h 412(h22

- hll

Thus, the mean curvature vector satisfies

IHI2

1 ((h131+h22)2+ (h411 +h22)2) 2

h3

112 h3 22

+

h4 11

h4

22

2

2

E 4

3

3

3

4

Ih1 21 Ih11 - h221 + Ih12I 1hll IGDI +G

.

+

(h12)2+ (h12)2+G 4

- h221 + G

G

§ 6. Surfaces in !R

241

Hence, we have

(6.13)

J

IHI2dV > f

G dV

IGDIdV + f

.

M

M

M

It is known that the integral of the Gauss curvature G and the integral of the normal curvature

2rr X(M)

2rr XD(M),

where

f

gives

denotes the Euler number of the normal

XD(M)

bundle (see, for instance, Little (1]).

(6.14)

GD

gives

Thus, (6.13) implies

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 immersion of a 2-sphere S2 into R4 . If (6.15) then S2

f

be an

IHI2dV < 8tr ,

is regularly homotopic to the standard imbedding of

f

into a linear 3-space

R3

If f :M -+ R4 M

R4

into

R4

- f(M)

,

is an imbedding of a compact surface the fundamental group irl(R4 - f(M)) of

is called the knot group of

of generators of knot group of of

f.

f

f.

The minimal number

is called the knot number

Wintgen obtained the following relation between total

mean curvature and knot number:

242

5. Total Mean Curvature Theorem 6.6 (Wintgen (1)).

Let

imbedding of a compact surface

M

R4

into

IH 12dV

(6.16)

be an

f : M -* R4

Then we have

.

47r p

M

where

p

denotes the knot number of

Proof.

We need the following simple lemma:

Lemma 6.1.

R4

f.

Let

be a height function of

ha

m

in

which has only non-degenerate critical points on

Then the number

(30(k)

of local minima satisfies

00/ha) -> p.

Without loss of generality we can assume that different values at the critical points written in the order induced from

ha.

M.

ha

takes

pi (i = 0,1,...,t) 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 < cj),

we have

'rl(Hj+l) p ,r1(Hj) + one generator, if

7T 1(Hj+1)

pj

is a local minimum;

,rI(H.) + one relation, if

irl(Hj+l) N Tr1(H.),

if

pj

pj

is a saddle point;

is a local maximum.

243

§6. Surfaces in JR" The lemma follows from these relations.

We denote by Since

A2(ha) =

A2(ha)

Moreover,

Ae

in

B1

p2(ha) Z p.

For

ha,

of

a

is semi-definite if

or local minimum at (p,e)

p

ha.

Lemma 6.1 implies

13 0(h-a),

each critical point

the number of local maxima of

p.

such that

Let Al

U

is normal to ha

U

at least

2p

at

p.

is either local maximum

denote the set of all elements

is semi-definite.

to above observation, we see that the unit sphere covered by

M

times under the map

Then according S3

is

v : B1 ..

S3

Thus, by a similar argument as given in the proof of Theorem 2.1, we obtain (6.16).

Remark 6.1.

(Q.E.D.)

For a surface in

Theorem 1.2 if knot number is

2 3

I23,

Theorem 6.6 improves

and, for a surface in

Theorem 6.6 improves Theorem 2.1 if the knot number is

it4

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.

5. Total Mean Curvature

244

Surfaces in Real-Space-Forms

*7.

Let

f :M -. FP(c)

M

oriented surface c.

be an isometric immersion of a compact

into a real-space-form of constant curvature

By Ricci's equation, the normal curvature tensor

RD

satisfies

RD(X,Y)g = h(X,A9Y)-h(A9X,Y)

(7.1)

X, Y

for

tangent to M and

g

We put

be an orthonormal tangent frame. i , j = 1 , 2 .

We define

normal to

Let

(X1,X2)

hij = h(Xi,X

as the endomorphism

a A b

(aAb)(c) = a-b

(7.2)

M.

.

Then (7.1) becomes

RD(Xl,X2) = (h11-h22) Ah 12

(7.3)

The mean curvature vector

.

and the Gauss curvature

H

G

given by

(7.4)

4IHI2 = Ih11 + h22I2

For each point p in (7.5)

If

Ih12I2

G = -

,

M.

We put

Ep = (h(X,X) IX E Tp(M), IXI = 1)

X = cos 0 X1+ cos 8 X2,

then

+c

are

245

§ 7. Surfaces in Real- Space- Forms

h(X,X) = H+ cos 2e h11-h22

is an ellipse in the normal space

This shows that

Ep

centered at

Moreover, as

H.

Tp(M)

goes once around the unit

X

We

goes twice around the ellipse.

h(X,X)

tangent circle,

+ sin 2e h12

notice that this ellipse could degenerate into a line segment on a point. at

p.

at

p.

we call this ellipse

The ellipse

RD # 0,

If

is degenerate if and only if

Ep

then

the ellipse of curvature

EP

h11-h22

and

h12

and we can define a 2-plane subbundle

are linearly independent

N

of the normal bundle

This plane bundle inherits a Riemannian connection

T.L(M).

Let

be an orthonormal oriented

from that of

T1'(M).

frame of

We define the normal curvature

TP(c)

N.

(e3.e4)

GD

of

M

in

by

(7.6)

G- =

Since M N'

RD = O

and

N

are oriented,

GD

is globally defined.

be the orthogonal complementary subbundle of

N

in

Let T1(N).

Then we have the following splitting of the normal bundle; TA. (N) - N ® Nl.

(7.7)

From the definition of

RD(X1.X2)S = 0

Let

a0 = a0(M)

of the tangent bundle

if

we have

N1.

P, E N1

.

denote the bundle of symmetric endomorphism T(M).

Define a map

4 :N -. a0 by

5. Total Mean Curvature

246

tr A A

(7.8)

Thus, because

RD

0

-

$ E N

2F

by assumption,

A

[A

e3.

(7.8) implies that

4

e

.

] ¢ 0.

Thus

We denote by

is an isomorphism.

X(N)

N

the Euler characteristic of the oriented 2-plane bundle over

We mention the following extension of a result of

M.

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

with nowhere vanishing normal

Mm(c)

curvature tensor, we have

X(N) = 2%(M)

(7.9)

be the bundle of symmetric

a0 = a0(M)

Let

Proof.

.

endomorphism endowed with the orientation induced by that of

N

via

Then because

4.

isomorphism, we have B(X)

let

is an orientation-preserving

4'

For each

X(N) = )((a0(M)).

be the element in

at

a0(M)

p

given by

B(X)(Y) = 2<X.Y>X - <X,X>Y

.

B(cos tX + sin tXl) = cos 2t B(X)+sin 2t B(X)l,

Then

Xl

X E Tp(M),

is a vector in

X, X1

Tp(M)

such that

give the orientation of

M.

IXjI = IXI,

where

X 1X1

and

Therefore, the index

formula for the Euler characteristic applied to a generic vector field proposition.

X

and to

B(X),

respectively, yields the (Q.E.D.)

§ 7. Surfaces in Real- Space- Forms

247

The following result is a generalization of Theorem 6.4. (Guadalupe and Rodriquez [11).

Theorem 7.1.

be an isometric immersion of a compact oriented

f :M -o Mmfc)

surface Mm(c).

Let

into an orientable m-dimensional real-space-form

M

Then we have

f

(7.10)

IH12dV

2 27 X(M) +

GD dVI -c vol(M)

If

.

M

M

GD

The equality holds if and only if

does not change sign

and the ellipse of curvature is a circle at every point.

From (7.1) and (7.6) we have

Proof.

D =

(7.11)

G

Ih11 - h22IIh12I

Thus, (7.4) and (7.11) imply

( Ih11 - h221 - 21h121 )2

0

Ihll =

-h2212+41h1212

-41h11 -h221 Ih121

Ih1112+ 1h22I2+21h1212-2G-4IGDI+2c IlhIl2-2G-4IGDI+2C

.

On the other hand, 41HI2

Ih11I2 + (h22I2+ 2

Ih11+ h22I2 =

1 h 1 1 1 2 + (h221

= 1Ih1I 2 + 2G

- 2c

2+ 21h1212+ 2G- 2c .

5. Total Mean CLrvature

248

Hence, we find

IHI2+ c _> G+ IGDI

(7.12)

with equality holding if and only if

2 (h11 - h22) = h12'

i.e., the ellipse of curvature is a circle.

M

over

gives (7.10).

if and only if

Integrating (7.12)

Moreover, the equality of (7.10) holds

does not change sign and the ellipse is

GD

always a circle.

Corollary 7.1.

be a compact oriented surface immersed in GD > 0

curvature

Let

(Guadalupe and Rodriguez [1)).

R4

M

If the normal

.

everywhere, then

(7.13)

IHI2dV

12ir

.

The equality holds if and only if the ellipse of curvature is always a circle. Proof.

Thus

M

If

GD > 0

X(N) = 2n

everywhere,

is homeomorphic to

Hence,

S2.

J

GD dV > 0.

X(N) = 2X(M) = 4,

which yields (7.13) by using (7.10).

Remark 7.1.

(Q.E.D.)

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.

curves in this bundle can also be projected down to to obtain examples of surfaces in

S4

Holomorphic S4

in order

with the property that the

ellipse is always a circle, hence giving equality in (7.10).

SUBMANIFOLDS OF FINITE TYPE

Chapter 6.

Order of Submanifolds

§1.

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 defined numbers

Here

p

p

and

q

M

Rm

of

associated with the submanifold

is a positive integer and

integer

S p.

manifold

M

We call the pair (Chen [151,22,25]).

to be of finite type if

q

to introduce two well-

q

[p,q]

is either

+ .

or an

the order of the sub-

The submanifold M

is finite.

M.

is said

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.

M be a compact Riemannian manifold and

M acting on

C+(M).

Then

A

A

Let

the Laplacian of

is a self-adjoint elliptic

operator and it has an infinite, discrete sequence of eigenvalues (cf. 43.2):

6. Submanifolds of Finite Type

250

0 = )`0 < al < %2 ... < lk < ...

(1 .1)

t

Let Vk = (f E C '(M) I Of = lkf} be the eigenspace of with eigenvalue

Then

Xk.

Vk

We

is finite-dimensional.

define as before an inner product

(f,g) = f

(1.2)

a

(

,

)

on

C (M)

by

fg dV

M

Then E 0 Vk by 0 Vk

is dense in

COO(M)

the completion of E Vk,

(in

L2-sense).

Denote

we have (cf. Theorem 3.2.2)

C (M) ='kVk f E C(M),

For each function of

ft

Vt (t = 0,1,2,...).

onto the suspace

f

let

be the projection Then we have the

following spectral decomposition

f = E ft,

(1.4)

(in

L2-sense)

t=O

Because V0 function

there is a positive integer

f E C *(M),

such that

fp 1 0

and

tap

f0 E V0

is a constant.

which are nonzero, we put integer

(1.6)

p z 1

f - fO= E ft

(1.5) where

1-dimensional, for any non-constant

is

q,

q a p,

If there are infinite

ft's

q = . Otherwise, there is an

such that

fq V 0

q

f - fo = E ft t=p

and

§ 1. Order of Submanifolds

If we allow

q

to be

251

we have the decomposition (1.6)

W,

m

for any f E C (M). M

Riemannian manifold

(1.7)

IRm,

we put

x = (xl,...,xm)

where in

into

xA

1Rm

.

of a compact

x :M 4 IRm

For an isometric immersion

,

M

A-th Euclidean coordinate function of

is the

For each

we have

xA,

qA xA -(xA)

(1.8)

= O

tF

For each isometric immersion

(1.9) A

is easy to see that or an integer and

q

p

z p.

q = q(x) = sAup(gA)

A

ranges among all

such that

is an integer

it

xA - (xA) 1

and

q

It

71 O.

is either

Moreover, it is easy to see that

p

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 more precisely, for each isometric immersion have a pair [p,q]

.

we put

x : M + ]Rm ,

p = p(x) = iAnf(pA},

where

A = 1,...,m

At

(xA)

=PA

associated with

M.

the order of the submanifold

M.

[p,q)

in

]Rm

(or,

x : M + ]Rm), we

We call the pair

By using (1.7), (1.8) and (1.9) we have the following spectral decomposition of

x

in vector form:

252

6. Submanifolds of Finite Type

q

x = x0 + E xt

(1.10)

t=p

in Mm

A compact submanifold M

Definition 1.1.

said to be of finite type if

q

is

Otherwise M

is finite.

is

of infinite type (Chen (22,25])".

A compact submanifold M

Definition 1.2.

said to be of

k nonzero

k-type

(k = 1,2,3,...)

(t t 1)

xt's

For a submanifold M

that M

if there are exactly

of order

(or of order

called a submanifold of order

we sometime say

[p,q), s q)

A submanifold of order

is not considered.

is

in the decomposition (1.10).

z p

is of order

Rm

in

if

q

(or

is also

[p,q]

p.

Let M be a compact submanifold of Rm .

Remark 1.1.

k-type in Rm

It is easy to see that M

is of

of infinite type in

if and only if M is of k-type

in any

Rm+m DJRm

Rm+m

Rm)

M

Lemma 1.1.

3Rm)

Let x : M -0 ]Rm

Proof.

M

(resp.,

(reap., of infinite type in any

be an isometric immersion

of a compact Riemannian manifold M the centroid of

into

Rm.

Then

x0

Rm.

in

Consider the decomposition

x = E xt

(1.11)

t=O We have

p)

Axt = atxt.

