Structures on Manifolds (Series in Pure Mathematics)

Series in Pure Mathematics

-

Volume 3

STRUCTURES ON MANIFOLDS

KENTARO YANO

MASAHIRO KON

Tokyo Institute of Technology Tokyo Japan

Hirosaki University Hirosaki, Japan

World Scientific

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

STRUCTURES ON MANIFOLDS

Copyright © 1984 by World Scientific Publishing Co Pte Ltd. All rights reserved. This book, or parts thereof. may not be reproduced in and form or by any means, electronic or mechanical. Including photo. copying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. ISBN 9971-966-15-8 9971-966-16-6 pbk

Printed in Singapore by Kim Iiup Lee Printing Co Pte Ltd.

V

PREFACE The theory of structures on manifolds is a very interesting topic of modern differential geometry and the differential geometric aspects of subnanifolds of manifolds with certain structures are vast and -very fruitful fields for Riemannian geometry.

The purpose of this book is to provide an introduction to the theory of various differential geometric structures on manifolds and to gather and arrange the results on subnanifolds of Riemannian manifolds with certain structures.

In Chapter I we have given a brief survey of differentiable manifolds, tensor fields, connections, fibre bundles and Riemannian curvature tensors. Chapter II contains the theory of submanifolds of Riemannian manifolds. We first of all state general formulas on sutmanifolds and the formulas of Laplacians of the second fundamental forms which will play very important roles in the following discussions. We also prove various theorems of submanifolds of Riemannian space forms. In Chapter III we provide differential geometric foundations for almost

complex manifolds and Hermitian metrics, in particular, complex manifolds and Kaehlerian metrics. We also discuss the theory of nearly Kaehlerian manifolds and quaternion Kaehlerian manifolds. Chapter IV is devoted to the study of submanifolds of Kaehlerian manifolds. We study complex submanifolds, anti-invariant submanifolds and CR sutmanifolds of Kaehlerian manifolds. In Chapter V we give the basic results of almost contact manifolds and contact manifolds. The main purpose of this chapter is to introduce the fundamental properties of Sasakian manifolds. We also consider the Boothby Wang fiberation and the Brieskorn manifolds. In Chapter VI we present the theory of submanifolds of Sasakian manifolds which include invariant su}manifolds, antiinvariant subnanifolds tangent to the structure vector field, antiinvariant submanifolds normal to the structure vector field and contact

vi

at submanifolds. Chapter VII is devoted to the study of f-structures on Riemannian manifolds. We also consider the hypersurfaces of framed manifolds. In Chapter VIII, we study the theory of product manifolds.

We give the fundamental formulas for product manifolds, and consider its submanifolds. We also discuss the Kaehlerian product manifolds and its submanifolds. In the last Chapter IX, we first give the fundamental formulas for submersions, and present some results of almost Hennitian submersions. In the last section of this chapter, we discuss the relations of Sasakian manifolds and Kaehlerian manifolds and submanifolds of these manifolds by using the theory of submersions.

The Exercises are intended to introduce the intereting results which are concerned with the subject of this book. The authors wish to express here their deep gratitude to Professor C. C. Hsiung who suggested that we include this book in "Series in Pure Mathematics". It is a pleasant duty for us to acknowledge -that World

Scientific Publishing took all possible care in the production of the book.

November 13, 1984

Kentaro Yano Masahiro Kon

vii

CONTENTS

CHAPTER I

RIEMANNIAN MANIFOLDS 1. Manifolds and tensor fields

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2. Connections and covariant differentiations

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3. Sectional curvature .

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4. Transformations

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5. Fibre bundles and covering spaces

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46

Exercises

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54

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18

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CHAPTER II

SUBMANIFOLDS OF RIEMANNIAN MANIFOLDS 1. Induced connection and second fundamental form .

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2. Equations of Gauss, Codazzi and Ricci .

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3. Laplacian of the second fundamental form . 4. Submanifolds of space forms

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5. Minimal sutxnanifolds

Exercises

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61 .

67 73

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78

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89

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CHAPTER III COMPLEX MANIFOLDS

1. Almost complex manifolds and complex manifolds .

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2. Examples of complex manifolds and almost complex manifolds

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3. Hermitian manifolds .

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124

4. Kaehlerian manifolds

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5. Nearly Kaehlerian manifolds

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. 129

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..144-

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6. Quaternion Kaehlerian manifolds .

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. 158

Exercises

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CHAPTER IV SUBMANIFOLDS OF KAEHLERIAN MANIFOLDS 1. Kaehlerian submanifolds

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180

2. Anti-invariant suL-manifolds of Kaehlerian manifolds-.

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3. CR submanifolds of Kaehlerian manifolds .

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214

R\ercises .

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252

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255

3. Torsion tensor of almost contact manifolds . 4. Contact distribution

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5. Sasakian manifolds .

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CHAPTER V CONTACT MANIFOLDS 1. Almost contact manifolds .

2. Contact manifolds

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6. Regular contact manifolds 7. Brieskorn manifolds

Exercises .

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263

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269

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272

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286

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306

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312

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CHAPTER VI

SUBMANIFOLDS OF SASAKIAN MANIFOLDS 1. Invariant submanifolds of Sasakian manifolds

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2. Anti-invariant submanifolds tangent to the structure vector field of Sasakian manifolds

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3. Anti-invariant submanifolds normal to the structure vector field of Sasakian manifolds 4. Contact CR submanifolds

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5. Induced structures on submanifolds

Exercises .

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344

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351

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366

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372

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CHAPTER VII

f-STRUCTURES

1. f-structure on manifolds . 2. Normal f-structure . . . 3. Framed f-structure . . .

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379

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392

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402

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ix 4. Hypersurfaces of framed manifolds

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408

Exercises

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412

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CHAPTER VIII

PRODUCT MANIFOLDS 1. Locally product manifolds .

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3. Sutmanifolds of product manifolds

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414

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418

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424

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2. Locally decomposable Riemannian manifolds .

4. Submanifolds of Kaehlerian product manifolds

Exercises

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429 ..

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CHAPTER IX SUBMERSIONS 1. Fundamental equations of submersions

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2. Almost Hermitian sutmersions

439 448

3. Submersions and sutmanifolds .

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455

Exercises

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BIBLIOGRAPHY

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INDEX

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495

1

CHAPTER I

RIEMANNIAN MANIFOLDS

In this chapter, we have given a brief survey of Riemannian geometry. In §1, we give the basic definitions and formulas on-di.ffe-

rentiable manifolds. §2 is devoted to the study of connections and covariant differentiation on manifolds. We introduce the torsion tensor field and the curvature tensor field and derive the structure equation of Cartan. Moreover, we define the Riemannian connection on a manifold. In §3, we define the sectional curvature of a Riemannian manifold -and

give examples of space forms, that is, Riemannian manifolds of constant curvature. In §4, we discuss transformations on a Riemannian manifold and give some integral formulas (cf. Yano [6]). In the last §5, we prepare some tesults of fibre bundles for later use.

In §§1, 2 and 3 we followed Chapter I of Helgason [1] and Kobayashi Nomizu [1]. In §5, we followed fairly closely KobayashiNomizu [1]. We also refer to Matsushima [1].

1. MANIFOLDS AND TENSOR FIELDS

Let M be a topological space. We assume that M satisfies the Hausdorff separation axian which states that any two different points in M can be separated by disjoint open sets. An open chart on M is a pair (U,4)) where U is an open subset of M and 0 is a haneomorphisn

of U onto an open subset of Rn, where Rn is an n-dimensional Euclidean

space.

2 Definition. Let M be a Hausdorff space. A differentiable structure on M of dimension n is a collection of open charts (Ui'+i)iEA on M

where YU1) is an open subset of EP such that the following conditions are satisfied:

(a) M = ieAUi.' (b) For each pair i,j e A the mapping 4,.$11 is a differentiable

mapping of Oi(Ui A U) onto 4j(Ui n

i);

(c) The collection (U1, 'i)iEA is a maximal family of open charts

for which (a) and (b) hold. A differentiable manifold (or (f-manifold or simply manifold) of dimension n is a Hausdorff space with differentiable structure of dimension n. If M is a manifold, a local coordinate system on M (or a local chart on M) is by definition a pair (1`i,4.). If p is a point in

Ui and ¢i(p) = (x1(p),...,xn(p)), the set Ui is called a coordinate

neighborhood of p and the numbers x .(p) are called local coordinates J

of p. The mapping Oi : q

> (x(q),...,x(q)), q E Ui, is often

denoted by (x1,...,xn}. We notice that the condition (c) is not essential in the definition of a manifold. In fact, if only (a) and (b) are satisfied, the family (Ui'oi)iEA can be extended in a unique way to a larger family of open charts such that (a), (b) and (c) are all fulfilled. This is easily seen by defining the larger family as the set of all open charts (V,O) on M satisfying: (i) O(V) is an open set in Rn; (ii) for each i E A,

is a diffeamrphisn of (V n U ) onto ¢i(V n Ui). i

An analytic structure of dimension n is defined in a similar fashion. In (b) we just replace "differentiable" by "analytic". In this case M is called an analytic manifold.

In order to define a complex manifold of (complex) dimension n we replace TO in the definition of differentiable manifold by n-dimensional complex number space Cn. The condition (b) is replaced by the condition that the n coordinates of

should be holomorphic

functions of the coordinates of p.

Given two manifolds M and M', a mapping f : M

> M' is said

to be differentiable (or C differentiable), if for every chart

3

of M' such that f(Ui) c v., the

(U.,¢i) of M and every chart mapping 1P

f-¢71 of ¢i(Ui) into

is differentiable. A differen-

tiable function on M is a differentiable mapping of M into R. If -M is

analytic, the function f on M is said to be analytic if for every chart is an analytic function on g1(Ui). The

(Ui,oi), the function

definition of a hoZomorphic (or complex analytic) mapping or function is similar.

Let M and N be two manifolds of dimension n and m, respectively. Let (Ui4i)ieA and (Va't'a)arA' be collections of open charts on bi and

N, respectively. For i e A, a E A', let 0i X a denote the mapping

> (Oi(p)'Yq))

(p,q)

of the product set Ui x Va into Rn'. Then

the collection (Ui X Va'4i X 'a)ieA aeA' of open charts on the product space M X N satisfies (a) and (b), and hence M X N can be turned into a manifold the product of M and N. By a differentiable curve in a manifold M, vie shall mean a diffe-

rentiable mapping of a closed interval [a,b] of R into M. We shall now define a tangent vector (or singly a vector) at a point p of M. Let 51(p) be the algebra of differentiable functions defined in a neighborhood of p. Let T(t) (a < t < b) be a curve such that T(t0)=:p.

The vector tangent to the curve T(t) at p is a mapping X : 3(p) -> R defined by

Xf = (df(T(t))/dt) t

.

0 In other words, Xf is the derivative of f in the direction of the curve T(t) at t = t0. The vector X satisfies the following conditions: (1) X is a linear mapping of 3(p) into R; (2) X(fg) = (Xf)g(p) + f(p)(Xg)

for f,g a $(P).

The set of mappings X of 3(p) into R satisfying the preceding two conditions forms a real vector space. Let x1...... n be local coordinates in a coordinate neighborhood U of p. For each i, (9/8xi)p is a mapping of gy(p) into R which satisfies (1) and (2). Given any curve

x(t) with p = T(t0), let xl = T'(t), i = 1,...,n, be its equations in terms of the local coordinates x1,...,x . Then

4

(df(-r(t))/dt) t

=

0

(af/axi)p(dri(t)/dt)t0,

i

which proves that every vector at p is a linear combination of

(a/axl)pn)p .

Conversely, given a linear combination

EE1(a/axl)p, consider the curve defined by

x1 = x1(p) + &it,

i = 1,...,n.

Then the vector tangent to this curve at t = 0 is EEi(a/axi)p. If vie

assume EE (a/axl)

= 0, then 0 =

for j = 1,...,n.

)p,...,(a/ax0)p

are linearly independent and hence

Therefore, (s/ax

these form a basis of the set of vectors at p. The set of tangent vectors at p, denoted by Tp(M), is called the tangent space of M at

p. The n-tuple of numbers E1,...,

are components of the vectors

W(a/axl)p with rrnspect to the local coordinates x,...,xn. We notice that on a Coo differentiable manifold M the tangent space TT(M) coincides with the space of -X : j(p)

> R satisfying

the conditions (1) and (2) above. A vector field X on a manifold M is an assignment of a vector Xp to each point p of M. If f is a differentiable function on M, then Xf is a function on M defined by (Xf)(p) = XP f. A vector field X is

said to be differentiable if Xf is differentiable for every differentiable function f. In terms of local coordinates x1,...,x", X may be

expressed by X =

where

E1

are functions defined in the

coordinate neighborhood, called the components of X with respect to xl,...,xn. X is differentiable if and only if its components E1 are

differentiable.

We denote by 3(M) the set of all differentiable vector fields on M. From now on we shall consider mainly manifolds of class C7, mappings

of class C and vector fields of class C. If X and Y are vector fields, define the bracket [X,Y] as a mapping from the ring of functions on M into itself by

ro

[X,Y]f = X(Yf) - Y(Xf).

Let X = E4i(a/axi) and Y = RFJ(8/axj). Then

[X,Y]f = I

i/axd)

i,j

- r1dmi/axj)(af/axi).

This means that [X,Y) is a vector field with carponents r (0(arti/axi) - 'nd(ai/axi)), i = 1,...,n. With respect to this bracket operation, 3(M) is a Lie algebra over R. For any vector fields X, Y and Z, we have the Jacobi identity:

[[X,YI,ZI + [[Y,ZI,XI + [[Z,X],Y1 = 0.

We may also regard J KM) as a module over the algebra I(M) of differen-

tiable functions on M as follows. If f is a function and X is a vector field on M, then fX is a vector field on M defined by (fX)p = f(p)Xp

for p c M. We also have

[fX,gY] = fg[X,Y] + f(Xg)Y - g(Yf)X.

Let T**(M) be the dual space of the tangent space p(M) of M at

p. An element of p(M) is called a covector at p. An assignment of a covector at each point p is called a 1-form (differential form of degree 1). For each function for M, the total differential (df) p of f

at p is defined by

<(df)pX> = Xf

for X e Tp(M),

where < , > denotes the value of-the first entry on-the second entry

as-a linear functional on T (M). Met (U;x) be a local coordinate

p

system at'p.'Then (dxl)p,...,(dxn)

form a basis for T*(M). They form

the dual basis of the basis (a/ax )p,...,(a/axn)p for Tp(M). In a

neighborhood of p, every 1-form w can be uniquely written as

6

w = fjdx-,

where f. are functions in U and are called the componenta of w with respect to xl,...,xn. The 1-form w is called differentiable if fi are differentiable. This condition is independent of the choice of a local coordinate systen. We shall only consider differentiable 1-forms. A 1-form w can be defined also as an 5F(M)-linear mapping of the 1(M)-module aE(M) into 3(M). The two definitions are related by

(w(X))p = <wp,X >,

X e 3:(M).

Let A be a commutative ring with identity element and E1,...,E5 be A-modules. Then E1 x ... x Ea is also an A-module. A mapping f

: E1 x ... x Es

> F, where F is an A-module, is said to be

A-maltilinear if it is A-linear in each argument. The set of all Arrultilinear mappings of E1 x ... x Es into F is again A-module. Suppose

that all the facters Ei coincide. The A-multilinear mapping is called

a:ternate if f(X1,...,Xs) = 0 whenever at least two i coincide. Let Tr denote the 9{M)-module of all 5t(M)-nnl.tilinear mapping of

3E(M) * x ... x f(M)* x

(M) x ... x 36(M)

(((M)* r times, ((M) s times)

into SM), where .(M)* is the dual 9(M)-module of 3E(M). We put 10 = Tr, TS = Ts and T- = 3r(M). A tensor field K on M of type (r,s)

is an element of T. This tensor field K is said to be contravariant of degree r, covariant of degree s. In particular, the tensor fields of type (0,0), (1,0) and (0,1) on M are just the differentiable functions on M, the vector fields and the 1-fours on M, respectively. If p is a point in M, we define Tr(p) as the set of all R-multilinear mappings of

T

T;(M) x .., x T**(M) x Tp(M) x ... x Tp(M) (TT*(M) r times, Tp(M) s times)

into R. The set TS (p) is a vector spece over R and is nothing but the

tensor product

Tp(M) 0 ... 0 Tp(M) 0 T**(M) 0 ... 0 p(M) (Tp(M) r times, T*p(M) s times)

or otherwise written

TS (P) = OrTp(M) 0 SgT* (M)

We also put TO(p) = R. Let 4j1,...,xn} be a system of coordinates valid on an open

neighborhood U of p. Then there exist vector fields X1,...d and 1-forms w1,...,wn on M and an open neighborhood V of p, p e V c U

such that on V

Xi = (a/axi),

w3(Xi) = dij

(1 < i,j < n).

On V we put

Zi = If ikXk'

ei = igjlwl

where fik, gjl E g(V). Then we have, for q E V, 1 r We ,...,e ,Z1,...,Zs)(q)

n

_

I

f

g11 "grl 1j=1,k 1 1 r lk1

l

..fsk K(w

s

lr

Xkl,.,Xk )(q) s

This shows that K(91,...,er,Z1,..., s)(p) = 0 if some 6J or some Zi

8

vanishes at p. We can therefore define an element I £ TS (p) by p((9p;...,(Ar)P(Zl)P...,(Zs)P) = K(91,...,Or,Z1,...,Zs)(P).

Thus the tensor field K gives rise to a family Kp, p e M, where

p

e TS (p). If

P

= 0 for all p, then K=O.

The element

P

depends

differentiably on p in the sense that if V is a coordinate neighbor-

hood of p and

(q a V) is expressed as above in terms of basis for q 3E(V)* and 3E(V), then the coefficients are differentiable functions on

V. On the other hand, if there is a rule p

> Kp which to each

p e M assigns a member K(p) of TS(p) in a differentiable manner, then there exists a tensor field K of type (r,s) such that KP = K(p) for all p E M. In the case when M is analytic it is clear how to define analyticity of a tensor field K.

Let T denote the direct sum of the RM)-nodules TS,

M T =

E T.

r,s O

Similarly, if p E M we consider

T(p) =

I

Tr(p).

r,s=O An element of T is of the form Er sKs, where KS a TS are zero except for a finite number of them at each p E M. The vector spece T(p) can be turned into an associative algebra over R as follows: Let

a=el0 ... 0er0 fl0... ® fs, b=ei® ... 0el0V 0... 0f'd where ei, e are members of a basis for Tp(M), fk, f'1 are members of a dual basis for T*p(M). Then a 0 b is defined by

a ®b = e1 0 ... 0 er 0 ej 0 ... 0 e ®f1 ® ... ® fs ® f' 1 ® ... ® f'd We put a 0 1 = a, 1 0 b = b and extend the operation (a,b) -> a 0 b to a bilinear mapping of T(p) x T(p) into T(p). Then T(p) is an

associative algebra over R. The multiplication in T(p) is independent of the choice of basis.

The tensor product 0 in T is defined as the (K,L)

(M)-bilinear mapping

>K 0 L of T x T into T such that

K C TS, L e Td.

(K 0 L)p = K p 0 LP,

This turnes the IM-module T into a ring satisfying

f(K0L)=fK0L=K0IL for f E 5(M), K,L E T. In other words, T is an associative algebra over Y(M). The algebras T and T(p) are called the mixed tensor algebras over M and Tp(M) respectively. The submodules cc

are subalgebras of T and the subspaces

T*(p) =

I Tr(p),

T*(p) =

r=0

I T (p)

s=

are subalgebras of T(p).

We now define the notion of contraction. Now let r, s be two integers > 1, and let i, j be integers such that 1 < i < r, 1 < j < s.

Consider the R-linear mapping

TS (p)

> Ts_1(p) defined by

...®fs) = <ei,fj>(el®..ei..®er®fl..fd..®fs),

where {ek} is a basis of Tp(M), {f1) is the dual basis of T*(M).

(The symbol " over a letter means that the letter is missing.) We note that

is independent of the choice of basis. There exists now

a unique 3(M)-linear mapping

C

. TS

(Ci w)p = Ci(Kp)

> T1-1 such that

10 for all K e Tr and all p e M. This mapping satisfies the relation

C(X1e...@X @j 10 ...OWS) _ <X1,wj>(X1e..Xi..e roj

..wJ..®WS)

for all X,,...,Xr a T1, wl,...,ws a T1. The mapping

is called the

contraction of the i-th contravariant index and the j-th covariant index.

For the basis {ei} for Tp(M) and the dual basis {fS} for Tp*(M),

every tensor K of type (r,s) can be expressed uniquely as K..1..1r

Kf-

Jle...ef is,

e e...0e of J1..js

£

11

lr

r where K.

J1--Js

are called the components of K with respect to the above

basis. In terms of components, the contraction Ci is represented by (CiK)i1..ir-1 - £Kil..k..ir-1 J

.. k..Js-19

J1..Js-1

k J1

where the superscript k appears at the i-th position and the subscript k appears at the j-th position. Let AT**(M) be the exterior algebra over T**(M). An r-form w is an

assignment of an element of degree r in AT*(M) to each point p of M. I P In terms of a local coordinate system x,...,x, n r-form w can be

expressed uniquely as 11

w =

f

r

di

i l..ir

...

dx lr

The r-form w is said to be differentiable if the components f. il..ir

are all differentiable. By an r-form we shall mean a differentiable r-fotm. An r-form w can be defined also as a skew-symmetric r-linear mapping over 3(M) of 3(M) x ... x .(M) (r times) into ,(M). The two definitions are related as follows. If w1,...,wr are 1-forms and

11

X1,...,Xr are vector fields, then (win ... A wr)(Xi,...,Xr) is 1/r! times the determinant of the matrix (wi(7))j,k=1,...,r of degree r. Let Dr = Dr(M) be the totality of r-forms on M for each r = 0,1,.. .,n. Then DO = RM). Each Dr is a real vector space and can be also

considered as an 3(M)-module: for f c 3(M) and w e Dr, fw is an rform defined by (fw)p = f(p)wp, p c M. We set n

D = D(M) =

E Dr(M).

r=0 With respect to the exterior product, D(M) forms an algebra over R. Exterior differentiation d can be characterized as follows: (1) d is an R-linear mapping of D(M) into itself such that

d(Dr) C Dr+1; (2) For f E D0, df is the total differential;

(3) If wi a Dr and w2 a Ds, then

d(wi n w2) = dwi A w2 + (_,)r

i A dw2;

(4) d2 = 0.

Let V be an m-dimensional real vector space. We define a V-valued r-form w on M as an assignment to each point p of M a skew--symmetric

r-linear mapping of Tp(M) x ... x Tp(M) (r times) into V. If we take a basis e1,...,em for V, we can write w uniquely as w = E3wJej, where

wi are usual r-forms on M. If wJ are all differentiable, w is said to be differentiable. The exterior derivative dw is defined to be Ei dw?ej, which is a V-valued (r+l)-form..

Let f be a mapping of a manifold M into another manifold M'. We define the differential f* : TT(M)

> Tf(p)(M') at p of f as

follows: For each X c Tp(M), choose a curve T(t) in M such that X is

the vector tangent to T(t) at p = T(t0). Then f*(X) is the vector tangent to the curve f(T(t)) at f(p) = f(T(t0)). If g is a function in a neighborhood of f(p), then (f*(X))g = X(g-f). When it is necessary

to specify the point p, we write (f*)p. Then there is no danger of confusion, we may singly write f instead of f*. The transpose of (f*) p

is a linear mapping of T (p)(M') into (M). For any r-form w' on M',

12

we define an r-form f*w' on M by

(f*w')(X1,...,Xr) = w'(f, Cr,...,f*Xr),

X1,.... Xr E TT(M).

We also have d(f *w') = f*(dm'), that is, d commutes with f*. A mapping f of M into M' is said to be of rank r at p E M if the dimension of f*(Tp(M)) is r. If the rank of f at p is equal to n = dim M, (f*)p is injective and dim M < dim M'. If the rank of f at p is equal to n' = dim M', (f*)p is surjective and dim M > dim M'. We notice that the following proposition is hold (see Chevalley [ ;pp. 79-80)).

PROPOSITION 1.1, Let f be a mapping of M into M' and p be a point of M.

(1) If (f*)

is injective, there exist a local coordinate system P {U;xl} of p and a local coordinate system {U';ya} of f(p) 'such that yl(f(q)) = xl(q)

for q E U, i = 1,...,n.

In particular, f is a homeowrphism of U onto f(U); (2) If (f*)

P

is surjective, there exist a local coordinate system

{U;x'} of p and a local coordinate system {U';ya} of f(p) such that ya(f(q)) = xa(q) In particular, f : U

for q e U, a

> M' is open;

(3) If (f*)p is a linear isomorphism of Tp(M) onto Tf(p)(M'), then f defines a heneanorphisn of a neighborhood U of p onto a neighborhood U' of f(p) and its inverse f -l : U'

> U is also

differentiable.

A mapping f of M into M' is called an immersion if (f*) p is

injective for every point p of M. Then M is called an immersed submanifold of M'. If the immersion f is injective, it is called an imbedding of M into M'. We say then that M (or the image f(M)) is an imbedded submanifold of M'. When there is no danger of confusion, we say simply that M is a submanifold of M' instead of that M is an immersed sutranifold of M' or an imbedded sutmanifold of M'.

13

A homeororphism f of M onto M' is called a diffeomorphism if both

f and f-1 are differentiable. A diffeomorphism of M onto itself is called a differentiable transformation (or simply a transformation) of M. A transformation 4) of M induces an autemorphism 0* of the algebra

D(M) of differential forms on M and, in particular, an autemorphisn of the algebra 9(M): (4)*f)(p) = f(4)(p)) for f E 3(M), p E M. From this we

Q),

have an autamorphism 0* of the Lie algebra 3E(M) by (4) X)p = where X E 3r(M), p = 4)(q). They are related by 4>*((4,,KX)f) = X(¢*f).

Although any mapping 0 of M into M' carries a differential form w' on M' into a differential form 4>*(w') on M, 0 does not send a vector field

on M into a vector field on M' in general. We say that a vector field

X and

X on M is ¢-related to X' on M' if (¢*)pXp =

Y are 4)-related to X' and Y' respectively, then [X,Y] is 4)-related to [X',Y'].

A 1-parameter group of (differentiable) transformations of M is a mapping of R x M into M, (t,p) E R x M

> ¢t(p) E M, which satis-

fies the following conditions: (1) For each t c R, 4t : p

> 0t(p) is a transformation of M;

(2) For all t, s c R and p e M, 0t+s(p)

= 0t(0s(p)).

A curve T(t) in M is called an integral curve of a vector field X if the vector XT(t) is tangent to T(t) for every t. For any point p of M, there is a unique integral curve

-r (t) of a vector field X,

defined for Itl < e for some E > 0, such that p = T(0). Each 1-parameter group of transformations 0t induces a vector field X as follows. For any point p E M, Xp is the vector tangent to the curve T(t) = ot(p), called the orbit of p, at p = 4)0(p). The

orbit t(p) is an integral curve of X starting at p. A local 1-parameter group of local transformations can be defined in the same way, except that 4t(p) is defined only for t in a neighborhood of 0 and p in an open set of M. A local 1-parameter group of local transformations defined on IE x U is a mapping of IE x U into M which satisfies the following conditions:

14

(1') For each t c IE, t : p

> rpt(p) is a diffeamorphism

of U onto the open set ot(U) of M; (2') If t, s, t+s e IC and if p, 0s(p) E U, then

Ot+sM=Ot(Os(p)).

As in the case of a 1-parameter group of transformations,

4t induces a vector field X defined on U. We also have the converse, that

is, we can prove the following: Let X be a vector field on M. For each point p0 of M, there exist a neighborhood U of p0, a positive number E and a local 1-parameter group of transformations Ot : U

> M,

t E IE, which induces the given X (see Kobayashi-Nanizu [1;p.13]). We say that X generates a local 1-parameter group of local transformations ¢t in a neighborhood of p0. If there exists a (global) 1-parameter group of transformations of M which induces X, then we say that X is complete. If ot(p) is defined on IE x M for some e, then X is canplete. Thus, if M is compact, every vector field X is complete.

We notice here that the differential * of a transformation p of M gives a linear isomorphism of T c

(M) onto TT(M). This linear (x)

isomorphism can be extended to an isomorphisn of the tensor algebra T(¢-1(x)) onto the tensor algebra T(x), which we denote by the same 4,.

Given a tensor field K, we define a tensor field K by

(OK)x = q(K -1 4,

).

(x)

In this way, every transformation 0 of M induces an algebra autcmorphism of T which preserves type and commutes with contractions. Let X be a vector field on M and 0t a global 1-parameter group of

transformations of M. For each t, 0t is an autommrphism of the algebra

T. For any tensor field K on M, we set (LXK)x = lim x t-+o t[

-

The mapping LX of T into itself which sends K into LxK is called the Lie differentiation with respect to X. It will be no difficulty in modifying the definition of Lie differentiation when X is not complete,

15

that is, when of is a local I -parameter group of transformations gene-

rated by X. For the tie differentiation we have PROPOSITION 1.2, Lie differentiation LX with respect to a vector field X satisfies the following conditions:

(a) lx is a derivative of T, that is, it is linear and satisfies LX(K 0 K') _ (INK) 0 K' + K 0 (LXK'),

K, K' c T;

(b) LX is type-preserving: LX(TS)C TS; (C) LX cornm tes with every contraction of a tensor field;

(d) LXf = Xf, f c

(M);

(e) LXY = [X,Y], Y C T(M).

Proof. It is clear LX is linear. Moreover, we have LX(K ® K') = lim t[K 0 K' - 0t(K 0 K')] t-+O

= lim t[K 0 K' - (0tK) 0 K'] 4 l

t[($tK) ® K' -

_ (li t[K -

tK) 0 (4tK'))

K'

+ limi(4tK) 0 (t[K' t-*O

tK')])

= (LXK) ® K' + K 0 (LXK').

Since 0t preserves type and camnites with contractions, so does L. If f is a function on M, then

(I f)(x) = lim t[f(x) - f(ot1(x)] = -lim t[f(ft'k) - f(x)].

We see that

Ot1

= ¢-t is a local 1-parameter group of local transfor-

mations generated by -X, and hence we have LXf = -(-X)f = Xf. Let f be a function on M. We can take a function gt such that f + t.gt and go = Xf (see Kobayashi-Namizu [1;p.15]). We put

p(t) = 4t1(p). Then

16

((¢t)*Y)pf =

(Yf)p(t)

+ t(Ygt)P(t)

and

lim 1[Y - ( )*Y] f = lim 1[(Yf)

t-*O t

t

p

t

p

- (Yf

p(t) ]

- lim (Y

lim

gt p(t) )

= Xp(Yf) - YpgO = [X,Y]pf. Therefore we have (e).

QED.

By a derivative of T, we shall mean a mapping of T into itself which satisfies conditions (a), (b) and (c) of Proposition 1.2. We now prepare some basic formulas for latter use. For the proof see Kobayashi-Nomizu [1].

Let X and Y be vector fields on M. Then

LMY] _ [LX,LY]. Let K be a tensor field of type (1,r). Then we have r

(LXK)(Y1,...,Yr) _ [X,K(Y1,...,Yr)] -

K(Y1,...,[X,Yi],...,Yr) 1=1

We notice that a tensor field K is invariant by ¢t for every t if and only if LXK = 0.

A derivation (resp. skew-derivation) of D(M) is a linear mapping A of D(M) into itself which satisfies

A(w A W) = Aw A w' + w A Awl,

w, w` a D(M)

(resp. A(w A w') = Aw A w' + (-1)rw A Awl, w e Dr(M), w' a D(M)).

A derivation or a skew-derivation A of D(M) is said to be of degree k if it maps Dr(M) into Dr+k(M) for every r. The exterior differentiation d is a. skew-derivation of degree 1 and the Lie differentiation LX is a

derivation of degree 0. Indeed, the formula

17

r wall ...,[X,Yi],...,Yr)

(LXw)(Y1,.... Yr) = X(w(Y1,...,Yr)) i=1

implies that I(DT(M))C Dr(M) and, for w, w' E D(M),

LX(w Awl) =y A w' + w A I w' . Moreover, I% camutes with d. If w is an r-form, then

r (dw)(XO,X1,...,Xr) = r+l.I (-1)i'Xi(w(Xa,...'Xi,...,Xr)

+ rl

(-1)l+jw([Xi,

Oci<j
where " means that the term is omitted. If w is a 1-form, then

2(d,))(X,Y) = X(w(Y)) - Y(w(X)) - w([X,Y])

If w is a 2-form, then

3(dw)(X,Y,Z) = X(w(Y,Z)) + Y(w(Z,X)) + Z(w(X,Y))

- w([X,Y],Z) - w([Y,Z],X) - w([Z,X],Y)

Remark. Sometimes, we define dw without the coefficients 1/(r+l) of the right hand side of the above equation to simplify the notation. In the next place, we give the definition of a distribution on a manifold. A distribution S of dimension r on U is an assignment to each point p of M an r-dimensional subspace SP of TT(M). S is called ..differentiable if every point p has a neighborhood U and r differen-

tiable vector fields, say, Xl,...,Xr, which form a basis of Sq at every q c U. The set X1,...,Xr is called a local basis for S in U. A vector field X is said to belong to S if Xp c Sp for all p e M. S is called invoZutive if [X,Y] e S for any X, Y c S.-By a distribution we shall always mean a differentiable distribution.

18

A connected suhmanifold N of M is called an

integral manifold of

S if f*(TT(N)) = Sp for all p e N, f being the imbedding of N into M.