If

t y/ 0.,

then Hopf lemma implies

is

253

§ 1. Order of Submanifolds

xt dv - -11

f

(1.12)

tM

M

Since

Ax t dV = O

is a constant vector in

x0

we obtain from

3tm,

(1.11) and (1.12) that x0 = f x dV / vol (M)

(1.13)

.

M

This shows that

is the centroid of

x0

M.

(Q.E.D.)

Lemma 1.1 shows that if we choose the centroid of (in

as the origin of

3tm)

3tm

M

then we have

,

q

(1.14)

x = E xt

t=p

Let

v1

and

v2

be two

Htm-valued functions on

We define the inner product of

(vl,v2) = f

(1.15)

and

vl

by

v2

< v1,v2 >dV

M.

,

M where

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

(xt,xs) = 0

(1.16)

where

xt

is the

for

t-th component of

into

t ¢ s

3tm.

,

x with respect to the

spectral decomposition (1.10). Proof.

Since

A

Than we have

is self-adjoint, we have

6. Submanifolds of Finite Type

254

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

255

Submanifolds of Finite Type

42.

First, we rephrase Proposition 4.5.1 of Takahashi in terms of order of submanifolds as follows: Proposition 2.1.

x :M -

Let

]m

be an isometric

M

immersion of a compact Riemannian manifold Then

x

is of

1-type if and only if

manifold of a hypersphere of

into

7Rm.

is a minimal sub-

M

]Rm

From this proposition, we see that if minimal submanifold of a hypersphere

M

SD-1(r)

is a compact centered at

the origin, then we have

(2.1)

for some constant

X X.

Because

Ax =

- nH

ap E 7R

.

(Lemma 4.5.1),

(2.1) implies

HH = X H,

(2.2)

In views of this, we give the following characterization of submanifolds of finite type (Chen (221). Theorem 2 . 1 .

Let

be an isometric immersion

x : M + 1 m

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

.

6. Submanifolds of Finite Type

256

In other words,

M

curvature vector

is of finite type if and only if the mean satisfies a differential equation of the

H

form:

AkH+c1Ak-1H+ ...+ck-lAH+ckH = O

(2.4)

for some integer

k ? 1

Let

Proof.

and some real numbers

x : M -0 IRm

c1....,ck.

be an isometric immersion of a

compact Riemannian manifold M

into

]Rm

.

Consider the

following decomposition

q x = x0 + E xt ,

(2.5)

Axt

t=p If

M

is of finite type, then

-nA1H =

(2.6)

t=p

q < ..

1 i+1 xt t

xtxt

From (2.5) we have

i - 0 , 1 , 2 ,...

,

q Let

E xt

cl

t=p (-1)q-p+l

lp

(2.7)

-

Xq.

c2

t<s ltls

cq-Pt 1

Then by direct computation, we find

AkH + c1 Ak 1H +

where k - q -p+ 1. k z 0,

'

... + ckH - 0 ,

Conversely, if H

then, because m

satisfies (2.7) for

is compact, we have k x 1.

the spectral decomposition (2.5).

Using (2.6) and (2.7), we

find

(2.8)

E It0t+c1xt'1+ ...+ek-llt+ek)xt - 0

t=1

Consider

§ 2. Submanifolds of FYnite Type

For each positive integer

t

at(Xk+clat-1+

1

Since

(x5.xt) =

J

<x s

(2.8) gives

s,

m

(2.9)

'x

t

257

... +ck) fM for

> dV = 0

0

t ¢ s

(Lemma 1.2),

we obtain (fig+c1ag-1+ ...+ck)11xs1j2

(2.10)

=0

where

Nall2 = (xs,xs)

(2.11) If

xs ¢ 0,

then

jjxsjj ¢ 0.

(2.12)

Thus (2.10) implies

+clas-1+

)

...+ck = 0 k

Since equation (2.12) has at most

real solutions and

equation (2.10) holds for any positive integer

k

of the

are nonzero.

xt's

s,

at most

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. Let

Theorem 2.2. Then

M

is of

is a polynomial

k

k-type P

M

be a compact submanifold of Mm. if and only if there

(k =

of degree k

such that

distinct positive roots and

(2.13)

P(A)H = 0

.

P(t)

has exactly

6. Submanifolds of Finite Type

258

By using exactly the same proof as Theorems 2.1 and 2.2, we may also obtain the following. Let

Theorem 2.1'.

x : M -4 ]Rm

of a compact Riemannian manifold

M

be an isometric immersion into

Then M

IRm.

is

of finite type if and only if there is a non-trivial polynomial

such that

P(t)

P(A) (x -x0) = 0

(2.14)

Let

Theorem 2.2'.

M

Then

is a polynomial

k

be a compact submanifold of

M

k-type

is of

k

Rm.

if and only if there

(k = 1,2,3,...)

of degree

P

.

such that

P(t)

has exactly

distinct positive roots and

P(A) (x-x0) = 0

(2.15)

.

From the proof of Theorem 2.1, we see that

Remark 2.1.

the positive roots of

in Theorems 2.2 and 2.2'

P(t)

fact eigenvalues of the Laplacian of

are in

M.

The following corollary is an easy consequence of Theorem 2.1.

Corollary 2.1. If

then

be a compact homogeneous space.

M

is equivariantly, isometrically immersed in

M M

vectors

iRm,

k-type with k s m.

is of

Proof.

m+ 1

Let

Let H,

u

be an arbitrary point of i

at

u

M.

Then the

are linearly dependent.

259

§ 2. Submanifolds of Finite Type

Thus, there is a polynomial P(6)H = 0

at

immersed in

u.

IRm,

P(t) = 0

with k s m.

M

Because

P(A)H = 0

by Theorem 2.1 we see that because

P(t)

M

has at most

(Q.E.D.)

of degree s m

such that

is equivariantly isometrically at every point of

M.

is of finite type.

k

roots,

M

is of

Thus,

Moreover,

k-type

6. Submanifolds of Finite Type

260

Examples of

03.

2-type Submanifolds

According to Proposition 2.1 of Takahashi, minimal submanifolds of hyperspheres of of

Rm.

are

R1Q

1-type submanifolds

Moreover, Corollary 2.1 shows that there exist many

important finite type submanifolds in Rm .

In this section,

2-type submanifolds in Rm

we will give many examples of

(Chen [22,25]) . Let

Example 3.1, (Product Submanifolds).

and Rm .

be two compact submanifolds of Rm M xM

Then the product submanifold only if both

M

both M and M M xM

and

M

are of

Moreover, if

For instance, con-

2-type.

Consider the isometric imbedding

x = x(r,s) _ (a cos

a,

is the lattice

A

(0,2rrb). Then

and

is isometric

T2

T2 = S1(a) xS1(b).

to the product of two plane circles;

(3.1)

respectively.

1-type, then the product submanifold

sider the flat torus T2 = R2 /A, where

(2rra,O)

and M

is of finite type if and

are of finite type.

is either of 1-type or of

generated by

M

x

of

T2

in R4

by

a sin a, b cos b, b sin b)

Then, by direct computation, we find

-(.+ -7) , ar as 2

(3.2)

(3.3)

(3.4)

H

2

x0 = (0,0,0,0)

I(acosa, asina,

AH = 7(- .cos S, a

1 a

fibSsinb) cos.

sing,

-1

b

cos b,

-

'sin b)

b

§ 3. Examples of 2-type Submanifolds

(3.5)

a

261

as ab bb

2

H =T 1(1a cos r, 1 sin r, 1 cos e, 1 sin s) 16

if and only if

Assume that

a = b.

1-type in

is of

T2

From (3.1) and (3.3). we see that

II24

From (3.3),

a 9d b.

(3.4) and (3.5), we obtain (3.6)

-1.7+ -1.f) AH + -.T1,sl

a2H

Thus, by Theorem 2.2,

a

b

a T2

is of

P(t) - t2

(3.7) Then

P(t)

(1,2)

Example 3.2.

in

1R4

a2;17

Thus, by using Proposi-

-17.

b tion 3.5.7, we can conclude that if a-7

is of order

Let

2-type.

a1 +b1)t+ and

has roots

=O

a > b >

then

T2

.

(A flat torus in

Again consider

1R6 .)

the flat torus

T2 =

(3.8)

with

A

generated by

IIt2 /A

,

Let x : T2 _. IIt6

((2Tra,O) , (0,2Trb)) .

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

.

6. Submanifolds of Finite Type

262

By a direct computation, we have

(3.11)

H=

(O, sins sin b, sins cos., O, cos s sin b ,

+

cos s cos

b

)

,

SH = (1 + - )H - a (sin s, O. O, cos s, O, O)

(3.12)

2b

b

(3.13)

= (1 +) 2H -(2

A

b

2b

+1b ) (sin s, O, O, cos s, O, O )

Consequently, we have

(3.14)

A2H

- (2 +

) GH + (1 + 2)H = 0

This shows that

T2

is of

2-type in

Example 3.3.

and x :M a ]Rm

IIt6

(Diagonal immersions.)

x :M -4 IRm

be two isometric immersions of a compact

Riemannian manifold M

into

and

]Rm

Then the normalized diagonal immersion defined by

Let

x'(p) = 1 (x(p),x(p))

respectively.

ltm,

x' :M + IItM+m

is of finite type if

2

and x are of finite type.

and only if both

x

ular, if both

and x are of

that

x'

x

is either of

1-type, then we can show

1-type or of

For example, consider the unit

(3.15)

S2 =

2-type.

2-sphere in

( (x,y,z) E IIt3 1 x2+ y2+ z2 = 1)

Define an isometric immersion

In partic-

u :S2 + 7R8

by

1R3

.

by

.

§ 3. Examples of 2-type Submanifolds

ul =

u2 =

,

(3.16)

u 4 = YE

- y2)

(x2

2

u6

v2

5

u7

u3 =

Y

263

,

2

u8 = 17 (x2 + y2 - 2z2)

,

2

Then, by a direct computation, we can see that

order

[1.2]

in

Example 3.4.

Ilt8

Thus,

.

S2

is of

is of 2-type in

S2

IIt8

H(2n +2; C)). Let S4n+3 C2n+2 = S4n+4 given by denote the unit hypersphere in S4n+3

=

(MM,n

in

2n+1

((z0,...,z2n+1)t E C2n+2

IzAI2

= 1)

A O In

S4n+3

we have the following generalized Clifford torus

M2n+1, 2n+1 = S

2n+1

1 (

)xS

2n+1

1

(

)

defined by

M2n+1,2n+1

(3.17)

2n+l

{(z0,...,z2n+1)tEC2n+2 It n IztI2

O Let

GC = (z E C (IzI = 1).

acting on

S4n+3

and on

Denote the quotient space

Mn,n

Then

.

is a group of isometries

GC

M 2n+1,2n+1

; t=z1 Izt'2 .)

by multiplication.

M2n+1,2n+1 /GC

by

Mn.n.

Then

admits a canonical Riemannian structure such that C

M2n+1,2n+1

> Mn,n

6. Submanifolds of Finite Type

264

becomes a Riemannian submersion with totally geodesic fibres S1

Moreover, we have the following commutative diagram:

.

1

M2n+1,2n+1

) S

4n+3

(3.18) QP2n+1

MT n,n

where

i

and

are inclusions.

i'

minimal in S4'3, Mn,n of Cp2n+1 Let

cp

:TP2n+1

standard imbedding of

Since

M2n+1,2n+1

is

is a minimal (real) hypersurface

denote the first

> H(2n +2; T) cP2n+1

into H(2n +2; C defined

by (cf. *4.6) ip(z) = zz*

(3.19) Then

:p

induces an isometric imbedding of

H(2n + 2; C).

of

Mean

(3.20)

Because,

Mean

into

By a direct long computation, we may prove

that, for any point

H

.

in

H(2n + 2; T)

H=

the mean curvature vector

A E cp(Mn,n),

T

AA - -(4n+ 1)H,

at

A

is given by

(2I - (4n+3)A -At)

(3.20) implies

(3.21)

AA = 2(4n+3)A.+2At-4I

,

(3.22)

AAt = 2(4n+3)At+2A-4I

.

§ 3. Examples of 2-type Submanifolds

265

(3.21) and (3.22) we may obtain

From (3.20),

P(A)H = 0 ,

(3.23)

P(t) = (t - 4(2n+ 1) ) (t - 4(2n+ 2)) . Consequently, by applying Theorem 2.2, we obtain the following

where

Proposition 3.1. H(2n+2; Q).

Moreover,

Example 3.5. unit hypersphere

is a 2-type submanifold of

Mn,n

4(2n+1), r(2n+2) E Spec (M11

(MQ,n

in in

S8n+7

H(2n +2; Q)). Q2n+2 = S8n+8

have the generalized Clifford torus

Consider the we

S8n+7

In

M4n+3,4n+3

defined by

M4n+3,4n+3 =

2n+1

((z0,...,z2n+1)tEb2n+21 nE Iz1,2= GQ = fz E Q Ijzi = 1).

Let

acting on

S

Bn+7

and on

GQ

M4n+3,4n+3

Denote the quotient space MQ

The

M

4n+3,4n+3

by multiplication. by y

/ GQ

given by

rp(z) = zz*.

n,n

Then

QP2n+1

Consider the first standard imbedding into H(2n + 2; Q)

)z12=)

is a group of isometrics

is a minimal real hypersurface of

n

1

j=n+l

i=O

cp

of

QP2n+1

Then, by a long

direct computation, we can prove that the mean curvature

vector

H

of

MQ,n

n

in H(2n+2; Q)

at

A E cp(MnQ,n)

given by

(3.24)

H=

8n+3 (21 - (8n + 7)A -At)

is

6. Submanifolds of Finite 7)'pe

266

Since

AA =

- (8n+3)H,

(3.24) implies P(A)H = 0

(3.25)

,

P(t) _ (t-4(4n+3))(t-16(t+l)).

where

Consequently, by

applying Theorem 2.2, we have the following. MQ,n

Proposition 3.2.

2-type submanifold in

is a

H(2n+2; Q) . Example 3.6.

(MQ,n,n

following product of three

in

H(3n +3; Q)).

Consider the Q3n+3

(4n +3)-spheres in

defined in an obvious way;

_ 4n+3

M=M4 +3 4 n+3 4 n n , ,

+3-S

1 1 1 (-) xS 4n+3 (-) xS4n+3 (-) 13

V3 CS12n+11(1) CQ3n+3

Then GO = (z E Q (Izi = 1)

in

2

QP

3n+2

Then Qp3n+2

12n+ll (1)

and on

M / GQ

by

MQ,n,n

Then

into

cp

induces an isometric

H(3n + 3; Q).

By a long computation, we may prove that the mean curvature vector A E

(3.26)

cP(M4,n,n)

H=

M

is a minimal submanifold of codimension MQ,n,n Consider the first standard imbedding cP of

into H (3n +3; Q)

imbedding of

S

Denote the quotient space

by multiplication. MQ non.

acts on

H

of

MQ,n,n

in

H(3n +3; Q)

at

is given by

n+

(32I - 96 (n + 1) A + 21 (A -At) 1

§ 3. Examples of 2-type Submanifolds

267

this implies P(n)H = 0, where P(t) = (t -24n - T) (t -24n - 24) . Consequently, by Theorem AA = -6(2n +1)H,

Because

2.2, we obtain the following. Proposition 3.3.

MQ

2-type submanifold in

is a

n,,

H(3n+3; Q). Example 3 .7 .

in H (n + 2; V). Let Cpn+l be

(Qn

the complex projective

(n + 1)-space with constant holomor-

phic sectional curvature geneous coordinates of

4.

z0....,zn+l

Let

be the homo-

Then the complex quadric

CPn+1

Qn

is defined by

n+I E Jzi12

Qn = ((z0,...,zn+l) E CPn+1

=

01

i=O Denote by

cp

H(n+ 2; (r).

the first standard imbedding of

A E cp (Qn)

(3.27)

Thus we have

into

Then, by a direct computation, we may prove Qn

that the mean curvature vector of

a point

CPn+i

in

H(n + 2; C)

at

is given by H=

n(I - (n+l)A -At)

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 manifold in H (n + 2; C) . Example 3 .8 .

(MI n

[ 2 ]) .

Qn

is a

in H (2n + 2, ]R))

following generalized Clifford torus in

.

2n+1

2-type sub-

Consider the S.

6. Submanifolds of Finite Type

268

Sn(1) xsn(

Mn,n =

V2

a S2n+1(1) C IR2n+2

/2

Denote by

defined in an obvious way.

G

the group of

isometries generated by the antipodal map. the quotient space

Mn,n /G.

Einstein hypersurface of

Then nM n

cp

H (2n + 2:

]RP2n+l

induces an isometric imbedding of Ilt)

.

MIR n

is a minimal

1RP?n+l = S2n+1 / G

the first standard imbedding of Then

Denote by

.

Denote by

cp

into H(2n+ 2; ]R) Mm n n

into

By a long computation as before, we many prove

that Mn n is a Remark 3.1.

2-type submanif old in H (2n + 2; ]R) . 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

Characterizations of 2-type Submanifolds

44.

In this section. we will give some characterizations of subIn order to do so, we need to recall the

manifolds of 2-type.

definition of allied mean curvature vector introduced in Chen [7) and to compute

t H.

M be an n-dimensional submanifold of an m-dimensional

Let

Riemannian manifold

en+1" ",em be mutually orthogonal

Let

N.

M

unit normal vector fields of

N

in

parallel to the mean curvature vector a normal vector field

a(H)

such that H

of

M

en+l in

N.

is

We define

by

m

a (H) = E tr (AH Ar) er.

(4. 1)

r=n+2

Then

M in

is a well-defined normal vector field (up to sign) of

a(H)

in

N.

N.

We call

a(H)

It is clear that

Definition 4.1. N

the allied mean curvature vector of M a(H)

is perpendicular to

in

Remark 4.1.

N

of a Riemannian manifold

if the allied mean curvature

N

is called an Q-submanifold of

vector of M

M

A submanifold

H.

vanishes identically.

For results on a-submanifolds, see for instance,

Chen [7), Houh [1], Rouxel [1), and Gheysens, Verheyen, and Verstraelen [1,2). M

Let

vector

(4.2)

H.

be a compact submanifold of Rm with mean curvature For a fixed vector

c

in Rm

fc = < H, C >.

we put

6. Submanlfolds of Finite Type

270

Then, for any tangent vector

of

X

we have

M.

Xfc = - + .

(4.3)

tangent to

X, Y

Thus, for vector fields

we find

M,

YXfc = - - + .

(4.4)

X

Thus, we obtain

n

n

A = E

(4.5)

Ei) - E EiEi

(vE

i=l

i=1

i Dn

< 6 H , c > + E and

AD

the

Laplacian of the normal bundle, that is,

n

(4.6)

ADH

= E (DV i=1

Because (4.5) holds for any

Ei

c

E

Regard

v AH and

ADH

i Ei

in

AH = ADH + E (h(Ei,AHEi)

(4.7)

H - DE D

i

(4.5) implies

Htm,

+ AD

Ei

H).

H

E

i + (VE 1AH) Ei).

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 H Y.

We put

(4.9)

V AH = V AH + ADH.

§ 4. Characterizations of 2-type Submanyolds

271

Then we have

n

tr (v AH)

(4.10)

We notice that if Let

(vE AH) Ei) . i Ei H Ei +

E (AD

i=1

we have

DH = 0,

En+l,...,em en+l

such that

32m

=

be an orthonormal normal basis of is parallel to

E h (Ei,AH Ei)

(4.11)

II An+1 II2 = tr (A2n+l )

where

v AH = V AH.

.

=

II An+l

H.

112

M

in

Then we have

H + a(H) ,

Combining (4. 7) , (4.10), and (4-11),

we obtain

Lemma 4.1.

Let

M

Htm.

be an n-dimensional submanifold of

Then we have

AH = CDH + IIAn+lII2H + a(H) + tr(vAH).

(4.12)

For the comparison with 2-type submanifolds, we give the following

Let

Proposition 4.1.

M

be a compact submanifold of

Ht m

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 is an C!-submanifold of I.

(3) M

Proof. If

M

DH = 0,

is of 1-type in

AH = b H.

(4.13)

Because

H2m,

we have

AD H = 0

there is a constant

Thus, by Lemma 4.1, we have

IIAn+l II2H + a(H) + tr(VAH) = bH.

and

b

v AH = v AH.

such that

6. Submanlfolds of Finite Type

272

H. a(H),

Since

and

are mutually orthogonal, we obtain

tr(v AH)

(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 M

Now, we assume that manifold of a hypersphere

Rm.

at the origin of vectors of

M

Rm

in

(Q.E.D.)

is an n-dimensional compact subof radius

Sm-1(r)

Denote by and

is of 1-type.

H

and

H'

in Rm

r

the mean curvature

respectively.

Sm-1(r),

centered

Then we have

(4.14) where

x

be the unit vector parallel to where

M

denotes the position vector of H'.

in

Rm.

Then we have

H' = a'

We choose an orthonormal normal basis

a' = IH'1.

en+1' -,em of M in Rm such that (4.15)

Let

en+l = H / a , en+2 = ( + a' x) / r a,

where 1

(4.16)

a = (HI = (a'2 +

Because Ax = - I,

we have

(4.17)

tr (AH An+2)

(4.18)

tr (AH Ar)

r2

)2

= a ' (IIA9112 -n (a') 2) / r a, = tr (AH , Ar) , r = n+3,. .. , m.

From these, we obtain

P

§ 4. Otaracterizations of 2-type Submanifolds

a (H) = a' (H') + r a'

(4.19) where

n (a')2 ) en+2' M

denotes the allied mean curvature vector of

a'(H')

Sm-1 (r).

in

For the normal vector field x

273

we have

x,

M

is parallel in the normal bundle of

any normal vector < DTI,x > = 0.

D'

where

in

M

Thus, for

Rm.

in

i.e.,

with < x, fl > = 0,

Htm

we have

From these, we find that aD

(4.20)

of

Tj

Dx = 0,

H=

QD '

H'

is perpendicular to x,

denotes the normal connection of M

in

Sm-1(r).

From (4.14) and (4.15), we also find (4. 21) a2 IIA

n+lH !12

= tr (A , + I 2) 2

=

r

(a )

2

IIAS II

2 +

2n (a") 2 + n

r2

r4

Therefore, by combining'(4.12), (4.14), (4.19), (4.20), and (4.21), we obtain the following. Lemma 4.2.

hypersphere

(4.22)

Let

Sm-1 (r)

M be an n-dimensional submanifold of a of radius

r

in

Mm.

Then we have

A H = AD' H' + a ' (H') n++ tr (v AH) + «' (IIAtII 2 + 2)

-

-r2 (x-c0), where

c0

denotes the center of

Sm-1(r).

We need the following. Definition 4.2.

metrically imbedded in

Let M be a symmetric space which is iso-

Rm

by its first standard imbedding.

Then

6. Submanifolds of Finite Type

274

of M

M

a submanifold

centroid (i.e., the center of mass) of

of

M

is called mass-symmetric in

M

Rm

in

if the

is the centroid

in Rm.

M

Let

Lemma 4.3.

hypersphere symmetric in Proof.

M be a compact minimal submanifold of a of radius

Sm(r)

Rm+1

in

r

M

Then

is mass-

Sm(r).

Because

A x = - n H ,

Hopf's Lemma implies

HdV=0. M

M

Since

is minimal in

Sm(r),

we have

H =

12

(c - x),

where

r

c

is the center of

Sm(r)

Thus we find

Rm+1.

in

c = f x dV / f dV. M

This shows that

M

is the centroid of M

c

in

Rm+1

(Q. E. D.)

Lemma 4.3 shows that compact minimal submanifolds of hyperspheres 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

pact, mass-symmetric submanifold of

type in

]Rm,

Sm-1(r).

If

M

is of 2-

then

(1) the mean curvature

and is given by

M be an-n-dimensional, com-

cx'

of

M

in

Sm-1 (r)

is constant

275

§ 4. Characterizations of 2-type Submanifolds

(a')2

(4.23)

=

(n)2 ( 2 - Xp) (aq

r

(2)

tr (v'A H.) = 0,

(3)

ADH' + a'(H')

and +

(jjAj!l2 + 2) H'

r

denotes the Weingarten map of

A'

where

- 2) r

VA 'H' + AD'

=

M

(),p + Xq) H', in

'H1.

Conversely, if (1), (2), and (3) hold, then in

center of

is of 2-type

Without loss of generality, we may assume that the Sm-1 (r)

Rm,

(4.24)

is the origin of

Eim.

If M

is of 2-type

then Theorem 2.2' and Lemma 4.3 imply

AD H' + a'(H') + tr (v AH) + a' (UAgEl2 -

n r

+ bH' for some constants M

M

Rm Proof.

in

and

Sm-1(r)

b

and other terms in

and

r2 c.

2

S - n2 r

x - x = 0, n Since

tr(v A H)

(4.24) are normal to

M,

is tangent to we have

On the other hand, because A H = A . + 2 I r we have tr (v'A H.) = 0. Furthermore, because

tr (v A H) = 0. DH = D'H',

normal to Sm-1(r) and other terms in (4.24) are tangent to Sm-1(r), we obtain from (4.24) that (4.25)

(a')2

On the other hand,

(4.26)

+ r12 = a2 = - nb - ncr2 (4.24) gives

P(s) (x) = 0.

and x

is

276

6. Submanifolds of Finite Type

where

P(t) = t2 + bt - n.

[p,q],

M

Since

is of 2-type with order

(4.26) implies b = - (lip + Xq)

and

c = Xp Xq.

Statement (3) follows

by (4.25), we obtain Statement (1).

and equation (4.24).

from Statements (1) and (2)

The converse

of this follows from Theorem 2.2' and Lemma 4.3.

If M is a hypersurface of

Thus,

(Q.E.D.)

then we have the

Sm-1(r),

following.

Theorem 4.2.

(Chen [25].)

symmetric hypersurface of

Rm+l

M be a compact, mass-

Let

Sn+1(r).

If

M

is of 2-type in

then (1)

the mean curvature

a of M

in

Iltn+2

is constant

and is given by

(4.27)

(Xp + ),q) - (n)2 lp lq o

a2 =

n (2)

the scalar curvature

(4.28)

n An+2

of M

(lp + ) q) - n -1) (nr

T=

(3)

T

is constant and is given

ap aq

the length of the second fundamental form

h

of M

in

is constant and is given by

(4.29) (4)

IIh!I =

and

tr(VAH) = 0.

Proof.