If there is no other integral manifold of S which contains N, N is called a maximal integral manifold of S. The classical theorem of Frobenius can be formulated as follows (cf. Chevalley [1;p.94]). PROPOSITION 1.3. Let S be an involutive distribution on a manifold M. Through every point p e M, there passes a unique maximal integral manifold N(p) of S. Any integral manifold through p is an open submanifold of N(p).

2. CONNECTIONS AND COVARIANT DIFFERENTIATIONS

First of all we give the definition of affine connection on a

manifold M. Definition. An affine connection on a manifold M is a rule V which assigns to each X e t(M) a linear napping VX of the vector space

(M)

into itself satisfying the following two conditions: (1)

VfX+gY = fVX + gVY;

(2)

Vx(fY) = fVXY + (Xf)Y

for f, g e g(M), X, Y e Y(M). The operator VX is called the covariant differentiation with respect to X.

LE A 2.1. Suppose M has the affine connection V and let U be an open sutmanifold of M. Let X, Y e VM). If X or Y vanishes identically

on U, then so does YProof. Suppose Y vanishes identically on U. Let p e U and g e 9(M). To prove that ((VXY)g)(p) = 0, we select f e 3:(M) such that

f(p) = 0 and f = 1 outside U. Then fY = Y and

(VXY)g = (VX(fY))g = (Xf)(Yg) + f(VXY)g

which vanishes at p. The statement about X follows similarly.

W.

19

By Lemma 2.1, an affine connection V on M induces an affine connection on an arbitrary open su}manifold U of M, which will be denoted by the same V. In fact, let X, Y be two vector fields on U. For each p e U there exist vector fields X', Y' on M which agree with X and Y in an.open neighborhood V of p. We then put (VXY)q = (VXOY')q for q e V. By Iemcm 2.1, the right-hand side of this equation is independent of the choice of X', Y'. It follows immediately that the > VX (X a .(U)) is an affine connection on U.

rule V : X

In particular, suppose U is a coordinate neughborhood with coordinate system {xl,...,xn}. To simplyfy the notation, we write Vi . We define the functions rid on U bg

instead of V a

axi

V.( a) = jrk.

(2.1)

l axj

k 3-j axk

If {y1,...,y } is another coordinate system valid on U, we get another set of functions Trab by

b) _ L

Va(

c

ay

c.

ay

From (1) and (2) in the definition of affine connection we find easily

(2.2)

Tab =

axl

axJ ay'

i,J.kaya ayb axk

azxl rk

ij

i ayayb axd

On the other hand, suppose that there is given an open covering of M by open coordinate neighborhoods U and in each U a system of functions rl. such that (2.2) holds whenever two of these neighborhoods overlap. Then we can define V. by (2.1) and thus we get an affine connection V in each U. We define an affine connection V on M as follows: Let X, Y c

(M) and p c M. If U is a coordinate neighborhood containing

p, let

(VXY)p (VX'YI)p

if X' and Y' are the vector fields on U induced by X and Y, respectively. Then V is an affine connection on M which on each U induces the

2D

connection

.

Let {x1,...,x"} be a coordinate system valid on U of p. On the set U we have X = Efi(a/axl) where fi a $(U) and fi(p) = 0 (1
so does YDefinition. Suppose V is an affine connection on M and f is a diffeemorphism of M. A new af fine connection- VI can be defined on M

by

V-XY = f,k'(Vf

*Y),

X, Y e 3(M).

The affine connection V is called invariant under f if V' = V. In this case f is called an affine transformation of M. Similarly we can define an affine transformation of one manifold onto another. Let y

:

t

> Y(t) (t a I) be a curve in M defined on an open

interval I e R. Differentiation with respect to the parameter will often denoted by a dot ('). Let J be a closed subinterval of I such that yJ : t

> Y(t) (t c J) has no double points and such that y(J)

is contained in a coordinate neighborhood U. We need the following LEI4IAA 2.3. Let g(t) be a differentiable function on an open

interval containing J. Then there exists a function G e 5(M) such that

G(y(t)) = g(t)

(t E J).

We put X(t) = y(t) (t E I). By Lemma 2.3 it is easy to see that there exist vector fields X, Y e 3(M) such that (Y(t) being an associated vector to t e I)

Xy(t) = X(t),

Y(t) = Y(t)

(t E J).

The family Y(t) (t e J) is said to be parallel with respect to yJ

21

(or parallel along y ) if

i

(VXY)y(t) = 0

(2.3)

for all t c J.

We show that this definition is independent of the choice of X and Y. We express (2.3) in the coordinates {x1,...,xn}. Then X and Y are given

by

X

a

Y = jYj i

axl

j

on U.

axj

For simplicity we put x'(t) = x'(y(t)), X1(t) = X1(y(t)) and Y1(t) _

Since

Y1(y(t)), t e J, 1 < i < n. Then X1(t) = .

a

y =

`

+

ak

I x'Yjrlj

k i

on U,

ax

i,i

ax

we obtain

dyk

(2.4)

k +ijIrijd =0 i

(t EJ).

This equation involves X and Y only through their values on the curvae. Consequently, condition (2.3) for parallelism is independent of the choice of X and Y. It is well now obvious how to define parallelism with respect to any finite curve segment yJ and finally with respect to the entire curve Y. Definition. Let y : t

> y(t) (t E I) be a curve in M. The

curve y is called a geodesic if the vector field X(t) = Y(t) defined along y is parallel with respect to y, that is, VXX = 0 for all t. A geodesic y is called maximal if it is not a proper restriction of any geodesic.

From (2.4), if yJ is a geodesic segment, then z (2.5)

dt

+

i

E I a = 0 isj

(t E J).

0 If we change the parameter on the geodesic and put t = f(s), (f'(s) 0

0), then we get a new curve s

yJ(f(s)). This curve is a geodesic

if and only if f is a linear function, as (2.5) shows. We notice that the two following propositions hold. PROPOSITION 2.1. Let p and q be two points in M and y a curve segment from p to q. The parallelism T with respect to y induces an

isomorphism p(M) onto Tq(M). PROPOSITION 2.2. Let M be a manifold with an affine connection.

Let p be any point in M and let X 0 0 in p(M). Then there exists a unique maximal geodesic y : t

> y(t) in M such that y(O) = p and

y(0) = X.

The geodesic with properties in Proposition 2.2 will be denoted by -;X. If X = 0, we put yX(t) = p for all t E R. We then have (cf.

Helgason [1])

THEOREM 2.1. Let M be a manifold with an affine connection. Let p be any point in M. Then there exists an open neighborhood NO of 0 in TT(M) and an open neighborhood Np of p in M such that the mapping > yX(1) is a diffecmorphism of NO onto Np.

X

The mapping X

> yX(1) descrived in Theorem 2.1 is called the

exponential mapping at p and will be denoted by Exp (or Expp). Let M be a manifold with an affine connection and p be a point in M. An open neighborhood NO of the origin TT(M) is said to be normal if

(1) Exp is a diffeomorphisn of NO onto an open neighborhood Np

:

of p in M; (2) if X E N0, 0 < t < 1, then tX E N0. A neighborhood NP of p in M is called a normal neighborhood of p if Np = ExpNO, where NO is a normal neighborhood of 0 in TT(M). Assuming this to be the case, and letting X1,.... Xn denote some basis

of TT(M), the inverse mapping

Exp(a1X1 + ... + anXn)

of

P

>

(al,...,an)

into Rn is called a system of normal coordinates at p.

0 We now give a useful refinement of Theorem 2.1 (cf. Helgason [1]). THEOREM 2.2. Let M be a manifold with an affine connection. Then

each point p e M has a normal neighborhood p which is a normal neighborhood of each of its points.

In the next place, we shall define covariant derivatives of arbitrary tensor fields. We first prove THEOREM 2.3. Let M be a manifold with an affine connection. Let

p c M and let X, Y be two vector fields on M. Assume p # 0. Let s

> O(s) be an integral curve of X through p = 4(O) and Tt the

parallel translation from p to 0(t) with respect to the curve 0. Then

(VXY)p = lim s-+0

s

Ts1Y,(s) - Yp).

Proof. Consider a fixed s > 0 and the family Zi(t) (0 which is parallel with respect to the curve

such that Z¢(0)

s)

= TsIYO(s).

We can put

Zi(t) - yzi(t)aXi)O(t)'

YO(t) -

lY1 (t) (XO(t)

and we have the relations

0

Zk(t) ±

(0 < t < s),

i,j

Zk(s) = Yk(s)

(1 < k < n).

By the mean value theorem Zk(s) = 2k(O) + sZk(t*) for a suitable number t* between 0 and s. Hence the k-th component of (1/s)(TSlY

(s)

YP) is

I Zk(0) - Yk(0))

= s Zk(s) =

- sZk(t*) - Yk(0))

s Yk(s) - Yk(0)).

24 As s

> 0 this expression has the limit

dyk +

ds

lk dxi Yj i ,J ijds

Let this last expression be denoted by Ak. It was shown earlier that

(OXY)p = JAk(ak)p. k ax This proves the theorem.

QED.

By using Theorem 2.3 it is now possible to define covariant derivatives of arbitrary tensor fields.

Let p and q be two points of M and y a curve sent in M from p to q. Let T be the parallel translation along y. If F E T*(M), we

p

define

for each A e Tq(M). If K is

e T**(M) by

a tensor field on M of type (r,s), we define

E Tr(q) by

s) = Kp(T-1F1,...,T-lFr,r-'Ai,...,T-lAs)

for Ai a Tq(M) and F3 - a T**(M). Let X e T(M) and p be any point in M

where p # 0. With the notation of Theorem 2.3 we put

(2.6)

(OXK)p = lim s-0

sTsKG(s)

- c).

For each point q e M where q = 0 we put (OXK)q = 0 in accordance with Learn 2.2. For a functionf on M we put

(vxf)p = lim 5 f(¢(s)) - f(p)),

if Xp # 0, otherwise we put (7f)p = 0. Then we have OXf = Xf. Finally OX is extended to a linear crapping of T into itself.

THEOREM 2.4. The operator OX has the following properties: (1) aX is a derivation of the mixed tensor algebra T; (2) VX preserves type of tensors; (3) V, erimiutes with contractions.

25 We now give the structure equations of Cartan. For this purpose we define the torsion tensor field and the curvature tensor field.

on a manifold M with an affine connection, we put

T(X,Y) = OXY - DI,X - [X,Y],

R(X,Y) = V)Vy - VYVx - V[X,Y],

where X, Y are vector fields on M. Note that T(X,Y) _ -T(Y,X) and R(X,Y) = -R(Y,X). It is easy to verify that T(fX,gY) = fgT(X,Y) and R(fX,gY)hZ = fghR(X,Y)Z for all f, g, h c > w(T(X,Y)) is an

mapping (w,X,Y)

.(M), X, Y, Z E Z(M). The

(M)-multilinear mapping

3E(M)* x X(M) x 3E(M) into $(M) and is an element of T2(M). This element

is called thetorsion tensor field and is also denoted by T. Similarly, > w(R(X,Y)Z) is an , (M)-multilinear mapping

the mapping (w,Z,X,Y)

.X(M)* x a°(M) x 3M x 3E(M) into ¢(M) and therefore is an element of T31 W. This element is called the curvature tensor field and is denoted by R. The tensor fields T and R is of type (1,2) and (1,3)' respectively. Let p be a point of M and suppose X1,.... Xn is a basis for the

vector fields in some open neighborhood N of p, that is, each vector field X on N can be written as X = EifiX1 where fi e 51(Np). We define the functions I'ij, Tij, Riij on Np by the formulas aX X. ° krijXk, i

T(X ,XJ) _ JTijXk,

k R(Xi,%)Xk = IR

jXl.

Let w', w (1 < i,j < n) be the 1-forms on Np determined by

wl(X.) =

i

wi _ Ir'kiwk k

26

That is, we put

wj(xk) = rki.

Thus the forms w determine the functions rI on Np and thereby the connection V. On the other hand, as the next theorem shows, the forms

w are descrived by the torsion and curvature tensor fields. THEOREM 2.5 (the structure equations of Cartan). T?kwJ

dwl = -Jwl A wJ +

(2.7)

J,k

J J

A W

J

dw = -jwk A w + k,l k

(2.8)

wl

Proof. If we define the functions cjk by [Xj,xk] = EicJkXl

obtain dwl(Xi,Xk)

-'cl

=

As for the right-band side of this equation, we have

T(Xj,Xk) = V

Xk - vXk Xi - [Xj,Xk]

Xj

=

-

ri

i

from which i

Ti

1

i

i

= rik - rkJ - c]k.

We also have

(-iwi A w1)(X,,Xk) = -ji[wi(Xj)wl(Xk) -

_(rk;

-

rr).

i., we ik.

si

Fran these equations we have (2.7).

Similarly, we find

Rjkl = J(rpjr P

- r rlp) + Xlrlj - X1rkj - `klrPj' lj

xlrkj `klrP.7),

-

-wI A mp(Xk,Xl) = -i(rpjrkp - rkjrlp

F From these equations we have (2.8).

QED.

We now define a Riemannian metric on M. It is a tensor field g of type (0,2) which satisfies the following two conditions: (i) It is symmetric: g(X,Y) = g(Y,X) for any X,Y a 35(M);

(ii) It is positive-definite: g(X,X) > 0 for every X e E(M) and g(X,X) = 0 if and only if X = 0.

A manifold M with Riemannian metric g is called a Riemannian manifold. A Rienmannian metric gives rise to an inner product on each

tangent space x(M) to M at x. Let

{x1,...,xn)

be a local coordinate

system in M. The components gij of g with respect to this local coordinate system are given by

gij = g(a/axi,a/axj),

i,j = 1,...,n.

We call gij the covariant components of g. The contravariant components g13 of g are defined by

i,j = 1,...,n.

giJ = g(dx',dx3),

We have then

gij

k_

i

1,

0,

k=i

k#i

where, here and in the sequel, if we need, we use the Einstein convention, that is, repeated indices, one upper index and the other

28

lower index, denotes summation over its range. If X1 are components of

a vector field X With respect to {xl,...,xn}, that is, X =

Xi(a/axi),

then the components Xi of the corresponding covector or the corresponding 1-form are related to X1 by

Xi = gijXJ.

X1 = g1JXj,

The inner product g in the tangent space Tx(M) and in its dual space TA(M) can be extended to an inner product, denoted also by g, in the tensor space Tr at x for each type (r,s). If K and L are tensors at

x of type (r,s) with components 11...1r

i1...i r

K.

and

i1"'is

i1-is

L

with respect to {xl,...,xn}, then the inner product g(K,L)'of K and L is defined to be

g(K,L) = gilkl...gi

j t j t 1 1...g s

k g

rr

ki...k

i 1... i

sK

rLt

J1"'Js 1"' trs

We put

g(K,K)1/2,

IKI =

which is called the length of the tensor field K with respect to g. On a Riemannian manifold M, the arc length of a differentiable curve

y: t

> y(t), a < t < b, is defined by

L(y) = fbg(Y(t),Y(t))1/2dt.

This definition can be generalized to a piecewise differentiable curve in an obvious manner. The distance d(x,y) between two points x and y of M is defined by the infinimum of the lengths of all piecewise differentiable curves joining x and y. Then we have

29

d(x,y) = d(y,x),

d(x,y) + d(y,z) > d(x,z),

d(x,y) > 0

d(x,y) = 0 if and only if x = y.

We can see that the topology defined by the distance function (metric) d is the sane as the manifold topology of M.

THEOREM 2.6. On a Riemannian manifold M there exists one and only one affine connection satisfying the following two conditions: (1) The torsion tensor T vanishes, i.e.,

T(X,Y) = VXY - VYX - [X,Y] = 0; (2) g is parallel, i.e., VXg = 0.

Proof. Existence: Given vector fields X and Y on M, we define OXY by setting

(2.9)

2g(VXY,Z) = Xg(Y,Z) + Yg(X,Z) - Zg(X,Y) + g([X,Y],Z) + g([Z,X],Y) + g(X,[Z,Y])

for any vector field Z on M. Then the mapping (X,Y)

> VXY defines

an affine connection on M. From the above definition of OXY, we have T(X,Y) = 0 and

Xg(Y,Z) = g(VXY,Z) + g(Y,VXZ),

which shows that VXg = 0, that is, V is a metric connection on M. Uniqueness: By a straightforward computation, we can see that, if OXY satisfies VXg = 0 and T(X,Y) = 0, then it satisfies the equation which defines VXY.

QED.

The connection V given by (2.9) is called theRiemannian connection (sometimes called the Levi-Civita connection).

Putting X = a/ax3, Y = a/axl and Z = a/axk in (2.9), the ccapo-

nents jk of the Riemannian connection with respect to a local

30

coordinate system {xl,...,xn} are given by

Ig 1

i(

agi. ax3

ag +

ax

- agk ax

Let M and M' be Riemannian manifolds with Riemannian metrices g and g' respectively. A mapping f : M

> M' is called isometric

at a point x of M if g(X,Y) = g'(f,X,f*Y) for all X,Y a Tx(M). In this case, f* is injective at x, because f,X = 0 implies X = 0. A mapping f which is isometric at every point of M is thus an immersion, which we call an isometric immersion. If, moreover, f is 1

:

1, then -it is

called an isometric imbedding of M into M'. If f maps M 1 : 1 onto M', then f is called an isometry of M onto M'. In this case, the differen-, tial of the isometry f comiutes with the parallel translation. Moreover, if f is an isometric immersion of M into M' and if f(M) is open in M', then the differential of f commutes the parallel translation and every geodesic of M is mapped by f into a geodesic of M'. A Riemannian manifold M or a Rienannian metric g on M is said to be complete if the metric function d is complete, that is, all Cauchy sequences converge. It is well-known that the following conditions on M are equivalent: (1) M is complete;

(2) Every bounded subset of M with respect to d is relatively

compact ; (3) All geodesic are can be extended in two directions indifinitely with respect to the are length. It is also well-known that any two points x and y in M can be joined by a geodesic arc whose length is equal to d(x,y) (Hopf-Rinow [1],

de Rham (2]). We can also see that every compact Riemannian manifold is complete.

31

3. SECTIONAL CURVATURE

Let M be an n-dimensional Riemannian manifold with metric tensor field g. We see that

R(X,Y) = OXOY - VYVx - vMY]

is an endcmorphism of Tx(M). We call this R(X,Y) the (Riemannian)

curvature transformation of T{(M) determined by X and Y. It follows that R is a tensor field of type (1,3) such that R(X,Y) = -R(Y,X), which will be called a (Riemannian) curvature tensor field (or simply, curvature) of type (1,3).

For any vector fields X, Y and Z, using T = 0, we obtain

R(X,Y)Z + R(Y,Z)X + R(Z,X)Y

= VX[Y,Z] - V[Y,Z]X + VY[Z,X] - V[Z,X]Y + vz[x,Yl - VMY]Z = [X,[Y,zll + [Y,[z,X]] + [Z,[X,YI] = 0

by the Jacobi identity. Thus we have

(3.1)

R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0.

We call (3.1), Bianchi's 1st identity. We have then, by a straightforward computation, the following Bianchi's 2nd identity:

(3.2)

(VxR)(Y,Z) + (VyR)(Z,X) + (VzR)(X,Y) = 0.

We now give algebraic preliminaries for a quadrilinear mapping on a real vector space.

Let V be an n-dimensional real vector space and B: V x V x V x V > R a quadrilinear mapping with the following three properties:

32

(a) B(v1,v2,v3,v4) _ -B(v2,vl,v3,v4), (b) B(v1,v2,v,1,v4) _ -B(v1,v2,v4,v3),

(c) B(vl,v2,v3,v4) + B(vl,v3,v4,v2) + B(vl,v4,v2,v3) = 0.

LEMMA 3.1. If B possesses the above three properties, thenit possesses also the following fourth property: (d) B(vi,v2,v3,v4) = B(v3,v4,v1,v2).

Proof. Putting the left hand side of (c) by S(v1,v2,v3,v4), we obtain

0 = S(vi,v2,v3,v4) - S(v2,v3,v4,v1) - S(v3,v4,v1,v2) + S(v4,v1,v2,v3) = B(vl,v2) v3,v4) - B(v2,vl,v3,v4) - B(v30v4,v1,v2)

+ B(v4,v3,vl,v2).

By applying (a) and (b), we have (d).

QED.

LEMMA 3.2. Let B and T be two quadrilinear mappings with the properties (a), (b) and (c). If

B(vi,v2,vi,v2) = T(v1,v2,v1,v2) for all vi, v2 a V,

then B = T.

Proof. We may assume that T = 0; consider B - T and 0 instead of

B and T. We asses therefore that B(vi,v2,vi,v2) = 0 for all vi, v2 of V. We have

0 = ?B(vi,v2+v4,vl,v2+v4) = B(vl,v2,vl,v4).

0 F'om this we obtain

0 = B(v1+v3,v2,vl+v3,v4) = B(vl,v2,v3,v4) + B(v3,v2,v1,v4).

Now, by applying (d) and then (b), we have

0 = B(v1,v2,v3,v4) + B(vi,v4,v3,v2) = B(v1,v2,v3,v4) - B(vl,v4,v2,v3).

Hence,

B(vl,v2,v3,v4) = B(vl,v4)v2,v3).

Replacing v2, v3, v4 by v3, v4, v2, respectively, we obtain

B(vl,v2,v3,v4) = B(v1,v3,v4,v2).

FY-an these equations we obtain

3B(v1,v2,v3,v4) = B(v1,v2,v3,v4) + B(vi,v3,v4,v2) + B(vl,v4,v2,v3),

where the right hand side vanishes by (c). Thus we have

B(vl,v2,v3,v4) = 0.

This proves our assertion.

QED.

Besides a quadrilinear mapping B, we consider an inner product on V, which will be denoted by (

,

). Let p be a plane, that is, 2-dimen-

sional subspace in V and let v1 and v2 be an orthonormal basis for p.

We put K(p) = B(v1,v2,vl,v2),

34 which is independent of the choice of an orthonormal basis for p. PROPOSITION 3.1. If vl, v2 is a basis (not necessarily orthonormal)

of a plane p in V, then K(p) = B(vl,v2,vl,v2)/[(vl,vl)(v2,v2)-(vl,v2)2].

Proof. We obtain the formula making use of the following orthonormal basis for p:

vl a[(vl,vl)v2 - (vl,v2)vl], (v1,v1)1/2 where we have put

a =

[(vl,v1)((vl,vl)(v2,v2)-(vl,v2)2)]1/2.

QED.

We put

R1(v1,v2,v3,v4) _ (vi,v3)(v2,v4) - (v2,v3)(v4,v1).

Then R1 is a quadrilinear mapping having the properties (a), (b) and (c). If vi, v2 is an orthonormal basis for p, we obtain

K1(p) = R1(vl,v2,vl,v2) = I.

PROPOSITION 3.2. Let B be a quadrilinear mapping with properties (a), (b) and (c). If K(p) = c for all planes p, then B = cRl.

Proof. By Proposition 3.1 we have

B(vi,v2,v1,v2) = cRl(vl,v2,vl,v2)

for all v1, v2 c V. Applying Lame 3.2 to B and cRl, we conclude

B = cRl. We apply these results to the tangent space TX(M) of an n-dinensional Riemannian manifold M.

QED.

36

The Riemannian curvature tensor field of covariant degree 4 (of type (0,4)) of M, which denoted also R, is defined by

R(X1,X2,X3,X4) = g(R(X3,X4)X2,X1), Xi a T,(M), i=1,..,4.

Obviously, R is a quadrilinear mapping Tx(M) x T{(M) x TT(M) x X(M)

> R at each x e M with properties (a), (b) and (c). For each plane p in the tangent space x(M), the sectional curvature K(p) for p is defined by

K(P) = R(X1,X2,X1,X2) = g(R(X1,X2)X2,X1),

where X1, X2 is an orthonormal basis for p. K(p) is independent of the choice of an orthonormal basis X1, X2. Lemur. 3.2 implies that the set

of values of K(p) for all plane p in X(M) determines the Riemannian curvature tensor at x. If K(p) is a constant for all plane p in Tx(M) and for all points x of M, then M is called a space of constant curvature. A Riemannian manifold of constant curvature is called a space form. Sometimes, a space form is defined as a complete simply connected Riemannian manifold of constant curvature. The following theorem due to Schur [1] is

well-known. THEOREM 3.1. Let M be a connected Riemannian manifold of dimension > 2. If the sectional curvature K(p) depends only on the point x, then M is a space of constant curvature. Proof. For any W, Z, X, Y e T{(M), we put

R1(W,Z,X,Y) = g(W,X)g(Z,Y) - g(Z,X)g(Y,W).

From Proposition 3.2 we have R = kR1, where k is a function on M. Since g is parallel, so R1. Hence

(VuR)(W,Z,X,Y) = (0uk)R1(W,Z,X,Y)

36

for any U c TX(M). Thus we have

((VuR)(X,Y))Z = (Uk)[g(Z,Y)X - g(Z,X)Y].

Therefore, Bianchi's second identity implies

0 = (Uk)[g(Z,Y)X - g(Z,X)Y] + (Xk)[g(Z,U)Y - g(Z,Y)U] + (Yk)[g(Z,X)U - g(Z,U)X].

For an arbitrary X, we choose Y, Z and U in such a way that X, Y and Z are mutually orthogonal and that U = Z with g(Z,Z) = 1. This is possi-

ble since n > 2. Then we obtain

(Xk)Y - (Yk)X = 0.

Since X and Y are linearly independent, we have Xk = Yk = 0. This shows

that k is a constant.

QED.

In the course of the proof of Theorem 3.1 we have

THEOREM 3.2. For a space of constant curvature k, we have

R(X,Y)Z = k[g(Y,Z)X - g(X,Z)Y].

If

and gij are the carponents of the curvature tensor and the

metric tensor with respect to a local coordinate system, then the canponents Rijkl of the Riemannian curvature tensor of type (0,4) are given by

Rijkl = If M is a space of constant curvature k, then

Rijkl = k(gikgjl - gilg k) or Rl ,7

jkl

= k(6ikgjl - 6i1

jk )

37

If we take an orthonormal frame field, the we have gij = 6ij at x of M, and hence

Rijkl = Rjkl = k(6ik6jl - 6il6jk).

In view of a local coordinate system, we obtain

Rijkl + Rjjkl = 0,

Rijkl + Rijlk = 0,

Rijkl + Riklj + Riljk = 0,

Rijkl - Rklij = 0.

If E1,...,En are local orthonormal vector fields, then

= 43.3)

n

S(X,Y)

g(R(Ei,X)Y,Ei) i=1

defines a global tensor field S of type (0,2) with local components

i

i Rjl = Rjil = g

jkil'

From the tensor field S we define a global scalar field n

(3.4)

r =

S(Ei,Ei) i=1

with local components

r = g13Rij.

The tensor field S and the scalar function r are called the Ricci tensor and the scalar curvature of M respectively. If n = 2, then G = -),Lr is called the Gaussian curvature.

If the Ricci tensor S is of the form

S = ag,

Rij = agij,

38

where a is a constant, then M is called an Einstein manifold.

We have the following THEOREM 3.3, Let M be a connected Riemannian manifold. If S = ag, where a is a function on M, then a is necessarily a constant provided that n = dim M > 2. Proof. From the Bianchi's second identity, we have

Rijkl;m + Rijlm;k + Rijmk;l = 0.

Multiplying by gik and gJl, summing up with respect to i, j, k and 1 and finally using the following formulas

Rijkl = -Rjikl = -Rljlk,

,

Ri.jkl = R i i

= 4gjl,

we obtain

(n-2)a

;m

= 0.

Hence a is a constant.

QED.

The following proposition is due to Schouten-Struik [1]. PROPOSITION 3.3. If M is a 3-dimensional Einstein manifold, then M is a space of constant curvature. If the curvature R vanishes, that is, M is a space of zero curvature, then we call such a Riemannian manifold M a locally flat space.

A Riemannian manifold M is called a locally symmetric space if its curvature tensor is parallel, that is, OR = 0. A complete locally symmetric space is called a symmetric space.

We define the Ricci operator Q of a Rienannian manifold M by

setting

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

for any vector fields X and Y on M. Q is a tensor field of type (1,1).

39

We now give examples of simply connected complete Riemannian mani-

folds of constant curvature. It is well-known that any two simply connected complete Rienannian manifolds of constant curvature k are isometric to each other (cf. Kobayashi-Ncmizu [1;p.265]).

In the following we shall construct, for each constant k, a simply connected complete space form with curvature k. Let Rn be the affine space of dimension n with cartesian coordinates x1,...,xn and let g be the Euclidean metric on Rn, that is,

g = (dx1)2 + ... + (dxn)2.

Then Rn with this metric g forms a space form of zero curvature. We call it the Euclidean n-space and we denote it by the same Rn. We put

{(x1,...,xn+1)

Mn(k) =

a

0+1:

kl((x1)2+...+(xn)2+sgn(k)(xn+1)2) - 2xn+1 = 0 xn+1 > 01

s

where sgn(k) = 1 or -1 according as k > 0 or k < 0. Then the Riemannian connection induced by

g = (dx1)2 + ... + (dxn)2 + sgn(k)((Jxn+1)2

on Rn+1 is the ordinaly Euclidean connection for each value k. In each case the metric tensor induced on Mn(k) is complete and of constant curvature k. Moreover, each Mn(k) is simply connected. Thus Mn(k) gives a model space of a simply connected complete space form of curvature k. A Riemannian manifold of constant curvature is said to be eZZiptic, hyperbolic or flat (or locally Euclidean) according as the sectional curvature is positive, negative or zero.

The hyperspheres in Mn(k) are those hypersurfaces given by quadratic equations of the form

40

(xl-al)2 +.... + (xn-an)2+ sgn(k)(xn+l-an+l)2 = constants,

where a = (a1,...,an+1) is an arbitrary fixed vector in

Rn+1.

In Mn(0),

there are just the usual hyperspheres. Among these hyperspheres the g.-_a: hj'persphere are those sections of hyperplanes which pass through fl- center

of Mn(k) in Rn+1, k # 0. For k = 0, we

cr,nsider the point at infinite on the Xn+1-axis as the center in

Rn+1.

Tit, intersection of a hyperplane through the center in Rn+1 is just a h;-perplane in Mn(O). All other hyperspheres in M5(k) are called small

k . - rsp;!eres. 4, TRANSFORMATIONS

L,t 1I be an n-dimensional Riemannian manifold with metric tensor g. Let

be a positive function on M. Then g* = peg defines a change

of metric on M which does not change the angle between two vectors at a point. Hence it is a conformal change of the metric. In particular,

if the function p is a constant, the conformal transformation is said to be hcnothetic. If p is identically equal to 1, the transformation is nothing but an isometry. If a Riemannian metric g is conformally related to a Riemannian metric g* which is locally flat, then the Riemannian manifold is said to be conformally flat or conformally Cuc'idean.

The Weyl conformal curvature tensor field of M is the tensor field C of type (1,3) defined by

(4.1)

C(X,Y)Z = R(X,Y)Z + 1n2[S(X,Z)Y - S(Y,Z)X + g(X,Z)QY - g(Y,Z)QX] -

(n-1)(n-2)[g(X,Z)Y

- g(Y,Z)X]

for any vector fields X, Y and Z on M. Moreover, we put

(4.2)

c(X,Y) = (oXQ)Y - (o 2)X - 2(1 -2)[(OXr)Y - (VYr)X].

41

The tensor field C of type (1,3) vanishes identically for n = 3. The Weyl conformal curvature tensor C is invariant under any conformal change of the metric. The following is a well-known theorem of Weyl

[11, [2] THEOREM 4.1. A necessary and sufficient condition for a Riemannian manifold M to be conformally flat is that C = 0 for n > 3 and c = 0

for n = 3. It should be noted that if M is conformally flat and of dimension n > 3, then C = 0 implies c = 0.

Let r be an affine connection on M. A vector field X on M is called an infinitesimal affine transformation of M if, for each x c M, a local 1-parameter group of local transformations Ot of a neighborhood U of x into M preserves the connection r.

An infinitesimal transformation X of a Riemannian manifold M is said to be conformal if LXg = pg, where p is a function on M. In this case, the vector field X is called a conformal Killing vector field. The local 1-parameter group of local transformations generated by an infinitesimal transformation X is conformal if and only if X is conformal. If p is constant, then X is bomothetic, and if p = 0, then X is isanetric.

A vector field X on M is called an infinitesimal isometry (or, a Killing vector field) if the local 1-parameter group of local transformations generated by X in a neighborhood of each point of M consists of local isanetries. An infinitesimal iscmetry is necessarily an infinitesimal a.ffine transformation. X is an infinitesimal isometry if and

only if LXg = 0. The condition LXg = pg and LXg = 0 can be rewritten as, respectively:

(LXg)(Y,Z) = g(VYX,Z) + g(VZX,Y) = g(Y,Z),

(LXg)(Y,Z) = g(VYX,Z) + g(VZX,Y) = 0,

for any vector fields Y and Z on M.

42 We shall give some integral form las. When a Riemannian manifold M is orientable, we can define the volume element of M by

*1 = /idxl A dx2 A ... A dxn,

where g = Iggij. Then we can consider the integral of a scalar function f, jDf(x)*l, over a dc*nain D of M.

We now consider a p-form i

dx W= i-wi p! 1 2." p Then the exterior differential d

cI

=

p+i)!

i

i

1 A dx 2 A ... A dx p

of w is the (p+l)-form given by

iwi1i2.- .ip - ail ii2...ip i

- ai2wilii3.- .i

- ... -

81p W

..i

i

)dxl A dx 1A...Adx p.

P

We see that dew= d(dw) = 0. The codifferential 6w of w is the (p-1)form with the local expression

.. i i 6w = - (P-1)! {gjiwii 2... i )dx 2 A ... A dx p. P

For a scalar function f, we put 6f = 0. We see that dzar- 6(6w) = 0.

We can also define the codifferential 6T of a more general tensor

field T. For example, let T be a tensor field of type (0,3). Then T is the tensor field of type (0,2):

6T : -gt3OtTjih = - VJTiiho

where we have put Vi = gtivt and Tjih are local components of T. A p-form w, or a skew-symmetric tensor field of type (O,p), is said to be harmonic if dw = 6w = 0. If we put

A=-6d-d6,

43

then we have Aw = 0 for a harmonic p-form w. For a vector field X with local components Xh there is associated 1-form F given by

= gj1Xldx1 = Xidx1.

The codifferential g of

is given by

V1-X1 = - g11V X1.

dC

We denote it by 6X. The famous Green's theorem can now be stated as follows.

GREEN'S THEOREM. In a ccnpact orientable Riemannian manifold M without boundary, vie have

fM(6X)*l = 0

for any vector field X on M, or

fM(viX1)*l = 0

or

fM(gj1V Xi)*1 = 0,

X1 and X1 being components of X.

On the other hand, the divergence of X, denoted by div X, is

given by

(div X)x = trace of the endamorphisn V -> VVX, V E TX(M).

Therefore, the integral equation in Green's Theorem can be stated as

fM(div X)*1 = 0.

If Re have a function f, we can form the codifferential ddf of the differential df of f. This bas the local expression

44

Af = gjiv V if = A if. i The differential operator A = gjlvjvi or vlvi is sometimes called the Lapiacian.

Applying Green's Theorem, we have THEOREM 4.2. In a compact and orientable Riemannian manifold M

without boundary, we have

f0f*1 = 0 for any function f on M.

Using this theorem, we have HOPF'S LEMMA, Let M be a compact orientable Riemannian manifold without boundary. If f is a function on M such that (or

f > 0 everywhere

f < 0 everywhere), then f is a constant function. Let M be an n-dimensional Riemannian manifold. We denote by {ei}

an orthonormal frame of M. For any vector field X of M, we have

div(VXX) - div((divX)X) = S(X,X) +

(divX)2,

where S is the Ricci tensor of M. On the other hand, we have

jj(g(VejX,ei) + g(velX,ej))2

= 2

(g(Ve.X,ei)g(veX'ej) + g(Ve X,ei)2). i

i,J i

J

Thus we have

g(Ve X,ei)g(VeX,ej) _ -IVXI2 + 11L1gJ2.

i,J J

i

45

We have therefore (Yano [1])

THEOREM 4.3. For any vector field X of a Riemannian manifold M,

we have

(4.3)

div(vxX) - div((divX)X)

= S(X,X) + jIIZ -

IVXI2 - (divX)2.

In the following, we shall give applications of Theorem 4.3. Let = Xidx1 be a 1-form associated to the vector field X. Then we have

I

12 =

I (g(Ve X,ei) - g(0e X,e3))2 i.3

1

J

= -2 1 g(Ve X,ei)g(Ve X,e3) + i,3

3

i

Thus, (4.3) can be rewritten as

(4.4)

div(VXX) - div((divX)X) C = S(X,X) - ildSI2 + IVXI2 -

If M is compact orientable, by (4.4), we have

jM[S(X,X) + IVXI2]*1 = 0 for a harmonic vector field X. Therefore, we have (Bochner [1]) THEOREM 4.4, If the Ricci tensor S of a compact orientable Riemannian manifold M satisfies S(X,X) > 0, then a harmonic vector field X in M has a vanishing covariant derivative. If the Ricci tensor S is

positive-definite, then a harmnic vector field other than zero does not exist in M.

The existence of harmonic forms in M is closely related to the topology of M. In fact we have (Hodge [1])

46

THEOREM OF HODGE. In a compact orientable Rienannian manifold, the number of linearly independent (with constant real coefficients) harmonic p-forms is equal to the p-th Betti number Bp of the manifold. Combing this with Theorem 4.4, we obtain (Bochner [1], Myers [1])

THEOREM 4.5. In a compact orientable Riemannian manifold M with positive Ricci tensor, the first Betti number vanishes. From (4.3) we also have (Bochner [1]) THEOREM 4.6. Let M be a compact and orientable Riemannian manifold. If the Ricci tensor S of M satisfies S(X,X) < 0, then a Killing vector field X in M has vanishing covariant derivative. If S is negativedefinite, then a Killing vector field other than zero does not exist in M.

Let X be a conformal Killing vector field such that Leg = 2pg, where p is given by p = -----6X. Then (4.3) implies

fM[S(X,X) - IVXIZ -

nri

6X)2]*l = 0,

from which we have (Yano [1]) THEOREM 4.7. Let M be a compact and orientable Rianannian manifold. If the Ricci tensor S of M satisfies S(X,X) < 0, a conformal Killing

vector field X in M has vanishing covariant derivative, and if S is negative-definite, a conformal Killing vector field other than the zero vector field does not exist in M.

5. FIBRE BUNDLES AND COVERING SPACES

Let G be a differentiable manifold with a countable basis. If G

is a group such that the group operation (a,b) e G x G -> ab 1 e G is a differentiable mapping, then G is called a Lie group.

A Lie group is clearly a locally compact group with a countable basis. Let GO be the connected component of G containing the identity element of G. We see that GO is a closed normal subgroup of G0.

47

Moreover, since G0 is locally connected, GD is an open sukmanifold of G. Go is also a Lie group.

We denote by La (resp. Ra) the left (resp. right) translation of G by an element a E G: Lax = ax (resp. Rax = xa) for every x E G. For

a E G, ad a is the inner autamrphism of G defined by (ad a)x = axa 1 for every x E G. A vector field X on G is called left invariant (resp. right invariant) if it is invariant by all La, i.e., (La),iX = X for all

a E G (resp. Ra, i.e., (Ra)*X = X for all a c G). Let % be the set of all left invariant vector fields on G. We call °l the Lie algebra of the

Lie group G. In fact, } is closed for the usual addition, scalr multiplication and bracket operation. As a vector space, c-is isomorphic

with the tangent space e(G) at the identity, the isomorphism being given by the mapping which sends X O Y into e, the value of X at e. Thus I is a Lie subalgebra of dimension n (= dim G) of the Lie algebra of vector fields

1r(G).

Every A E % generates a (global) 1-parameter group of transformations of G.

We say that a Lie group G is a Lie transformation group on a manifold M or that G acts on M if the following conditions are satisfied: (1) Every element a E G induces a transformation of M, denoted by

x

> xa, where x c M;

(2) (a,x) c G x M

> xa c M is a differentiable mapping;

(3) x(ab) = (xa)b for all a,b c G and x e M.

We also write ax for xa and say that G

M on the right. If we

write ax and assume (ab)x = a(bx) instead of (3), we say that G acts on M on the left and write Lax for ax also. Note that Rab =

Lab =

and

Frcm (3) and from the fact that each Ra or La is 1 : 1

on M, it follows that a and e are the identity transformation of M. We say that G acts effictively (resp. freely) on M if Rax = x for all x e M (resp. for same x c M) implies that a = e. Definition. Let M be a manifold and G a Lie group. A (differentiable) principal fibre bundle over M with group G consists of a manifold P and an action of G an P satisfying the following conditions:

48

(1) G acts freely on P on the right: (u,a) e P x G > ua=Rau a P; (2) M is the quotient space of P by the equivalence relation induced by G, M = P/G, and the canonical projection it : P

M is differentiable; (3) P is locally trivial, that is, every point x of M has a neighborhood U such that n-1(U) is iscmorphic with U x G in the sense that there is a diffeornorphism 4) : n1(U)

> U

x G such that -

4)(u) = (1r(u),4)(u)) where 0 is a mapping of 7r 1(U) into G satis-

fying 4)(ua) = (4(u))a for all u e it(U) and a e G. A principal fibre bundle will be denoted by P(M,G,ir), P(M,G) or

simply P. We call P the total space or the bundle space, M the base space, G the structure group and it the projection. For each point x of M, n 1(x) is a closed sutxnanifold of P, called the fibre over x. If u

is a point of 7r-1 (x), then n-1(x) is the set of points ua, a e G, and is called the fibre through u. Every fibre is diffeemorphic to G.

Given a Lie group G and a manifold M, G acts freely on P = M x G on the right as follows. For each b e G, Rb maps (x,a) e M x G into (x,ab) E M x G. The principal fibre bundle P(M,G) thus obtained is called trivial.

From local triviality of P(M,G) we see that if W is a submanifold of M, then 7-1(W)(W,G) is a principal fibre bundle. We call it the restriction of P to W and denote it by PIW. Given a principal fibre bundle P(M,G), the action of G on P induces a hcmomorphism a of the Lie algebra g of G into the Lie algebra

VP) of vector fields on P. a can be defined as follows: For every u,

let au be the mapping a e G -> ua a P. Then (ou)*Ae = (aA)u. For each A E

, A* = a(A) is called the fundamental vector field corresponding

to A. Since the action of G sends each fibre into itself, Au is tangent to the fibre at each u E P. As G acts freely on P, A* never vanishes on P if A # 0. The dimension of each fibre being equal to that of 17, the

mapping A -> (A*)u of 7 into Tu(P) is a linear isomorphism of % onto the tangent space at u of the fibre through. We also see that for each a E G, (Ra)*A* is the fundamental vector field corresponding to

(ad(a 1))A a °l^.

49

We now give the concept of transitive functions. For a principal fibre bundle P(M,G), we can choose an open covering {U .} of M, each i

7T-1(Ui) provided with a diffemmrphism u -> (n(u),4i(u)) of n-1(Ui)

Will UJ.), then onto U. x G such that 01 (ua) = (gi(u))a. If u e 1r 1(Ui Oj(ua)(Oi(ua))-1

0i(u)(Oi(u))-1

= Oi(u)(¢i(u))-1, which shows that

depends only on n(u) not on u.

We define a mapping phi : Ui n U > G by p..(4(u)) _ Y u)(Oi(u))-1. The family of mappings ji are called

transitive

functions of the bundle P(M,G) corresponding to the open covering {Ui} of M. It is easy to verify that

(5.1)

for x e 1 A U A U.

ki (x) = cyki

Conversely, we have (cf. Kobayashi-Nomizu [1])

PROPOSITION 5.1. Let M be a manifold, {Ui} an open covering of M and G a Lie group. Given a mapping ip

: Ui A UJ -> G for every

Ji non-empty Ui n U , in such a way that the relation (5.1) are satisfied, i

we can construct a (differentiable) principal fibre bundle P(M,G) with

transitive functions

ji.

A homomorphism f of a principal fibre bundle P'(M',G') into another principal fibre bundle P(M,G) consists of a mapping f'

:

P' ->

P and a homormrphism f" : G' -> G such that f'(u'a') = f'(u')f"(a)

for all u' c P' and a' E G'. For the sake of simplicity, we shall denote f' and f" by the same letter f. Every homomorphism f

: P' ->

P maps each fibre of P' into fibre of P and hence induces a mapping of M' into M, which will be also denoted by I. A homomorphism f : P'(M',G')

-> P(M,G) is called an imbedding or injection if the induced mapping f-

: M' > M is an imbedding and if f : G' > G is a monocmrphism.

By identifying P' with f(P'), G' with f(G') and M' with f(M'), we say that P'(M',G') is a subbundle of P(M,G). If, moreover, M' = M and the

induced mapping f : M' > M is the identity transformation of M, f : P'(M',G') > P(M,G) is called a reduction of the structure group G of P(M,G) to G'. The subbundle P'(M,G') is called a reduced bundle.

Given P(M,G) and a Lie subgroup G' of G, we say that G is reducible to

50

G' if there is a reduced bundle P'(M',G'). Let P(M,G) be a principal fibre bundle and F a manifold on which

G acts on the left : (a,C) F. G x F >.ag a F. We shall construct a fibre bundle E(M,F,G,P) associated with P with standard fibre F. On the product manifold P x F, we let G act on the right as follows: a e G maps (u,E) e P x F into (ua,a 1C) e P x F. The quotient space of P x F

by this group action is denoted by E = P x GF. The mapping P x F > M which maps (u,E) into n(u) induces a mapping irE, called the projection,

of E onto M. For each x e M, nE(x) is called the fibre of E over x. Every point x of M has a neighborhood U such that 7r

(U) is isomorphic

to U x G. Identifying n-1 (u) with U x G, we see that the action of G on n 1(U) x F on the right is given by

(x,a,E) -> (x,ab,b 1E)

for (x,a,F) e U x G x F and b e G.

It follows that the isomorphism T r

N U x G induces an isomorphism

nE1(U):z: U x F. We can therefore introduce a differentiable structure

in E by the requirement that nE1(U) is an open submanifold of E which is diffeonorphic with U x F under the isomorphism nE1(U) 2 U x F. The projection TrE is then a differentiable mapping of E onto M. We call

E(M,F,G,P) or simply E the fibre bundle over the base space M, with

standard fibre F and structure group G, which is associated with the principal fibre bundle P. We recall here some results on covering spaces. Given a connected, locally arcwise connected topological space M, a connected space E is

called a covering space over M with projection p : E > M if every point x of M has a connected open neighborhood U such that each connected ccuponent of p-1 (U) is open in E and is mapped haneanorphically

onto U by p.

Two covering spaces p : E > M and p'

: E' -> M are isanorphic

if there exists a haneormrphisn f : E > E' such that

p : E -> M is a universal covering space if E is simply connected. If M is a manifold, every covering space has a unique structure of manifold such that p is differentiable.

51

PROPOSITION 5.2. (1) Given a connected manifold M, there is a unique (unique up to an isomorphism) universal covering manifold, which will be denoted by M. (2) The universal covering manifold M is a principal fibre bundle over M with group ir1(M) and the projection p : M

> M, where n1(M)

is the first hanotopy group of M.

For the proof, see Steenrod [1;pp.67-71]. PROPOSITION 5.3. Let M be a Riemannian manifold with metric g.

Let p : E

> M be a covering manifold of M. Then p*g is a Riemanni-

an metric on E. Moreover, E is complete if and only if M is complete.

ExampZe 5.1. Bundle of linear frames: Let M be an n-dimensional manifold. A linear frame u at a point x of M is an ordered basis

X1,...,X of TT(M). Let L(M) be the set of all linear frames u at all points of M and let 1r be the mapping of L(M) onto M which maps a linear

frame u at x into x. The general linear group GL(n;R) acts on L(M) on the right as follows. Let a =

a GL(n;R) and u = (X1,...,X.) be

a linear frame at x. Then ua is the frame

at x defined by

Yi = Eja3Xj. It is clear that GL(n;R) acts freely on L(M) and w(u) ,r(v) if and only if v = ua for some a c GL(n;R). Let (xl,...,xn) be a

local coordinate system in a coordinate neighborhood U in M. Every frame u at x e U can be expressed uniquely in the form u = (X1,...,X )

with X. = F(8/2xk), where (Xi) is a non-singular matrix. This shows that n 1(U) is in 1 : 1 correspondence with U x GL(n;R). We can make L(M) into a differentiable manifold by taking (xi) and (Xk) as a local

coordinate system in n(U). It is easy to verify that L(M)(M,GL(n;R)) is a principal fibre bundle. We call L(M) the bundle of linear frames over M. A linear frame u at x c M can be defined as a non-singular linear mapping of Rn onto Tx(M). The two definitions are related to each other as follows. Let e1,...,en be the natural basis for R°: e1 = (1,0,...,O),...,en = (0,...,0,1). A linear frame u = (X1,...,Xn) at x can be given as a linear napping u : IP

> X(M) such that

uei = Xi for i = 1,...,n. The action of GL(n;R) on L(M) can be accordingly interpreted as follows. Consider a =

a GL(n;R) as a linear

0 transformation of Rn which maps e

into Eiaie

Then ua : Rn

> TA(M)

i

is the composite of the following two mappings:

Rn a > Rr u > TX(M). Example 5.2. Tangnet bundle: Let GL(n;R) act on Rn as above. The tangent bundle T(M) over M is the bundle associated with L(M) with

standard fibre R. The fibre of T(M) over x E M may be considered as Tx(M).

Example 5.3. Tensor bundle: Let Tr be the tensor space of type (r, s) over the vector space Rn. The group GL(n;R) can be regarded as

a group of linear transformations of T. With this standard fibre TS, we obtain the tensor bundle Tr(M) of type (r,s) over M which is associated with L(M). It is easy to see that the fibre of TS(M) over x E M may be considered as the tensor space over TX(M) of type (r,s). Example 5.4. Vector bundle: Let F be either the real number field R or the complex number field C, Fn the vector spa-,e of all n-tuples

of elements of F and GL(n;F) the group of all (n,n)-non-singular matrices with entries from F. GL(n;F) acts on Fn on the left in a natural

manner; if a = (al)e GL(n;F) and E =

a Fn, then a& =

a Fn. Let P(M,G) be a principal fibre bundle and p a representation of G into GL(n;F). Let E(M,F",G,P) be the associated bundle with standard fibre Fn on which G acts through p. We call E a real or complex vector bundle over M according as F = R or F = C. Each fibre 1r-1(x), x E M, of E has the structure of a vector space such that

every u E P with n(u) = x, considered as a mapping of Fn onto nE1(x), is a linear isomorphism of Fn onto 1rE(x). We give examples of universal covering manifolds. Exwnple 5.5. Let el,...,en be any basis of Rn, and let G be the subgroups of Rn generated by el,...,en: G = {Emiei : mi integers}. The action of G on Rn is properly discontinuous and Rn is the universal covering manifold of Rn/G. The quotient manifold Rn/G is called an n-dimensional torus.

0 Example 5.6. Let SP =

{(xl,...,xn+1) c

Rn+1

:

E(xl)Z = 11 and G

be the group consisting of the identity transformation of Sn and the transformation of Sn which maps (x1) into (-x'). Then Sn (n > 2) is the universal covering manifold of Sn/G. The quotient manifold Sn/G is called the n-dimensional real projective space.

54

EXERCISES

A. CONCIRCULAR CURVATURE TENSOR: Let M be an n--dimensional

Riemannian manifold with metric tensor field g. We denote by R, Q and r the Riemannian curvature tensor, the Ricci operator and the scalar curvature of M respectively. We put

Z(X,Y)W = R(X,Y)W - n(n 1)(g(Y,W)X - g(X,W)Y),

GX=QX - nx for any vector fields X, Y and W on M. We call Z the concircular curvature tensor of M, which represents deviation of the manifold from constant curvature. We also see that G = 0 if and only if M is

an Einstein manifold. We have

IZI2

= IRI2

n(n-1

2,

2

IGI2 = IQI2 -

n We put

P(X,Y)W = R(X,Y)W - ng(Y,W)QX - g(X,W)QY).

We call P the Weyl's projective curvature tensor. The vanishing of

P is equivalent to the fact that M is of constant curvature. We have

IpI2 = IRI2 - n21IAI 2

Let C be the conformal curvature tensor of M. Then vie have _

IC12 = IRIZ

4 2 n-2IQI2 + (n-1)(n-2

2

55

B. CONFORMALLY FLAT RIEMANNIAN MANIFOLDS:

The conditions for

a conformally flat Rienannian manifold to be a space of constant

curvature are given by the following (Goldberg-Oklanura [1] ) THEOREM 1. Let M be an n-dimensional con pact conformally flat

Rienannian manifold with constant scalar curvature r. If the length of the Ricci tensor is less than r/(n-1)1/2, n > 3, then M is of constant curvature.

THEOREM 2. In an n-dimensional compact conformally flat Riemannian manifold M, if the length of the Ricci tensor is constant and less than r/(n-1)1/2,

then M is of constant curvature.

Theorem 1 is a generalization of the theorem of Goldberg [4].

C, NOMIZU'S PROBLEM: If a Riemannian manifold is locally symmetric, then its curvature tensor R satisfies

(*)

R(X,Y)R = 0 for all tangent vectors X and Y,

where the endoimrphism R(X,Y) operates on R as a derivation of the tensor algebra at each point of M. Conversely, does this algebraic condition (*) on the curvature tensor field R imply that M is locally symmetric?.

H. Takagi [1] showed that, in a 4-dimensional Euclidean space R4,, there exists an irreducible and complete hypersurface M which

satisfies the condition (*) but is not locally symmetric.

Let (x,y,z,w) be a Cartesian coordinate system in R4. We consider the hypersurface M defined by

w = (x2z - y2z - 2xy)/2(z2 + 1)

or

2z2w - x2z + y2z + 2w + 2xy = 0,

which satisfies the non-linear partial differential equation

56

wX - wy + 2w = 0.

Then M is a desired manifold. For Riemannian manifolds satisfying the condition (*) see Sekigawa [ ]. D. EQUATION OF 1T1AURER-CARTAN: A differential form w on a Lie

group G is called left invariant if (La)*w = w for every a e G. The vector space 7* formed by all left invariant 1-forms is the dual space

of the Lie algebra "T: if A E °l and w e 7*, then the function w(A) is constant on G. If w is a left invariant form, then so is dw. We have the equation of Mourer-Cartan:

dw(A,B) = -4w([A,B])

for w e of*, A,B E 01.

The canonical 1-form 0 on G is the left invariant 0;-valued 1-form uniquely determined by B(A) = A for A E al.

E. CONNECTION: We define a connection in a principal fibre bundle P(M,G) (see Kobayashi-Nomizu [1]). For each u e P, let Tu(P) be the

tangent space of P at u and u the subspace of T1(P) consisting of vectors tangent to the fibre through u. A connection r in P is an assignment of a subspace Qu of Tu(P) to each u E P such that

(a) Tu(P) = u + Qu (direct sum); (b) Qua = (Ra)*Qu for every u E P and a e G; (c) Qu depends differentiably on u.

Condition (b) means that the distribution u

>

'u

is invariant

by G. We call G11 the vertical subspace and Qu the horizontal subspace of Tu(P). A vector X E T1(P) is called vertical (resp. horizontal) if

it lies in u (resp. Qu). By (a), ary vector X of P at u can be uniquely written as

X=Y+Z,

YEG1,ZEQu.

57

We call Y (resp. Z) the vertical (resp. horizontal) canponent of X.

Given a connection r in P, we define a 1-form w on P with values in the Lie algebra 0J of G as follows. For each X c Tu(P), we define w(X) to be the unique A e °d such that (A*)u is equal to the vertical

component of X, where A* is the fundamental vector field corresponding to A. It is clear that w(X) = 0 if and only if X is horizontal. The form w is called the connection form of the given connection T. The projection n : P

> M induces a linear mapping n : TA(P)

>

Ts(M) for each u e P, where x = n(u). When a connection is given, n

maps Qu isomorphically onto T(M). The horizontal lift or simply, lift of a vector field X on M is a unique vector field X* on P which is horizontal and n(X*) = X7(u) for every u c P.

THEOREM 1. Given a connection in P and a vector field X on M, there is a unique horizontal lift X* of X. The lift X* is invariant by Ra for every a c G. Conversely, every horizontal vector field X* on P invariant by G is the lift of a vector field X on M. Let P(M,G) be principal fibre bundle and A a subset of M. We say that a connection is defined over A if, at every point u c P with n(u) c A, a subspace Qu of T1(P) is given in such a way that conditions (a) and (b) for connections are satisfied and Qu depends differentiably on u in the following way. For every point x c A, there exist an open neighborhood U and a connection in PjU = W-1 (U)

such that the horizontal subspace at every point u e it(A) is the given space Qu. We have THEOREM 2. Let P(M,G) be a principal fibre bundle and A a closed subset of M (A may be empty). If M is paraconpact, every connection defined over A can be extended to a connection in P. In particular, P admits a connection if M is paraconpact.

F. CANONICAL FLAT CONNECTION: Let P= M x G be a trivial principal fibre bundle. The canonical flat connection in P is defined by taking the tangent space to M x {a} at u = (x,a) e M x G as the horizontal subspace at u. Let e be the canonical 1-form on G and

f: M X G

> G be the natural projection. Then w = f*9 is the

58

connection form of the canonical flat connection in P. From the Maurer-Cartan equation of 0 we see that the canonical flat connection has zero curvature. A connection in any principal fibre bundle P(M,G) is called flat if every point x of M has a neighborhood U such that the induced connection in PJU = 7r 1(U) is isomorphic with the canoni-

cal flat connection in U x G. More precisely, there is an isomorphism f

: i1(U)

> U x G which maps the horizontal subspace at each

u e 7r 1(U) upon the horizontal subspace at f(u) of the canonical flat

connection in U x G. THEOREM 1. A connection in P(M,G) is flat if and only if the curvature form vanishes identically. THEOREM 2. Let r be a connection in P(M,G) such that the curvature

vanishes identically. If M is paraccact and simply connected, then P is isomorphic with the trivial bundle M x G and I' is isomorphic with the canonical flat connection in M x G.

G. STRUCTURE EQUATION: Let P(M,G) be a principal fibre bundle with a connection r. Let h : Tu(P)

> Qu be the projection and

w be the connection form of r. We put Dw = (dw)h. Iii is called the curvature form of w, which will be denoted by SZ. Then

THEOREM (Structure equation). Let w be a connection form and Q its curvature form. Then

dw(X,Y) = -1[w(X),w(Y)] + Q(X,Y),

X,Y a T1(P).

(See Ambrose-Singer [1].)

H. LINEAR CONNECTION: Let L(M) be the bundle of linear frames over a manifold M and GL(n;R) the general linear group, n = dim M.

The caninical form 6 of L(M) is the 0-valued 1-form on L(M) defined by

6(X) = u-1 WX))

for X e u(P),

where u is considered as a linear mapping of Rn onto Tn(u)(M)

59

(cf. Exanple 5.1). A connection in L(M) over M is called a Zinear connection of M. Let U be a coordinate neighborhood in M with local coordinate system (x'). We put Xi = a/2x1, i = 1,...,n, in U. Every linear frame at a point x of U can be uniquely expressed by

(EX (Xi)x,...,EXh(Xi)x), 1

det(Xi) 0 0.

We take (x1,Xk) as a local coordinate system in 7T-1(U) C L(M). Let (Yk) be the inverse matrix of (Xk) so that EJXiY. = EjYiXj = di. Let e1,...,en be the natural basis for Rn and set

a = 161ei,

which is the canonical form. Then 6 is expressed by

61 = J

Let w be the connection form of a linear connection r of M. With respect to the basis {Ei) of the Lie algebra .L(n;R), we put

w= EWEi

i,j

Let a be the cross section of L(M) over U which assigns to each

x E U the linear frame ((X1)x'...,((dx). We set

wU = a*w.

Then wU is a DL(n;R)-valued 1-form defined on U. We define

(Tlkdxi)EI

wU = ,j,k

The functions rJk are called the canponents (or Christoffel's symbols) of the linear connection r with respect to the local coordinate system (x1). We obtain

so

Xj vxi = k &k

k (see the definition of affine connection in §2).

Traditionally, the words "linear connection" and "affine connection" have been used interchangeably. But, in the book of KbbayashiNan;_"u [1], these words are made a logical distinction.

61

CHAPTER 11

SUBMANIFOLDS OF RIEMANNIAN MANIFOLDS

In this chapter we give the fundamental results and some theorems concerning geometry of subnanifolds which will be needed for the later treatment of submanifolds.

In §1, we discuss the induced connection on submanifolds and second fundamental forms of immersions. We give the basis formulas for submanifolds which are called the Gauss and Weingarten formulas. In §2, we prepare equations of Gauss, Codazzi and Ricci which give the relationship between sutmanifolds and ambient manifolds. In §3, we compute the Laplacian of the second fundamental forms of immersions and give the Simons' type formula on submanifolds. §4 is devoted to the study of submanifolds of space forms. In the last §5, we discuss minimal submanifolds of space forms. Moreover, in §§4 and 5, we also discuss sutmanifolds with flat normal connections or with parallel second fundamental forms.

For the general theory of sutmanifolds we refer to Chen [1] and Kobayashi Nomizu [2].

1. INDUCED CONNECTION AND SECOND FUNDAMENTAL FORM Let M be an n-dimensional manifold iscmetrically immersed in an m-dimensional Riemannian manifold R. We put m =,n+p, p > 0. Since the discussion is local, we may assume, if we want, that M is imbedded in M. If the manifold M is covered by a system of coordinate neighborhoods {V;u } and M is covered by a system of coordinate neighborhoods {U;xh}, where, here and in the sequel, the indices A, B, C,... run

62

over the range 1,2,...,m and i, j, k,... run over the range 1,2,...,n. Then the sutmanifold M can be represented locally by

uA = uA(xh). In the following, we shall identify vector fields in M and their images under the differential napping, that is, if i denotes the immersion of M into M and X is a vector field in M, we identify X and i*(X). Thus, if X is a vector field in M with local expression X = Xhah (ah = a/axh), then X also has the local expression X = BhAXhaA BhA in M (aA = aI3uA), and = auA/ax , where we used the Einstein convention, that is, repeated indices, with one upper index and one lower index, denote summation over its range.

We denote by g the Rienannian metric tensor field of M. Then the submanifold M is also a Rienannian manifold with the Riemannian metric h given by h(X,Y) = g(X,Y) for any vector fields X and Y in M. The Rienannian metric h on M is called the induced metric on M. We see that gj i = gBBj BEiA with h = gjidxxdxl and g = gBAduBduA.

Throughout this chapter, the induced metric It will be denoted

by the same g as that of the ambient manifold M to simplyfy the notation because it may cause no confusion. If a vector V of 6I at a point x of M satisfies g(X,V) = 0 for

any vector X of M at x, then V is called a normal vector of M in R at x. A unit normal vector field of M in M is sometimes called a normal section on M.

Let T(MY' denote the vector bundle of all normal vectors of M in M. Then the tangent bundle of M, restricted to M, is the direct sum of the tangent bundle T(M) of M and the normal bundle T(M)1 of M in M. We denote by G the operator of covariant differentiation on R.

LEI41A 1, 1, Let X and Y be vector fields on M and let X and Y be extensions of X and Y, respectively. Then [X,Y]IM is independent of the extensions and [s,Y]IM = [X,Y], and (GXY)IM is also independent of the extensions.

Proof. Let IA(u) and VA(u) be extensions of BhAXh(u) and BhAYh(U),

0 respectively. Then

XA(u(x)) =

BhY(u),

YA(u(x)) = BhAYh(u).

Thus we have

[X,?]AIu=u(x) ° (XBaBYA -

yBaB

_ (Bh$xhaBYA

= Xhah(BiAyi

I

) -

(x)

Bh'asXA)lu=u(x) - Yhah(BiAX)

= BlA(XhahY1 - YhahXl) = BiA[X,Y]1.

This shows that [X,Y]IM does not depend on the extensions X and Y of X and Y and is equal to [X,Y].

In the next place, the components of vXY are given by

OoBY-c + rcYA) , from which

XB(aBYc + raaYA)lu=u(x)

=B

(aBY + rBABAYj)lu=u(x)

= Xi(ai(BjCY3) + rCB1BBj.AYJ).

Therefore OXY does not depend on the extensions. We denote (VXY)IM by VXY. We put

(1.1)

vxY = vXY + B(X,Y),

where VXY is the tangential component of VXY and B(X,Y) the normal

QED.

64

canponent of CXY. Then V is the operator of covariant differentiation

with respect to the induced metric on M. We

all prove this fact in

the following:

Let a and b be functions on M. Men

'7aXbY = aaXbY = a[ (Xb)Y + bbxY]

= [a(Xb)Y + abVXY] + abB(X,Y),

from which

VaXbY = a(Xb)Y + abVXY

and

B(aX,bY) = abB(X,Y).

The first equation shows that V defines an affine connection on M and the second equation shows that B is bilinear in X and Y, since additivity is trivial. We next prove V has no torsion and Vg = 0. Since ® has no torsion, we have

0 = flXY - 7yx - [X,Y]

= VXY + B(X,Y) - VC - B(Y,X) - [X,Y].

Comparing the tangential and normal parts of the equation above, we

find VXY - VI,X - [X,Y] = 0

and

B(X,Y) = B(Y,X).

as

These equations show that V has no torsion and B is symmetric. Moreover, since ig = 0, we see that

Vx(jg(Y,Z)) = VX(g(Y,Z))

= g(OXY,Z) + g(Y,DXZ) = g(VXY,Z) + g(Y,OxZ),

which means that Vg = 0.

We call the Riemannian connection V the induced connection and B the second fundcvnentat form of M (or of the immersion i). For each

poin x of M, B(X,Y) at x depends only Xx and X. Next, let V be a normal vector field on M and X be a vector field on M. We put

VXV = -AVX + DxV,

(1.2)

where -AV% and DXV are, respectively, the tangential carponent and the

normal canponent of Y. It is easily verified that the vector fields AVX and DXV are differentiable on M. For any functions a and b on M we have

VaxbV = aVXbV = a[ (Xb)V + bVXV]

= a(Xb)V - abAVX + abDXV,

from which AbV(aX) = abAVX

and

Dax(bV) = a(Xb)V + abDxV.

66

Since the additivity is trivial, AvX is bilinear in V and X and AVX at x depends only on

and Xx. We give a relation between B and A. Let

x on M and V be a vector field normal to M. Then X, Y be vector fields 0 = g(VXY,V) + g(Y,VXV) = g(B(X,Y),V) - g(Y,AVX),

from which

g(B(X,Y),V) = g(AVX,Y).

Consequently, AV is a symmetric linear transformation of X(M), that is, A is an element of Han(T(M)L,S(M)), where S(M) is the bundle whose fibre at each point is a symmetric linear transformation of TA(M).