Sn+l(r).

ap + Xq ,

Let

If M

M be a compact mass-symmetric hypersurface of is of 2-type, then Theorem 4.1 implies that the

§ 4. Characterizations of 2-type Submanifolds

mean curvature

of

a'

M

M

Since the codimension of vature vector that is,

(a')2 A 'H,

+

H'

M

of

r2

=A

H

tr(v A H) = 0.

we also have

is a non-zero constant.

Sn+l (r)

is therefore parallel,

Sn+1(r)

DD H' = 0

is one, the mean cur-

too.

Because,

a2 =

equation (4.23) implies (4.27). Since we have

,

-

in

in

Thus

D'H' = 0.

Sn+l (r)

in

277

2 I,

statement (2) of Theorem 4.1 implies

r

M

Now, because a'(H') = 0.

is a hypersurface of

Sn+1(r),

Thus, by statement (3) of Theorem 4.1,

we obtain

(4.30)

IIAg112 +

Because

2r

11h112 = 11Aj112 + 2 ,

r

= lp + aq.

(4.30) implies (4.29). Equation

(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].)

symmetric hypersurface of curvature

a

tr(v A H) = 0, Proof.

center of and

T

Let M be a compact mass-

Sn+1(r).

If

M

has constant mean

and has constant scalar curvature then

M

'r,

and if

is either of 1-type or of 2-type.

without loss of generality, we may assume that the Sn+1 (r)

is the origin of R

are constants.

by Lemma 4.2, we find

Then we have

n+2

DD H'

Assume that = a'(H')

= 0.

a Thus,

278

6. Submanifolds of Finite Type

O H = a'( IIA

(4.31)

2

II2 + 2)

nr 2 x

r

= .'11h Since

and

a

r

Because

stants.

r

are constant,

a'

H = a'g - x/r2,

A H - 11h 112 H +

(4.32)

- nag X.

112 C

112

and

(4.31) implies

x = O.

(na2-IIhII2)

rZ

are also con-

11h

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.

M be a compact. mass-symmetric submani-

j,g,

fold of a hypersurface

parallel mean curvature vector

2-type if and only if (1)

H'

IIA H II

in

Proof.

Let

M

If

Sm-1(r),

is constant,

is an Q-submanifold of Sm-1 (r)

and (3) M

Sm-1(r)

Rm.

of

Sm-1 (r)

has non-zero

M

then

is of

(2) tr (V A H .) = 0,

.

M be a compact mass-symmetric submanifold of M

such that

is of 2-type.

zero parallel mean curvature vector

Assume that H'

in

M

has non-

Sm-1(r).

Then, by

Theorem 4.1, we have

(4.33) Because Since A

H

IIASII2 + '

r2

a'

a' (H') = 0.

is constant, this implies that

a'(H') = 0,

= A'H ,

= ap + lq,

+ I/r2

M

and

implies tr(VAH,) = 0.

is an Q-submanifold of

D'H' = 0,

IIA H.II

is constant.

Sm-1(r).

Because

statement (2) of Theorem 4.1

§ 4. Characterizations of 2-type Submanifolds

279

Conversely, if IIA H, H is constant, tr (V A H ) = 0 and a' (H') = 0, then, we have tr (v A H) = 0. And moreover, by Lemma 4.2, we also have

(4.34)

AH = a'(IlAsII2 + a'(IIAsll2 + r2 )

where

H = a'C - x/r2,

(4.35)

AH-

Since

a2 = (a')2 +

and

Because

(IIASII2

r2)

6

-

n

r

2r

2

X.

are constants.

(4.34) implies

+2)H+ r

r2

(na2

- CIA;II2 - 2) x = 0.

r

(4.35) and Theorem 2.2' imply that M

H' # 0,

2-type in Rm.

is of (Q. E. D.)

For surfaces in

S3(r),

we have the following classification

theorem (Chen [25].)

Let M

Theorem 4.5.

of

S3 (r)

in

R4

be a compact, mass-symmetric surface

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) , Proof.

hypersphere

Let

a 71 b. M be a compact mass-symmetric surface of a

S3(r)

in

3R4.

assume that the center of

without loss of generality, we may 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 Conversely, if

M

M

is of 2-type in

is of 2-type in

3R4,

3R4.

then by Theorem 4.2,

M has constant mean curvature and constant scalar curvature. Moreover, we also have

6. Subma :ifolds of Finite Type

280

(4.36)

tr(7AH') = 0.

by virtue of D'H' = 0.

Let

El, E2 be the eigenvectors of AH

Then we have (4.37) where T

i = 1, 2,

A H , Ei = .1i Ei, _l, -2

are the eigenvalues of

are constants,

.-l,

-2

Because,

A H

and

are constants.

We pu t

(4.38)

2

V E1 =

E..

J

j=1

3

Then we find

(vE1 AH,) E1 = (-

(4.39)

wl2 (E1) E2

Similarly, we also have

vE AH) E2

(4.40)

2-..1) w2 (E2) E1.

2

Because obtain

M

u

is of 2-type, = 0.

Thus, by

-2.

tr(v A H,) = 0, we

From these, we may conclude that

the product of two plane circles.

Because,

M

M

is in fact

is of 2-type, the

radii of these two plane circles must be different. Remark 4.2.

In general, if

with A H E i = M E .

,

only if

(4.41)

Eµ= i i jji

Remark 4.3.

M

i = 1, ... , n,

iji

(u-µ) w(Ei)

is a submanifold of

(Q.E.D.)

Sm-1(r)

then tr (17 A H ) = 0 if and

1.... ,n.

Recently, A. Ros [2,3] has applied the concept

of 2-t rpe SubutaniJolds

4.

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

.

be a minimal isometric immersion

Sm-1(r)

: M -4

of an n-dimensional, compact. Riemannian manifold IItm

centered at

the Euclidean coordinates of

X1,.... xm

into a

Denote by

0.

Sm-1(r)

in

be the row matrix given by

(xi .Oxm)

x =

Let

of

Sm-1(r)

hypersphere

M

IItm.

xl,...Oxm.

m-1

(r) into H (m ; ]R) Define an isometric immersion f of S Then f is an order 2 immersion of Sm-1 (r) by f (x) = x t x . H(m

into

M

An isometric immersion of

IR).

in

is

Sm-1(r)

is not contained in any totally geodesic

called full if

M

submanifold of

Sm-1(r).

The results obtained by A. Ros [2,3)

are the following. Let

Theorem 4.6.

:M -4 Sm-1 (r)

of a compact Riemannian manifold immersion M

tr (A A')

Sm-1(r),

of

in

M

where

is a constant and

Sm-1 (r)

Let

M

A'

is Einsteinian

S,

of

M

in

is the Weingarten map

be a compact, Kaehler submanifold of

such that the immersion is full.

M

fo$ of

.

standard imbedding of

Then

M

is of 2-type if and only if

k

such that the

Sm-1(r)

= kg(,") for all normal vectors

Theorem 4.7. CPm

into

Then the immersion

is full and minimal.

H(m ;IR)

into

and

r

M

be an isometric immersion

SPm

is of 2-type in

into

Denote by

H(m + 1 ;C)

H (m +1 ;C)

Einsteinian and the Weingarten map of

cp

defined in §4.6.

if and only if M

in

the first

CPm

M

is

satisfies

6. Subnwnifolds of Finite Type

282

tr (A9'S A')

= k g (S , rl)

for all normal vectors

S, -n

of M in

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.

£Pm,

§ 5. Closed Curves of Finite Type ¢5.

283

Closed Curves of Finite Type In this section we shall study closed curves of finite

Im

type in

In order to do so, we first recall the Fourier

.

series expansion of a periodic function. Let

Then

2'rr.

be a periodic continuous function with period

f(s)

has a Fourier series expansion given by

f(s)

a

f(s) _ -2+a1 cos(y) +a2 cos(2r) +

+b1 sin(r) +b2 sin(2r ) + where

and

ak

are the Fourier coefficients given by

bk

(5.2)

ak =

T

(5.3)

bk =

-r

pTr r J

---r

pTr r

J

f(s) cos (ki)ds,

k = 0,1,2,

f(s) sin (ks)ds,

k = 1,2,---

-Try

In terms of Fourier series expansion, we have the following

(Chen [221) Theorem 5.1. Then

C

Let

be a closed smooth curve in

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. IItm

]Rm

Assume that

C

such that the length of

the arc length of

C.

We put

is a closed smooth curve in C

is

2Trr. .

Denote by s

284

6. Subinanifolds of Finite Type

(5.4)

x(j) =

d]x ds]

Because

- d2 - in this case, we have

0 =

ds

A H = (-1)jx(2j+2),

(5.5)

If

C

is of finite type in

j =

IRm,

0,1,2,...

then Theorem 2.1

implies that each Euclidean coordinate function in

IRm

xA

of

C

satisfies the following homogeneous ordinary differ-

ential equation with constant coefficients:

x(2k+2)

(5.6)

A

for some integer

+c1x(2k) + ...+ckx(2) A A =0

k z I

and some constants

cl,...,ck.

Because the solutions of (5.6) are periodic with period each solution

xA

2-r,

is a finite linear combination of the

following particular solutions: n.s 1, cos( r )

(5.7)

Therefore, each

(5.8)

xA

qA

x = c., + E

t= pA

,

PA, q

function

xA

A

;

,

ni,mi E a

is of the following forms:

a_ (t) cos (ts) +b_ (t) sin (ts) r 1

for some suitable constants integers

m.s sin( r )

aA(tl, bA(t), cA

A = 1,...,m.

and some positive

Therefore, each coordinate

has a Fourier series expansion which has only

finite nonzero terms.

285

§ S. Closed Curves of Finite Type

Conversely, if each

has a Fourier series expansion

xA

which has only finite nonzero terms, then the position vector x

C

of

in

takes the following form:

]Rm

q

x = c+ E {at cos(ts) +bt sin(tr)

(5.9)

t=p

for some constant vectors a, bt, c

in

and some

IRm

2

Since A = - - ,

p, q.

integers

(5.9) implies

ds q

2

Ax = E

(5.10)

(-xt,)

t=p

(at cos(ts) +bt sin(ts) )

Let xt = at cos (ts) +bt sin(ts) . x = c+ Et

that x

for

type.

c

in

p

Itm

xt

is in fact the spectral decomposition of Since

.

Then (5.9) and (5.10) show

q

is finite,

c

is of finite

(Q .E .D . )

From the proof of Theorem 5.1, we obtain the following.

Let

Corollary 5.1. in

2Trr

IItm.

If

C

be a closed curve of length

C

is of finite type, then we have the

following spectral decomposition:

(5.11)

q

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)

Let

C

be a closed curve in

k-type, then

C

lies in a linear

Proposition 5.1. If

C

is of

]RA1

O-subspace

286

IItS

6. Submanifolds of Finite Type

of

IItm

Since

Proof. of

of

C

Let

IR2k .

IRm.

Then we have

E t2(IatI2 + IbtI2] = 2r2

E

t+t =k

tt'(-)

E

t+t =k

E

t-t =k

where

1 1 k 3 2q,

t-tE=k

tt'(+) = 0

tt' +

for

there is a

t. p

+2 (5.14)

S-subspace

be a closed curve of length

C

is of finite type in

(5.12)

(5.13)

k,

k-type which lies fully in

Proposition 5.2.

is con-

xt

(Q.E.D.)

For each positive integer

closed curve of

If

Since each

must lie in a linear

s 1 2k.

with

Remark 5.1.

2rrr.

C

Span(at,bt1, ]Rm

k-type, there exist exactly k

is of

C

which are nonzero.

xp,...,xq

tained in

IIt8

with S s 2k.

tt'(-) = 0

at,bt; p s t 9 q

are vectors in

m IIt

given by (5.11).

Conversely, if there exist

at,bt; p 3 t s q,

such that (5.12), (5.13) and (5.14) hold then

x(s) = Et

p

(at cos(ts) +bt sin(tr))

type closed curve in

)Rm.

for

in

IRm

1 s k s 2q,

defines a finite

§ 5. Closed Curves of Finite 7Ype

287

From (5.11) we have

Proof.

q

x(s) = xo+ E [at cos (tr) +bt sin (tr )

(5.15)

t= P

Thus we find from

q

= 1

.

that ,

= t,E

r2

(5.16)

< x'(s),x'(s) >

1

( < atat, > sin (r s) sin (trs) =p

+ cos (i)s cos (trs)

- 2 < at.bt, > sin (rS) cos (trs) l From this we find

2r2

(5.17)

=

t E, ( t [ cos (t rt)s -cos (trt)s) + [ cos (t rt)s+cos (trt,)s)

- 2 [ sin (t Since

rt/)s+sin

1, cos(y), sin(r) ,...,cos(). 2sin(?-q-)

are inde-

pendent, (5.17) implies (5.12), (5.13) and (5.14). verse of this follows from Theorem 5.1.

The con-

(Q.E.D.)

Using Proposition 5.2, one may classify closed curves of finite type.

Theorem 5.2. If

C

of

IRm

(5.18)

is of ,

C

Let

C

2-type in

be a closed curve of length IItm

,

2irr.

then, up to a Euclidean motion

takes the following form:

x (s) = (a cos (

)

, a sin

13 cos (v) , 13 sin (as) , O, ... O)

6. Submanifolds of Finite Type

288

where

a

and

are nonzero constants such that

S

(pa)2+

(qB)2 = r2.

Proof.

If

is of

C

2-type in

3Rm

,

then by Proposition

5.2, we have

2r2

(5.18)

=

P2(Iap12+ lbp12) +g2(IagI2+ IbgI2)

(5.19)

IapI = lbpI.