We call A the associated second fundamental form to B or simply the second fundamental form of M. LFM1A 1,2, D is a metric connection in the normal bundle T(M}'' of M in M with respect to the induced metric on T(M)--.

Proof. We easily see that D defines an affine connection in the

normal bundle T(M)`. Moreover, for any vector fields V and U in T(M)

,

we have

g(DXV,U) + g(V,DXU) = g(VXV,U) + g(V,VXU) = VXg(V,U) = DXg(V,U),

which shows that D is metric for the fibre metric in T(M)`, namely, the restriction of g to the normal spaces.

QED.

We have the first set of basic formulas for subm nifolds, na.-nely,

VXY = VxY + B(X,Y)

and

VXV = -AVX + DXV.

The first formula is called the Gauss formula and the second formula is called the Weingarten formula.

67

A normal vector field V on M is said to be parallel in the normal bundle, or simply parallel, if DXV = 0 for all vector fields X tangent to M. A submanifold M is said to be totally geodesic if its second fundamental form vanishes identically, that is, B = 0 or equivalently A = 0. For a normal section V on M, if AV is everywhere proportional to the identity transformation I, that is, Av = aI for some function

a, then V is called an umbilical section on M, or M is said to be umbilical with respect to V. If the submanifold M is umbilical with respect to every local normal section of M, then M is said to be totaZZy umbilical.

Let e1,...,en be an orthononnal basis in T{(M). The mean curvature vector p of M is defined to be p = (1/n)(TrB), where TrB = EiB(ei,ei),

which is independent of the choice of a basis. We obtain u = Ea[(TrAa)/n1ea, where en+l,...,em is an orthonormal basis in X(M t,

and we denote Ae by A. for simplicity, a = n+l,...,m. If p = 0, then a M is said to be miminaZ. We notice that any sutmanifold M which is minimal and totally umbilical is totally geodesic.

2. EQUATIONS OF GAUSS, CODAZZI AND RICCI

For the second fundamental form B we define its covariant

derivative VXB by

(VXB)(Y,Z) = DXB(Y,Z) - B(VXY,Z) - B(Y,VXZ)

for any vector fields X, Y and Z tangent to M, which is defined

equivalently by putting

(VXA)VY = VX(AVY) - A VY - AVOXY

for any vector fields X, Y tangent to M and any vector field V normal to M. If VXB = 0 for all X, then the second fundamental form of M is

said to be parallel, which is equivalent to VA = 0 for all X.

68

Let R and R be the Rinni an curvature tensor fields of M and M respectively. Then the Gauss and Weingarten foimilas imply

R(X,Y)Z = VxV Z - VYVXZ - 0[XlY]Z

= VX(VYZ+B(Y,Z)) - y(VXZ+B(X,Z)) - (V[X,YJZ*B([X,Y],Z)) = VXVyZ + B(X,VyZ) + (VXB)(Y,Z) + B(VXY,Z) + B(Y,VXZ) - VYVXZ - B(YyVXZ) - (VYB)(X,Z) - B(VyX,Z) - B(X,VYZ) - V[X,y]Z - B([X,Y],Z)

AB(Y,Z)X + AB(X,Z)Y = R(X,Y)Z - AB(Y,Z)X + AB(X,Z)Y + (VXB)(Y,Z) - (VyB)(X,Z),

from which

(2.1)

R(X,Y)Z = R(X,Y)Z - AB(Y,Z)X + AB(X,Z)Y + (VXB)(Y,Z) - (V B)(X,Z)

for any vector fields X, Y and Z tangent to M. For any vector field W tangent to M, (2.1) gives equation of Gauss

(2.2)

g(R(X,Y)Z,W) = g(R(X,Y)Z,W) -g(B(X,W),B(Y,Z)) + g(B(Y,W),B(X,Z)).

Taking the normal component of (2.1), we obtain equation of Codazzi

(2.3)

(R(X,Y)Z)` = (VXB)(Y,Z) - (VYB)(X,Z).

69

We now define the curvature tensor RL* of the normal bundle of M

by

(2.4)

RL.(X,Y)V = DXDyV - DyV - D[X,Y]V.

For any vector fields X, Y tangent to M and any vector field V normal to M, from the Gauss and Weingarten formulas, we have

R(X,Y)V = YyV - pYpXV - 0[X'Y] V = DX(-AJY+DyV) - Dy(-AVX+DXV) - (-AV[X,Y]+D[X,Y]V)

= -VX(AVY) - B(X.AVY) - A vX + DXDyV +VY(AVX) + B(Y,AVX) + AD VY - DyDXV

X + AV[X,Y] - D[XIY]V = RL(X,Y)V - B(X,AVY) + B(Y,AVX) - (VXA)VY + (VYA)VX,

that is,

(2.5)

R(X,Y)V = R`(X,Y)V - B(X,AVY) + B(Y,AVX) - (VxA)VY + (V

)VX.

Let U be a vector field normal to M. Then we have

-g(B(X,AVY),U) + g(B(Y,AVX),U)

= -g(AUX,AVY) + g(AUY,AVX) = g([AU,AV]X,Y),

where [AU,AV] = AUAV - AVAU. Thus (2.5) implies equation of Ricci

(2.6)

g(R(X,Y)V,U) = g(R-(X,Y)V,E) + g([AU,AV]X,Y).

70 If R" vanishes identically, then the normal connection: of M is said to

be flat (or trivial). When (R(X,Y)V)` = 0, the normal connection of M is flat if and only if the second fundamental form of m is commutative, that is, [AV,AU] = 0 for all U, V. Particularly, if M is of constant curvature, (R(X,Y)V f = 0, and hence the normal connection of M is flat if and only if the second fundamental form of M is caunutative. If R(X,Y)Z is tangent to M, equation of Codazzi (2.3) reduces to

(vxB)(Y,Z) = (VyB)(X,Z),

(2.7)

which is equivalent to

(2.8)

(VXA)VY = (VYA)VX.

Particularly, if M is of constant curvature, R(X,Y)Z is tangent to M. If M is of constant curvature c, then equation of Gauss reduces to

(2.9)

g(R(X,Y)Z,W) = c[g(Y,Z)g(X,W) - g(X,Z)g(Y,W)] + g(B(Y,Z),B(X,W)) - g(B(X,Z),B(Y,W)).

On the other hand, vie have

g(B(Y,Z),B(X,W)) - g(B(X,Z),B(Y,W)) _ 1[g(B(Y,Z),ea)g(ea,B(X,W)) - g(B(X,Z),ea)g(ea,B(Y,W)] a

_ j[g(AY,Z)g(Aa,W) - g(Aa,Z)g(AaY,W)]. a

Thus (2.9) can be rewritten as

(2.10)

g(R(X,Y)Z,W) = c[g(Y,Z)g(X,W) - g(X,Z)g(Y,W)] + )[g(Aa ,Z)g(Aa ,W) - g(AaX,Z)g(AaY,W)]. a

71

We can give a similar equation to (2.10) for (2.2) by using Aa. Let S be the Ricci tensor of M. Then (2.10) gives

(2.11)

S(X,Y) = (n-1)cg(X,Y) + ITrAag(AaX,Y) - jg(AAX,AaY).

Therefore, the scalar curvature r of M is given by

(2.12)

r = n(n-1)c + I(TrAa)2 - jTrA2, a a

Ea Aa is the square of the length of the second fundamental form of M, which will be denoted by JA12. We also have

IB12

=

I g(B(ei,ej),B(ei,ej)) = JAI2. i,j

We shall give the structure equations of an n-dimensional submanifold M of an m-dimensional Riemannian manifold M. We choose a local field of orthononnal frames el,...,em in 2 in such a way that,' restricted to M, the vectors el,...,en are tangent to M and

hence n+1'"' 'em are normal to M. With respect to this frame field of 2, let w1,...,win be the field of dual frames. We shall make use the following convention on the ranges of indices: A,B,C,... = 1,...,m;

i,j,k,... = 1,...,n;

a,b,c,... = n+1,...,m.

Then the structure equations of 2 are given by dMA=-wBA B ,

wB+wA=O,

dwB=-4CAMB+0B,

11Bj

wCA D

We restrict these forms to M. Then Ma = 0. Since 0 = dwa = -wan Ml, by Cartan's lemma, we obtain

.(2.13)

wi = hijMj,

h J =

hi.

72

We can see that hij = g(Aaeiej). Thus hl

are components of the

second fundamental form a with respect to ea. Therefore Aa can be considered as a symmetric (n,n)-matrix Aa = (hii). Moreover, we have the following equations:

dill=i

(2.15)

dw1 _ _Wk n

(2.16)

(2.17)

i+ wk= 0'

k-

kAw'

(2.14)

wk

w + Std

i

S2

k n wl,

c)

(hikhjl - hi.lhjk)

Rjkl = RjkI +

a = -wc n wb + S2b,

du. ;

sZb =

Rbklwk

n wl

b'

(2.18)

Rbkl

Rbkl + i(hAl - hilhA). C

G.

Equation (2.16) is the local expression of the Gauss equation (2.1) and equation (2.18) is the local expression of the Ricci equation (2.6). The forms (uj

define the Riemannian connection of M and the

forms (i,>b) define the connection induced in the normal bundle of M.

The second fundamental form of M is represented by hajJwJea and is sometimes denoted by its components ha.. The second fundamental form of M is cotmutative if and only if

0 for all

a, b, i and k. If Rbkl = 0 for all a, b, k and 1, then the normal connection of M is flat. The mean curvature vector µ of M is given by (1/n)(Ekha ea). If hlj = (1/n)(Ekhbk)5ij for all a, then M is totally umbilical, and if Ekhbk = 0 for all a, then M is minimal. The covariant. derivative hijk of ha is given by

hijkw = dhij

- hilwj - hljwi + hijwb

a 1 a 1 a 1 a hijklw = dhijk - hljkwi - hilkwj

a

1 + hijk%. b a

- hijlwk

73

If hick = 0 for all indices, then the second fundamental form of M is parallel.

3, LAPLACIAN OF THE SECOND FUNDAMENTAL FOR1

Let M be an n-dimensional sukmanifold of an m-dimensional Rienannian manifold M. Let e1,.. .,en denote an orthonormal basis of

X(M). We denote by the same letters local, orthonormal vector fields on M which extend el,...,en, and which are covariant constant with

m be an orthonormal basis in Tx(M)-.

respect to V. Let en+l,

We now compute the Laplacian of the second fundamental form of M. First of all, from equation of Codazzi (2.3), we have &)c (3.1)

j-17N, )Cn,U)

(OZB)(X,Y) = INi V eiB)(X,Y) = X[(R(ei,X)B)(ei,Y) + (VX(R(ei,Y)ei)`)1 i

+ (Ve (R(ei,X)Y)1)1] + DXDy(TrB), i

where TrB = EiB(ei,ei). On the other band, we obtain

(3.2)

j(R(ei,X)B)(ei,Y) = Y[RL(ei,X)B(ei,Y) - B(R(ei,X)ei,Y)

i

i

- B(ei,R(ei,X)Y)],

(3.3)

1(VX(R(ei,Y)ei)l f = I[(OXR)(ei,Y)ei + R(B(X,ei),Y)ei + R(ei,B(X,Y))ei + R(ei,Y)B(X,ei)]

(3.4)

- IB(X,R(ei,Y)ei)T

i(Vel(R(ei,X)Yf f = 1[0. R)(ei,X)Y + R(B(ei,ei),X)Y + R(ei,B(e;,,X))Y + R(ei,X)B(ei,Y)]' -

B(ei,(R(ei,X)Y)T).

For any vector V normal to M, (3.1), (3.2), (3.3) and (3.4) imply

74 (3.5)

g((V2B)(X,Y),V) = g(D

y(TrB),V) +

[g((OXR)(ei,Y)ei,V)

- g((oe R)(ei,X)Y,V)] + l(g(R-(ei,X)B(ei,Y),V)

i

i

+ 2g(R(ei,Y)B(X,ei),V) + g(R(ei,B(X,Y))ei,V) + g(R(B(ei,ei),X)Y,V) + g(R(ei,X)B(ei,Y),V) - g(AVX,R(ei,Y)ei) - g(AVei,R(ei,X)Y)]

[g(R(ei,X)ei,AVY) + g(R(ei,X)Y,AVei)], 1

where we used the first Bianchi identity. Moreover, we have

(3.6)

- l[g(R(ei,X)ei,AVY) + g(R(ei,X)Y,AVei)] i

[g(R(ei,X)ei,AVY) + g(R(ei,X)Y,AVei)]

- I[g(AVAaX,Y) - TYAag(AVAa,Y) + TYAaAVg(AX,Y) a

- g(Aa a ,Y)], (3.7)

Zg(Rl(eiX)B(ei,Y),V) = Zg(R(ei,X)B(ei,Y),V) + J[g(AaAVAa ,Y) - g(AaAVX,Y)].

a Substituting (3.6) and (3.7) into (3.5), we get

(3.8)

g((V2B)(X,Y),V) = g(yy(TrB),V) + J[g((VXR)(ei,Y)ei,V) i

- g((4e R)(ei,X)Y,V)] + 1[2g(R(ei,Y)B(X,ei),V)

i

i

+ 2g(R(ei,X)B(ei,Y),V) - g(AVX,R(ei,Y)ei) - g(AVY,R(ei,X)ei) + g(R(ei,B(X,Y))ei,V) + g(R(B(ei,ei),X)Y,V)

- 2g(AVei,R(ei,X)Y)] + j[Trag(A A ,Y) - 'ISrAAVg(AaY.,Y) a + 2g(AaAVAaX,Y) - g(AaAVX,Y) - g(AVA2a,Y)]

73

PROPOSITION 3.1. Let M be a sutmanifold of a locally symmetric Riemannian manifold M. If the mean curvature vector of M is parallel, then

g((v2B)(X,Y),V) = 1[2g(R(ei,Y)B(X,ei),V) + 2g(R(ei,X)B(Y,ei),V)

(3.9)

1

- g(AVX,R(ei,Y)ei) - g(AVY,R(ei,X)ei + g(R(ei,B(X,Y))ei,V) + g(R(B(ei,ei),X)Y,V) - 2g(AVei,R(ei,X)Y)J

+ J['IrAg(AVAa ,Y) - TrAaAVg(AaX,Y) + 2g(AaAVAa,Y) a - g(AaAVX,Y) - g(AVAza ,Y)].

If M is minimal, then

(3.10)

g((V2B)(X,Y),V) _ Z[2g(R(ei,Y)B(X,ei),V) x

+ 2g(R(ei,X)B(Y,ei),V) - g(AVX,R(ei,Y)ei) - g(AVY,R(ei,X)ei) + g(R(ei,B(X,Y))ei,V)

- 2g(AVei,R(ei1X)Y)] + )[2g(AaAVAa,Y) - g(AaAVX,Y) a - g(AVAaX,Y)].

We now assume that R(X,Y)Z is tangent to M for any vector fields X, Y and Z tangent to M. Then (3.1) reduces to

(3.11)

(V2B)(X,Y) = J(R(ei,X)B)(ei,Y) + DRDy(TrB). i

Fran (3.2) and (3.11) we have PROPOSITION 3..2. Let M be a submanifold of a Riemannian

manifold 2. If R(X,Y)Z is tangent to M for any vector fields X, Y and Z tangent to M, then

76 (3.12)

(OZB)(X,Y) = [R`(ei,X)B(ei,Y) - B(R(ei,X)ei,Y) i

- B(ei,R(ei,X)Y)] + D

(TrB).

We now give a necessary and sufficient condition for a sutmanifold to have a flat normal connection (cf. Chen [1I). PROPOSITION 3.3. Let M be an n-dimensional sutmanifold of an in-dimensional Riemannian manifold M. Then the normal connection D of M in M is flat if and only if there exist locally rn-n mutually

orthogonal unit normal vector fields ea such that each of the ea is parallel.

Proof. Suppose that there exist locally m-n mutually orthogonal. unit normal vector fields ea such that Dea = 0. Then (2.4) gives R1(X,Y)ea = 0, which shows that the normal connection of M is flat. Conversely, suppose that the normal connection of M is flat.

Then we have

DXDye

X'ea - D[X,Y]ea = 0

a

for any m-n mutually orthogonal unit vector fields ea normal to M.

If we put DXea = ma(X)eb,

then we see that wa = - t

and

0 = [XLa(Y) - Y a(X) - a([X,Y])]eb + [wa(Y) (X) - ma(X)

that is,

dwb='c4b ^ a c'

a

wb=_ a a

%-

c(Y)]eb,

77

Thus we know that there exists an (m-n,m-n)-matrix A = (aa) of functions satisfying

to=A1,

dA=-A2,

where S2 = (wa). This equation has the local expression

dac = JAbwb = -

a

bac

Ab c

bawb.

We put ea = Aaeb. Then ea are also m-n mutually orthogonal unit normal vector fields of M and we have

ab

a

awb

'

where w'a are defined by Dea = w'aeb. This shows that each of ea is parallel in the normal bundle. This proves our assertion.

QED.

Let M be an n-dimensional submanifold of an m-dimensional Rianannian manifold M. For x E M, the first normal space N1(x) is the orthogonal canplement in T{(M)1 of the set

N0(x) = {V a T{(M) : Ay = O.

If, for any vector field V with x e N1(x), we have DXV c N1(x) for any vector field X of M at x, then the first normal space N1(x) is said to be paraileZ with respect to the normal connection. We have the following reduction theorem of codimension (cf. Erbacher [1]). THEOREM 3.].. Let M be an n-dimensional suhnanifold of an m-dimensional complete simply connected space form Aln(c). Suppose

the first normal space N1(x) has constant dimension k, and is parallel with respect to the normal connection. Then there is a totally geodesic (n+k)-dimensional su M.

nifold Mn+k(c) of FRO which contains

78

4. SUBMANIFOLDS OF SPACE FORMS

In this section we shall consider sulmanifolds of space forms and give some examples.

First of all, we prove PROPOSITION 4.1. A totally umbilical submanifold M of a Riemannian manifold M of constant curvature c is also of constant curvature.

Proof. Since M is totally umbilical, we have B(X,Y) = g(X,Y)p

for any vector fields X and Y tangent to M, where p is the mean curvature vector of M. Thus (2.9) implies

g(R(X,Y)Z,W) = (c+jpj2)[g(Y,Z)g(X.W) - g(X,Z)g(Y,W)].

This shows that M is of constant curvature c+JpI2for dim M > 2. If dim M = 2,

1p 2is a constant by equation of Codazzi (2.7).

Consequently, M is of constant curvature.

QED.

We now give applications of the expression for the Laplacian of the second fundamental form.

PROPOSITION U. Let M be an n-dimensional sutmanifold of a space form 1(c). Suppose that the normal connection of M is flat and the mean curvature vector of M is parallel. Then

(4.1)

Kij(Xa X )2,

JAIA12= JVAI2+ # a,i,j

where Kij denotes the sectional curvature of M for the section spanned by ei and ej and Xi denote the eigenvalues of k -with respect to the basis (eil. Proof. Fran the assumption and (3.12) we have

(V2B)(X,Y)

l[B(R(ei,X)ei,Y) + B(ei,R(ei,X)Y]. i

79

On the other hand, we have

g((V2B)(X,Y),V) = g((V2A)VX,Y)

for any vector field V normal to M. Thus we get

g((V2A)VX,Y)

[g(AVY,R(ei,X)ei) + g(AVei,R(ei,X)Y)], i

from which

(4.2)

g((V2A)aei,Aaei)

g(V2A,A) _ a,i

[g(Aaej,R(ei,e.)e.)+g(Aae1,R(ei,ej)Aae.)]. a,i,j

Since the normal connection of M is flat, we see that the second fundamental form of M is carmutative. Thus we can choose el,...,e such that Aaei = Aiei for all a. Therefore, (4.2) becomes

g(V2A,A) _

Kij(Aa-x a,i,j

a)

a,i,j

Since we have

g(V2A,A) _ - IVA12 + fAIAI2,

we obtain (4.1) from the equation above.

QED.

As applications of Proposition 4.2 we have the following propositions.

PROPOSITION 4.3. Let M be an n-dimensional submanifold of

gm

(c)

with flat normal connection and parallel mean curvature vector. If

M is compact and M has non-negative sectional curvature, then the second fundamental form of M is parallel.

80

PROPOSITION 4.4. Let M be an n-dimensional submanifold of b1°(c)

with flat normal connection and parallel mean curvature vector. If IA12 is constant on M, and M has non-negative sectional curvature,

then the second fundamental form of M is parallel.

We now give a few exanples of n-dimensional sutmanifolds discussed above in an m-dimensional Euclidean space Rm with usual inner product (x,y) (cf. Yano-Ishihara [3]). ExcmtpZe 4.1. For integers pl,...,p > 1, pl+ ... + pN = n,

consider Rm given

p1+1

Rm=R

pN

x ... x R'

,

N=m-n.

We put pi+1 Pi S (ri) _ {xi E R

:

(xi)xi) = ri},

i = 1,...,N.

Then the pythagorean product

flSpi(ri) = S 1(r1) x ... x S'N(rN)

_ ((x1) .... xN ) E Rm : xi e Spl(ri),

is an n-dimensional subnanifold Mn of essential codimension m-n in

F. The mean curvature vector u of Mn is given by

u = n p1r12x1 + ... + pNrN2N) at (x1,...,x1) E Mn, which is parallel in the normal bundle of Mn

and the function IAI2 is given by

1A12 = 1/r2 + ... + 1/rN2,

which is constant on Mn. Moreover, the normal connection of Mn is flat.

81

For integers p1,.... pN,p such that p1,...,pN,p > 1,

pl + ... + pN + p = n, consider Rm given pi+l

Rm=R

pN+l

x ... x R

xRP,

N=m-n.

Then the pythagorean product

SP1(rl) x ... x SPN (rN) x RP

_ {(xl,...,xN,x) E Rm

xi E SPi(r

i = 1,...,N, x E Rp)

is an n-dimensional sutmanifold Mn of essential codimension N = m-n

in R. Mn has parallel mean curvature vector and flat normal connection. Moreover IAI2 is'constant.

Exa*ple 4.2. In Rm+l we consider an m-dimensional sphere of radius a > 0:

Sm(a) = {x E Rm+l : (x,x) = a2).

n

For mutually orthogonal unit vectors b1,...,b

in ell a submani-

fold En (r) in e(a) defined by

En(r) = {x a Sm(a)

:

(x,bt) = dt, t = 1,...,m-n)

is called an n-dimensional small sphere of Sm(a) with radius r if (dl,...,dn-n) # (0,...,0), where r2 = a2 - di - ... -

n > 0 and

1 < n < m. 1'(r) is called an n-dimensional great sphere of Sm(a) if

(d1,...,dLn) = (0,...,0), that is, if r = a. If r # a, then E(r) is a totally umbilical suhmanifold of essential codimension m-n in Sm(a), and the mean curvature of En(r) is given by

Iu1 = d/(a(a2--d2)1/2),

,

d2 = dl + ... + dm-n (d > 0).

82

A great sphere L'(a) is totally geodesic in S"(a).

Exmnple 4.3. For integers p1,...pN such that p1,...,pN

It

pl + ... + pN = n, consider

pN+11

R=1 Rp1+1 x ... x R (4.3)

Spl(r1) x ... x S

(rN)

= {(x1,...IXN) a

pi where S (

r

i

= m-n+l.

el

xi a SPi(ri), i = 1,...,N},

pi+l (i = 1,...,N), is an n-dimensional sulinanifold ) C R

Mn of essential codimension m-n imbedded in Sm(a) if

ri = a2. The

mean curvature vector u of Mn relative to Sm(a) is given by

P = n plr12x1 + ... + pNrN2xN) - a 2(x1+

xN)

at (x1,.:.,xN) e-P, which is parallel in the normal bundle of Mn relative to Sm(a). We also have

JA12 = r2(r12 - a 2)2 + ... + r2N(r-2 - a 2)2 + N(N-1)a 2,

which is constant in Mn. It is easily verified that the connection of the normal bundle of Mn is flat. Let E1°-1(r) be an (m-l)-dimensional small sphere of Sm(a) (0 < r < a). For integers p1,...,pN,such that p1, ...,p1, > 1,

p1 + ... + pN, = n, N' = m-n, in Em-1(r) consider an n-dimensional

submanifold Mn of the form

(4.4)

Epi(r1) x ... x EPNI(rN,)

C

p where r2 + ... + r2, = r2 < a2,

1(ri) (i = 1,...,N') is a

pi-dimensional sphere with radius ri and t is constructed in in the same way as that used in constructing in Sm(a) a submanifold

Mn of the form (4.3). Then 0 is an n-dimensional submanifold of

83

essential codimension m-n-1 in

EM-1(r)

and therefore m-n in Sm(a). The

mean curvature vector of ban relative to Sp(a) is parallel,

I A12

relative to SJD(a) is constant and the normal connection of IP relative to Sm(a) is flat.

We now prove the following lcmmia (Erbacher [2], Yano-Ishihara [3]). LEM'1A 4.1. Let M be an n-dimensional submanifold of &"(c) with

flat normal connection and parallel mean curvature vector. Suppose that M has non-negative sectional curvature, and M has constant scalar curvature or M is compact. Then for every point x of M, there exist orthonormal vector fields v1,.... vp (p = m-n) defined in a neighbor-

hood U of x such that (1) va are all parallel in the normal bundle on U; (2)

Aa =

I

where Ik is the (pa,pa) identity matrix, the zero matrix in the upper left-hand corner is of degree p1+...+Pa 1 and Aa's are expressed with respect to their common orthonormal eigenvectors e1,...,en. Note that Aa = 0 if p1+.;.+pa-1 = n and we may assume that Aa = 0 implies Ab = 0

for b > a; (3) Each Aa is constant on U.

Proof. First of all we notice that JA12 is constant because of the scalar curvature of M is constant and the mean curvature vector of M is parallel.

Since the normal connection of M is flat, there exist orthonormal vector fields vi,.... vp (p = m-n) such that Dva = 0 in U. Then we see

that (VXA)a = VX(Aa) = 0 by Propositions 4.3 and 4.4. Thus the eigenvalues of At are constant. If vb = EaOabva, (Oab) an orthogonal matrix

with constant entries, then Dvb = 0 and AL = EaOabAa, where Ab is the second fundamental tensor corresponding to vb. In what follows we will

begin with any a such that Dva = 0 in U and slow that there exists an orthogonal matrix (Oab) with constant entries such that Ab have the

84

desired property (2). The claim is clearly true if all the k's = 0 at x (and therefore by constancy of the eigenvalues a = 0 in U). If this is not the case we distinguish three cases:

(i) all sectional curvatures of M > 0 at x, (ii) all sectional curvatures of M = 0 at x, (iii) at least one non-zero sectional curvature at x and at least one sectional curvature that is zero at x. Suppose that va and U have been chosen such that (1) is satisfied. Thus each Aa has constant eigenvalues in U.

Case (i): From Proposition 4.2 we see that Aa = kaI. We may assume that Al # 0. Let

vi = (Eaava)/(Eaa)1/2 a a and

vb = (Alvb - lbvl)/(a1 + ab)1/2

for b > 1. Then Ai = XI, A # 0 and Ab = 0 for b > 1, vb 1 vi. Use the Cram-,Schmit orthogonalization process on v2,..., p to obtain

vi,...,vp. Then AA = 0 for b > 1.

Case (ii): Let aai

AA =

Aan j expressed with respect to the connon eigenvectors el,...,en of the Aa's. We may assume that A11 # 0. Let

Vj =

a

j

)1/2t

va = (ailva - Aalvl)/(aal + aal)1/2

for a > 1.

85

Again, v2 1 vl. Use the Gram-Schmit orthogonalization process on

v2,..., p to obtain vi,...,v, Then, for a > 2,

(0

0 ...0

0 A' _ a

0

and all # 0. Thus, we may assume that Aal = 0 for a > 1, all # 0. for i > 1, we have Alj = 0 for j > 1. = EaXalXaj = "llXlj If one of the As, for a > 2, is not zero, we may assume that it is Since 0 = Klj

A2 and apply the above argument to v2,...,vp and A2,..., A restricted to the span {e2,...,en}. We obtain A2j = 0 for j > 2 and Aa2 = 0 for a > 2. It is now clear that an induction argument will work.

Case (iii): Order el,...,en so that Klt > 0 for 2 < t < pl, and Klt = 0 for t > pl. Then (4.1) implies that aal = Aat for 1 < t < pl. Define va as in case (ii). We see that we may assume that Aat = 0 for 1 < t < pl, 2 < a < p. Then Klt

0 for t > pl and thus = A1111t = alt = 0 for t > pl. If Kij # 0 for same i,j > pl, we repeat the above argument applied to v2,...,vp and A2...,Ap restricted to the span {ep +1,...,ea}. If Kij = 0 for all i,j > p1, we apply the argument of case (ii) to v2,...,vp and A2,...,Ap restricted to the span {epl+l,...,en}. In either case we obtain the desired foam for Al and

A2. It is clear that an induction argument will work.

QED.

We shall prove the following theorems (Erbacher [2], Yano-Ishihara [3]).

THEOREM 4.1. Let M be an n-dimensional canplete suteanifold of Rtn with non-negative sectional curvature. Suppose that the normal

connection of M is flat and the mean curvature vector of M is parallel. If the scalar curvature of M is constant, then M is a sphere Sm(r), n-dimensional plane Rn, a pythagorean product of the form

(4.5)

S 1 (rl) x ... x 5

(rN),

E pi = n, 1 < N < m-n,

as or a Pythagorean product of the form

(4.6)

S l(r1) x ... x S (rN) x RP,

EPi+p = n, 1 < N < m-n.

Proof. Let va be chosen as in lemma 4.1. We may assure that Ab # 0 for 1 < b < c-1 and Ab = 0 for b > c (it all X. = 0, then M is totally geodesic).

Define distributions

TV... ITk by

Tb(x) = {X a TA(M) : AbX = XbX}

for b < c,

Tc(x) = {X E TA(M) : AaX = 0, 1 < a < m-n}.

Let pa = dim Ta (pa may be zero). Assume that M is simply connected and complete. Then each Ta is globally defined (for v e TX(M)r, the parallel translate of v with respect to the normal connection is independent of path if the normal connection is flat and M is simply connected). Each Ta has constant dimension and is differentiable (the eigenspaces of Aa have constant dimension, and thus we may find differentiable orthonormal eigenvectors). The Ta's are orthogonal to each other and

Tx(M) = T1(x) ® ... S 7C(x)

(orthogonal direct sum).

Let Y be in Ta. For any vector field X tangent to M we have

AaVXY = VX(Aa ) - (oXA)a = aaVXY.

Thus we have VXY E Ta. Therefore, each T. is parallel. For each point x of M, let M a be the maximal integral submanifold of Ta through x. Then M is the Riemannian product

M1 X ... X

c

137

If Pa = 1, then the image of M a is a circle since we have assumed that M is spimply connected and complete. If pa > 1, then the curvature

tensor of M a is the restriction of the curvature tensor of M, since

M a is totally geodesic in M. Hence, the sectional curvature of M a is PC PC. constant and equal to A. On the other hand, we see that M = R Thus, M is a product of small spheres and possibly a great sphere. Clearly, the corresponding local result is true if we dopa not assume completeness since we only used completeness to obtain M

as the

entire sphere.

The second fundamental forms and the normal connection forms of our submanifold with respect to va, chosen as in Lemma 4.1, are the same as Example 4.1. Thus, by the fundamental theorem of sutmanifolds (cf. Kobayashi-Nemizu [ ;p.45] for the case p = 1), we see that the sutmanifold M is of the form as in the theorem up to a rigid motion. If M is not simply connected, let M be its simply connected Riemannian covering manifold and let IT be the covering map. Then the composition mapping p of M into fin under n and the immersion of M

satisfies the assumptions of the theorem and, by the above, there exists an isometry 0 of Rm such that ip = '°i, i being the immersion of Example 4.1.. Tbus, M is immersed as the suhnanifold of the form as in the theorem. This completes the proof of the theorem.

QED.

Remark. If M is a Pythagorean product of the form (4.5) or (4.6), then M is of essential codimension N.

THEOREM 4.2. Let M be an n-dimensional canpact sutsnanifold of R1D with non-negative sectional curvature, and suppose that the normal connection of M is flat. If the mean curvature vector of M is parallel, then M is a sphere S'(r) or a Pythagorean product of the form (4.5), which is of essential codimension N. If the ambient manifold 2 is a sphere SjD(a), we just imbedded

Sm(a) in R°l as Example 4.2. Then M regarded as a submanifold of

el having the properties in Theorem 4.1 by the following lemma, which will be proved easily.

Be

LEMMA 4.2,

Let M be an n-dimensional subnwifold of SS(a) such

that (1) JA12is constant on M; (2) The mean curvature vector u of M is parallel; (3) The normal connection of M is flat.

Then, if we consider Sm(a) as isometrically immersed in Ftl f conditions (1), (2) and (3) are also satisfied. (of course, JAI 2, 11

and the normal connection are now taken with respect to M in Rel.) From Theorem 4.1 and Leama 4.2 we have (Erbacher [2], YanoIshihara [3]) THEOREM 4.3. Let M be an n-dimensional complete sulinanifold of

Sm(a) with non-negative sectional curvature. Suppose that the mean curvature vector of M is parallel and the normal connection of M is flat. If the scalar curvature of M is constant, then M is a small

sphere En(r), a great sphere En(a) or a pythagorean product of a certain number of spheres. Moreover, if M is of essential codimension m-n, then M is a pythagorean product of the form (4.3) with r2 + ... + rN = a, N = m-n+l, or of the form (4.4) with r2 + ... + rN, = r2 < a2,

NO = m-n. If M is a pythagorean product of the form (4.4) with ri + ... + rN = r2 < a, N = m-n, then M is contained in a snail sphere

of Sm(a). THEOREM 4,4. Let M be an n-dimensional canpact sutmanifold of Sm(a) with non-negative sectional curvature. Suppose that the normal connection of M is flat and the mean curvature vector of M is parallel. If M1 is of essential codimension m-n, then we have the same conclusion

as that in Theorem 4.3. Remark. If Mn in Sm(a)

atisfies the condition of Theorem 4.3

or 4.4 and if Mn is of essential codimension s less than rn-n, then

Mn is contained in a great sphere Sn+s(a) of Sm(a) (see Theorem 3.1).

5. MINIMAL SUBMANIFOLDS

Let M be an n-dimensional subnanifold of Sm(a) and satisfy the conditions in Theorem 4.3 or 4.4. Then the mean curvature vector of M is given by

P = n plul + ... + PNUN)

where y1I...' N are distinct vectors of eigenvalues, and p1,...'

N

the multiplicities of p1,...,uN respectively. Thus M is minimal if

and only if

P =p1111+... +PNUN=O.

(5.1)

Therefore, the vectors p1,.... UN are linearly dependent. Hence we see

that M is of essential codimension N-i if M is a pythagorean product of the form (4.3). Since we have

(5.2)

.1

"

0 = n plr12x1 + ... + pNrN2XN) - a 2(x1 + ... + xN),

where ri = a(pi/n) 1/2 (i = 1,...,N). Therefore, Theorems 4.3 and 4.4 imply the following theorems (Yano-Ishihara [3]).

THEOREM 5.1, Let M be an n-dimensional complete minimal sutmanifold of Sm(a) with non-negative sectional curvature, and suppose that the normal connection of M is flat. If the scalar curvature of M is constant, then M is a great sphere of Sm(a) or a pythagorean product of the form

(5.3)

SP1(r1) x ... x SPN(rN),

Epi = n, 1 < N < m-n+1

with essential codimension N-1, where ri = a(pi/n) 1/2 (i = 1,...,N).

so THEOREM 5.2. bet M be an n-dimensional compact minimal suksaanifold of S1n(a). If M has non-negative sectional curvature and the

normal connection of M is flat, then we have the same conclusion as that in Theorem 5.1. Example 5.1. Let n and p be two positive integers such that n > p. We put

M P,n-P We imbed

p,n-p

= S1((P/n)1/2) x SP-P(((n-p)/n)1/2).

into Sn+l =

Sn+1(1)

as follows. Let (xl,x2) be a point

of Mp,n-p where x1 (resp. x2) is a vector in RP+1 (resp. Rn-P+1) of length (p/n) 1/2 (resp. ((n-p)/n) 1/2 ). We consider (xl,x2) as a unit Rp+l x Rn-p+1. vector in Rn+2 = Then p,n-p is a minimal hypersurface

of

Sn+1,

p,n-p is called a Clifford minimal hypersurface. In

particular, if n = 2 and p = 1, M1 1 is a flat minimal 'surface of S We call this minimal surface the Clifford torus.

Example 5.2. Let (x,y,z) be the standard coordinate system in R3 and (u1,u2,u3,u4,u5) be the standard coordinate system in R5. We consider the mapping defined by

ul =

(1/3)1/2yz,

u2 = (1/3)1/2zx,

u4 = (1/12)1/2(x2 - y2),

u3 = (1/3)1/2xy,

u5 = (1/6)(x2 + y2 - 2z2).

This defines an isometric immersion of

into S4 = S4(1). Two

points (x,y,z) and (-x,-y,-z) of S2(v) are mapped into the same point of S4, and this mapping defines an imbedding of the real projective plane into S4. This real projective plane imbedded in S4 is called the Veronese surface. It is minimal surface of S4.

Let M be an n-dimensional minimal submanifold of

(c) (m-n = P)

Then (3.10) implies

(5.4)

JAIA12 = ncIA12 -

E (TrAaAb)2 + I Tr[Aa,Ab]2 + IVA12. a,b a,b

91

We need the following lemma (Chern-do Ca

-Kobayashi [1]).

Lrrw 5.1. Let A and B be symmetric (n,n)-matrices. Then

-Tr(AB - BA)2 < 2TrA2TrB2,

and the equality holds for non-zero matrices A and B if and only if A and B can be transformed sinnltaneousely by an orthogonal matrix into scalar multiples of A and B respectively, where

01

1

0

0

A=

1

l

0

s=

,

0' 0J

0

-1

0

0

Moreover, if A1, A2 and A3 are (n,n)-symmetric matrices and if

-T.r(AiA. - A.Ai)2 =

1 < i,j < 3,

then at least one of the matrices Ai must be zero. Proof. We may assume that B is diagonal and we denote by b1,.... bn the diagonal entries in B. 7ben we have

-Tr(AB - BA)2 =

a2 (bi - bk)2, i#k

where A = (ai3). Since (b1 - N)2 < 2(bi + bk), we obtain

-Tr(AB - BA)2 =

E a2 (b i#k

A i

- bk)2< 2 E a x(b2 + q2) i#k

< 2( al2.k)(lbi) = 2TrA2TrB? i i,k

Now, assume that A and B are nonzero matrices and that the equality holds. From the second equality in the inequalities above, we have

a,, _ ... = aan = 0 and bi + bk = 0 if i k # 0. Without loss of generality, we may assume that a12 # 0. Then bl = -b2. From the third

92

equality, we now obtain b3 = ... = bn = 0. Since B # 0, we must have bi = -b9 # 0 and we conclude that aik = 0 for (i,k) # (1,2). Tb prove the last statement, let A1, A2, A3 be all nonzero symmetric matrices. Fran the second statement we have just proved, we see that one of these matrices can be transformed to B a scalar multiple of A as well as to a scalar multiple of B by orthogonal matrices. But this is impossible

since A and B are not orthogonally equivalent.

QED.

From Lemma 5.1 we obtain

(5.5)

1 (TrA Ab)2 - I Tr[A,Ab]2 < I(TrA2)2 + 2 TrAa2TrAb a a a a,b a a,b a b = 2(1TrA2a)2 - J(TrAa)2 a a

_ (2 - 1/p)(JTrA2)2 -

a

a

1

(TrA2 - TrAb)2

pa>b 1

a

(2 - 1/p)IA,4,

where p = m-n and we used a frame {ea} of TX(V. for which TrAaAb = 0 if a 0 b, that is, the symmetric (p,p)-matrix (TrAaAb) can be diagonalized. From (5.4) and (5.5) we have

IVAI2 - JAIAI2 < [(2 - 1/p)IAI2 - nc]IAI2.

(5.6)

Consequently, we have (Simons 11])

THEOREM 53, Let M be an n-dimensional compact minimal submanifold of 9

(c). Then

0 < IMIVAI2*1 < IM[(2 - 1/p)IAI2 - nc]IA12*1,

where p = min.

THEOREM 5,4, Let M be an n-dimensional compact minimal sulmanifold of &

(c) (c > 0). Then either M is totally geodesic, or

IAI2 = nc/q, or at some point of M, IAI2

nc/q, where q = 2 - 1/(m-n).

93

Proof. Suppose that JA!2 < nc/q everywhere on M. Then Theorem 5.4 implies that the second fundamental form of M is parallel and hence IA12 is constant. Thus IAI 2 = 0 and M is totally geodesic or IA!2 =

nc/q. Except for these possibilities, IA!2 > nc/q at some point of M. QED.

We shall consider the case that 1AI2= nc/q. In this case the second fundamental form of M is parallel. Therefore, we first study the submanifolds with parallel second fundamental form (Yano-Kon[ll}).

THEOREM 5.5, Let M be an n-dimensional sukmanifold of KP(c) (c > 0) with flat normal connection. If the second fundamental form of M is parallel, then the sectional curvature of M is non-negative. Proof.

Since the normal connection of M is flat, the Aa's are

simultaneously diagonalizabe at each poin of M. Let ai (1 < i < n, n+1 < a < m) be the eigenvalues of Aa corresponding to eigenvectors e1,...,en. We can choose a local field of orthonormal frames en+1,.... m in the normal bundle such that Dea = 0 (a = n+l,...,m).

Since the second fundamental form of M is parallel, we have

AaR(ei,e.)ei = R(ei,ej)Aaei = X R(ei,e.)ei.

Thus the eigenspace Ta corresponding to as has the property that

R(ei,e.)Ta C Ta

for all a = n+l,...,m.

If Xa # a for some a, then we have

g(R(ei,e3)ei,ej) = 0,

which means that the sectional curvature K. of M with respect to the section spanned by ei, ej vanishes. On the other hand, fran equation

of M is given by

of Gauss (2.9), the sectional curvature Kii

KiJ = c +

tai a a

94

If ai # a for some a, then Kij = 0, and if Xa = a for all a, then Kij > 0. Thus the sectional curvature Kij with respect to the section spanned by ei, ej is non-negative. In the following, we prove that the sectional curvature g(R(X,Y)Y,X) for any orthonormal vectors X and Y is non-negative.

Fran (2.10) we have

g(R(X,Y)Y,X) = c + Jg(AaX,X)g(AaY,Y) - jg(Aa,Y)2.

a

a

We now put X = Eiaiei and Y = Eiaiei. Then we see that

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

aiajsisjaal

I a,i,j

a,i,j

= c + 1I (K.. - c)als -

'j

1j

(Kij - c)aiajsisj.

1J '

Since we have Ei,jaia = 1 and EiaiBi = 0, the equation above becanes

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

JKij aifij -

lJK ija

ajsisj

Kij(aisj - ji)2.

_ i>j

We have already seen that K.. > 0, and hence g(R(X,Y)Y,X) > 0. Therefore, the sectional curvature of M is non-negative.

QED.

PROPOSITION 5.1. Let M be an n-dimensional minimal sukmanifold of ho(c) with parallel second fundamental form. (1) If c < 0, then M is totally geodesic; (2) If C > 0 and 1A12 > n(m-n)c, then Ri = 0. Proof. Since the second fundamental form of M is parallel, we see that JA12is constant. Thus (5.4) implies

0 < -

I Tr[Aa,Ab]2 = ncIA12 - I (TrAaAb)2. a,b a,b

We consider the symmetric (m-n,m-n)-matrix (TrAaAb). Then we can choose a suitable basis en+l,.... em for which the matrix (TrAaAb) can be

95

assumed to be diagonal. Thus we have

(5.7)

0 < a,b

Tr[Aa,Ab]2 = ncIA12 - m-n ITrAa)2 a

mI <

E (TrAa - TrAb)2 a>b

n[n(m-n)c - JAI2]JAI2.

If c < 0, (5.7) implies that 1A12 = 0 and hence M is totally geodesic. If c > 0 and JAJ2 > n(m-n)c, then (5.7) shows that [Aa,Ab] = 0 for all a and b, which means that the normal connection of M is flat, that is,

R`=0.

QED.

We prove the following theorem (Yano-Kon [1l]).

THEOREM 5,6. Let M be an n-dimensional complete minimal sutmanifold of Sm with parallel second fundamental form. If IAI2 > n(m-n), then M is a Pythagorean product of the form

SP1(r1) x ... x SPN(rN),

rt = (Pt/n)1/2 (t = 1,...,N),

where 1 < p1,...,pN < n, p1 + ... + pN = n, m-n = N -1. Proof. Fran Theorem 5.5 and Proposition 5.1, M has non-negative sectional curvature and flat normal connection. Thus Theorem 5.1 implies that the essential codimension of N is n-i. On the other hand, by (5.7), we have TrAa = TrAi for all a, b. Since 1AI2 = n(m-n), M is not totally geodesic and hence TrAa = IA12/(m-n) 0 0. Thus we must have m-n = N-1. The other statements are trivial consequences of Theorem 5.1.

QFD.

Frcm Theorem 4.3 and Theorem 5.5 we have the following theorem

(Yano-Kon [y] ). THEOREM 5.7. Let M be a complete n-dimensional submanifold of SP with flat normal connection. If the second fundamental form of M is parallel, then M is a small sphere, a great sphere or a pythagorean product of a certain numbers of spheres. Moreover, if M is of

98

essential codimension m-n, then M is a pythagorean product of the form

S 1(r1) x ... x S

Eri = 1, m-n = N-1,

(rN),

or a pythagorean product of the form

Sp1(r1)

x ... x

SPN'

(rN,) C 5m-1C Sm,

Eri = r2 1, m-n = N1.

Remark. The local version of Theorem 5.6 and Theorem 5.7 is also true (see the proof of Theorem 4.1). See also Waerden [1].

Let M be an n-dimensional minimal submanifold of Sm with I A12 = n/(2 - l/p), p = m-n. Then the second fundamental form of M is parallel.

Moreover, (5.5) implies

(5.8)

TrA2 = TN

for all a, b.

From Lemma 5.1 we may assume that

(5.9)

Aa = 0

for a = n+3,...,m.

Since M is not totally geodesic, (5.8) and (5.9) imply that p = m-n < 2. Let p = 1. Then M is a hypersurface of Sm. Then the normal

connection of M is flat and 1A12 = n. Thus, from Theorem 5.6, M is locally a Clifford minimal hypersurface

St((t/n)1/2)

Mt,n-t =

x Sn-t(((n-t)/n)1/2).

We next consider the case that p = 2. Fran Lemma 5.1 we have

An+1 = X',

An+2=UB,

X, U#0,

where A and B are defined in Lemma 5.1. In other words,

97

w1+1 = awl,

wi+1 = 0 for i = 3,...,n,

w2+1 = Aw1,

n+2 = uw 1

n+2

- -Vw

2

wn+2 = 0 for i = 3,...,n.

Since the second fundamental form of M is parallel, (2.19) implies

dhij = h

(5.10)

a w + hkjw1

- hbjwb.

Setting a = n+1, i = 1 and j = 2, we see that dA = dh121 = 0, that is, A is constant. Setting a = n+1, i = 1 and j > 3, we see that

w

= 0

for j > 3.

Setting a = n+1,. i = 2 and j > 3, we see that

w

= 0

for j > 3.

Similarly, setting a = n+2, and i = j = 1, we see that p is a constant.

Thus, if j > 3, we obtain

0 = dw _ -wk A w + w1 A wj = w1 A wi.

Since w1,...,wn are orthonormal, w1 A wj = 0 implies w3 = 0 for j > 3.

This shows that dim M = 2 and hence dim M = m = 4. On the other hand, from (5.8), it follows that A2 = j12. Since 4/3 = 1AI2 = 4A2, we have

A2 = u2 =1/3. Thus we may assume that -A = u = (1/3)1/2. Setting a = 3 and i = j = 1, we obtain

w4 = (2A/u)wi = -2wi.

The curvature of M is given by

f22

= w1 A w2 + W3 A w2

W4

+

A w2 =

(1-Az-u2)w1

A w2 =

3 1A w2.

99

From these considerations, the connection form (wB) of 1, restricted

to M, is given by

-

1

0

w2

1

uw2

w2

0

uw

awl

Awl

0

2w1

-awl

awl

2wi

0

l

V

2

-a = u = (1/3) 1/2

This coincides the form of the Veronese surface in Example 5.2 (see Chern-do Carmo-Kobayashi [1]). Therefore, M coincides locally with the Veronese surface. Consequently, we obtain the following theorem (Chern-do Carmo-Kobayashi [1]).

THEOREM 5.8. The Veronese surface in S4 and the Clifford hypersurface Mt,n-t in Sn+l are the only compact minimal suYmanifold of dimension n in Sn+p satisfying JA12= n/(2 - 1/p). The corresponding local result also holds. If the normal connection of a minimal sut=nanifold M of Sm is

flat, then we have

W.AJA12 = njAI2 -

E (TrAaAb)2 + JDA12. a,b

If M is compact, then fM[IAl2 - n]IA12*1 > 0.

Using this, Kenmotsu [4] proved THEOREM 5.8. Let M be an n-dimensional compact minimal sutxnani-

fold of Sm with flat normal connection. If 1A12 = n, then there exists an (n+l)-dimensional sphere Sn+1 containing M as a Clifford minimal hypersurface.

N EXERCISES

A. TOTALLY UMBILICAL SU31ANIFOLDS: let Ml(k) be a singly connected complete space form of constant curvature k (see §3 of Chapter I). We have (cf. Chen [1])

THEOREM. A totally umbilical submanifold II in If(k) is either totally geodesic in 1P(k) or contained in a hypersphere of an (m+l)dimensional totally geodesic suYmanifold of Mn(k).

B. SUBMANIFOLD WITH R(X,Y)R = 0: Nomizu [2] studied the effect of the condition

(*)

R(X,Y)R = 0 for any tangent vectors X and Y

for hypersurfaces M of the Euclidean space

Rn+l,

where R denotes the

Riemannian curvature tensor and R(X,Y) operates on the tensor algebra at each point as a derivation, and prove the following THEOREM 1, let M be a connected and conplete hypersurface of

Rn+1

so that the type number is greater than 2 at least at one point. SkxRn-k, If M satisfies the condition (*), then it is of the form M =

where Sk is a hypersphere in Rk+1 of Rn+1 and Rn-k is a Euclidean subspace orthogonal to

Rk+1.

Ryan [1] treated the same condition for hypersurfaces of spaces of non-zero constant curvature. Moreover, Tanno [3] discussed the effect of the condition

(**)

R(X,Y)Q = 0 for any tangent vectors X and Y

for hypersurfaces of the Euclidean space, where Q denotes the Ricci operator. Obviously, the condition (*) implies the condition (**).

If the ambient manifold is of non-zero constant curvature, the condition (*) and (**) are equivalent. Tanno-Takahashi [1] proved

100

THEOREM 2. Let M be a connected and coaplete hypersurface of a sphere Sn+1(c) (n 5 4) of curvature Z. Then M satisfies the condition (**) if and only if M is one of the following: (1) M = Sf(c), great sphere; (2) M = Sn(c), small sphere, where c > c; (3) M = SS(cl) x Sn-p(c2), where p,n-p > 2 and cl > c, c2 > c

such that

c-1 + c21 =

c

1

C. EINSTEIN HYPERSURFACES: Fialkow [1] proved a local classification theorem for Einstein hypersurfaces of space forms. THEOREM, Let M be a hypersurface of a space form Mn+l(c) (n > 2).

Let M be Einstein: S = pg. If p > (n-1)c, then M is umbilical and M is of constant curvature p/(n-1). If p = (n-1)c, then M is of constant curvature c. If p < (n-1)c, then c > 0, p = (n-2)c and M is locally Mp(P-2 ) x Mn-p( nnp2lc), where 1 < p < n-1.

(See also Thomas [1], Cartan (see Thomas [2]), Ryan [1].)

D. MINIMAL IfMERSIONS: Let M be a Riemannian manifold and x : M

> Rn+p be an isometric immersion of M in a Euclidean

(n+p)-space Rn+p. Then we have (Tsunero Takahashi [1])

THEOREM 1, An isometric immersion x : M

> Rn+p is minimal

if and only if Ax = 0. THEOREM 2. If an isometric immersion x : M

> Rn+p satisfies

Ax = Xx for some constant A # 0, then A is necessarily positive and x realize a minimal immersion in a sphere

Sn+p-l

of radius (n/a)1/2

in Rn+p. Conversely, if x realizes a minimal immersion in a sphere of radius a in Rn+p, then x satisfies Ax = Xx up to a parallel displacement in Rn+p and A = n/a2. For minimal submanifolds of space forms, the Simons' type formula and its applications were studied by many authors. Yau [1] proved the following

101

THEOREM 3, Let M be an n-dimensional minimal submanifold of a space form Mn+p(1). If M has non-negative curvature and JA12 = np, then M is an open piece of the product IIS 1((mi/n)1/2),

I

m i = n.

i=1

Matsuyama [3] proved the following THEOREM 4, Let M be an n-dimensional (n > 4) minimal submanifold of SJII which has at most two principal curvatures (If exactly two are

distinct, then we assume these multiplicities > 2) in the direction of any normal. Then the second fundamental form is parallel and JA!2 holds IA12 = 0 or n < 1A12 < n2/4.

ISOPERIMETRIC SECTION: Let M be an n-dimensional submanifold of an m-dimensional Riemannian manifold M. A parallel section

(# 0)

of the normal bundle of M is called an isoperimetric section if

TrA = constant # 0 and a minimal section if TrA = 0. If M admits an isoperimetric section, then we can consider the Laplacian for A

2 and Snyth [3] proved THEOREM 1, Let M be an n-dimensional compact irreducible submani-

fold of a space formm(c) with non-negative sectional curvature. If the normal bundle of M admits an isoperimetric (resp. minimal) section then M lies in a small (resp. great) hypersphere orthogonal to this section.

THEOREM 2. Let M be an n-dimensional compact irreducible sutznani-

fold of a space form IP+p(c) with non-negative sectional curvature. If the mean curvature vector of M is non-zero parallel, then M must lie minimally in a hypersphere

Sn+p-1

of

+p(c) of constant positive +P-l.

curvature. When p = 2, M is imbedded as a great hypersphere of SP

F. PRODUCT IMMERSION: Let Mi be a compact connected Riemannian manifold of dimension ni > 2 (1 < i < p), and M the Riemannian product manifold M1 x M2 x ... x Mp. Fbr isometric immersions of M into a

Euclidean mrspace Rm, Moore [1] proved

102 THEOREM 1. For 1 < i < p, let Mi be a connected non-flat Riemannian manifold of dimension ni, and let M. be a connected flat Riemannian

Let I x Ml x ... x M be the Riemannian

manifold of dimension

Xa

product manifold, anEuclidean space of dimension m = no + Eni + p. Then any isometric immersion

f . MOxM1x...xM.p

>Rm

is a product immersion, that is, there exist isometric immersions n.+1 i+1 (1 < i < p) and a decomposition of Rm into a fi : M; > R +1 n +1

Riemannian product Rim = RU X R

x ... x R p

so that

f(m0,ml,...,mp) = (fO(cxb),fi(ml),...,fp(m,)), mi e Mi (0 < i < p), n

f0 : M0

> R 0.

THEOREM 2. For 1 < i < p, let Mi be a compplete connected Riemanni-

an manifold of dimension ni > 2, M= M1 x ... x

P

the Riemannian

product, and Rml a Euclidean space of dimension m = Eni + p. Then

any isometric immersion f : M

> Rm satisfies at least one of the

following conditions: (a) It is a product of hypersurface immersions; (b) It carries a complete geodesic onto a straight line in Rm. Alexander-Maltz [1] consider the following condition:

(*) No Mi contains an openlsubmanifold which is isometric to the

Riemannian product R 1

x (-E,E)

Then THEOREM 3. Let M1,...,Mk be connected complete non-flat Riemannian manifold satisfying condition (*). Then any k-codimensional isometric immersion of the Riemannian product M = Ml x ... x Mk in Euclidean space is a product of hypersurface immersions.

G. PLANER GEODESIC IMMERSIONS: Let M be an n-dimensional connected complete Riemannian manifold and M an m-dimensional connected complete Riemannian manifold. An isometric immersion of M into 2 is called a planer geodesic immmersion if every geodesic in M is mapped locally

103

into a 2-dimensional totally geodesic suhi nifold of M. Then the immersion is isotropic in the sense of O'Neill [1], that is, the second fundamental form B of the immersion satisfies IB(X,X)12 = X2 for all unit vector X where A is a function, and the second funda-

mental form is parallel. Hong [1] studied planer geodesic immersions into Euclidean space and proved THEOREM 1. Let f : Mn

> Rn+P be a planer geodesic immersion.

Then the sectional curvature of M is 1/4-pinched except for the totally geodesic case and moreover if M has constant positive sectional curvature, then f(M) is an n-dimensional sphere or a Veronese manifold.

Sakamoto [2] studied the planer geodesic immersions into any space forms and constructed models of planer geodesic immersions.

Moreover, the classification theorem of planer geodesic immersions was obtained. In [2] Sakamoto showed that THEOREM 2. Let f : Mn

> Mn+p(c) be a planer geodesic

immersion. Then Mn is isometric to a symmetric space of rank one or an Euclidean space and the immersion is rigid.

H. ISOTROPIC IMMERSIONS: An isotropic immersion is an isometric immersion such that IB(X,X)12 = A2 for all unit vector X where A is a function. If the second fundamental form B satisfies (0xB)(Y,Z) _ (V B)(X,Z), then A is constant.

Itoh-Ogiue [1] proved the following theorems. THEOREM 1. Let M be an n-dimensional space form of constant curvature c, and rd be an (n+4n(n+1)-1)-dimensiona1 space form of

constant curvature E. If c < c, and M is an isotropic sulmanifold of M with parallel second fundamental form, then c = (n/2(n+l))c, and the immersion is rigid.

THEOREM 2. Let M be an n-dimensional space form of constant curvature c, and M be an (n+jn(n+1)-1)-dimensional space form of constant curvature E. If c < c,'and M is an isotropic submanifold of M, then c = (n/2(n+1))c, and the immersion is rigid provided that n<4.

104

CHAPTER III

COMPLEX MANIFOLDS

In this chapter, we study the various almost complex manifolds and complex manifolds. In §1, we first prepare some algebraic results on real and complex vector space, and we give the fundamental properties of almost complex manifolds and complex manifolds. In §2, we give

sane examples of almost complex manifolds and complex manifolds. §3 is devoted to the study of Hermitian manifolds. We give the definitions of Kaehlerian manifolds and nearly Kaehlerian manifolds. In §4, we discuss Kaehlerian manifolds. We define the holomorphic sectional curvature of a Kaehlerian manifold and give the typical examples of

complex space forms, that is, Kaehlerian manifolds of constant holomorphic sectional curvature. In §5, we consider nearly Kaehlerian manifolds. We give the condition for a nearly Kaehlerian manifold to be Einstein. The main theorem of this section is that there does not exist non-Kaehlerian nearly Kaehlerian manifold of constant holomorphic sectional curvature when the dimension of the manifold is not 6. In the last §6, following S. Ishihara [3], we study quaternion manifolds by using tensor calculas. In this chapter, we refered to Goldberg [1]. Kobayashi-Nomizu [2],

Matsushima [1] and Yano [5]. 1. ALMOST COMPLEX MANI FOLDS AND COMPLEX MANI FOLDS

First of all, we prepare some linear algebraic results on real and complex vector spaces, which will be applied to tangent spaces of

manifolds.

105

Let V be a vector space over R. The compZexification Vc of V is the set of all symbols X + iY (i = r--I), X,Y C V satisfying the follow-

ing conditions: We suppose that X + iY = X' + iY' if and only if X = X' and Y = Y'. The sum of X + iY and X' + iY' is defined to be

(X + iY) + (X' + iY') = (X + X') + i(Y + Y').

The product of a + ib e C and X + iY is defined by

(a + ib)(X + iY) = (aX - bY) + i(bX + aY).

Then if becomes a vector space over C. By identifying X + i0 of if with

X of V we may consider V C if. For Z =X+ iY a Vc we put Z = X - iY, which will be called the conjugate of Z with respect to V. The eompZes conjugation in if is a conjugate linear map from if to if defined by

>Z, i.e.,Z+W=Z+W andaZ=TZ(AcC).

Z

We assume that V is of n-dirrensional. Let {e1,...,en) be a .basis

of V over R. For any vectors X, Y of V we put X = Ea3ej, Y = EbJej.

Then

X + iY = j(adej + ibdej) = 3AJej,

where we have put A3 = a3 + ib3, j = 1,...,n. If we consider el,...,en as elements of if, they are linearly independent over C. In fact, if

EaJej = 0 , then EaJej = 0 and EbJej = 0. Thus we have aJ = bJ = 0 for all j, and hence A3 = 0 for all j. Therefore e1..... en are linearly

independent over C. From this we see that {el,...,en) is a basis of

if. A linear endamorphisn J of real vector space V satisfying J2 =-I is called a complex structure on V, where I stands for the identity transformation of V.

Let V be a real vector space with a complex structure J. We can define the product AX of a complex number A = a + ib and an element X

of V by

AX= (a+ib)X=aX+bJX.

106

Then we can consider V as a vector space over C. Clearly, the real dimension of V must be even. Conversely, given a complex vector space V of complex dimension n, let J be the linear endomorphism of V defined by JX = iX for all X e V. If we consider V as a real vector space of real dimension 2n, then J is a complex structure of V. Now let V be a real vector space with a complex structure J. 'Then,

we can extend J to a complex linear endoarphism of if, denoted also by J, by setting

J(X + iY) = JX + iJY.

Clearly, J2 = -I. In a 2n-dimensional real vector space V with complex structure J, there exist elements X1,...,Xn of V such that (X1,...,Xn,JX1,...,JXn}

forms a basis of V. We set

Zk = '(Xk - iJXk),

Then {Z1,.

Zk = '(Xk + iJXk),

k = 1,...,n.

Zk,Z1..... Zn} forms a basis of if, and we have

JZk = iZk,

JZk = -iik,

k = 1,...,n.

Hence if we set

V1'0 = {Z a if : JZ = iZ},

V0'1 = {Z a if : JZ = -iZ},

then we have the complex vector space direct sum:

if = V1'0 + VO'l.

We notice that V1'0 = VC'l. For any Z e if we have

Z = }(Z - iJZ) + #(Z + iJZ).

107

The first term of the right hand side of this equation belongs to V1'0 and the second belongs to V°'1. Therefore, we easily see that

VO'1 = {X + iJX : X C V).

V1'O = {X - iJX : X C V},

We denote by V* the dual space of a real vector space V. Then we can construct the ccmplexification V*c of V*. Naturally, we can identify V*c and the dual space Vc* of the ccaplexification Vc of V. A complex structure J on V induces a complex structure on V*,

denoted also

by J, by setting

<JX,X*> = <X,JX*>,

X C V, X* C V*.

Then we have the following decocrposition

c V* = V1'O + V0,1,

where

U = {X* E V*c : <X,X*> = 0 for all X E VO'1},

V1

VO1 = {X* E V*c : <X,X*> = 0 for all X E V1'0J. ,

Let M be a real differentiable manifold. A tensor field J on M is called an almost complex structure on M if, at every point x of M,

J is an endomorphism of the tangent space X(M) such that J2 = -I. A manifold M with a fixed almost ccuplex structure J is called an almost complex manifold. Every almost complex manifold is of even dimensional. We shall prove that every complex manifold M carries a natural almost complex structure. Let (z1,...,zn) be a ccuplex local coordinate system on a neighborhood U of a point p of M. We put z3 = xi + iy3, j = 1,...,n. We define an endomorphism J of Tp(M) by

(1.1)

JO/3x3) = 3/3yj,

J(3/3y3) = -(3/3xi),

j = 1,...,n.

108 We prove that the definition of J does not depend on the choice of the complex local coordinate system. Let TT(M) be the complexification of

T(M). We extend J to Tp(M) and we have

J(a/az3) = i(a/az3),

(1.2)

J(a/aEJ) = -i(a/az3),

j = 1,...,n,

where

(a/az3) = 1{(a/ax3)-i(a/a y3)},

(1.3)

(a/azj) = z{(a/axj)+i(a/ayj)}.

Hence if an element Z of Tp(M) is a linear combination of (a/az3) (j = 1,...,n) only, then JZ = iZ, and if Z is a linear combination of (a/az3) (j = 1,...,n) only, then JZ = -iZ. Now if (w1,...,wn) is another complex local coordinate system on

U at p, and if wk = uk + ivk, k = 1,...,n, then define an endamrphism J' of Tp(M) by

J'(a/avj`? = a/avk,

J'(a/avk) = -(a/auk),

k = 1,...,n.

J,(a/awk)

k = 1,...,n.

Extending J' to Tpc(M), we obtain

J'(a/awk) = i(a/awk),

On the other hand, at p e M, a

awk

= 1 k p) jaw

Hence a/awk and

a, azj

ak = -a k(p)-a., k = 1,...,n. aw

jaw

az3

are linear combinations of a/aza and a/azj

respectively. Thus we have

J(a/awk) = i(a/awk),

J(a/awk) =

Consequently, we see that J and J' coincide at p e M, and hence J does not depend on the choice of the complex local coordinate system in the

109

neighborhood of p. It is clear that J2 = -I. Thus J is an almost

complex structure on M. Let M and M' be almost complex manifolds with almost complex structures J and J' respectively. A mapping f: M to be almost complex if

> M' is said

f* J.

PROPOSITION 1,1, Let M and M' be complex manifolds. A mapping f: M

> M' is holamrphic if and only if f is almost complex with

respect to the complex structures of M and M'.

Proof. Let J and J' be almost complex structures of M and M', respectively. Let (z1,...,zn) and (wl,...,w1n) be complex local coordinate systems on neighborhoods of p e M and f(p) a M', respectively, and set zk = xk + iyk, wJ = u3 + ivi. If we put

f*u3 = ai(xl,...xn,yl,...,yn), f*vJ = Si(xl,...,xn,y1,...,yn),

then

f*(a/ax ) = Y{(aaj/a)k)(p)(a/auj)+(nj/ayk)(p)(a/avj)}, j f*(alayj`)

_ y{(aaj/ayk)(p)(a/auj)+(asj/ayk)(p)(a/avj)}. j

Comparing f*(J(a/axk)) with J'(f*(a/axk)), and f*(J(a/ayk)) with .7'(f*(a/ayk)), we see that f is almost complex if and only if

(aa3/axk)(p) = (as3/ayk)(p),

(aa3/ayk)(p) = -(asi/axk)(p)

for all j, k. But this is the Cauchy-Riemann equation of f*wJ = f*u3 + if*v3 = a3 + i0j. Thus f is almost complex if and only if f is

holoorphic.

QED.