(5.20)

a p,bp,aq,bq

IagI = lbgI

are orthogonal

,

,

.

Thus, by (5.18), (5.19) and (5.20) we obtain the theorem.

(Q.E .D. ) Remark 5.2.

bt, p s t s q if

if of

C

From (5.20) we see that the vectors

are orthogonal if k-type with

k ? 3,

are not orthogonal in general. closed curve in

iR6

is of

C

is of

then

2-type.

at,

However,

at, bt, p s t s q,

For example, the following

3-type but

.

6 Submanifolds of Finite Type

290

Thus, by using (5.22) and (5.23), we see that bq-1

a

q-1

and

take the following forms:

aq-1 = (uq-l'vq-1) ,

(5.24)

bq_1 = ( -vq_l,uq_1)

From (5.22), (5.23) and (5.24) we obtain

< at.bt' = 0, t = q -1,q

latl = Ibti (5.25)

= 1.

For (7.4) we consider

q (7.8)

n2(dAkH,dAkH) = n2(AkH,Ak+1H) =

2k+3IIxtII2

t= p By using (7.6) and (7.8), we obtain (7.4) for

by applying Holder's inequality, we obtain (7.4) for The equality cases can be easily verified.

Thus,

r = 1.

r > 1.

(Q.E.D.)

From Proposition 7.2, we obtain immediately the following.

302

6. Submanifolds of Finite Type Corollary 7.1.

submanifold of

7Rm

(7.9)

f

(7.10)

6kH112rdV '

(

M

2r

or (7.10) holds for some order

2k+1 1n ) r vol (M)

2k+2 I dV a

(

-n) r vol (M)

k = 0,1,2, .

and

n-dimensional compact

Then we have

.

fM jjdAkH11

r z 1

for

M be an

Let

The equality sign of (7.9)

and

r

k

if and only if

M

is of

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

compact submanifold

M

Proposition 7.3. submanifold of

]Rm

(7.11)

S

.

in

Let

k =

with

IRm

M

there is no

be an

AkH = 0.

n-dimensional compact

Then we have

116kH112dV s

Xq

vol (M)

M

2k+2 (7.12)

f

IIdAk 1II2dV ` (-gn

) vol (M)

M

for

k = 0,1,2, .

holds for some

k

The equality sign of (7.11) or (7.12)

if and only if

M

is of order

We omit the proof of this proposition.

q.

§ 8. Some Applications to Spectral Geometry ¢8.

303

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.

surface which admits an order ]Rm.

M

Let

Theorem 8.1 (Chen [16)).

isometric imbedding into

1

Then, for any compact surface

conformally equivalent to

(8.1)

M

into

(8.2)

]Rm

.

IItm

is of order

be an order

Let x : M -+ iRm M

in

which is

a ll vol (M)

equality holding if and only if

imbedding of

M

we have

M,

X1 vol (M)

Proof.

be a compact Riemannian

1

1.

isometric

Then, by Theorem 6.2, we have

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

6. Submanifolds of Finite Type

304

fM

(8.4)

_

a

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 Therefore, by applying Theorem 6.2,

(8.4) holds.

also of order

in

1

M

is

The converse of this follows

IItm.

immediately from Theorem 6.2 and Proposition 5.3.5.

Let

M

(Q.E.D.)

be a Clifford torus (or a square torus).

we obtain from §3.5 that

to be of order

1.

The standard

X1 vol (M) = 472.

imbedding of the Clifford torus in

M

A compact surface

is known

(m ? 4)

IItm

Then

is called

IRm

in

a conformal Clifford torus or a conformal square torus if

differs from M

Rm

by conformal mappings of

M

.

From Theorem 8.1 we obtain immediately the following Corollary 8.1 (Chen [16]).

If

M

is a conformal

Clifford torus, then we have

al vol (M) s 4v2

(8.5)

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- . Veronese imbedding of IRP2 into ]R5 is an order isometric imbedding.

A compact surface

M

in

3R

TO

The 1 is

called a conformal Veronese surface if it is conformally

§ 8. Some Applications to Spectral Geometry

equivalent to the Veronese surface.

305

From Theorem 8.1 we

obtain the following (Chen (161). Corollary 8.2.

If

M

is a conformal Veronese surface,

then we have

X1 vol (M) 9 12'r ,

(8.6)

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.

into

IItm ,

For any isometric immersion of

we have IHIndv

r

(8.7)

IRPn

2(nn 1)

n 1

n

equality holding if and only if the immersion is of order Similarly, if we denote by

QPn

and

QPn

the complex and

n-spaces, respectively, with the

quaternion projective

standard metrics, then, by Proposition 3.5.5, 3.5.6 and Theorem 6.2, we have the following. Corollary 8.4.

into (8 .8)

IRm ,

For any isometric immersion of

we have

CPn

IH'2ndV s (2(n+1)7r)n nn n

1.

CPn

6. Submanifolds of Finite Type

306

equality holding if and only if the immersion is of order 1. Corollary 8.5.

into

IItm ,

For any isometric immersion of

we have IHI4ndV x

(8.9)

QPn

IQPn

n+

.

equality holding if and only if the immersion is of order Corollary 8.6. OP2

plane

into

1.

For any isometric immersion of the Cayley

Iltm,

we have 7

IHI16dv a (72

(8.10)

)w

OP 2

equality holding if and only if the immersion is of order

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.

1,

§9. Spectra of Submanifolds of Rank-one Symmetric Spaces

307

Spectra of Submanifolds of Rank-one Symmetric Spaces

§9.

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.

M

Let

(Chen [22].)

pact submanifold of a hypersphere

be an n-dimensional comof radius

Sm(r)

r

Rm+1

in

Then

M

if

(1)

is mass-symmetric in

xp = 2 if and only if

M

Sm(r),

is minimal in

then

Sm(r)

X 1i xp 9E - 2 r

and

and hence

r

M

is of 1-type.

if

(2)

and only if Proof.

of M

in

is of finite type, then xq z 2 and Xq = 2 if

M

M

is of 1-type.

Denote by

H

and in

Itm+l

(9. 1)

H'

and Sm(r),

the mean curvature vectors respectively.

Then we have

IHI2 = IH'I2 + r-2.

Hence, by applying Theorem 6.3, we find x

(r )

(9-2)

This shows that imply

H' = 0,

X

q

z n/r2. M

that is,

a result of Takahashi, clear.

fM

vol (M)

M

Thus, Statement (2)

I H 12 dV

q

(n) vol (M) .

If xq = n/r2, then (9.1) and (9.2) is minimal in is of 1-type.

Sm.

Consequently, by

The converse of this is

is proved.

For Statement (1), we assume that the centroid of

center of S. m

M

is the

Without loss of generality, we may assume that

Sm

6. Submanifolds of Finite Type

308

From Lemma 4.5.1 and Proposition 6.1,

is centered at the origin.

we have

n (x , H) = (x , A x)

n vol (M)

(9.3)

q

Ep at llxtll2 ? ap M

Since

lies in

we find

Sm,

ilxll2.

11x112 = r2 vol (M).

Thus, by

(9.3), we obtain (9.4)

-

r2

xp.

If the equality of (9.4) holds, then the inequality of (9.3) becomes equality.

M

Thus,

is of 1-type.

The converse

of this is clear.

(Q.E.D.)

From Theorem 9.1, we obtain immediately the following.

(Chen

[221.)

M

If

Corollary 9.1.

is a compact, n-dimensional mass-

symmetric submanifold of a unit hypersphere al 9 n,

equality holding when and only when M Let

Corollary 9.2.

manifold with imbedding of q

Sm

M

Rm+l

then

is of order 1.

M be an n-dimensional compact Riemannian

for some integer

n

Xq

in

q.

into a unit hypersphere

is an imbedding of order

Then every isometric

Sm 2

Rm+l

of order

q.

Now, we need the following. Lemma 9.1.

Let

imbedding of R Pm

p

into

R Pm 4 H(m+l; R) H(m+l ; R).

be the first standard

Then an n-dimensional minimal

§9. Spectra of Submanifolds of Rank-one Symmetric Spaces submanifold of

12 Pm

geodesic R Pn

in

M

is of 1-type if and only if If this case occurs,

IEtPm.

M

309

is a totally is of order 1

in H (m+l ; P) . Proof.

Assume that

is of 1-type in

M

H(m+l; H2).

Then

by Takahashi's result;

M

is minimal in a hypersphere

S(r)

Denote by

C

the center of

C

H (m+l ; IR) .

S (r) .

Since

of

is a

symmetric matrix, by choosing a suitable coordinates of H(m+1;JR), we may assume that

C

For any A E cp(M),

is diagonalized.

mean curvature vector of

M

H(m+l ;3R)

in

at

A

the

is given by

H = 2 (A-C) . r

(9. 5)

Because

M

is minimal in

We put

AC = CA.

Thus, we have

H E TA (Ht PI) .

IR Pm,

L = (Z E H(m+1 ;IR) ,ZC = CZ).

Then

M c L.

Let O

C1I

CiIi

C =

(9.6)

0

where A2 = A

Ii

and

are identity matrices. tr A = I.

fl L =

For each

A E cp(1

Ai = Ai and A

tr Ai = 1

r

From this, we obtain the following disjoint union:

r

cp (H2 Pm)

where

m),

O

Ai 0

I

k k

Thus, we find

Al T (IR Pm)

C

n L = U wi, i=1

we have

310

6. Submanifolds of Finite Type

O

O

Ai

Wi = O

,

O

It is clear that each

W.

R Pm

Thus, each

H(m+l ; R).

in

Ai = A1 and tr Ai = 1

I

is a totally geodesic submanifold of is a real projective space

W.

k.

R P

1

H(ki+l; R).

in

M

we see that

M

Since

lies in a

is connected and

m)

n L,

1 1 i s r.

Since

M

is totally geodesic in

R Pm,

M

for some

Wi

M c ca (R P

i,

k.

is minimal in

R Pm

and R P

1

k.

is minimal in R P 1

which lies in

lies in the hypersphere of

H(ki+l; R).

H(ki+l ;1R)

radius

r

as a minimal submanifold.

vector

H

of

for

in H (ki+l ; R)

M

M

Moreover,

centered at

c I i

with

i

Thus, the mean curvature

is given by H = -2 (A -a r

Thus, by Lemma 4.6.4, we have

A E cp(M).

2(nnl)r4 = 2 - ai - 2 (ki+1) ai. On the other hand, because

mH1 = r,

we also have

r2 = 1/1H12

Thus, we find

n/2 (n+l) .

(n+l) (ki+l) a2 + 2 (n+l) ai - 1 = O. Since

Thus, we find that R Pm.

M

a O.

is real, the discriminate of this equation is

ai

is

n 3 ki. R P

Therefore,

1.

Since

Thus,

M

M

M

lies in

this implies

is a totally geodesic

is of order 1 in

verse of this is clear.

k. I RP 1,

H(m+l ; R).

R Pn

in

The con(Q.E.D.)

311

§9. Spectra of Submanifolds of Rank-one Symmetric Spaces For CR-submanifold of

we also have the following

CPm,

result of Ros [1). Lemma 9.2.

of

where

CPm,

L

M be a minimal, n-dimensional CR-submanifold

CPm

is imbedded in

standard imbedding.

M

Then

by its first

H(m+1; C)

is of 1-type in

if and

H(m+l ;C)

only if one of the following two cases holds: a)

M

is a totally geodesic complex submanifold of

b)

M

is a totally real submanifold of a complex n-dimen-

sional totally geodesic complex submanifold of

Then

Q

M

We suppose that

Proof.

denote the center of

S(r),

CPm.

is of 1-type in

is minimal in a hypersphere

M

S(r)

of

CPm,

H(m+l; C).

H(m+l ;C).

we can suppose that

Q

Let

is a

diagonal matrix, othersise we can use an isometry of H(m+l ;C) of the type

Since

matrix.

vector

H

A E T (M) . Thus,

A '+ PAP-1,

M

,ZQ = QZ).

of

M

with P E U(m+l),

is minimal in

in

M

S(r),

to obtain a diagonal the mean curvature

satisfies

H (m+l ; C)

H = (A - Q) /r2,

we have AQ = QA

Since H ETA (CPm),

is contained in the linear subspace We put 0

alll

Q =

a

arIr

O

Then we have the following disjoint union:

r

cp (CPm)

f1 L = U Wi, i=1

for

for

A E cp (M) .

L = (Z E H(m+l ; C)

312

6. Submanifolds of Finite Type

where o

o

E H (m+l ; C)

A.

Wi = O

I

Ai = Ai and

tr Ai =1

O

Each of these components is evidently a totally geodesic complex submanifold of

CPm (it is a

submanifold of a component

k s m)

CPk,

M

and

Consequently, the problem is

CPk.

reduced to the study of minimal CR-submanifolds of

are minimal in some hyperaphere of H(k+1;C) al,

a E ]R,

is a minimal

and whose radius is

CPk

which

whose center is

r.

From Lemma 4.9.3, we have

IHI2 = r-2 =

(9.7)

C a = dimk

n (n2+n+2a),

On the other hand, we have

H = - 2 (A - a I).

Thus, we find

r

r2 = IHI2r4 = < A - a I , A - a I > = 2 (k+l) a2 - a + 2

(9.8)

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

n2 s (n + 2a) k.

Because

implies that either 2a = n,

n2

that is,

(n + 2a) k,

And if M

that is,

M

n = 2k.

This

is totally real or

is a complex submanifold of

we find that if M

is complex,

we get

k a n - a, we get a (2a - n) ? O.

a = 0,

M

? 0,

CPk.

Because

is totally real, then

If the first case occurs,

k = n.

M

is

§9. Spectra of Submanifolds of Rank-one Symmetric Spaces

a totally real submanifold of a totally geodesic M

If the second case occurs,

in

aPn

is a totally geodesic

313

CP

QPm

k

with

n = 2k.

Conversely, if

then

M

M

is a totally geodesic

is of order 1 in

any orthonormal basis

C) c H (m+l ; a) .

H (k+l

totally real submanifold of

aPk

aPn,

in

CPm

If

M

is a

then for any A E T(M)

E1,...,En

of

TA M,

form an orthonormal basis of TA (aPn).

and

E1....,En, J El,...,J En

Therefore, by Theorem

4.6.1 and (4.6.26), we obtain H = 2(1 - (n+l) A)/n. This implies that

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].)

minimal submanifold of tional curvature 1.

M

Laplacian of

R Pm,

M be a compact, n-dimensional,

Let

where R Pm

is of constant sec-

Then the first non-zero eigenvalue

11

of the

satisfies

%1 s 2 (n+l) ,

(9.9)

equality holding if and only if

M

is a totally geodesic

it Pn

in

IIt Pm

Proof.

Let

fold of P Pm.

M

be a compact, n-dimensional, minimal submani-

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

6. Submanifolds of Finite Type

314

applying Lemma 9.1, we conclude that in

IIiPn

is a totally geodesic

(Q.E.D.)

The converse of this is clear.

R Pm.

Theorem 9.3. (n a 2),

M

(Chen [24].)

M be an n-dimensional

Let

compact, minimal submanifold of

CPm,

of constant holomorphic sectional curvature 4.

(9.10)

QPm

where

Then we have

%1 s 2(n+2),

equality holding if and only if (1)

n

is even,

M

(2)

is a

n

CP2

and (3)

M

is a complex totally geodesic submanifold of

CPm

Proof. Let

QPm

be isometrically imbedded in

its first standard imbedding.

minimal submanifold of

CPm,

(9.11)

IHI 2

by

is a compact, n-dimensional,

then, by Lemma 4.6.5, we obtain 2 n+2 n

equality holding if and only if submanifold of

M

If

H(m+l ;C)

n

is even and

M

is a complex

By combining (9.11) with inequality (6.7)

LPm.

of Reilly, we obtain (9.10).

If the equality sign of (9.10) holds, then the equality sign of (9.11) holds.

submanifold of

n

is even and

M

is a complex

On the other hand, we also have

CPm. p

(9.12)

Thus,

J

k

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 in

315

Therefore, by applying Lemma 9.2, we conclude

H(m+l ;C).

n

M

that

is a

which is imbedded in

CP2

QPm

Conversely, if

geodesic complex submanifold.

as a totally

M

is a

n

then we have

CP2,

Remark 9.1.

Yang and Yau [1] showed that if

holomorphic curve in

CPm,

holding if and only if

M

M

is a

Moreover, Ejiri [2)

is a compact, n-dimensional

M

Kaehler submanifold of

then

a1 a 2(n+2),

equality

is a totally geodesic complex sub-

CPm.

Theorem 9.4. sional

a1 1 8.

then

CPm,

and Ros [2] proved that if

manifold of

(Q. E. D.)

X1 = 2(n+2).

(n ? 4),

(Chen [24].)

Let

M be a compact, n-dimen-

minimal submanifold of

QPm,

QPm

is of

Then we have

constant guaternion sectional curvature 4.

(9.13)

where

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.

manifold of

(9.14)

Let QPm.

M be a compact, n-dimensional, minimal subThen, Lemma 4.6.6 implies IHI2 s 2(n+4)

n

Therefore, by combining (9.14) with Theorem 6.2, we obtain (9.13).

6. Submanifolds of Finite Type

316

Now, if the equality sign of (9.13) holds, then (9.14) becomes equality.

Thus, by Lemma 4.6.5,

is a quaternionic submanifold of

n

is a multiple of 4 and M Thus, by a result of

QPm.

n

Gray [1], we conclude that

M

is a totally geodesic

The converse of this is clear.

QPm.

Remark 9.2.

QP4

in

(Q.E.D.)

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 is of maximal sectional curvature 4.

OP2, where

OP2

Then we have

x1 s 4n.

(9.15)

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) where

x1 a

s 2 (n2 + n + 2a) /n,

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 Proposition 9.2.

M be a compact, n-dimensional,

Let

minimal CR-submanifold of

QPm.

Then we have

ll s 2(n2+n+12a)/n,

(9.17) where

317

a

is the guaternionic dimension of the guaternion dis-

tribution. Theorem 9.6.

of

IR Pm,

where

M

Let

is imbedded in H (m+l ; It)

I2 Pm

standard imbedding.

be an n-dimensional, compact submanifold

M

If

is of finite type in

by its first H(m+l ; I2),

then

we have

(9.18)

X

a 2(n+l),

q

equality holding if and only if in

I2Pm.

Proof.

If this case occurs,

M

is a totally geodesic P Pn

q = p = I.

From Theorem 6.3, we have

(9.19)

p J

H j

2

d

q

I

3 (n) vol (M).

M

Moreover, from Lemma 4.6.4, we also have IHI2 a 2(n+l)

(9.20)

n

Combining (9.19) and (9.20), we obtain (9.18). If the equality of (9.18) holds, then (9.19) and (9.20) become 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

Thus, by Lemma 9.1, we obtain the theorem.

H (m+l ; ]R) . (Q.E.D.)

6. Submanifolds of Finite Type

318

of

CPm,

where

dard imbedding.

M be an n-dimensional, compact submanifold

L

Theorem 9.7.

is imbedded in

CPm

M

If

H(m+l ; C)

by its first stan-

is of finite type, then we have

Xq x 2 (n+l) .

(9.21)

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.

Then, by Theorem 6.3, we see that the mean curvature vector

CPm. H

Let M be an n-dimensional, compact submanifold of

of

M

in

H (m+l

satisfies

C)

a

(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 in

CPm

and M

is of 1-type in

M

is totally real and minimal

H(m+l ;C).

Thus, by using

Lemma 9.2, we obtain the theorem.

(Q.E.D.)

Similarly, we also have the following.

of

QPm,

where

QPm

standard imbedding.

(9.24)

M be a compact. n-dimensional submanifold

L

Theorem 9.8.

is imbedded in If

M

H (m+l ; Q)

by its first

is of finite type, then we have

Xq 6 2 (n+l) ,

99. Spectra of Submanifolds of Rank-one Symmetric Spaces

equality holding if and only if

M

submanifold of a totally geodesic

319

is a minimal totally real QPn

in

QPm

6. Subnwnifolds of Finite Type

320 §10.

Mass-symmetric Submanifolds From Theorem 9.1 and Corollary 9.1, we have a best possible

estimate of

for mass-symmetric submanifolds of a hypersphere.

X 1

In this section, we shall study

Xp

for mass-symmetric submani-

folds in projective spaces. Theorem 10.1.

be isometrically imbedded in

L g t R Pm

by its first standard imbedding.

H(m+l ;]R)

If M

n-dimensional, mass-symmetric submanifold of R Pm,

is a compact, then

2n m+1

(10.1)

m

1

equality holding if and only if

Since R Pm

Proof.

n = m

and

M = R Pn

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

of

radius

r = [m/2(m+1)]1/2. R Pm

centroid of

Thus, by Lemma 4.3, we see that the

is the center of

M

the centroid of

S(r)

S(r).

Thus, by the hypothesis,

is the center of the hypersphere

There-

S(r).

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

M

Therefore,

we obtain

and

M

is also minimal in R Pm.

we conclude that x1 = 2(n+l).

H(m+l ; R)

M

and

M = R Pn.

S(r).

By applying Lemma 9.1,

is a totally geodesic R Pn

On the other hand, we have

n = m

is minimal in

in R Pm. Hence

al = 2n(m+l)/m.

Thus,

The converse of this is clear. (Q. E. D. )

§ 10.

M

symmetric submanifold of

be an n-dimensional. compact, massIF Pm,

isometrically imbedded in imbedding.

H(m+l;IF)

s ap

1

%p = 2n(m+l)/m

IF Pm

is

by its first standard

2n m+1 m

if and only if

minimal totally real submanifold of

to be of order Proof.

where

Then we have

(10.2) Moreover,

= Q or Q,

IF

[p , q]

Since

in

IF Pm

m = n

and

Where

IF Pn.

M

M

is a

is assumed

H (m+l ; IF) . 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

of

r = rm/2(m+1)]1/2.

radius IF Pm

is the centroid of

S(r)

Thus, by Lemma 4.3, the centroid of

in H (m+l ; IF) .

S (r)

hypothesis, the centroid of

M

Hence, by the

is also the centroid of

S(r).

Therefore, by applying Theorem 9.1, we obtain inequality (10.2). if

%p = 2n(m+l)/m,

minimal in M

then Theorem 9.1 implies that

S (r) and hence

is also minimal in

is of 1-type in

M

IF Pm.

IFPm

manifold of a totally geodesic case occurs, then to (10.2). find

n = m

X

l

is

H (m+l ; IF) .

Thus,

By applying Lemma 9.2 and its quater-

nionic version, we conclude that invariant submanifold of

M

M

is either a totally geodesic

or a totally real minimal subIF Pn

in

IF Pm.

= 2(n+d), d = 2 or 4.

If the second case ocuurs, by our assumption.

If the first

This contradicts

ap = 2(n+l).

Thus, we

6. Submanifolds of Finite Type

322

M

Conversely, if

is a totally real minimal submanifold of

then, by (4.6.26) and the fact that

]F Pn,

submanifold of the hypersphere [n/2(n+1)]1/2,

vector

H

IF Pn

-IL = 2 (n+l)

.

r

with radius

S(r)

is a minimal r =

we may conclude that M has mean curvature H = H,

satisfying

vector of

iF Pn

where

ft

is the mean curvature

Thus, we obtain

in H (n+l ; iF) .

ap = (Q.E.D.)

Similarly, by using Remark 4.6.2 and Theorem 9.1, we have the following.

Let

Theorem 10.3.

M be a compact, n-dimensional

mass-symmetric submanifold of ly imbedded in

H(3 ;Cay)

OP2,

Where

OP2

(n z 2),

is isometrical-

by its first standard imbedding.

Then we have xl a 3n,

(10.3)

equality holding if and only if surface of

OP2.

M

is a minimal, totally real

Here, by a totally real surface of

OP2,

A&

mean a surface whose tangent planes are totally real with respect 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

Submanifolds

§ 10.

as a mass-symmetric submanifold for

IR Pm

Proof.

m' n.

can be isometrically imbedded in

IR Pm

Et Pm

as a mass-symmetric submanifold, then Theorem 10.1 implies

m >n,

l

If

323

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

as a mass-symmetric submanifold in a natural way,

be isometrically imbedded in submanifold.

cannot

as a mass-symmetric

This result is a special case of the following.

Corollary 10.2.

M

L

Riemannian manifold with

be a compact, n-dimensional,

>l ? 2(n+l).

isometrically imbedded in unless

IF Pm, m >n,

P Pn

]F P n

m = n, al = 2(n+l)

real minimal submanifold in

IF Pm

and

Then

M

cannot be

as a mass-symmetric submanifold M

can be imbedded as a totally

IF Pn.

This Corollary follows immediately from Theorem 10.2.

BIBLIOGRAPHY

Abe, K. [1]

Applications of Riccati type differential equation to Riemannian manifolds with totally geodesic distribution, Tohoku Math J., 25 (1973), 425-444.

Adem, J. [1]

Relations on iterated reduced powers, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 636-638.

Asperti, A.C. [1]

Immersions of surfaces into 4-dimensional spaces with nonzero normal curvature, Ann. Mat. Pura Appl. 125 (1980), 313-328.

Asperti, A.C., Ferus, D., and Rodriguez, L. [1]

Surfaces with non-zero normal curvature tensor, to appear.

Atiyah, M.F., and Lawson, H.B. [1]

In preparation.

Atiyah, M.F., and Singer, I.M. [11

The index of elliptic operators, III, Ann. of Math., 87 (1968), 546-604.

Banchoff, T.F. [1]

The two-piece property and tight n-manifolds-with boundary in Em, Trans. Amer. Math. Soc. 161 (1971), 259-267.

Barros, M., Chen, B.-Y., and Urbano, F. [1]

Quaternion CR-submanifolds of quaternion manifolds, Kodai

Math. J., 4 (1981), 399-417.

Barros, M. and Urbano, F. [11

Submanifolds of complex Euclidean space which admit a holomorphic distribution, Quart. J. Math. Oxford, (2) 34 (1983), 141-143.

[2]

Topology of quaternion CR-submanifolds, Boll. Un. Mat. Ital., (6) 2-A (1983), 103-110.

326

Bibliography

Bejancu, A. [1]

CR-submanifolds of a Kaehler manifold, I, Proc. Amer. Math. Soc., 69 (1978), 135-142.

[2]

CR-submanifolds of a Kaehler manifold, II, Trans. Amer. Math. Soc, 250 (1979), 333-345.

[3)

On the.geometry of leaves on a CR-submanifold, An. St. Univ. "Al. I. Cuza" Iasi, 25 (1979), 393-398.