We call TX(M)the complex tangent space of-M at x. An element of I (M) is called a complex tangent vector at x. Let M be an almost complex manifold with almost complex structure J. Then

110

TX(M) = TX'°(M) + To l(M),

where Ti'0(M) and To'1(M) are the eigenspaces of J corresponding to the eigenvalues i and -i respectively. A complex tangent vector (field) is said to be of type (1,0) (resp. (0,1)) if it belongs to Ti'0(M) (resp. TX'1(M)). We see that a complex tangent vector Z is of type (1,0) (resp. (0,1)) if and only if Z = X - iJX (resp. Z = X + iJX) for some X c Tx(M).

Let M be a real 2n-dimensional complex manifold with the natural almost complex structure J. Let (z1,...,zn) be a complex local coordinate system. Then, from (1.2), we see that {a/az1,...,a/azn} is a basis of Tx1'0(M) and {a/al,...,3/3zn} is a basis of TX'1(M). Moreover,

{a/az...,a/azn,a/azl,...,a/azn} is a basis of TX (M). We denote by TYC(M) the complexification of the dual space TX(M) of TL(M). We put

zi = xi + iy3, j = 1,...,n, and set

(1.4)

dz3 = dx3 + idyl,

dV = dxJ - idyl,

j = 1,...,n.

Then {dzl,...,dzn,dzl,...,dzn} forms a basis of TXC(M). We also have

dz3(a/azk) = dzJ(a/azk) = dk, dzi(a/azk) = dzj(a/azk) = 0,

j,k = 1,...,n.

Let Dr(M) be the space of r-forms on a manifold M. The ccaplexificat ion Cr(M) of Dr(M) is the set of all w = wl + iw2, where wl and

w2 are real r-forns. We call w e Cf(M) a complex r-form on M. A complex r-form w on M gives an element of Ar ,c(M) at each x c M, that is, a skew-sy mletric r-linear mappings T-(M) x ... x TX(M)

> C at each

x E M. More generally, we can define the space of complex tensor fields on M as the complexification of the space of real tensor fields. Such operations as contractions, brackets, exterior differentiations, Lie differentiations, etc. can be extended by linearlity to complex

111

tensor fields or canplex forms.

Let V be a vector space over R. Then we have seen that V*C = V10 0

and AV0 1 can be considered as 0 subalgebras of AV*C. Let Ap'gV*c be the subspace of AV*C spanned by + V0 1. The exterior algebras AV1

p, A S, where a c ApV1,0 and 6 E AgV01. Then we have the decomposition

AV*C =

ArV*c r=0

with

ArV*c =

Ap'gV*c

E

p+q--r

The complex conjugation in V*C gives a real linear isomorphism between AP'gV*c and Aq'pV*c. Applying these results to TXC(M), we have the decomposition

C(M) _

I Cr(M) =

r=0

Cp'q(M),

E

C'(M) _

Cp'q(M),

p+q=r

p,q=O

where C(M) is the space of carplex differential forms on M. An element of Cp'q(M) is called a (complex) form of degree (p,q). A complex 1form w is of degree (1,0) if and only if w(Z) = 0 for all complex vector fields Z of type (0,1), and w is of degree (0,1) if and only if w(Z) = 0 for all Z of type (1,0). Let {wl,...,wn} be a local basis of C1'O(M). Then its complex

conjugate {W1,...,wn} is a local basis of CO'1(M). It follows that the

k set of f o r m s W

A ... A wJp A

W-

A ... A w q, 1 < j1 < ... < j

p

< n

and 1 < kl < ... < k < n, is a local basis of Cp'q(M). Since C(M) is locally generated bygC°'0(M), C1'O(M) and CO'1(M), we have

dC0'0(M) C C1'0(M) + CO'1(M), dC1'O(M) C C2'O(M)+C1'1(M)+CO'2(M),

dCO'1(M)C C2'0(M) + C1'1(M) + CO'2(M).

lrcm these inclusions we obtain

dCp'q(M) C CP+2,q-l(M) + C 1'q(M) + &,q+l(M) + Cp-1'q+2(M).

Here we define the torsion tensor field N of type (1,2) of an

112

almost complex structure J by (1.5)

N(X,Y) = [JX,JY] - J[X,JY] - J[JX,Y] - [X,Y]

for any vector fields X and Y on M. THEOREM 1.1, For an almost complex manifold M, the following conditions are equivalent: (a) If Z and W are complex vector fields of type (1,0),-so is [Z,W];

(b) If Z and W are complex vector fields of type (0,1), so is [Z,W];

(c) dCl'O(M) C C2'0(M)+C1'1(M), dC0'1(M) C C1'1(M)+CO'2(M); (d) dCp'q(M) C

Cp+l,q(11)

+ Cp'q+l(M)

for p,q = 0,1,...,n;

(e) The almost complex structure has no torsion.

Proof. For any complex vector fields Z = X + iY and W = X' + iY',

the bracket [Z,W] is given by

[Z,W] = ([X,X'] - [Y,Y']) + i([X,Y'] + [Y,X']).

Thus we have [Z,W] = [Z,W], which gives the equivalence of (a) and (b). Let w e Cl'0(M). If Z and W are complex vector fields of type (0,1), then (b) implies

2dw(Z,W) = Z(w(W)) - W(w(Z)) - w([Z,W]) = 0.

Hence dw does not have the component of degree (0,2). Similarly, we see that if w e CO'1(11), then dw does not have the component of degree (2,0). To prove the converse (c)

let Z and W be vector

fields of type (1,0). Then u([Z,W]) = 0 for all forms of w of degree (0,1). Hence, [Z,W] is of type (1,0). The proof of (c)

> (b) is

similar. Since C(M) is locally generated by CO-O(M), C1'O(M) and C9 '1(M), (c) implies (d). The converse (d)

> (c) is trivial.

We next prove the equivalence of (a) and (e). Let X and Y be real vector fields and put Z = [X-iJX,Y-iJY]. Then (a) holds if and only if

113

Z is of type (1,0) for all X and Y. On the other hand, we have

Z + iJZ = -N(X,Y) - iJN(X,Y).

Since Z + iJZ = 0 if and only if Z is of type (1,0), (a) holds if and only if N(X,Y) = 0 for all X and Y.

QED.

If the condition (a) holds, we say that the almost complex structure J is integrable. That is, J is integrable if and only if N vanishes identically.

We now give the conditions that an almost complex manifold M to be a complex manifold. THEOREM 1.2. Let M be an almost complex manifold with almost complex structure J. Then J is a complex structure if and only if J has no torsion. Proof. Let (x',...,x2n) be a local coordinate system in M. With respect to this the components Njk of N is given by 2n

Njk =

JkahJ - JhajJk + JhakJ

where J denote the components of J. If J is a complex structure on M, then, by (1.1), we see that the components of J are constant. Thus we have Njk = 0. The converse is beyond the scope of this book. For the complete proof of the converse is given by Newlander-Nierenberg [1]. (See also QED.

Kobayashi-Nomizu [2], Matsushima [1].)

THEOREM 1.3. Let M be a real 2n-dimensional almost complex manifold with almost complex structure J. Suppose that there exists an open covering {U} of M satisfying the following conditiond: There is a local coordinate system (xl,...,x

n) on each U, such that

for each point of U, J(a/axi) = 2/ay3,

Then M is a complex manifold.

J(a/aye) = -(a/9x3),

j = 1,...,n.

114

Proof. Let (x&,yi) and (u3,v1) be local coordinate systems on U and V respectively satisfying the condition. On U

Ui

=

aj(xk,y ),

vi

=

r%

V # (6 we set

BJ()(k,Sk)

Then we have

a/axj = F{(aak/2x0 (a/auk) + (2sk/ax3)(a/avk)},

k a/ayi =

{(aak/ayJ)(2/au ) + (2sk/3yJ)(a/avk)}.

k Applying J on the both sides of the equations above, we obtain

2/ayi = I{(aak/ax3)(a/av) - (ask/axj)(a/auk)}, k a/axi = -1{(aak/ayj)(a/avk) + OBk/ayj)(a/auk)},

k From these equations we have

aak/axj = ask/ayi,

aak/ayJ = -(ask/axj),

j,k = 1,...,n.

We put z' = xi + iy', w3 = uu + iv3, then (z1,...,zn) and (w1,...,wa) are complex local coordinate systems of U and V respectively. We have

wk = fk(zl,...,zn),

fk = ak + isk,

k = 1,...,n.

Since fk is holonnrphic, M is a complex manifold.

If a vector field X on M satisfies LXJ = 0, LX being the Lie differentiation with respect to X, then X is called an infinitesimal automorphism (analytic vector field) of an almost canplex structure J. For any vector fields X and Y on M we have

(LXJ)Y = LXJY - JLY = [X,JY] - J[X,Y].

QED.

115

Thus we have PROPOSITION 1,2. A vector field X on M is an infinitesimal automorphism of an almost complex structure J on M if and only if [X,JY] _ J[X,Y] for all vector fields Y on M.

THEOREM 1.4, bet Ot be a 1-parameter group on a complex manifold M and X be the infinitesimal transformation of 4t. Then X is an infinitesimal autcmorphism of an almost complex structure J on M if and only if Ot is a holanorphic isamrphisn of M for each t.

Proof. For any vector field Y on M we have

[X,JY]x - Jx[x,Y]x = lim

t[(JY)x - (fit)*(JY) -1

0t (x)

t_ +O

- x(Yx - (fit)*Y

1

t

=

t[Jx((¢t)*Y -1

li t4o

- (fit)*(JY)

t (x)

<x)

-1

]

t (x)

at each point x of M. Thus, from Proposition 1.1, if t is a holomorphic isomorphism, then [X,JY] = J[X,Y]. Hence X is an infinitesimal

autcmorphisn. Conversely, we suppose that X is an infinitesimal autamrphism. We put

0(t;Y) = J(ot)*Y - (0t)*JY

for any vector field Y on M. Then we have [d4)(t;Y)x/dt]t=0 = 0. On the

other hand, we see that

O(s;(¢t)*Y) + (¢s)*Vt;Y). Hence we

-obtain

lli 8+0

- '(t;Y)X] = -[X,O(t;Y)]x i[O(t+s;Y)x S

As a function of t, 0(t;Y) x is a solusion of the ordinary differential

equation

116

dd(t;Y)x dt

, = -[X4(t;Y)] x.

We notice that VO;Y) = 0. Since the vector field which is constantly zero is a solusion of the equation above, the uniqueness of solusions implies that b(t;Y)x = 0. Therefore, Ot is a holarmrphic isomorphism.

QD PROPOSITION 1.3. Let X be an infinitesimal automorphism of an almost complex structure J on M. Then JX is also an infinitesimal autormrphism of J if and only if N(X,Y) = 0 for all vector fields Y on M.

Proof. From Proposition 1.2 we obtain

N(X,Y) = [JX,JY] - J[JX,Y]

for any vector field Y on M. From this we have our assertion.

QFJ.

Let M be a complex manifold and (z1,...,zn), zJ = xJ + iy3, be a complex local coordinate system of M. We define

d'

&'q(1f) :

> C11'q(M)

d" : Cp'q(M)

>

Cp'q+l(M)

by

d2= d'w + d"w

for w E Cp'q(M).

We obtain

(d' )2 = 0,

(d")2 = 0,

d'd" + d"d' = 0.

A form w of degree (p,0) is said to be holomorphic if d"w = 0. We now express w in terms of the local coordinate system (z1

W = IfJi...jp dz1 n

... A dzP.

zn

117

Then d"w = 0 if and only if d"f.

= 0. For any function f on M

...j

1

we obtain

P

df = )(af/azd)dzd + j(af/az-J)dzJ,

from which

d'f = j(af/az3)dz3,

Therefore,

d"f = j(af/azJ)dzj.

see that d"w = 0 if and only if af. /a-z3 = 0 for J1...3P

j = 1,...,n. Hence w is holcmorphic if and only if the coefficient f.

J1 ... jP

of w are all holomorphic.

A complex vector field Z of type (1,0) on a complex manifold M is said to be hoZomorphic if Zf is holomorphic for every locally defined holomorphic function f. We put

Z = IfJ(a/az3).

Then Z is holomorphic if and only if f3 are all holomorphic functions. PROPOSITION 1.4. Let M be a complex manifold with almost complex structure J. If X is an infinitesimal automorphisn of J on M, then X - iJX is a holomorphic vector field.

Proof. We can set

i(X - iJX) = If3(a/az3),

?r(Y - iJY) = Igl(a/azJ).

We easily see that [X,JY] = J[X,Y] if and only if Eg(afu/azk) = 0 for j = 1,...,n. Since Y is an arbitrary vector field, we see that afj/8z = 0 for all j, k. Thus X - iJX is holomorphic.

QED.

Let Z be a holcmorphic vector field on M. We have

Z = lfj(a/azj) = 7l{aj(a/axj)+RJ(a/ayj)]+-1,,il{sJ(a/axj)-aj(a/ayj)),

j

j

118

where f3 = aj + isj. If we put X = E{a3(3/ax3) + 8j(a/ayj)}, then Z = J(X - iJX). X is independent of the choice of coordinate neighbor-

hoods. Since af3/ax = 0 for all j, k, we have [X,JY] = J[X,Y] for all Y. Thus X is an infinitesimal automorphism. We take a mapping 8

: X

> #(X - iJX) form the Lie algebra of infinitesimal automor-

phisms of,J to the Lie algebra of holomorphic vector fields. For any infinitesimal automorphisms X and Y of J we have

8([X,Y]) = [BX,eY]

by using J[X,Y] = [JX,Y] = [X,JY] and [JX,JY] = -[X,Y]. Claerly, 0 is one-to-one. Therefore, 8 is an isomorphism.

2. EXAMPLES OF COMPLEX MANIFOLDS AND ALMOST COMPLEX MANIFOLDS

In this section we give some examples of complex manifolds and almost complex manifolds. First of all, we give examples of complex structures on vector spaces. Example 2.1. Let Cn be the complex vector space of all n-tuples of complex numbers (zl,...,zn). If we set zz =

k

+ iyk, xk, yk a R,

k = 1,...,n, then Cn can be identified with the real vector space R2n

the identification will be given by (zl,...,z ) > (xl,...,xn,yl,..., yn). The complex structure of R2n defined by (xl,...,xn,yl,...,yn) > (y1I...,yn,_xl,...,-xn) is called the canonical complex structure of R2n. In terms of the natural basis of R2n, it is given by the matrix

JO =

Example 2.2. A Lie algebra °3. over C is called a complex Lie alge-

bra. Considering °j as a real vector space, we define a complex structure J on 1: by JX = iX. The complex structure J satisfies [JX,Y] = J[X,Y] = [X,JY] for all X,Y E °°, that is,

for all X.

119

Conversely, let °d be a Lie algebra over R with a complex structure J such that ad(X).J = J.ad(X) for all X e °V. Then, defining (a + ib)X

= aX + bJX, we have

[(a+ib)X,Y] = a[X,Y] + b[JX,Y] = a[X,Y] + bJ[X,Y] = (a+ib)[X,Y).

Thus we see that 7-is a complex Lie algebra.

We next give some examples of complex manifolds. Excmrple 2.3. A complex Lie group G is a group which is at the

same time a complex manifold such that the group operations (a,b) c G x G

> a b e G and a e G

> a1

c G are both holarnr-

phic. We give some examples of complex Lie groups: (a) G(n,C) is a complex Lie group;

(b) Cn is a complex Lie group with respect to addition; (c) Direct product of two complex Lie groups is a complex Lie group;

(d) Every even-dimensional commutative Lie group is a complex

Lie group. If the Lie group G is a complex Lie group, then the Lie algebra

9-of G is a prrplex Lie algebra. Conversely, if the Lie algebra °d. of a real Lie group G is a complex Lie algebra, then G has the structure of a complex Lie group (see Matsush;ma [ ]). Excv7rple 2.4. Let M be a complex manifold. We consider a covering

space M over M with projection p : M

> M. Let {Ui} be an open

covering of M such that for every i Ui is mapped holamrphically onto p(Ui) by p. Such a covering of M aleays exists. We denote by J a carplex structure of M. We denote by pi the restriction of p to Ui. We put

Ji =

On the intersection U. n U3, Ji and Ji coincide since

is the

identity mapping on Ui A U3. Thus, the operatou' on M having the Ji as its restrictions defines a complex structure on M. FYran the construc-

tion we see that the projection p is bolemorphic with respect to this

120 complex structure.

Example 2.5. Consider Cu as a real 2n-dimensional vector space. Let {a1,...,a2n} be a basis of Cu over R. We put

2n

r =

J!{m..

.

: m. integers}.

We consider Cu as an abelian group with respect to the addition. Then r is a subgroup of Cu. We denote by T the quotient group Cn/t of Cu by > T be the natural mapping. Let U be the subset of

r. Let it : Cu

T. Defining U to be an open set of T if it-1(U) is open in Cu, we intro-

duce a topology on T. Then each point n(a) of T (a E Cu) has a neighborhood homeornorphic to a neighborhood of a it Cu. Thus we can see that

T is a complex manifold of complex dimension n. We call T a complex

,9rts. Example 2.6. In the set Cn+1 - {0} we define an equivalence rela-

ticn as follows: two points z = (zk) and w = (w) of

Cn+1

- {0} are,

equivalent if there is a nonzero complex number X such that Z = Xw,

that is, zk = Awk (k = 0,1,...,n). The set of equivalence class obtained by the equivalence relation defined above is called an n-dimensional complex projective space, which will be denoted by CPn. The topology of CPn is defined by the natural quotient topology. For each j, j = 0,1,...,n, we put

Ui 'f _ {z : zj it 0} C

(Cn+1 - {0}).

Let U. be the image of U under the natural projection

Cn+1

- {0}

CP'1. We define a crapping 0. from U. to Cu which maps an equivalence class of z with

0 onto (z0/zj,J..,zj 1/zj,zj+1/zj,...,zn/zj)

of Cn. Then {(U.,O )}, 0 < j < n, is a complex coordinate system of J

The local coordinate system (z

0/zjj-1/zj j+1/z' ,...,z ,z

n ,...,z

is called the inhomogeneous coordinate system of CPn and the coordinate system (zO,zl,...,zn) is called the homogeneous coordinate system of Cpn.

We denote by C* the multiplicative group of nonzero complex

121

numbers. C* acts freely on Cn+1 - {0} by

(X,Z) E C* x (Cn+1 - {0}) -> aZ E Cn+1 - {0}. Cn+l

- {O} is a principal fibre bundle Cn+l - {0} over CPn with group C*. We denote by it the projection Then CPn = (Cn+l - {O})/C*.

->

CPn. Then the local triviality Wj : 7T-1(Ui) : Ui x C* is given by

*i(z) = (n(z),zd) E

J x

z = (z3) E Cn+1 - {O}.

C*,

> C* by

We define the trnsition functions kj : Uj /1 Uk

kj =

zk/zd

for a homogeneous coordinate system z0,...,zn. Let S2n+l be the unit sphere in

Stn+1 = {(z0,...,zn)

The unit circle S1 = {z E C :

Cn+1,

that is,

n E Cn+1 .

k

IZkJ2 = 1}.

IzI = 1} may be considered as a multi-

plicative group and identified with the additive group R/Z (the real numbers modulo 1) by e2n16

A =

E

S1

>

2ni logy = 6 E R/Z.

We see that Stn+1 is a principal fibre bundle over CPn with group S1 which is a subbundle of Cn+l - {0}. We denote by it the projection

Stn+1

> CPn. Then the local trivialities

and the transition functions rykj.: Uj n Uk

7T_1 (U )

> S1 are given respec-

tively by

* j(z) = (ir(z),zd/(zdI) E Uj x S1,

J'kj = (zklz3j)/(zjlzkJ).

U. x S

z E S2n+l

i If ve consider S2n+1 as a principal fibre beadle over Cpn with group R/Z, then the transition functions *kj : IIj A Uk

> R/Z are given

by

lkj

21ri

iogkj

=

ti

log zk/zj - loglzkl/lzjl).

S2p+1 Example 2.7. Let and 52q+1 be two unit spheres. Then the S2p+1 product manifold x 52q+1 (p,q > 0) admits a complex structure

(Calabi-Eclanann [ 1] ) . Especially, the fact that 52p+1 x S1 admits a complex structure was discovered by Hopf [1]. ExcanpZe 2.8. Let C denote the algebra of Cayley numbers with a basis {I,e0,el,.... e6}, I being the unit element of C. The multipli-

cation table is given by the following:

U # j), i,j = 0,1,...,6,

ei = -I,

the other

e2,

e4,

e6,

e5,

e6,

e5,

-e6,

being given by permuting the indices cyclically. The

algebra C is non-associative.

Any element of C may be written as

xI + X,

x e R,

where

6

X= E xei,

xl e R, i = 0,1,...,6.

i=0

If x = 0, the element is called a purely imaginary Cayley number. All

purely imaginary numbers form a 7-dimensional subspace E7 C C. Let

X= ti-e ei, Y = E6 0yiei a E7. Then

ikK X x Y,

where 6

<X,Y> =

i i E Xy i=0

is the scalar product in E7, and

X x Y =

I x1y3ei.ej

i#j

is the vector product of X and Y.

The operation of vector product is bilinear, and X x Y is orthogonal to both X and Y, that is, <X,X x Y> = = 0. We have noreover X x Y = -Y x X.

Consider the unit 6-dimensional sphere S6 in E7:

S6 = {X a E7 : <X,X> = 11.

The scalar product in E7 induces the natural tensor field g on S6. The tangent space TX(S6) at X c S6 can naturally be identified with the subspace of E7 orthogonal to X. Define the endanDrphism JX on TX(S6) by

JXY = X x Y

TX(S6).

for Y E

Then

JXY=JX(Xx Y)=Xx(Xx Y) = =

x Y) =

<X,Y>X

X,X>Y = Y,

124

from which JJ = -I. Thus the correspondence X

> JX defines a

tensor field J such that J2 = -I. On the other hand, we have

g(JXY,JXZ) = g(Y,Z),

Y,Z E TX(S6).

Consequently, S6 admits an almost Hermitian structure (J,g) (see §3).

.3. HERMITIAN MANIFOLDS

Let M be an almost complex manifold with almost complex structure J. A Hermitian metric on M is a Riemannian metric g such that

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

for any vector fields X and Y on M.

An almost complex manifold with a Hermitian metric is called an

almost Hermitian manifold and a complex manifold with a Hermitian metric is called a Hermitian manifold.

PROPOSITION 3,1, Every almost complex manifold admits a Hermnitian

metric provided it is paracompact. Proof. Since the manifold M is paracotmpact, we can take a Riema-

nnian metric g. If we set

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

for any vector fields X and Y on M, then h is a Hermitian metric on M.

QED.

We assume that all manifolds M are always paracommpact. We can

easily see that a Hermitian metric g on an almost complex manifold M can be extended uniquely to a complex symmetric tensor field of covariand degree 2, also denoted by g, such that

(3.1) g(Z,W) = g(Z,W) for any canplex vector fields Z and W;

925

(3.2) g(Z,Z) > 0 for any non-zero complex vector Z;

(3.3) g(Z,W) = 0 for any vector field Z of type (1,0) and any vector field W of type (0,1).

Conversely, every complex symmetric tensor g satisfying (3.1), (3.2) and (3.3) is the natural extension of a Hermitian metric g on M.

Let M be an almost Hermitian manifold with almost complex structure J and Hermitian metric g.The fundmnental 2-form (D of M is defined

by

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

for all vector fields X and Y on M. Then we have

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

Since g is positive-definite and J is non-singular at each point, it follows that

P _ (D A ..... A (D

(p times), 1 < p < n, 2n = dim M,

is non-zero at each point. Thus we have PROPOSITION 3.2. Let M be a real 2n-dimensional almost complex manifold. Then M is orientable.

THEOREM 3.1. Let M be an almost complex manifold with almost complex structure J. Then M is a complex manifold if and only if M admits a linear connection V such that VJ = 0 and T = 0, where T denote the torsion of V.

Proof. Suppose that there exists a linear connection V on M satisfying VJ = 0 and T = 0. Then we have

[X,Y] = VXY - VYX,

VXJY = JVXY

128 for any vector fields X and Y on M. Fran these equations we obtain

N(X,Y) = [JX,JY] - J[X,JY] - J[JX,Y] - [X,Y] = JVJXY - JVJYX - J2VXY + JVJYX - JvJXY + J2VYX - VXY + VYX = 0.

Therefore, from Theorem 1.2, we see that M is a complex manifold. Conversely, we suppose that M is a complex manifold, that is, N = 0. First of all, we can take a linear connection V with vanishing

torsion T = 0. We put

A(X,Y) = (VxJ)Y - (VyJ)X,

S(X,Y) = (VXJ)Y + (VYJ)X.

If we set

V'XY = VXY + ;[A(X,JY) - JS(X,Y)],

then we can prove that VI 'is the desired connection. From the defini-

tion V' is obviously a linear connection on M. We shall prove that

V'J = 0 and T' = 0, where T' is the torsion of T. For any vector fields X and Y on M we have

(V'XJ)Y = V'XJY - Jv'XY = VXJY - JVXY - }[A(X,Y) + JS(X,JY) + JA(X,JY) + S(X,Y)] = (VXJ)Y - l[(VXJ)Y + J(VXJ)JY]

=

[(VXJ)Y - J(VXJ)JY].

On the other hand, we obtain

J(VXJ)JY = JVXJ2Y - J2VXJY = -,JVXY + (VXJ)Y + JVXY = (VXJ)Y.

127

Thus we have V'xJ = 0.

We next prove that T' = 0. Since T = 0, we obtain

A(JX,Y) + A(X,JY) = -[X,Y] + [JX,JY] - J[JX,Y] - J[X,JY] = N(X,Y),

fran which

T'(X,Y) = V'xY - V'YX - [X,Y] = T(X,Y) + a[A(JX,Y) + A(X,JY)] = IN(X,Y).

Consequently, if N = 0, then T' = 0.

Qom.

LEMMA 3,1, Let M be an almost Hermitian manifold with almost

complex structure J and Hermitian metric g. Then the covariant derivative V of the Riemannian connection defined by g, the fundamental 2-form 0 and the torsion N of J satisfy

2g((VxJ)Y,Z) - g(JX,N(Y,Z)) = 3d'(X,JY,JZ) - 3dO(X,Y,Z)

for any vector fields X, Y and Z on M. Proof. First of all, we have

(VxJ)JY = -J(VxJ)Y,

(Vx(p)(Y,Z)

N(Y,Z) = (VJYJ)Z - (VJZJ)Y + J(VZJ)Y - J(VYJ)Z.

On the other hand, we obtain

3dO(X,Y,Z) = X(O(Y,Z)) + Y(O(Z,X)) + Z(O(X,Y))

- '([X,Y],Z) - '([Y,Z],X) - 0([Z,X],Y) = g(Y,(V J)z) + g(z,(VYJ)x) + g(x,(VZJ)Y),

128

3d$(X,JY,JZ) = g(JY,(VxJ)JZ)+g(JZ,(v,J)X)+g(x,(V

T)JY).

From these equations we have

3d0(X,Y,Z) - 3d0(X,JY,JZ) = -2g((OxI)Y,Z) + g(Z,(V J)X)

+ g(x,(VZJ)Y) - g(JZ,(V

J)x) - g(X,(V, J)JY)

= -2g((OxJ)Y,Z) - g(J(V J)Z,JX) + g(JX,J(DZJ)Y)

+ g((V

J)z,Jx) - g((V, J)Y,Jx)

= -2g((OxJ)Y,Z) + g(JX,N(Y,Z)).

Thus we have our equation.

Qom:

THEOREM 3.2. Let M be an almost complex manifold with almost

complex structure J and

Hermitian metric g. Let V be the covariant

differentiation of the Riemannian connection defined by g. Then the following conditions are equivalent: (a) VJ = 0; (b) V

= 0;

(c) The almost complex structure has no torsion and the fundamental 2-form (D is closed, that is, N = 0 and d0 = 0.

Proof. We have seen that (Vx@)(Y,Z) = g(Y,(VJ)Z) for any vector fields X, Y and Z on M. Thus VJ = 0 if and only if DO = 0. Hence (a) is equivalent to (b).

We suppose (b). Then 0 = 0 obviously. Moreover, by Lemma 3.1,

we have N = 0. Conversely, we suppose (c). Then Lemma 3.1 implies VJ = 0 and hence o0 = 0. Hence (b) is equivalent to (c).

QED.

A Hermitian metric g on an almost complex manifold M is called a Kaehlerian metric if the fundamental 2-form 0 is closed. An almost complex manifold M with a Kaehlerian metric is called an almost Kaehlerian manifold. A complex manifold M with a Kaehlerian metric

12

is called a KaehZerian manifold. In view of Theorem 3.2; a Hermitian manifold M is a Kaehlerian manifold if and only if VJ = 0. An almost Hermitian manifold M with almost complex structure J is called a nearly Xaehlerian manifold (almost Tachibana manifold, K-space) if

(VXJ)Y + (VYJ)X = 0

for any vector fields X and Y on M or equivalently

(VXJ)X = 0

for any vector field X on M.

If we have (VxJ)Y + (VJXJ)JY = 0

.for any vector fields X and Y on M, then M is called a quasi-Xaehlerian manifold. Let {el,.... en,Jel,...,Jen} be a basis of M. Then 64)(X)

n

_ - i i{(De10)(ei,X) + (VJe4))(Jei,X)J.

If 6

= 0, M is called an almost semi-Xaehlerian manifold and moreover

if N = 0, M is called a semi-Xaehlerian manifold.

A 2n-dimensional manifold M with a 2-form (resp. a closed 2-form) 0 which is non-degenerate at each point of M is called an almost symplectec or almost Hamiltonian manifold (resp. a sympZectec or Hamiltonian manifold).

4. KAEHLmiAN MANIFOLDS

Let M be a real 2n-dimensional Kaehlerian manifold with almost canplex structure J and Kaehlerian metric g. We denote by R and S the Riemannian curvature tensor and the Ricci tensor of M, respectively.

130

PROPOSITION 4.1, For a Kanhlerian manifold M we have the following

properties: (a) R(X,Y)J = JR(X,Y) and R(JX,JY) = R(X,Y) for any vector fields

X and Y on M; (b) S(JX,JY) = S(X,Y) and S(X,Y) = #(Trace of JR(X,JY)) for any vector fields X and Y on M. Proof. (a) Since J is parallel, the first equation is clear. We prove the second equation. Fbr any vector fields X, Y, Z and W on M

we have

g(R(JX,JY)Z,W) = g(R(W,Z)JY,JX) = g(JR(W,Z)Y,JX) = g(R(W,Z)Y,X) = g(R(X,Y)Z,W).

Thus we have R(JX,JY) = R(X,Y). (b) Let {el,...,e2n} be an orthononnal basis of M. Then

S(JX,JY) _ _

g(R(ei,JX)JY,ei) = g(R(Jei,JX)JY,Jei) g(R(ei,X)JY,Jei) = Fg(JR(ei,X)Y,Jei) g(R(ei,X)Y,ei) = S(X,Y).

Thus we have the first formula of (b).

Using Bianchi's first identity, we obtain

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

Yg(JR(ei,X)JY,ei)

i

=

[g(JR(X,JY)ei,ei) + g(JR(JY,ei)X,ei)]

=

[g(JR(X,JY)ei,ei) + g(JR(JY,Jei)X,Jei)]

=

[g(JR(X,JY)ei,ei) + g(R(Y,ei)X,ei)]

= (Trace of JR(X,JY)) - S(X,Y). Hence we obtain the second formula of (b).

QED.

131

PROPOSITION 4.2, The Ricci tensor S of a Kaehlerian manifold M satisfies

(VZS)(X,Y) = (VXS)(Y,Z) + (V

S)(JX,Z).

Proof. From the second equation of (b) in Proposition 4.1 and Bianchi's second identity we have

(OZS)(X,Y) = i1g(J(O0)(X,JY)ei,ei) i

_ iYg(J(VXR)(Z,JY)ei,ei) + iyg(J(V, R)(X,Z)ei,ei)

i

_ (VXS)(Y,Z) + (V,

i

S)(JX,Z).

QED.

PROPOSITION 4.3. Let M be a real 2n-dimensional Kaehlerian manifold. If M is of constant curvature, then M is flat provided n > 1. Proof. If M is of constant curvature c, then

R(X,Y)Z = c[g(Y,Z)X - g(X,Z)Y]

for any vector fields X, Y and Z on M. From the second formula of (a) in Proposition 4.1 we obtain

R(X,Y)Y = c[g(Y,Y)X - g(X,Y)Y] = c[g(JY,Y)JX - g(JX,Y)JY] = R(JX,JY)Y,

from which (2n-1)cX = cX. Hence we have 2(n-1)c = 0. Since n > 1, we

obtain c = 0.

QED.

In view of Proposition 4.3, the notion of constant curvature for Kaehlerian manifolds is not essential. So we introduce the notion of constant holomorphic sectional curvature for Kaehlerian manifolds. To this purpose we prepare some algebraic results for quadrilinear mappings on a real vector space with almost complex structure. Let V be a 2n-dimensional real vector space with a complex

132

structure J. We consider a quadrilinear mapping B : V x V x V x V > R with the following four conditions:

(a) B(X,Y,Z,W) = -B(Y,X,Z,W) = -B(X,Y,W,Z), (b) B(X,Y,Z,W) = B(Z,W,X,Y), (c) B(X,Y,Z,W) + B(X,Z,W,Y) + B(X,W,Y,Z) = 0, (d) B(JX,JY,Z,W) = B(X,Y,JZ,JW) = B(X,Y,Z,W).