Berger, M. [1]

Geometry of spectrum, I, Proc. Symp. Pure Math. Vol. 27, Part 2,

[2]

(1975), 129-152.

Sur lea premieres valeurs propres des varietes riemanniennes, Compositio Math., 26 (1973), 129-149.

Berger, M., Gauduchon, P., and Mazet, E. [1]

Le Spectre d'une Variete Riemannienne, Math. 194 (1971), Springer, Berlin.

Lecture Notes in

Besse, A.L. [1]

Manifolds all of whose Geodesics are Closed, Ergeb. Math. Grenz. 93 (1978), Springer, Berlin.

Blair, D.E., and Chen, B.-Y. [1]

On CR-submanifolds of Hermitian manifolds, Israel J. Math., 34 (1979), 353-363.

Blaschke, W. [1]

Vorlesungen fiber Differentialgeometrie III, Springer, 1929, Berlin.

Sleeker, D., and Weiner, J. [1]

Extrinsic bounds on X1 of A on a compact manifold, Comm. Math. Helv., 51 (1976), 601-609.

Borsuk, K. [1]

Sur la courbure totale des courbes, Ann. Soc. Math. Polon., 20 (1947), 251-256.

Calabi, E. [1]

Isometric imbedding of complex manifolds, Ann. of Math., 58 (1953), 1-23.

Bibliography

327

Cartan, 9. [l]

Sur une classe remarquable d'espaces de Riemann, Bull. Soc. Math. France, 54 (1926), 214-264 and 55 (1927), 114-134.

[2]

Lecons sur la Geometrie des Espaces de Riemann, GauthierVillars, Paris, 1928.

Carter, S. and West, A. [1]

Tight and taut immersions, Proc. London Math. Soc., 25 (1972), 701-720.

Cecil, T.E. and Ryan, P.J. [1]

Tight sperical embedding, Lecture Notes in Math., 838 (1981), 94-104.

Chen, B.-Y. [1)

On an inequality of T.J. Willmore, Proc. Amer. Math. Soc., 26 (1970), 473-479.

[2)

On total curvature of immersed manifolds, I, Amer. J. Math., 93 (1971), 148-162.

[3)

On a variational problem of hypersurfaces, J. London Math. Soc., 6 (1972), 321-325.

[4)

G-total curvature of immersed manifolds, J. Differential Geometry, 7 (1972), 373-393.

[5]

On the total curvature of immersed manifolds, II, Amer. J. Math., 94 (1972), 899-907.

[6]

On an inequality of mean curvature, J. London Math. Soc., 4 (1972), 647-650.

[7]

Geometry of Submanifolds, Dekker, 1973, New York.

[8]

An invariant of conformal mappings, Proc. Amer. Math. Soc., 40 (1973), 563-564.

[9]

On the total curvature of immersed manifolds, III, Amer. J. Math., 95 (1973), 636-642.

[10]

Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital., (4) 10 (1974), 380-385.

[11]

Mean curvature vector of a submanifold, Proc. Symp. Pure Math. Vol. 27, Part I, (1975), 119-123.

Bibliography

328 [12]

Some relations between differential geometric invariants and topological invariants of submanifolds, Nagoya Math. J., 60 (1976), 1-6.

[13]

Total mean curvature of immersed surfaces in Amer. Math. Soc., 218 (1976), 333-341.

[14]

Totally umbilical submanifolds of quaternion-space-forms, J. Austra. Math. Soc. (Series A), 26 (1978), 154-162.

[15]

On total curvature of immersed manifolds, IV, Bull. Inst. Math. Acad. Sinica, 7 (1979), 301-311.

[16]

Conformal mappings and first eigenvalue of Laplacian on surfaces, Bull. Inst. Math. Acad. Sinica, 7 (1979), 395-

Em, Trans.

400. [17]

Geometry of Submanifolds and Its Applications, Sci. Univ. Tokyo, 1981, Tokyo.

[18]

Differential geometry of real submanifolds in a Kaehler manifold, Monatsch. fiir Math., 91 (1981), 257-274.

[19]

On the total curvature of immersed manifolds. V. Bull. Inst. Math. Acad. Sinica, 9 (1981), 509-516.

[20]

CR-submanifolds of a Kaehler manifold, I, II, J. Differential Geometry, 16 (1981). 305-322 and 493-509.

[21]

Cohomology of CR-submanifolds, Ann. Fac. Sc. Toulouse Math. (Series V) 3 (1981), 167-172.

[22]

On total curvature of immersed manifolds, VI, Bull. Inst. Math. Acad. Sinica, 11 (1983), 309-328.

[23]

Review of "Total Curvature in Riemannian Geometry" by Willmore, Bull. London Math. Soc. 16 (1984).

[24]

On the first eigenvalue of Laplacian of compact minimal submanifolds of rank one symmetric spaces, Chinese J. Math., 11 (1983).

[25]

On submanifolds of finite type, Soochow J. Math., 9 (1983).

[26]

Some non-integrability theorems of holomorphic distributions, Alg. Diff. Topology-Global Diff. Geom. Appl., Teubner-Verlag, 1984, Leipzig.

Chen, B.-Y. and Houh, C.S. [1]

On stable submanifolds with parallel mean curvature, Quart.

J. Math. Oxford,

(3)

26 (1975), 229-236.

Bibliography [2]

329

Totally real submanifolds of a quaternion projective space, Ann. Mat. Pura Appl., 120 (1979), 185-199.

Chen, B.-Y., Houh, C.S., and Lue, H.S. [1]

Totally real submanifolds, J. Differential Geometry 12 (1977) ,

473-480.

Chen, B.-Y., Ludden, G.D., and Montiel, S. [1]

Real submanifolds of a Kaehler manifold, (preprint).

Chen, B.-Y. and Nagano, T. [1]

Totally geodesic submanifolds of symmetric spaces, I, II, Duke Math. J., 44 (1977), 745-755; 45 (1978). 405-425.

[2]

Harmonic metrics, harmonic tensors and Gauss maps, J. Math. Soc. Japan, 36 (1984).

Chen, B.-Y. and Ogiue, K. (1]

On totally real submanifolds, Trans. Amer. Math. Soc., 193 (1974), 257-266.

[2]

Two theorems on Kaehler manifolds, Michigan Math. J., 21 (1974), 225-229.

Chen, B.-Y. and Vanhecke, L. [1]

The spectrum of the Laplacian of Kahler manifolds, Proc. Amer. Math. Soc., 79 (1980), 82-86.

[2]

Differential geometry of geodesic spheres, J. Reine Angew. Math. 325 (1981), 28-67.

Chen, B.-Y. and Verheyen, P. [1]

Totally umbilical submanifolds of Kaehler manifolds, II, Bull. Soc. Math. Belg. (to appear).

[2]

Sous-varietes dont les sections normales sont des geodesiques, C.R. Acad. Sc. Paris, 293 (1981). 611-613.

Chen, B.-Y. and Yano, K. [1]

On the theory of normal variations, J. Differential Geometry, 13

(1978), 1-10.

Cheng, S.Y. [1]

Eigenfunctions and eigenvalues of Laplacian, Proc. Symp. Pure Math. Vol. 27, Part 2, (1975), 185-194.

330

Bibliography

Chern, S.S. (1)

A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math., 45 (1944), 747-752.

[2)

Differentiable Manifolds, University of Chicago, 1959.

[3)

Minimal Submanifolds in a Riemannian Manifold, Univ. Kansas, Tech. Rep. No. 19 (1968).

[4]

Complex Manifolds without Potential Theory, Springer, 1979, Berlin.

[5)

Selected Papers, Springer, 1978, Berlin.

Chern, S.S., doCarmo, M., and Kobayashi, S. [1]

Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields, Springer (1970), 59-75.

Chern, S.S. and Hsiung, C.C. [1)

On the isometry of compact submanifolds in Euclidean space, Math. Ann. 149 (1962/63), 278-285.

Chern, S.S. and Kuiper, N. [1)

Some theorems on the isometric imbedding of compact Riemann manifolds in Euclidean space, Ann. of Math., 56 (1952), 422-430.

Chern, S.S. and Lashof, R.K. [1]

On the total curvature of immersed manifolds, Amer. J. Math., 79 (1957), 306-318.

[2]

On the total curvature of immersed manifolds, II,

Michigan Math. J., 5 (1958), 5-12.

Dajczer, M. [1]

Doctoral Thesis, IMPA, 1980.

doCarmo, M.P. and Wallach, N.R. [1]

Minimal immersions of spheres into spheres, Ann. of Math., 93

(1971), 43-62.

Eells, J., Jr. and Lemaire, L. [1)

A report on harmonic maps, Bull. London Math. Soc., 10 (1978), 1-68.

Bibliography

331

Eells, J. Jr. and Sampson, H. [1]

Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160.

Ejiri, N. [1)

A counter example for Weiner's open question, Indiana Univ. Math. J., 31 (1982), 209-211.

[2]

The first eigenvalue of 6 for compact minimal submanifolds in a complex projective space, to appear.

[3]

Totally real minimal immersions of n-dimensional real space form into n-dimensional complex space forms, Proc. Amer. Math. Soc., 84 (1982), 243-246.

Erbacher, J. [1]

Reduction of the codimension of an isometric immersion, J. Differential Geometry, 5 (1971), 333-340.

Escobales, R.H. [1)

Riemannian submersions with totally geodesic fibers, J. Differential Geometry, 10 (1975), 253-276.

Fary, I. [1]

Sur la courbure totale d'une courbe gauche faisant un noeud, Bull. Soc. Math. France, 77 (1949), 128-138.

Fenchel, W. [1]

Uber die Krilmmung and Windung geschlossenen Raumkurven, Math. Ann., 101 (1929), 238-252.

Ferus, D. [1]

Totale Absolutkriimmung in Differentialgeometrie and Topologie, Lecture Notes in Math., 66 (1968), springer, Berlin.

[2]

Symmetric submanifolds of Euclidean space, Math. Ann., 247 (1980), 81-93.

Gheysens, L., Verheyen, P., and Verstraelen, L. [1)

Sur les surfaces Q ou les surfaces de Chen, C.R. Acad. Sc. Paris, 292 (1981), 913-916.

[2]

Characterization and examples of Chen submanifolds, J. Geometry, 20 (1983), 47-62.

Bibliography

332

Goldberg, S.I. [1]

Curvature and Homology, Academic Press, 1962, New York.

Gray, A. [1]

A note on manifolds whose holonomy group is a subgroup Sp(1), Michigan Math. J., 16 (1969), 125of Sp(n) 128.

Guadalupe, I.V. and Rodriguez, L. [1]

Normal curvature of surfaces in space forms, Pacific J. Math., 106 (1983), 95-103.

Haantjes, J. [1]

Conformal representations of an n-dimensional euclidean space with a non-definite fundamental form on itself, Proc. Kon. Ned. Akad. Amsterdam, 40 (1937), 700-705.

Heintze, E. and Karcher, H. [1]

A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sc. Ecole Norm. Sup., 11 (1978), 451-470.

Helgason, S. [1]

Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978, New York.

Hersch, J. [1]

Quatre proprietes isoperimetriques de membranes spheriques homogenes, C.R. Acad. Sc. Paris, 270 (1970), 1645-1648.

Husemdller, D. [1]

Fibre Bundles, McGraw-Hill, 1966, New York.

Houh, C.S. [1]

On spherical

c7-submanifolds,

Chinese J. Math., 2 (1974),

128-134.

Hsiung, C.C. [1]

Some integral formulas for hypersurfaces, Math. Scand. 2 (1954), 286-294.

[2]

Isoperimetric inequalities for 2-dimensional Riemannian manifolds with boundary, Ann. of Math., 73 (1961), 213-220.

Bibliography

333

Klingenberg, W. [1]

Lectures on Closed Geodesics, Springer, 1978, Berlin.

[2]

Riemannian Geometry, Walter de Gruyter, 1982, Berlin.

Kobayashi, S. [1]

Imbeddings of homogeneous spaces with minimum total curvature, Tohoku Math. J. 19 (1967), 63-70.

[2]

Isometric imbeddings of compact symmetric spaces, Tohoku Math. J. 20 (1968), 21-25.

[31

Transformation Groups in Differential Geometry, Springer, 1972, Berlin.

[4]

Recent results in complex differential geometry, Jber. Deut. Math. Verein, 83 (1981), 147-158.

Kobayashi, S. and Nomizu, K. [1]

Foundations in Differential Geometry, I, II, WileyInterscience, 1963 and 1969, New York.

Kuiper, N.H. [1]

Immersions with minimal total absolute curvature, Colloq. Geom. Diff. Globale (Bruxelles, 1958), 75-88.

[2]

Tight imbeddings and maps, submanifolds of geometrical class three in EN, Chern Symposium, (1979), 97-146, Springer, Berlin.

Kuiper, N.H. and Meeks, W.H. [1]

Total absolute curvature for knotted surfaces, to appear.

KUhnel, W. [1]

Die totale mittlere Kr6mmung gewisser Hyperflachen im Monatsch. fur Math., 87 (1979), 123-131. 3R 5,

[2]

A lower bound for the ith total absolute curvature of an immersion, Colloq. Math., 41 (1979), 253-255.

Langevin, R. and Rosenberg, H. [1]

On curvature integrals and knots, Topology, 15 (1976), 405-416.

Lashof, R.K. and Smale, S. [1]

On the immersions of manifolds in Euclidean spaces, Ann. of Math., 68 (1958), 562-583.

334

Bibliography

Lawson, Jr., H.B. [1]

Rigidity theorems in rank-1 symmetric spaces, J. Differential Geometry, 4 (1970), 349-357.

[2]

Complete minimal surfaces in

S3,

Ann. of Math.,

(1970). 335-374.

Li, P. and Yau, S.T. [1]

A new cohformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math., 69 (1982), 269-291.

Lichnerowicz, A. (1)

Equations de Laplace et espaces harmoniques, Premier Colloque Equations Derives Partielles, Louvain, 1953, 9-23.

Little, J. [1)

On singularities of submanifolds of higher dimensional Euclidean spaces, Ann. Mat. Pura Appl., 88 (1969). 261336.

[2]

Manifolds with planar geodesics, J. Differential Geometry, 11 (1976). 265-285.

Little, J. and Pohl, W.F. [1]

On tight immersion of maximal codimension, Invent. Math., 13 (1971), 179-204.

Maeda, M. [1]

The integral of the mean curvature, Sci. Rep. Yokohama Nat. Univ., Sect. I. 25 (1978), 17-21.

Masal'cev, L.A. [1]

The first eigenvalue of the Laplacian of a compact orientable hypersurface of a Euclidean space, Ukrain. Geom. Sb. Vyp., 19 (1976). 104-108.

McKean, H.P. and Singer. I.M. [1]

Curvature and the eigenvalues of the Laplacian, J. Differential Geometry, 1 (1967). 43-69.

Milnor, J.W. [1]

On the total curvature of knots, Ann. of Math., 52 (1950). 248-257.

Bibliography

335

[2]

On manifolds homeomorphic to the 7-sphere, Ann. of Math., 69 (1956), 399-405.

[3]

Morse Theory, Ann. Math. Studies, 51 (1963), Princeton Univ. Press, Princeton.

[4]

Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 542.

Moore, J.D. [1]

Isometric immersions of Riemannian products, J. Differential Geometry, 5 (1971), 159-168.

Morse, M. [1]

The Calculus of Variations in the Large, Amer. Math. Soc., 1934, New York.

Morvan, J.-M. [1]

Topology of a submanifold and external curvatures, Rend. Mat. 13 (1980), 1-20.

Mostow, G.D. [1]

Equivariant embeddings of Euclidean space, Ann. of Math., 65 (1957), 432-446.

Nagano, T. [1]

On fibered Riemannian manifolds, Sci. Papers Coll. Gen. Ed. Univ. Tokyo, 10 (1960), 17-27.

[2]

On the minimal eigenvalues of the Laplacians in Riemannian manifolds, Sci. Papers Coll. Gen. Ed. Univ. Tokyo, 11 (1961), 177-182.

[3]

Homotopy invariants in differential geometry, memoir Amer. Math. Soc., 100 (1970).

Naitoh, H. and Takeuchi, M. [1]

Totally real submanifolds and symmetric bounded domains, Osaka J. Math., 19 (1982), 717-731.

Nakagawa, H. [1]

On a certain minimal immersion of a Riemannian manifold into a sphere, Kodai Math. J., 3 (1980), 321-340.

Nash, J.F. [1]

C1-isometric imbeddings, Ann. of Math., 60 (1954), 383-396.

Bibliography

336

Nomizu, K. and Yano, K. [1]

On circles and spheres in Riemannian geometry, Math. Ann., 210 (1974), 163-170.

Obata, M. [i]

Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan, 14 (1962), 333340.

Ogiue, K. [1]

Differential geometry of Kaehler submanifolds, Adv. in Math., 13 (1974), 73-114.

[2]

Positively curved totally real minimal submanifolds immersed in a complex projective space, Proc. Amer. Math. Soc., 56 (1976), 264-266.

O'Neill, B. [1]

The fundamental equations for a submersion, Michigan Math. J., 13 (1966), 459-469.

Osserman, R. [1]

A Survey of Minimal Surfaces, Van Nostrand, 1969, Princeton.

Otsuki, T. [1]

On the existence of solutions of a system of quadratic equations and its geometrical applications, Proc. Japan Acad., 29 (1953), 99-100.

[2]

On the total curvature of surfaces in Euclidean spaces, Japan, J. Math., 35 (1966), 61-71.

[3]

Surfaces in the 4-dimensional Euclidean space isometric to a sphere, Kodai Math. Sem. Rep., 18 (1966), 101-115.

Palais, R.S. [1]

Imbeddings of compact differentiable transformation groups in orthogonal representations, J. Math. Mech., 6 (1957),

673-678. [2]

Seminar on the Atiyah-Singer Index Theorem, Ann. Math. Studies, 57 (1965), Princeton.

Patodi, V.K. [1]

Curvature and the fundamental solution of the heat operator, J. Indian Math. Soc., 34 (1970), 269-285.

337

Bibliography

Reckziegel, H. [1]

On the problem whether the image of a given differentiable map into a Riemannian manifold is contained in a submanifold with parallel second fundamental form, J. Reine Angew. Math., 325 (1981), 87-104.

Reilly, R.C. [1]

Variational properties of mean curvatures, Proc. 13th Biennial Sem. Canad. Math. Congress, Vol. II, (1973), 102-114.

[2]

On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comm. Math. Helv., 52 (1977), 525-533.

Ros, A. [1]

Spectral geometry of CR-minimal submanifolds in the complex projective space, Kodai Math. J., 6 (1983), 88-99.

[2]

On spectral geometry of Kaehler submanifolds, to appear.

[3]

Eigenvalue inequalities for minimal submanifolds and P-manifolds, to appear.

Rouxel, B. [1]

Q-submanifolds in Euclidean space, Kodai Math. J., 4 (1981), 181-188.

[2]

Ruled Q-surfaces in Euclidean space Math., 6 (1980), 117-121.

En,

Soochow J.

Sakai, T. [1]

On eigenvalues of Laplacian and curvature of Riemannian manifolds, Tohoku Math. J., 23 (1971), 589-603.

Sakamoto, K. [1)

Planar geodesic immersions, Tohoku Math. J., 29 (1977), 25-56.

Shiohama, K. and Takagi, R. [1]

A characterization of a standard torus in ential Geometry, 4 (1970), 477-485.

E3,

J. Differ-

Simons, J. [1]

Minimal varieties in Riemannian manifolds, Ann. of Math., 88 (1968), 62-105.

Bibliography

338

Smale, S. [1)

The classification of immersions of spheres in Euclidean spaces, Ann. of Math., 69 (1959), 327-344.

Spivak, M. [1)

A Comprehensive Introduction to Differential Geometry, Vol. I-V, Publish or Perish, 1970, Berkeley.

Sternberg, S. [1)

Lectures on Differential Geometry, Prentice-Hall, 1964, New Jersey.

Sunday, D. [1)

The total curvature of knotted spheres, Bull. Amer. Math. Soc., 82 (1976), 140-142.

Tai, S.S. (1)

Minimum imbeddings of compact symmetric spaces of rank one, J. Differential Geometry, 2 (1968), 55-66.

Takahashi, T. [1)

Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380-385.

Takeuchi, M. and Kobayashi, S. [1)

Minimal imbeddings of R-spaces, J. Differential Geometry, 2 (1968), 203-215.

Tanno, S. [1)

Eigenvalues of the Laplacian of Riemannian manifolds, Tohoku Math. J., 25 (1973), 391-403.

Thomsen, G. [1]

Ober konforme Geometrie I: Grundlagen der konformen Flichentheorie, Abh. Math. Sem. Univ. Hamburg, (1923),

31-56. Wallach, N.R. [1)

Minimal immersions of symmetric spaces into spheres, Symmetric Spaces, 1972, Dekker, 1-40.

Weiner, J.L. [1)

On a problem of Chen, Willmore, et al., Indiana Univ. Math. J., 27 (1978), 19-35.

339

Bibliography

White, J.H.

A global invariant of conformal mappings in space, Proc. Amer. Math. Soc., 38 (1973), 162-164.

[1)

Willmore, T.J. [1]

Note on embedded surfaces, An. St. Univ. Iasi, s.I.a. Mat., 11B (1965), 493-496.

[2]

Mean curvature of immersed surfaces, An. St. Univ. Iasi, s.I.a. Mat. 14 (1968), 99-103.

[3)

Mean curvature of Riemannian immersions, J. London Math. Soc., 3 (1971), 307-310.

[4)

Tight immersions and total absolute curvature, Bull. London Math. Soc., 3 (1971), 129-151.

[5]

Mean curvature of immersed manifolds, Topics in Differential Geometry, Academic Press, 1976, 149-156.

[6)

Total Curvature in Riemannian Geometry, John Wiley and Sons, 1982, Chichester.

Willmore, T.J. and Jhaveri, C. An extension of a result of Bang-Yen Chen, Quart. J. Math. Oxford, 23 (1972), 319-323.

[1]

Wintgen, P. [1]

On the total curvature of surfaces 39 (1978), 289-296.

[2]

Sur l'inequalite de Chen-Willmore, C.R. Acad. Sc. Paris, 288 (1979), 993-995.

in E4,

Colloq. Math.,

Witt, E. [1]

Eine Identitat zwischen Modulformen zweiten Grades, Abh. Math. Sem. Univ. Hamburg, 14 (1941), 323-337.

Wolf, J.A. [1)

Elliptic spaces in Grassmann manifolds, Illinois J. Math., 7 (1963), 447-462.

[2)

Spaces of Constant Curvature, McGraw-Hill, 1967, New York.

Yamaguchi, S. [1)

Remarks on the scalar curvature of immersed manifolds, Kodai Math. Sem. Rep., 26 (1975), 240-244.

Bibliography

340

Yang, P.C. and Yau, S.T. [1]

Eigenvalues of the Laplacian of compact Riemann surfaces and Minimal submanifolds, Ann. Scuola Norm. Sup. Pisa, 7

(1980), 55-63.

Yano, K. [1]

The Theory of Lie Derivatives and its Applications, NorthHolland, 1957, Amsterdam.

[2]

Differential Geometry on Complex and Almost Complex Spaces, Pergamon, 1965, New York.

[3]

Integral Formulas in Riemannian Geometry, Dekker, 1970, New York.

[4]

Selected Papers, Amsterdam.

(Ed. by Obata), North-Holland, 1982,

Yano, K. and Bochner, S. [1]

Curvature and Betti Numbers, Ann. Math. Studies, 32 (1953), Princeton.

Yano, K. and Chen, B.-Y. [1]

Minimal submanifolds of a higher dimensional sphere, Tensor, 22 (1971), 369-373.

[2]

Pseudo-umbilical submanifolds in a Riemannian manifold of constant curvature, Differential Geometry, in honor of K. Yano, Kinokuniya, 1972, 61-71, Tokyo.

Yano, K. and Kon, M. [1]

Anti-invariant Submanifolds, Dekker, 1978, New York.

Yau, S.T. [1]

Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J., 25 (1976), 659-670.

AUTHOR INDEX

Abe, K. , 324

Adem, J., 195. 324 Asperti, A.C., 246, 324 Atiyah, M.F., 24B, 324

Banchoff, T.F., 324 Barros, M., 180, 181, 324 Bejancu, A., 172, 173. 176, 325 Berger, M., 25-. 100, 325 Besse, A.L., 325 Blair, 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., 32E Chen, 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., 332 Kobayashi, 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 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., 316 Masol'cev, L.A., 295, 333

Mazet, E., 95, 1QQ, 322.5 McKean, H.P.

99 333

Meeks, 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, 1fi4 Morvan, J.M., 334 Mostow, G.D., 334

Nagano, T., 115, 328. 334 Naitoh, H., 334

Nakagawa, H 334 Nash, J.F., 120. 1$Z, 334 Nomizu, K., 4 32. 12.3_. 332, 335

343

Author Index

344

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., 3116 Pleijel, 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, 331 Ros, A., 141, 180, 196, 267, 280, 281, 290, 311. 315.

316, 336 Rosenberg, H_, 332

Rouxel, B., 269. 3316 Ryan, P.J., 326

Sakai, T., 99, 3.316 Sakamoto, K., 141, 154. 156. 336

Sampson, H., 220 314

Santos, F.G., 316 Sard, A., 20

154

Shiohama, K., 184, 336 Simons, J., 33¢ Singer, I.M., 99, 324, 333

Smale, S . , 239. 241. 332, 331 Spivak, M., 337 Springer, T.A., 123

Author Index Sternberg, S., 24. 331 Sunday, D., 241. 3-32

Tai, S.S., 141. 145, 156, 33.7 Takahashi, 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, 31 331 Weiner, J.L., 225. 296. 325 332 West, A., 32L White, 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, 23 adjoint, 79 affine connection, 46 allied mean curvature vector, 20 associated vector field, 56

associated 1-form, 56 asymptotic 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, 22 Fredholm'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 3.9 Hopf 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, 2fx Reeb 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 State University. He has held visiting appointments at many universities,

including the Catholic University of Louvain. National Tsinghua University of Taiwan, Science University of Tokyo, University of Notre Dame, and University of Granada. Dr Chen's research interests focus on differential geometry, global analysis and complex manifolds. He is the author of numerous articles and two books Dr Chen received his B.S. degree in 1965 from Tamkang University. his M.S. degree in

1967 from Tsinghua University and his Ph.D. in 1970 from the University of Notre Dame. He is a member of the American Mathematical Society.

9971-966-03-4 pbk