We easily see that the Riemannian curvature tensor R of a Kaehlerian manifold satisfies the above conditions (a), (b), (c) and (d) (see §3 of Chapter I). LEM?1A q, 1. Let B and T be two quadrilinear mappings satisfying

the conditions (a), (b), (c) and (d). If

B(X,JX,X,JX) = T(X,JX,X,JX)

for all X E V, then B = T.

Proof. We may assume that T = 0; consider B - T and 0 instead of B and T. We assume therefore that B(X,JX,X,JX) = 0 for all X e V.

Replacing X by X + Y, we obtain

(4.1)

2B(X,JY,X,JY) + B(X,JX,Y,JY) = 0.

On the other hand, (a), (c) and (d) imply

B(X,JX,Y,JY) - B(X,Y,X,Y) - B(X,JY,X,JY) = 0.

This, combined with (4.1), yields

(4.2)

from which

3B(X,JY,X,JY) + B(X,Y,X,Y) = 0,

133

(4.3)

3B(X,Y,X,Y) + B(X,JY,X,JY) = 0.

From (4.2) and (4.3) we obtain B(X,Y,X,Y) = 0. By Lemma 3.2 of Chapter I, we have B = 0.

QED

Let g be a Hermitian inner product on V. We set

BO(X,Y,Z,W) = 4[g(X,Z)g(Y,W) - g(X,W)g(Y,Z) + g(X,JZ)g(Y,JW) - g(X,JW)g(Y,JZ) + 2g(X,JY)g(Z,JW)].

Then BO satisfies (a), (b), (c) and (d). We also have

BO(X,Y,X,Y) = a[g(X,X)g(Y,Y) - g(X,Y)2 + 3g(X,JY)2],

BO(X,JX,X,JX) = g(X,X)2.

Let p be a plane in V and X, Y be orthonormal basis for p. We set

K(p) = B(X,Y,X,Y).

Then K(p) depends only on p and is independent of the choice of an orthonormal basis for p. We assume that p is independent by J, that

is, Jp = p. Then X,JX is an orthonornal basis for p for any unit vector X in p. From Lemna. 4.1 we have

LEMA 4.2, Let B be a quadrilinear mapping satisfying (a), (b), (c) and (d). If K(p) = c for all J-invariant planes p, then B = cBO.

We shall now apply these algebraic results to the Riemannian curvature tensor R of a Kaehlerian manifold M with almost complex structure J and metric g.

For each plane p in the tangent space X(M), the sectional curvature K(p) is defined to be K(p) = R(X,Y,X,Y) = g(R(X,Y)Y,X),

134

where {X,Y} is an orthonormal basis for p. If p is invariant by J, then K(p) is called the holornorphic sectional curvature by p. The

holomorphic sectional curvature K(p) is given by

K(p) = R(X,JX,X,JX) = g(R(X,JX)JX,X),

where X is a unit vector in p. From Lemma 4.1 we see that the holamrphic sectional curvature K(p) for all J-invariant planes p in 7x(M) determine the Riemannian curvature tensor R at x of M. If K(p) is a

constant for all J-invariant planes p in X(M) and for all points x E M, then M is called a space of constant hoZomorphic sectional curvature-or a complex space fore. Sometimes, a cc Alex space form is

defined to be a singly connected carplete Kaehlerian manifold of constant holomorphic sectional curvature.

We prove the following theorem which is a Kaehlerian analogue of Schur's theorem.

THEOREM 4.1. Let M be a connected Kaehlerian manifold of ccmplex dimension n > 1. If the holcmorphic sectional curvature K(p), where

p is a J-invariant plane in x(M), depends only on x, then M is a complex space form.

Proof. We put

RO(X,Y,Z,W) = a(g(X,Z)g(Y,W) - g(X,W)g(Y,Z) + g(X,JZ)g(Y,JW) - g(X,JW)g(Y,JZ) + 2g(X,JY)g(Z,JW)].

Then Lemma 4.2 implies that R = cR0, where c is a function on M. Then the Ricci tensor S of M is given by

S = i(n+l)cg.

From Theorem 3.3 of Chapter I we see that c is constant provided n > 1.

QED.

935

In the course of the proof for Theorem 4.1 we have THEOREM 4.2. A Kaehlerian manifold M is of constant holocrorphic

sectional curvature c if and only if

R(X,Y)Z = }[g(X,Z)Y-g(Y,Z)X+g(JX,Z)JY-g(JY,Z)JX+2g(JX,Y)JZ]

for any vector fields X, Y and Z on M.

PROPOSITION 4.4. If a Kaehlerian manifold M is of constant holomorphic sectional curvature, then M is an Einstein manifold. Let M be a complex n-dimensional Kaehlerian manifold with cccrplex

structure J. We can choose a local field of orthonormal frames e1,..., en,ei*=Jei,...,en*=Jen in M. With respect to the frame field of M

chosen above, let wl,...,wn,w1*..;,wn* be the field of dual frames. i,j = 1,...,2n, be the connection form of M. Then we

Let w =

have

(4.4)

a __ wb

a=

a*

wb*

wb*,

a*

-wb ,

a_

b

wb = -wa'

a_ b

wb* = wa*,

where a,b = 1,...,n. We denote by 0 = (9the curvature form and write

Ri wk n wl k,l jkl

01=z

J We set

Ea = j(ea ' iea*),

Ea =

9a = wa + iwa*,

9a = wa - iwa*.

(ea + iea*),

Then {Ra) form a conplex basis of TIL'0(M) and {la} form a carplex

basis of ''1(M). The Kaehlerian metric g is given by

g

136

Moreover, we set

a ='oab + iw a* eb b a

b

b

a 9ba=%_

+ iSZ a*

b *

b - Rb

b

Then we obtain

deb

= - Tea A C +

Ta = F IZ,.ae c A

`Ya,

c

0

,

,

%cd - `IRbcd +

Rba

*cd*

+ i

We see that M is of constant holomorphic sectional curvature c if and only if

"bcd = lc(dacabd + dabacd)' or

Y'b = jc(ea A eb + Sab 0CA 0C). c We now take a complex local coordinate system [z

n,

z

in M.

We set

Za = 8/3za

Za = Za = 8/3za

a = 1,...,n.

We extend a Hermitian metric g to a complex symmetric bilinear form in

X(M). We put

gAB = g(ZA,ZB),

A,B = 1,...n,l,...n.

Then we have gab = giB = 0 and (gas) is an (n,n)-Hermitian matrix. We write

137

ds2 = 2 1 g a-tadzb a,b for the metric g. Let 4D be the fundamental 2-form of M. For any ccq lex

vectors Z, W we may write

Z = I(dza(Z)Za + dza(Z)ZaW = b(dzb(W)Zb + dzb(W)Zb).

Then

gab(dza(Z)dzb(W) - dza(W)dzb(Z)),

4)(Z,W) = -i

a,b

fran which

t7 = -2i I ga-dza /1 dzb. a,b Since the fundamental 2-form ' is closed, we have

agab/azc =

agab/azc = agar/azb

agcb/aza,

The covariant differentiation V, which is originally defined for real vector fields, is extended by ccn lex linearity to the act on complex vector fields. Setting

Z = rOA' VZC we obtain

I'ABC

with the connection that

-i.Zaand VJ=0we have

=

I'ABC

a. FY'an the fact that JZa = 'Za' JZa =

138

a

a

rBC = rBC = o.

Since J has no torsion we find

rbc = r

rjjc- = rcb,

b,

and other r

= 0.

Thus we see that the rAC's are determined by the metric as follows:

Eg jr a

Igaarbc = agab/azc,

= agdB/ai -

We set

R(ZC,ZD)ZB =

KAB. = g(R(ZC,ZD)ZB,ZA),

KABCD = Ig

E'

Fran Proposition 4.1 we see that R(ZC,ZD) coccnntes with J and hence

= 0,

KabCD = KEM = 0,

0,

KABcd = KABcd = 0.

from which

K

Bcd = KBcd

-

Therefore, only carponents of the following types can be different from zero:

Kbca' t.-d'

Ka5c

a,

Kay, Ka, K

Putting X = Zc, Y = Za and Z = Zb in R(X,Y)Z = CVX,VY]Z we obtain

d.

- V(x Y]Z'

139

= -arbc/az ,

K

from which

K

_

__

efagaf agbe

azgab

where (ga5)

azc az

e,fg

az°az

is the inverse matrix to (gaS) so that Ebg6gcb = da.c'

The canponents KAB of the Ricci tensor of M are given by

Kim=-jars /9zb,

K@3 =KaF)

c

Kab=K5b- =O.

We denote by G the determinant of the matrix (gam), that is, G2 19ABI =

Iga5I2.

Then we have

as=GIg ag6 az

b,c

.

az

Thus we obtain

c Erg

alog G

aza

from which

azaaz We now define the Ricci form

associated to the Ricci tensor S of M

by

p(%,Y) = S(X,JY)

for all vector fields X and Y on M. We then have the following

expression p=-2iIKaAdz-b

a,b

140 or

p = 2id'd"log G.

We can see that every Kaehlerian metric can be locally written as

ds2 = 2Egasdzadzb with gas = (82f/azaazb) for same real function f a ,b

(cf. Morrow-Kodaira [1; p.86]). We shall now give some examples of Kaehlerian manifolds. ExconpZe 4.1. Consider the complex n-space Cn with the metric

CL,-2 =

n I dzjdz3. j=1

The fundamental 2-form 4) in this case is given by

n

IdziAdz3.

j=1

Clearly, cD is closed, and so the metric defines a Kaehlerian structure on Cr'. Thus Cn is a complete, flat Kaehlerian manifold.

ExcnpZe 4.2. Let CPn be the n-dimensional complex projective

space with homogeneous coordinate system [z0,z1,...,zn}. For every index j, let Ui be the open subset of CPn defined by zJ # 0. We set

t On each Uj, we take

= zk/zi,

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

t indicates that t is

deleted) as a local coordinate system and consider the function

Then

fj = F

fj =

= k-k .. n i-i k-k E tjtj = I (tktk)tjt j = fktjtj i=0 i=0 n

on U. n Uk. Thus we have

log fj = log fk + log t + log t

11 Since t is holcmorphic in U A Uk, we have d"log t = 0, d'log t = d"log t = 0. From d'd" = -d"d' we obtain

d'd"logfi = d'd"log fk

on Ui n Uk.

By setting

(P = -4id'd"log fi

on Ui ,

we obtain a globally defined closed (1,1)-form (D on CPn. On the other

hand, we have

n

n

f0 = E tRd = 1 + E tats, j=0 00 a-1 n all fodta A dts

_ -4i

E

a, a=1 ataats

where we have put ta = t0, a = 1,...,n. Thus we have

§, = -4i EdtaA

dta + EtotaEdta n dto - Etadta A Etadta (1 + Etata)2

The associated metric tensor g is given by

ds2 =

4(1 + Etata)(Edtadta) - (Etadta)(Etadta) (1 + Et(Xta)2

This metric is sometimes called the Fubini-Study metric.

Excmple 4.3. Let EP be the open unit ball in Cn defined by

Il = {(z1,...,z) ::zaia < 1}.

We set

142

o = 4id'd"(1 - Jzav}. The associated metric g is given by

dS2 = 4(1 - Ez°i'z )(Edzadz-a)

+ (EZ dza)(Ezadz-01')

(1-E7z)2 DP is obviously a Kaehlerian manifold. Example 4.4. Let Cpn be the complex projective space with homo-

geneous coordinates z0,z1,...,zn. The complex quadric Qn-1 is a carplex hypersurface of CPn defined by the equation

(z0)2 + (z1)2 + ... + (zn)2 = 0.

Then Qn-1 is a Kaehlerian manifold. Moreover, Qn-1 is an Einstein manifold. (For complex sukmanifolds of Kaehlerian manifolds see §1 of Chapter IV.) In the next place, we give examples of carplex manifolds which do not admit any Kaehlerian metric.

Example 4.5. We notice that the even-dimensional Betti numbers of a carpact Kaehlerian manifold M are all positive (cf. Hodge [1]).

For the complex manifold S2p+1 Betti number B

x

52q+1 in Eranple 2.7 we see that the

(1 < k < p+q) vanish. Therefore,

S2p+1

x

5q+1 cannot

carry any Kaehlerian metric except for p = q = 0 (Calabi-Eckmann [1]).

THEOREM 4.2. (1) For any positive number c, the caiplex projective space CPn carries a Kaehlerian metric of constant holomorphic sectional curvature c. With respect to an inhomogeneous coordinate system z1,...

zn the metric is given by

ds' = 4 (1 + Ei V)(Edzadz-a) - (Ez-adza)(Ezadz-a) C (1 + Ezai ) 2

143

(2) For any negative number c, the open unit ball EP in Cn carries a complete Kaehlerian metric of constant holomorphic sectional curvature c. With respect to the coordinate system z1,...,zn of GP the metric is given by

4 (1 - Ezza)(EdzadzV) + (Ezadza)(EzadV) C

ds2

(1 - Ezaza)2

Proof. First of all, we see that (gs) of (1) and (2) are given

by

(+Jc)(1 +EzYzY)2gas = (1 + EzYzY)das + z zs,

respectively. We differentiate this identity with respect to a/azY and a2/az1zd and set z1 = ... = zn = 0. We then have

agog/azY = 0,

gas = (± c)das' 2

azYazd

c dasdY6 + dab 81

Thus we have

K - - = asYd

Vgas azYazd

_

C

ETagaE a4T

L g T,E

azY DO

- 'c(gasgyd + gadgsY)'

from which we see that the metric ds2 is of constant holomorphic sectional curvature c at the origin z1 = ... = zn = 0. Since both

CP and UP admit a transitive group of holcmorphic isometric transformations, ds2 is of constant holomorphic sectional curvature c and is complete.

QED.

Now we can prove the following theorem (Hawley [1], Igusa [1]).

144

THEOREM 4.3. A simply connected complete Kaehlerian manifold M of constant holomorphic sectional curvature c can be identified with

the complex projective space (

, the open unit ball

Dn in C

or Cn

according as c > 0, c < 0 or c = 0. Proof. From Theorem 4.2 it sufficies to show that any two simply connected complete Kaehlerian manifolds of constant holomorphic sectional curvature c are holomorphically isometric to each other. Let M and M' be two such manifolds. We choose a point o in M and a point o'

in M'. Then any linear isomorphism F : Z0(M)

> Z0,(M') preserving

both the metric and the almost complex structure maps the curvature tensor of M at o into the curvature tensor of M' at o'. Then we can see that there exists a unique affine isomorphism f of M such that

f(o) = o' and the differential of f at o is F. For any point x of M we take a curvae T from o joining to x. We put x' = f(x) and T' = f(T). Since the parallel displacement along T

corresponds to *that along T

under f and since the metric tensors and almost complex structures of M and M' are all parallel, the affine isomorphism f maps the metric tensor and the almost complex structure of M into those of M'. .breover, Proposition 1.1 shows that f is holomorphic.

5. NEARLY KAEiLERIAN MANIFOLDS

Let M be an almost Hermitian manifold with almost complex structure J and Riemannian metric tensor field g. Then

J2 = -I,

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

for any vector fields X and Y on M. We denote by V the operator of covariant differentiation with respect tog in M. If the alncst crariv-

lex structure J of M satisfies

(VXJ)y + (41,J)X = 0 for any vector fields X and Y on M, then the manifold M is called a

145

nearly Kaehlerian manifold (Tachibana space, K-space). The condition above is equivalent to

(VxJ)X = 0.

We assume that M is a nearly Kaehlerian manifold. Then

(V J)Y + (V,J)JY = 0,

(VX.J)JY = -J(VxJ)Y.

Let N be the torsion tensor field of J defined by (1.5). By a simple corrputation we have

N(X,Y) = -4J(VX.J)Y,

and

N(X,Y) + N(Y,X) = 0,

N(JX,JY) + N(X,Y) = 0,

Let T be

g(N(X,Y),Z) + g(N(X,Z),Y) = 0,

g(N(X,JY),JZ) + g(N(X,Y),Z) = 0.

field of type (0,2). If T satisfies T(JX,JY)

= T(X,Y) for any vector fields X and Y

T..), then T is 31 said to be hybrid (with respect to j and i), and if T(JX,JY) = -T(X,Y) 1

(T JjJr = -Tji), then T is said to be pure (with respect to j and i). We see that

Jjh = g(N(ejei),eh) is skew-symmetric and pure with

respect to j, i and i, h. It is well known that if a tensor field T

_

is pure and a tensor field S = (Sji) is hybrid with respect

to j and i, then T jiSJ' = 0.

First of all, we have PROPOSITION 5.1. If the Nijenhuis torsion N of a nearly Kaehlerian manifold M vanishes, then M is a Kaehlerian manifold. Let R be the Ri nannian curvature tensor field of M. Then we have

g(R(X,Y)Z,W) = g(R(X,Y)JZ,JW) - g((VXJ)Y,(Ve)W).

10 Thus we also have

g(R(X,Y)Y,X) = g(R(X,Y)JY,JX) + g((VXJ)Y,(VX.T)Y),

g(R(X,Y)Z,W) = g(R(JX,JY)JZ,JW).

We denote by Q the Ricci operator of M, that is, QX = EiR(X,ei)ei,

{ei} being an orthonormal basis of M. We put

g(Q*X,Y) = -Jlg(R(X,JY)ei,Jei). i

We call Q* the Ricci * operator of M. We now define the tensor field S by

S=Q-Q*. The we have

g(SX,Y) =

g((VXJ)ei,(VYJ)ei),

TrS = constant > 0.

In the following we put Rijkl = g(R(ei,ej)ek,el) = g(R(ek,el)ei,ej that is, Rijkl are the components of R. But Rijkl differ from the components of the curvature tensor of Chapter I by sign. We also put Rij = g(Qei,ei), R*ij = g(Q*ei,ej) and Sij = g(Sei,ej). Then Rij' R*i,7

and Sij are symmetric and hybrid with respect to i and j. We now give the equations above by using the components.

We have

JiJi

i

= -,

V .Jh + JbJaV j i b Jh a = 0,

gji,

ij a

J V .Jh

J i + Vi.J = 0,

VJ.Ji = 0,

aJi

where V. denotes the operator of covariant differentiation. The Nijenhuis torsion Njih = it(atJi - 3iJt) - Ji(3tJ - a -Jt) is given by

147

Jih

= -4(VjJi)JT.

Moreover, we obtain

(5.1)

Rijkl =

(5.2)

Rijkl =

(5.3)

Sij = RIj - R*ij =

(5.4)

TrS = ViJjh(V1JJh) = r - r* = constant > 0,

RJiJbJkJd, abed

R*ij = -JRiabc`1jJ

where r = TrQ = g1JRij, that is, r is the scalar curvature of M and r* = TxQ* = g1JR*... Since we have

0 for any skew-

symmet-ric tensor Tijk, we obtain

(V'JJk)ViRjklh = 0.

(5.5)

From V .r* = 2VJR* 1

(5.6)

J1

we we have

VJ(Rij - R*ij) = iVi(r - r*) = 0.

LEM A 5,1, In a nearly Saehlerian manifold M, we have

SJ1(Rkjih - 5RkjbaJ?Jh) = 0.

Proof. From (5.1) and (5.2) we have

2(VhJJl)fist

= -SbVSJtr

Applying Vh to this equation, we obtain, by (5.5) and (5.6),

148

2(0

ji

hJ )R ist -_ VN

tr'

or by Ricci s identity and S'OsphJtr = 0,

= Sh(Rh

jilt

satJar

-

Rh

tJa).

sar

Transvecting the equation above with Jk, we have

2Jk(ahV

Rhsta kjr)'

hJJl)Rjist

Since we have

V VtJj = Jhl(Rjl - R*lj

(5.7)

the above equation reduces to

2JkJaSaiRjist

Shr (Rhskr

RhstaJkJr

or by Bianchi's identity,

2JkJaSa1(Ritjs

Using

JaSal

+ Rtjis) =

br(Rhskr

- Rhsta jkJr)'

= -JaSaj, we have

4`jtJaRtjisSai

= Shr (Rhskr RhstaJkJr)'

Therefore, we have our equation.

QED.

We prove the following theorem (Watanabe-Takamtsu [1]). THEOREM 5.1. Let M be a nearly Kaehlerian manifold. If M is non-

Kaehlerian and if Sji = Rji - R*ji = agji, then M is an Einstein manifold.

149

Proof. From IRama 5.1 we have Ri =

Thus we have r = 5r*.

Since r - r* = na, we see that Rji = (r/n)gJ.i. Consequently, M is an Einstein manifold.

QED.

In the following-we put

Q _

- J

(did

)

T

,

*

(6m6 +

_

Then we obtain

Oij R*

O! j'Rab = 0,

= 0,

*O

DaJb = 0.

From these equations we obtain

Ri)OhJjl = 0. Applying Vh, we find

R i) = 0.

Thus (5.7) implies the following LE!1A 5,2, In a nearly Kaehlerian manifold M, we have

- R*ji)(5R*J1 - R3') = 0.

(R

ii

We also have (see Takamatsu [1])

(5.8)

R*ji(RJl - R*J1) = R

ih0tzhk3ts.

From learn 5.2 and (5.8) we get

(5.9)

R*ji)(RJ1 - R*J1) = 4R

ih0tI

We now put, for arbitrary given constants a and b,

ISO

(5.10)

Tkjih = Rkjjh - a(gkhSji - gjhSki + gjiSkh - gkiSjh) + b(r - r*)(gkhgji - gjhgki)

and t Ukjih = Oih kjts'

(5.11)

LEMMA 5,3, In an n-dimensional nearly Kaeblerian manifold M, we

have

UkJ ihU

ih = Rk)lvih

+ 2[(n-4)a2 - 2a]Sjisji

+ [2a2 + 2b - 4(n-2)ab + n(n-2)b2](r - r*)2

for arbitrary constants a and b. Proof. First of all we have =

ihkjts

UkjihUkj

ag , '0ihTkjts + QhS OihTkjts

lh0ihkjts

lsihkjts

-

+

ag

SJhOihkjts

+ b(r-r*)g gJl0ihkjts Rkj'hOt

-

4aglOihkjts

+

On the other hand, we have

ihkits -

jts -

2aSjiS`31 + b(r - r*)2

151

-ag1'0ihlots

=

i[(n-4)a2-a]SjiSjl + }[a2-(n-2)ab](r - r*)2.

QED.

From these equations we obtain our equation.

From (5.9) and Lemma 5.3 we find

UkJihUkjih

(5.12)

= ASjiSjl + B(r - r*)2,

where

A = 2(n-4)a2 - 4a + 1,

B = 2a2 + 2b - 4(n-2)ab + n(n-2)b2.

LEM 5,4. In an n-dimensional nearly Kaehlerian manifold M, if A + nB = 0, then

Ukj 1Ukjih = A[Sj i - (rnr*) gii][Sii - (rnr*) gii]

We notice that there exist real numbers a and b satisfying A + nB

0ifandonly ifn>6. When n = 6,, there exist constant a and b, for example a = 1/2,

b = 1/8, satisfying A + nB = 0. If we put n = 6, a = 1/2 and b = 1/8, then L7kjib

Ukjih

4[Sji - (rnr*) gji][Sj1

(rnr*) g01].

Consequently, we obtain (Takamatsu [1])

THEOREM 5.2. In a 6-dimensional nearly Kaehlerian manifold M, we have

Ski =

Ji - R*ji = 6 r - r*)gji.

From Theorems 5.2 and 5.2 we have (Matsumoto [1])

162 THEOREM 5,3. A 6-dimensional nearly Kaehlerian manifold M is an Einstein manifold. PROPOSITION 5,2, Let M be an Einstein nearly Kaehlerian manifold. Then the scalar curvature r of M is a positive constant.

Proof. From Leama 5.2 we have

4RjiS1

,

from which

Using (5.4), we see that r > 0.

QED.

In the sequel, we shall study a nearly Kaehlerian manifold M of

constant holomorphic sectional curvature. The holamrphic sectional curvature H(X) with respect to a unit vector X is given by

H(X) = g(R(X,JX)JX,X).

We put

B(X,Y,Z,W) = g(R(X,Y)Z,W) - t[g((OXJ)W,(DYJ)Z) - g((VXJ)Z,(VVJ)W) - 2g((V J)Y,(VJJ)W)].

Then B is a quadrilinear mapping satisfying the four conditions (a), (b), (c) and (d) in §4. We set

BO(X,Y,Z,W) = l[g(Y,Z)g(X,W) - g(X,Z)g(Y,W) + g(Y,JZ)g(X,JW) - g(X,JZ)g(Y,JW) - 2g(X,JY)g(Z,JW)].

153

Then

BO(X,JX,X,JX) = g(X,X)2.

Suppose that M is of constant holcmorphic sectional curvature c. Then

we have

B(X,JX,X,JX) = g(R(X,JX)JX,X) = c

for any unit vector X. Thus Lemma 4.2 implies that B = cBO. Hence we

have (Sawaki-Watanabe-Sato [1] ) THEOREM 5.3. Let M be a nearly Kaehlerian manifold. If M is of constant holomorphic sectional curvature c, then

g(R(X,Y)Z,W) = ;c[g(Y,Z)g(X,W) - g(X,Z)g(Y,W) + g(Y,JZ)g(X,JW) - g(X,JZ)g(Y,JW) - 2g(X,JY)g(Z,JW)]

+ 4[g((VXJ)W,(VYJ)z) - g((VXJ)z,(VYJ)w) - 2g((VXJ)Y,(VZJ)w)].

We now prove the following theorem (Takamatsu-Sato [1]).

THEOREM 5.4. There does not exist any dimensional, except 6-dimensional, non-Kaehlerian nearly Kaehlerian manifold of constant holomorphic sectional curvature.

To prove Theorem 5.4, we prepare some lemmas. Let M be an n-dimensional nearly Kaehlerian manifold of constant holomorphic sectional curvature c. We suppose that M is non-Kaehlerian. From the assumption we have

'c(gkhgji - gkigjh + JkhJji - JkiJjh - ZJkjJih)

+

VkJr(,.j 3.

J hr )

-

164 Thus we have

Rji + 3R*ji = (n+2)cgji.

Since R7i = 5R*ji, we have Rji = (r/n)gji- Thus M is Einstein. We also have c = (8r)/(5n2+10n) and hence c > 0. By a straightforward -computation we have

LEMMA 5.5, Let M be an n-dimensional non-Kaehlerian nearly Kaehlerian manifold of constant holcmorphic sectional curvature. Then ih - (6n+44) r2 = constant. 25n(n+2)

RkjihR

LE" 5.6. We have the following equations:

(5 13)

JbJaRkji 1 h

(5.14)

J

.

= (-2n+28) r 2

h

b JhJ

4 r2 , 25n '

' k b

=

ih

"kbia

,

25n(n+2)

ih

j

(5 . 15)

kiba

(5n-6) r2 25n(n+2)

.

Proof. We prove (5.13) only. From (5.1) and (5.3) we have

dc) =

a)(Rk3ih - Jij

(Rlj ih

SrsSrs,

from which 3ih

RkjihRk

b kjih = 8 r2 25n - Ji h jbaR

Fran this and Lemma 5.5 we have (5.13).

QED.

LEMr1A 5.7. We have the following equations:

(5.16)

ka(VsJ.b)VtJbh(V t

V5J

Jal)Rls).

ih

_-

128

s

125'2 n n+2) r'

155

(5.17)

otJbh(OtJal)RkJihRkjba

iffn-64 (2n+2)

r3'

b(Vtjja)Rkjih_

(5.18)

is -

125n9(n+2) r3,

Proof. We prove (5.16) only. First of all we have

VsJka(VSJJb)VtJbh(Vtjai)R'ih

= -41C(6

6

- 31dh + JhJi - JiJh - 2JkjJ1 ) x VSJka(VSJjb)VtJbh(VtJai)

+

[DrJh(VrJi) - VrJi(VrJh) - 2VrJkj (VrJ

)]

x VSJka(VSJjb)VtJbh(VtJai).

In the equation above, calculating term by term, we get the following identities:

dhdiVsJ

(VSJjb)VtJbh(VtJa1) = VSJka(VsJjb)VtJbk(Vtja3) = 0,

because, VtJbk(VtJai) is hybrid with respect to k, a, but VSJ

is

pure with respect to k, a,

-VSJka(VsJjb)VtJbj(VtJak)

SstSst JbJiVsJ

(VSJjb)VtJbh(Vtj i) = Vsj

16=-25nrz

(VSJbi)VtJbh(VtJa1) = 0,

-SstSst = - 216 -JiJhVSJka(VS Jjb)VtJbh(VtP) _ 5n r2,

IN

-2

JihVsJka(VSJjb)vtJbh(vtJa') = -2SajSai

25 ra,

-VrJ (VrJh)VsJl(Veib)VtJbh(VtJai) = 0,

-2VrJ'j(VrJih)VsJka(VSJjb)VtJbh(VtJaz) = 0

and

Vr.Jh(VrJi)VsJka(VSJjb)VtJbh(VtJai) = 0,

because, VrJh(VrJi)VSJka(VSJjb)Vtjbh is symmetric with respect to i, a,

but VtJ is skew-symmetric with respect to i, a. From these equations we obtain (5.16).

QED.

Proof of Theorem 5.4. From lemma 5.5 we obatin

(5.20)

VbRkjih(vbRk0ih) +

Rkj

bVbRkjih = 0.

Using Bianchi's and Ricci's identities, we have

RkjihvbvbRkjih

= =

-Rkiihvb(VkRjbih

+ vjRbkih)

2RkjihVbvkR -

= 2Rk

jih

bjih b

b

jih - R kbaRajih - R

(vkV

kjaRbaih

- Rbki %jah - Rbkh %jia)Since we have VbRbjih = 0, we obtain, by Bianchi's identity,

(5.21)

RkiihvbvbRk j

=

2r 1i7 il-

ih

+ 4RkJl On the other hand, we have

+ R' ihR

ba baih 'kbiaRbja.

167

j'h

(5.22)

R1

- 2JbaJ - 2V

J _ c(dhdi - didh + JhJi - JbiJh + *(VtJh(VtJi) - VtJb(OtJh)

)Rk3

tJba(7tJih)]Rkj

Niba'

We calculate each term of the equation above. By Ienma 5.5 we have 2(6n+44) 2 r ' 25n(n+2)

i hjba = -2Rkj

(dbda - dbda)RkJ

h i

From Iem

5.6 we obtain JbaJ )Rkj ih

(JbJa - JbJa h i i h

J hjba

= 2(2n-28) r 2 - 8 r 2 25n(n+2) 25n

.

We also have, by Lenma 5.7,

VtJh (7t`Ji )R -OtJb ( D

t

kjib

64

125n (

Rkjba

Jh )Rk3l

r3 '

64 jba

125n2(n+2)

2VtJba(OtJTh)RJ0ba

=

r3 ,

4Str(OrJki)StsVsJkj 32 r 3 125n2

.

(bnsequently, we obtain

(5.23) J

ba jbaR ih

(8n2+96n+416) 3 125n (n+2)2 r .

Similarly, we have

(5.24)

jjh-

aRb i

jah

= - (n2+164n+260) r3

.

125n (n+2)

Substituting (5.23) and (5.24) into (5.21), and using Lemma. 5.5, we

obtain

158

(5.25)

R ib bbv

bRk ih

.

48(n-6)

-125n-r(-n-+27.

r3.

From (5.20) and (5.25) we get

ih(vaRk

v

) + n48(n-6) (

r3 = 0.

Thus, if n > 6, then VaRkiih = 0 and n = 6 because of r > 0. When n = 4, we can prove the following (A. Gray [1]) THEOREM S.S. Any 4-dimensional nearly Kaehlerian manifold is a Kaehlerian manifold.

Proof. In (5.12), we put n = 4, a = 1/8 and b = -1/16. Then we have Ski = Rji - R*ji = 0, and hence r = r*. Thus (5.4) implies that V.Jih = 0. This means that the manifold is Kaehlerian.

QED.

If n = 2, we easily see that the torsion N of J vanishes. From these considerations we have Theorem 5.4.

QED.

Excnple 5.1. Let S6 be the 6-dimensional unit sphere defined in Example 2.8. Then S6 admits a nearly Kaehlerian structure (FukamiIshihara [1]). Any 6-dimensional non-Kaehlerian nearly Kaehlerian manifold of constant holomorphic sectional curvature is of constant curvature (Tanno [7]).

6. QUATERNION KAEHLERIAN MANIFOLDS

In this section we shall study quaternion manifolds by using tensor calculas. The results in this section have been proved by S. Ishihara [3].

Let M be an n-dimensional manifold with a 3-dimensional vector bundle V consisting of tensors of type (1,1) over M satisfying the following condition:

(a) In any coordinate neighborhood U of M, there exists a local basis {F,G,H} of V such that

159

F2 = -I,

G2 = -I,

H2 = -I,

(6.1)

GH=-HG=F,

HF=-FH=G,

FG=-GF=H,

I denoting the identity tensor of type (1,1) in M.

Such a local basis {F,G,H) is called a canonical local basis of the bundle V in U. Then the bundle V is called an almost quaternion structure in M, and M with V an almost quaternion manifold, which will be denoted by (M,V). An almost quaternion manifold M is of dimension n = 4m (m > 1). In an almost quaternion manifold M, we take coordinate neighborhoods U and U' such that U A U' # 6. Let

{F,G,H} and {F',G',H'}

be canonical basis of V in U and U' respectively. Then

F' = s11F + s12G + s13H, G' = s21F + s22G + s23H,

H' = s31F + s32G + s33H'

where sab (a,b = 1,2,3) are functions in U A U'. From (6.1) we see that (sab ) is an element of the proper orthogonal group 80(3) of dimension 3.

Thus every almost quaternion manifold M is orientable. If, in an almost quaternion manifold (M,V), there is a global basis {F,G,H) of V which satisfies (6.1), then (M,V) is traditionally called an almost quaternion manifold. Such a global basis {F,G,H} of V is called a canonical global basis of V.

Let (M,V) be an almost quaternion manifold with a canonical local basis of V in a coordinate neighborhood U. We now assume that there exists a system of coordinates (x ) in each U with respect to which F, G and H have components of the form

OE00 (6.2)

F=

E000 000E 00E0

OOEO G=

000E E000

0 -E 0 0

0

00E

H=0 0 E 0 OE00

E000

1GO

where E denotes the identity (m,m)-matrix. In such a case, the given almost quaternion structure V is said to be integrable. In any almost quaternion manifold (M,V), there is a Riemannian

metric tensor field g such that

g(FX,Y) + g(X,$Y) = 0

for any cross-section $ and any vector fields X, Y of M. An almost quaternion structure V with such a Riemannian metric g is called an almost quaternion metric structure. A manifold M with an almost quaternion metric structure {g,V} is called an almost quaternion metric manifold, which will be denoted by (M,g,V).

Let {F,G,H} be a canonical local basis of V of an almost quaternion manifold (M,g,V). Since each of F, G and H is almost Hermitian with respect to g, putting

@(X,Y) = g(FX,Y),

'V(X,Y) = g(GX,Y),

O(X,Y) = g(HX,Y)

for any vector fields X and Y, we see that 0, ' and 0 are local 2-forms. Then

S2=0n 0+T AT+OA 0 is a 4-form defined globally in M. Moreover, we see that

A=F0F+G0G+H0H is a global tensor field of type (2,2) in M.

We now assume that the Riemannian connection V of (M,g,V) satisfies the following conditions: (b) If $ is a cross-section (local or global) of the bundle V, then VX$ is also a cross-section of V, X being an arbitrary vector field in M.

From (6.1) we see that the condition (b) is equivalent to the

161

following condition: (b') If

F,G,H

is a canonical local basis of V, then

r(X)G - q(X)H,

VXF = VxG = - r(X)F

(6.3)

VXH =

+ p(X)H,

q(X)F - p(X)G

for any vector field X, where p, q and r are certain local 1-forms. If an almost quaternion metric manifold M satisfies the condition (b) or (b'), then M is called a quaternion XaehZerian manifold and an almost quaternion structure of M is called a quaternion KaehZerian structure.

THEOREM 6.1. An almost quaternion metric manifold M is a quaternion Kaehlerian manifold if and only if VS2 = 0 or VA = 0.

Proof. If M is a quaternion Kaehlerian manifold, then (6.3) implies VS2 = 0 and VA = 0. Conversely, if Vn = 0 or VA = 0 we see that F, G and H satisfy (6.3).

QED.

Let M = (M,g,V) be a quaternion Kaehlerian manifold of dimension n = 4m with canonical local basis {F,G,H} of V. Let R be the Riemannian curvature tensor of M. From (6.3) we have

(R(X,Y)F)Z = ((VXVY

-

VYVX

- V[X,Y])F)Z

= dr(X,Y)GZ + (p A-r)(X,Y)BZ - dq(X,Y)BZ + (p A r)(X,Y)GZ _ (dr(X,Y) + (p n q)(X,Y))GZ - (dq(X,Y) + (r A p)(X,Y))HZ,

from which

in (6.4)

(R(X,Y)F) = R(X,Y)F - FR(X,Y) = [R(X,Y),F] = C(X,Y)G - B(X,Y)H,

where we have put

B=dq+r^p, C=dr+pAq. Similarly, we obtain

(6.5)

(R(X,Y)G) = [R(X,Y),G] _ -C(X,Y)F + A(X,Y)H,

(6.6)

(R(X,Y)H) = [R(X,Y),H] = B(X,Y)F - A(X,Y)G,

where we have put

A=dp+qnr. Fran (6.4), (6.5) and, (6.6) we have g(R(X,Y)FZ,FW)-g(R(X,Y)Z,W) = c(X,Y)g(Z,HW)+B(X,Y)g(Z,GW), (6.7)

g(R(X,Y)GZ,GW)-g(R(X,Y)Z,W) = A(X,Y)g(Z,FW)+C(X,Y)g(Z,HW), g(R(X,Y)HZ,HW)-g(R(X,Y)Z,W) = B(X,Y)g(Z,GW)+A(X,Y)g(Z,FW).

Fran the second equation of (6.7) we find

4m 2

g(R(X,Y)ei,Fei) = 4mA(X,Y), i=1

where {ei} is an orthonormal basis of M. Thus we have

(6.8)

A(X,Y) -=hrPR(X,Y), 2m

B(X,Y)2mr (X,Y),

C(X,Y) ;6HR(X,Y),

where Tr is the trace. From (6.8) and the Bianchi's 1'st identity

lg(R(X,ei)Fei,Y) = 11[g(R(X,ei)Fei,Y)-g(R(X,Fei)ei,Y)] = il[g(R(X,ei)Fei,Y)+g(R(Fei,X)ei,Y)]

= zlg(R(Fei,ei)X,Y) = -Jlg(R(X,Y)ei,Fei) = -mA(X,Y).

Similarly, we have

(6.9)

lg(R(X,ei)Fei,Y) = -mA(X,Y),

7,g(R(X,ei)Gei,Y) = -mB(X,Y),

lg(R(X,ei)Hei,Y) = -mC(X,Y).

We denote by S the Ricci tensor of M. Then (6.7) and (6.9) imply

S(X,Y) = -mA(X,FY) - B(X,GY) - C(X,HY), (6.10)

S(X,Y) = -A(X,FY) - mB(X,GY) - C(X,HY), S(X,Y) = -A(X,FY) - B(X,GY) - mC(X,HY).

rhus, if m = 1, then

(6.11)

S(X,Y) = -A(X,FY) - B(X,GY) - C(X,HY),

and if m > 1, then

(6.12)

S(X,Y) = -(m+2)A(X,FY) = -(m+2)B(X,GY) = -(m+2)C(X,HY).

From (6.7) and (6.12) we find, for m > 1,

g(R(X,Y)FZ,FW)-g(R(X,Y)Z,W) 2[g(Z,HW)S(X,HY) +g(Z,GW)S(X,GY)],

164

g(R(X,Y)GZ,GW)-g(R(X,Y)Z,W) -2[g(Z,FW)S(X,FY)

(6.13)

+g(Z,HW)S(X,HY)],

g(R(X,Y)HZ,HW)-g(R(X,Y)Z,W)

2[g(Z,GW)S(X,GY) +g(Z,FW)S(X,FY)].

Since A, B and C are all skew-symmetric, using (6.12) we find, for m > 1,

(6.14)

S(F!,FY) = S(X,Y),

S(GX,GY) = S(X,Y),

S(HX,HY) = S(X,Y),

from which

(6.15)

S(FX,Y) _ -S(X,FY),

S(GX,Y) = -S(X,GY),

S(HX,Y) _ -S(X,HY).

On the other hand, using (6.3), we find

(6.16)

(VxS)(Y,FZ) + (VxS)(Z,FY) = 0.

Thus, if m > 1, then

(6.17)

(VX.S)(Y,Z) = (VxS)(FZ,FY).

LEM" 6.1. Fbr any quaternion Kaehlerian manifold M of dimension n = 4m (m > 1), the Ricci tensor S of M is parallel.

Proof. Fran (6.3) and (6.12) we have (VXS)(Y,FZ) = (m+2)[(4 i)(Y,Z)+q(X)C(Y,Z)-r(X)B(Y,Z)],

from which

(6.18)

(VxS)(Y,FZ) + (VYS)(Z,FX) + (VZS)(X,FY)

= (m+2)(dA + q n C - r A B)(X,Y,Z) = 0. Thus we have

-(VxS)(Y,Z) + (VYS)(FZ,FX) + (V

SX(x,FY) = 0.

Substituting (6.17) into the equation above, we find

(VYS)(Z,X) - (VxS)(Y,Z) = -(V

S)(X,FY).

Since we have (VxS)(Y,Z) = (VxS)(GY,GZ), we obtain

(VVS)(Z,x) - (VxS)(Y,Z) = (VyS)(GX,HY).

Similarly we obtain

(VYS)(Z,X) - (VxS)(Y,Z) = (VGZS) (HX,FY) = (VHZS)(FX,GY).

Combining these equations we have

(6.18)

(V

S)(GX,HY) = (VGZS)(HX,FY) = (VjjZS) (FX, GY).

In particular, we have (VS)(GX,HY) = (VS)(FX,GY). Changing Z, X, Y by GZ, HX, FY respectively, we obtain

(6.19)

(VHZS) (FX,GY) = -(V

From (6.18) and (6.19) we see that (V

S)(GX,HY).

S)(GX,HY) = 0 and hence

(VZS)(X,Y) = 0. Consequently, the Ricci tensor S of M is parallel. QED.

166

LEhMA 6.2, Let (M,g,V) be a quaternion Kaehlerian manifold of dimension > 8. Then the Riemannian manifold (M,g) is irreducible if (M,g) has non-vanishing Ricci tensor.

Proof. Suppose that M is reducible and has non-vanishing Ricci tensor. Since M is not flat and the Ricci tensor is parallel, we can take a coordinate neighborhood U of M such that the Riemannian manifold (U,g) is decomposed into a Riemannian product of certain number of Riemannian manifolds (W1,gl),...,(W gp), p > 2, in such a way that p

p

P

-

g(X,Y) =

S(X,Y)

gA(nAX,7TAY),

A=1

=

cAgA(nAX1nAY), A=1

where cA are constants such that cl < ... < cp and 7A: U > WA are the natural projections which denote at the same time their differential mappings. Since g(FX,FY) = g(X,Y) and S(FX,FY) = S(X,Y), we have

j1A(1TAFx1tAFY) = P

P

TTAFY) = AI CAgA(nAX,nAY). From these equations we obtain

gA(nAFX"TAFY) = gA(nAX,xAY),

A = 1,...,p.

From now on for simplicity we assume that p = 2, i.e., that (U,g) _ (W1,g1) x (W2,g2). Let (ye) = (y1,...,yr) and (zA) = (zr+l, ...,z°) be coordinate systems in W1 and W2 respectively. Then (xh) _ are naturally coordinates in U. With respect to (xh) = (YC,zX), g has components of the form

(gji) =

g,(s 0 0

g11v

where (g_(3) and (gu,)) are respectively the components of g1 and g2,

(g,,s) and (guv) being independent of the variables zx and y°` respec-

tively (see Chapter VIII). We then have Fa = 0 and hence

167

AA = B x = ua = 0. Similarly, we find Asa = BSa = Csa = 0 and

Na

a

= 0. Consequently, we obtain A = B = C, which implies

that S = 0 by (6.12). This is a contradiction. Therefore, M is irre-

ducible.

QED.

From Lemmas 6.1 and 6.2 we have the following theorems.

THEOREM 6.2. Any quaternion Kaehlerian manifold of dimension > 8 is an Einstein manifold.

THEOREM 6.3. If a quaternion Kaehlerian manifold M of dimension > 8 has non-vanishing scalar curvature, then M is irreducible.

THEOREM 6.4. If a quaternion Kaehlerian manifold M of dimension > 8 has zero scalar curvature, then M is locally a Riemannian product

of a flat quaternion Kaehlerian manifold and irreducible quaternion Kaehlerian manifolds with vanishing Ricci tensor.

In the next place, we assume that a quaternion Kaehlerian manifold (M,g,V) is of constant curvature c. Then we have

g(R(X,Y)Z,W) = c[g(Y,Z)g(X,W) - g(X,Z)g(Y,W)].

Prom this we have g(R(X,Y)FZ,FW) = c[g(Y,FZ)g(X,FW) - g(X,FZ)g(Y,FW)].

Using (6.8), we obtain

A(X,Y) = -

B(X,Y)

(FX,Y),

g(GX,Y),

C(X,Y)

g(HX,Y).

Substituting these equations into the first equation of (6.7), we have

c[g(Y,FZ)g(X,FW)-g(X,FZ)g(Y,FW)]-c[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)] C

from which

[g(HX,Y)g(Z,HW)+g(GX,Y)g(Z,GW)],

ISO

c(2m+1)(m-1)g(Y,Z) = 0.

Thus, if c # 0, we get m = 1. Qonsequently, we have THEOREM 6.5. If a quaternion Kaehlerian manifold M is of non-zero constant curvature, then M is of dimension 4. Since any quaternion Kaehlerian manifold M is Einstein, if M is conformally flat, then it is of constant curvature. Thus, from Theorem 6.5 we have THEOREM 6,6, If a quaternion Kaehlerian manifold of dimension > 8 is conformally flat, then it is of zero curvature. In the following we shall define Q-sectional curvatures in a

quaternion Kaehlerian manifold. We define the 4-dimensional subspace Q(X) of the tangent space

Tx(M) of M at x by

Q(X) = {Y : Y = aX+bFX+cGX+dHX}

for a vector X of M, a, b, c and d being arbitrary real numbers. Q(X) is called the Q-section determined by X. We denote by K(X,Y) the sectional curvature of M with respect to the section spanned by X and Y. Since M is Einstein, using (6.13), we obtain, for a unit vector X,

K(X,FX)=g(R(X,FX)FX,X) =

(6.20)

K(X,GX)=g(R(X,GX)GX,X) =

4m(m+k

2) -

k 4m(m+2)

g(R(X,FX)GX,HX),

- g(R(X,GX)HX,FX),

k being the scalar curvature of M. Now we suppose that for any Y,Z a Q(X) the sectional curvature K(Y,Z) is a constant p(X), which will be called the Q-sectional curvature of M with respect to X at x e M. Then, putting Y = X, Z = FX,

Y = X, Z = GX and Y = X, Z = HX, we have respectively

K(X,FX) = K(X,GX) = K(X,HX) = P(X)-

(6.21)

Since K(Y,Z) is constant for any Y,Z E Q(X), (6.20) and (6.21) imply

(6.22)

(6.23)

k P(X) = 4m(m+2)' g(R(X,FX)GX,HX) = g(R(X,GX)HX,FX) = g(R(X,HX)F%,GIX) = 0,

where we have used the Bianchi's l'st identity. Next, from the assumption we find K(X,aFX+t X) = p(X), a2+b2 # 0, which together with (6.21) implies g(R(X,FX)X,GX) = 0. Similarly,

(6.24)

g(R(X,FX)X,GX) = g(R(X,GX)X,HX) = g(R(X,HX)X,FX) = 0.

Thus, using (6.23) and (6.24), we find

(6.25)

R(Y,Z)Y - p(X)Z E Q(X)

for any Y,Z a Q(X), where Q-(X) denotes the orthogonal complement of Q(X) in Tx(M).

Conversely, if we assume that (6.25) holds for any Y,Z E Q(X) at x E M, then the sectional curvature K(Y,Z) is constant for any Y,Z e Q(X). In the case above, p(X) is called the Q-sectional curvature of M with respect to X at x, and the Q-section is said to have Q-sectional curvature p(X).

Let (M,g,V) be a quaternion Kaehlerian manifold and assume that each Q-section Q(X) at each point x of M, X being an arbitrary vector of M at x, has a Q-sectional curvature p(X). Moreover, if we suppose that the Q-sectional curvature p(X) is a constant c = c(x) independent of X at each point x, then we asy that M is of constant Q-sectional curvature c(x). Since M is Einstein, from (6.22), we see that c(x) is

constant on M. In the sequel we shall determine the form of the curvature tensor

170

R of a quaternion Kaehlerian manifold of constant Q-sectional curvature c. Let (M,g,V) be a quaternion Kaehlerian manifold of constant Q-sectional curvature c of dimension > 8. In the following, we shall calculate equations by using components of tensors. But, we notice that

the components Rijkl = g(R(eiej)ek,el) of the curvature tensor R differ from that of Chapter I by sign. Since M is Einstein, we may put

A

k7

= -4aFkj,

Bkj = -4aGkj,

Ckj = -4aHkj

a being a certain constant. Substituting these equations into (6.7),

we get

(6.26)

-

-RkjtsFte +

jih = -4a(GkjGih + HkjHih),

from which

(6.27)

-RvjshFkFi - RvjsiFkFh = 4a(GkjGih + Hk 1H h

(6.28)

-RktshFjFi

RktsiFjFh = -4a(G Gih + Ii .H

From the assumption we have K(X,FX) = c for any vector X, taking account of the definition of sectional curvatures, we get

RvjshFk'X'XXX'Xh = -cgjhgkiXkXjX

,

Xh being the components of X. Since X is arbitrary tatcen, from the equation above we obtain

RvjshFkFi + RvishFjFk + RvkshFiFj + RvjsiFk h + RvhsiFjFk + RvksiFhFj + RvkshFjFi + RvishFkFj

+ RvjshFiFk

+ RvksiFjFh +

RVhsiFkF + RvjsiFhFk

= -4c(gkjgih + gjigkh + gikgjh),

171

which, together with (6.27), implies

RvjshFkFF + RvishFJFk + RvkshFiFJ

_ -c(gkjgih + gjigkh + gykgjh) - 4a(GGih + HkjHih) -

2a(C'kjGih + G

j iGkh

+ G

ilcG,7.h

+

Hk jHih

If we transvect the above equation with

v

_

+ H

j iHkh

+ H

iltH,7.h).

then we obtain

v

v uqh pFj - pvuh qFj

Rjpgh

_ -c
ph qhjp + gqp jh gpHjh).

- 2a(%j Ph + ghGjp + GgpGjh + qj ph + 9hhjp + Substituting

RvughFQFj = Rjpqh + 4a(JPqh + HjpHgh

which is a consequence of (6.26), in the above equation we get

(6.29)

Rgiph - Rjpgh -

RpwhFFU

- 4a(GjpGgh + H. Hgh)

_ -c(qj ph - Fqh jp + ggpgjh)

- 2a( 4j Ph + qh jp + gpGjh

+ HqjHph + Hqh H.p + HqpHjh.

).

On the other hand, (6.26) implies (6.30)

pvuh qFj - RpuvhFVFj = (pvuh - puvh)FFFj

_ -RvuphFQFj = -R

gjph - 4a( q.7Ph + qj

Ph).

Taking the skew-symmetric parts of the both sides of (6.29) with

172

respect to q and j, and substituting (6.30) in the resulting equation

we obtain 9JPh - RJ

h + qpjh

+ qjph + 4a(Ggj

- 4a(GJP qh

Ph

+ qj ph)

Jh + JP qh - HgpR h)

_ -c(2FgJ Ph - Fqh Jp + JhFqp +

ggpgjh - gjpggh)

- 4a( 9j Ph + qj Ph), from which

gjph =

(gghgjp

+ a
- gjhggp + FqhFjp - Fjh gP -

21qj ph)

Jhgp - 2gjlih + QhHJp

Jh Qj - 2H jHi).

If we now take account of K(X,FX) = c, then we have a = Ic and hence

(6.31)

Rkjih = c(gkhgji - gjhgki + F Fji - FjhFki - 2F Fih

+ G Gji

jhGki

2GJGih

+ HkhHji

jhHki - 2H. Hih).

Summing up, we have THEOREM 6.7. A quaternion Kaehlerian manifold M of dimension n>8 is of constant Q-sectional curvature c if and only if

R(X,Y)Z = }c[g(Y,Z)X-g(X,Z)Y+g(FY,Z)FX-g(FX,Z)FY-2g(FX,Y)FZ +g(GY,Z)GX-g(GX,Z)GY-2g(GX,Y)GZ+g(HY,Z)HX-g(HX,Z)HY -2g(HX,Y)HZ]

for any vector fields X, Y and Z on M. Excmtple 6.1. Let HPm be a quaternion projective space of dimen-

sion 4m. HPm is of constant Q-sectional curvature 4. Let S3 be a

173

unit sphere of dimension 4m+3. Then we can consider the Hopf fibering S4m+3 S4m+3 admits a Sasakian 3-structure. > HIP.

174

EXERCISES

A. CONSTANCY OF THE HOLOMORPHIC SECTIONAL CURVATURE: Let M be a Kaehlerian manifold with almost complex structure J. Nomizu [5] gave the condition for constancy of the holomorphic sectional curvature of the curvature tensor of a Kaehlerian manifold M.

THEOREM 1, The curvature tensor R at a point of a Kaehlerian manifold M has constant holomorphic sectional curvature if and only if it has the following property:

(A) If g(X,Y) = g(JX,Y) = 0, then g(R(X,JX)JX,Y) = 0. A subspace S of the tangent space of M is holomorphic if JS = S. S is said to be totally real if it satisfies the condition: g(JX,Y) = 0 for all X,Y E S.

(B) Axiom of holomorphic 2k-planes. For any 2k-dimensional holomorphic subspace S of Tx(M), there exists a 2k-dimensional

totally geodesic suh nifold V of M containing x such that Tx(V) = S.

(C) Axiom of totally real k-planes. For any k-dimensional totally real subspace S of Tx(M), there exists a k-dimensional totally geodesic submanifold V of M containing x such that Tx(V) = S. As applications of Theorem 1, Nomizu [5] proved the following theorems.

THEOREM 2. If a Kaehlerian manifold M of dimension 2n satisfies the axiom of hold orphic 2k-planes for some k, 1 < k < n-i, then M has constant holonorphic sectional curvature.

THEOREM 3. If a Kaehlerian manifold M of dimension 2n satisfies the axiom of totally real k-planes for some k, 2 < k < n, then M has constant holomorphic sectional curvature.

For constancy of the holomorphic sectional curvature, see also Chen-Ogiue [1], Harada [2] and Yamaguchi-Kon [1].

175

B. COMPACT KAEHLERIAN MANIFOLD WITH R(X,Y)R = 0: On a Kaehlerian manifold M, Ogawa [1] studied the condition R(X,Y)R = 0 and prove the

following THEOREM Let M be a coapact Kaehlerian manifold with constant scalar curvature. If M satisfies the condition R(X,Y)R = 0, then M is locally symmetric.

Tb prove the theorem above, we used the integral formula of Lichnerowicz [4].

C. BOCHNER CURVATURE TENSOR: Let M be a complex n-dimensional Kaehlerian manifold. We denote by R, Q and r the Riemannian curvature tensor, the Ricci operator and the scalar curvature of M respectively. The Bochner curvature tensor B of M is defined by

B(X,Y)Z = R(X,Y)Z - 1[g(Y,Z)QX-g(QX,Z)Y+g(JX,Z)QJX-g(QJX,Z)JY +g(QY,Z)X-g(X,Z)QY+g(QJX,Z)JX-g(JX,Z)QJY-2g(JX,QY)JZ (2n+2)(r

-2g(JX,Y)QJZ] +

2n+4)[g(Y,Z)X-g(X,Z)Y

+g(JY,Z)JX-g(JX,Z)JY-2g(JX,Y)JZ].

Matsumoto-Tanno [1] proved the following theorems.

THEOREM 1, Let M be a Kaehlerian space with parallel Bochner curvature tensor. Then M is a locally symmetric space or a space with vanishing Bochner curvature tensor.

THEOREM 2. If a Kaehlerian space with vanishing Bochner curvature tensor has constant scalar curvature, then M is one of the following spaces:

(1) M is a space of constant holamrphic sectional curvature; (2) M is a locally product space of two spaces of constant holomorphic sectional curvatures c and -c.

178

CURVATURE: A. Gray [4] proved the

D.

following

THEOREM 1. Let M be a compact Kaehlerian manifold with nonnegative sectional curvature. If M has constant scalar curvature, then M is locally symmetric.

It is impossible to replace "locally symmetric" by "symmetric" in the conclusion to Theorem 1. Indeed, Auslander [1] has constructed compact flat Kaehlerian manifold which are not homogeneous. Furthermore we have by Theorem 1'the following THEOREM 2. If in addition to the hypotheses of Theorem 1, M has positive Ricci curvature, then M is globally symmetric: More generally, if M satisfies the hypotheses of Theorem 1 and has positive first Chern class, then M is globally symmetric. We also have the following (Berger [1], Bishop-Goldberg [1]) THEOREM 3. Let M be a compact Kaehlerian manifold with positive sectional curvature and constant scalar curvature. Then M is isometric to a complex projective space.

E. COMPLEX SPACE FORMS AND CHERN CLASSES: Let M be a Kaehlerian manifold of complex dimension n. We denote by ck the k-th Chern class of M. Let ml,...,c? be a local field of unitary coframes. Then the

Kaehlerian metric of M is written as g = E(ma ®w + ma 0 wa) and the Wa A V. Let fl = ERaY 1 A wa fundamental 2-form is given by -D = I be the curvature form of M. If we define a closed 2k-form rk by

(-1)k

rk

(2T v

)kk!

aal...a,c B1

B1...

Bk a1

...

'\nOk

ak

then the k-th Chern class ck of M is represented by rk. In particular,

cl and c2 are represented by

rl=2 a,

r2=-

;LE(nCLAns-staAS2a),

177

respectively. Chen-Ogiue [3] proved the following theorems. THEOREM 1, If M is an n-dimensional complex space form (n > 2), then c2 = i(n/(n+l))ci.

THEOREM 2, Let M be an n-dimensional compact Kaehlerian manifold

(n > 2). If (a) c2 = i(n/(n+l))ci,

(b) M is Einstein,

then M is a complex space form.

THEOREM 3. Let M be an n-dimensional compact Kaehlerian manifold (n > 2). If (a) c2 = i(n/(n+1))ci,

(b) the Bochner curvature is zero,

then M is a complex space form.

F. NEARLY KAEHLERIAN MANIFOLDS: Let M be a real n-dimensional nearly Kaehlerian manifold. Then we have (1) If n = 4, then M is Kaehlerian (A. Gray [1], Takamatsu [1]). (2) If n = 6, then M is Einstein.

(3) If n = 8, and if M is complete and simply connected, then

M is M1 x M2, where M1 is a 6-dimensional Einstein nearly Kaehlerian manifold and M2 is a 2-dimensional Kaehlerian manifold (A. Gray [ ]). When n = 10, Mats um to [3] proved the following

THEOREM, Let M be a 10-dimensional, connected, non-Kaehlerian, nearly Kaehlerian manifold. Then M is one of the following manifolds: (a) M, which is complete and simply connected, is M, X M2, where

M1 is a 6-dimensional Einstein nearly Kaehlerian manifold and M2 is a 4-dimensional Kaehlerian manifold; (b) M is an Einstein manifold;

(c) M satisfies 5Q - Q* = 8 I, which is not Einstein. See also, Matsumoto [2]. G. ALMOST QUATERNION MANIFOLDS:

It is well known for two given

tensor fields F and G of type (1,1) in a differentiable manifold the expression

178

[F,G](X,Y) = [FX,GY] - F[X,GY] - G[FX,Y] + [GX,FY] - G[X,FY] - F[GX,Y] + (FG+GF)[X,Y]

defines a tensor field [F-,G] of type (1,2), and that the tensor field [F,F] plays a very important role in the discussion of integrability conditions of an almost complex structure defined by F. We call [F,G]

the Nijenhuis tensor of F and G. Let M be an almost quaternion manifold with almost quaternion structure (F,G,H). If there exists a coordinate system in which the components of F, G and H are all constant, the almost quaternion structure is said to be integrable. Yano-Ako [i] obtained THEOREM, A necessary and sufficient condition that an almost quaternion structure (F,G,H) on M be integrable is that [F,G] = 0 and R = 0, where R is the curvature tensor of an affine connection on M.

For an affine connection on M see also Yano-Ako [1].

179

CHAPTER IV SUBMANIFOLDS OF KAEHLERIAN MANIFOLDS

In this chapter, we study submanifolds of Kaehlerian manifolds, especially those of carplex space forms. §1 is devoted to the study of Kaehlerian suYmanifolds (invariant submanifolds) of Kaehlerian

manifolds. We compute the laplacian for the second fundamental form and the Ricci tensor of a Kaehlerian submanifold of a complex space form, and give sore integral formulas. As applications of these integral formulas we prove the classification theorem of Kaehlerian hypersurfaces with parallel Ricci tensor of complex space forms, and we also study Kaehlerian hypersurfaces with constant scalar curvature. In §2, we discuss anti-invariant suhnanifolds of Kaehlerian manifolds

(see Yano-Kon [-1]). We give the basic formulas for anti-invariant sutmanifolds of canplex space forms and- sane examples of anti-invariant submanifolds of complex space forms. We then prove same theorems which characterize these examples. In the last §3, we discuss CR sutmanifolds

of Kaehlerian manifolds (see Yano-Kon [18]). We construct sane examples of CR submanifolds of a complex Euclidean space and complex projective space, which have parallel mean curvature vector, and flat normal connection or semi-flat normal connection. We also prove some theorems which characterize the above examples.

180

1. KAENLFRIAN SUBMANIFOLDS

Let 2 be a Kaehlerian manifold of complex dimension m (of real dimension 2m) with almost complex structure J and with Kaehlerian metric g. Let M be a ca plex n-dimensional analytic submanifold of

M, that is, the immersion f: M

> 2 is holamorphic, i.e.,

f*J, where f* is the differential of the immersion f and we denote by the same J the induced complex structure on M. Then the Riemannian metric g, which will be denoted by the same letter of 2, induced on M is Hermitian. It is easy to see that the fundamental 2-form with this Hermitian metric g is the restriction of the fudamental 2-form of 2 and hence is closed. This shows that every complex analytic such nifold M of a Kaehlerian manifold M is also a Kaehlerian manifold with respect to the induced structure. We call such a submanifold M of a Kaehlerian manifold M a Kaehlerian submanifold. In other words, a Kaehlerian submanifold M of a Kaehlerian manifold M is an invariant submanifold under the action of the complex structure J of 2, i.e., JTS(M) C Tx(M) for every point x of M. Then we see that JJX(M)L C X(M)4- for every point x of M. We denote by o (resp. V) the operator of covariant differentiation with respect to the connection in M (resp. M). For any vector fields X and Y on M we have

VXJY = VXJY + B(X,JY)

and

OXJY = JOXY = JOXY + JB(X,Y).

Both the tangent space TT(M) and the normal space Tx(M)' being invariant by J, we obtain

OXJY = JOXY

and

B(X,JY) = JB(X,Y).

The first identity shows once more that V is Kaehlerian. From the second identity and from symmetry of B(X,Y) in X and Y, we have

181

The second fundamental form B of a Kaehlerian submani-

LENMA 1.1.

fold M satisfies B(JX,Y) = B(X,JY) = JB(X,Y),

or equivalently

JAVX = -AVJX = AJVX.

PROPOSITION 1.1. Any Kaehlerian sutmanifold M is a minimal submanifold.

Proof. Fbr each x(M) we can choose an orthonormal basis el,...,en,Jel,...,Jen. Then Leuma 1.1 implies

TrB = l[B(ei,ei) + B(Jei,Jei)] = 0, i

which shows that M is a minimal sub;mnifold.

QED.

Frcm equations of Gauss and Lerma 1.1 we have PROPOSITION 1.2. Let M be a Kaehlerian submanifold of a Kaehlerian manifold M and let R and R be the Riemannian curvature tensors of M and M respectively. Then

g(R(X,JX)JX,X) = g(R(X,JX)JX,X) - 2g(B(X,X),B(X,X))

for all vector field X on M.

As a direct consequence of Proposition 1.2, we obtain PROPOSITION 1.3. Any Kaehlerian suhmanifold M of a complex space form M(c) of constant holanorphic sectional curvature c is totally geodesic if and only if M is of constant holomorphic sectional curvature c.

We now suppose that M is a ccaplex n-dimensional Kaehlerian suhmanifold of a complex m-dimensional space form IF(c) of constant

182

holcmorphic sectional curvature c. Then we have equations of Gauss and Codazzi respectively:

(1.1)

R(X,Y)Z = ic[g(Y,Z)X-g(X,Z)Y+g(JY,Z)JX-g(JX,Z)JY+2g(X,JY)JZ] + AB(Y,Z)X

- AB(X,Z)Y,

(VXB)(Y,Z) = (VyB)(X,Z).

(1.2)

Moreover, equation of Ricci is given by

(1.3)

g(R (X,Y)U,V) + g([AV,AU]X,Y) = zcg(X,JY)g(JU,V).

From (1.1) the Ricci tensor S and the scalar curvature r of M are respectively given by

(1.4)

S(X,Y) = i(n+1)cg(X,Y) - lg(B(X,ei),B(Y,ei)), i

r = n(n+l)c -

(1.5)

I g(B(e1 ej),B(eie i,j

From (1.4) and (1.5) we have PROPOSITION 1.4. Let M be a complex n-dimensional Kaehlerian sutrnanifold of a complex space form f°(c). Then

(1) S - i(n+1)cg is negative semi-definite; (2) r < n(n+1)c.

In the next place, we consider the Ricci tensor S of a Kaehlerian

submanifold M of a complex space form S(c). We have already seen that S satisfies (see Proposition 4.2 of Chapter III)

(OZS)(X,Y) = (VxS)(Y,Z) + (V 1S)(JX,Z).

LEMMA 1.2, If a Kaehlerian manifold M has the constant scalar curvature, then

2n

(O2S)(X,Y) = 2

(R(ei,X)S)(ei,Y). i=1

iuK Proof. Since the scalar curvature r of M is constant, the Ricci tensor S of M satisfies E(ve S)(ei,X) = 0. Thus we have i

(O2S)(X,Y)

IVe ((De S)(X,Y))

i i

i

i

CC 4[Ve.((VXS)(ei,Y)

=

1

i

+ Ve ((DjyS)(ei,JX))]

_ f[(R(ei,X)S)(ei,Y) + (R(ei,JY)S)(ei,JX)] 1

= 2j(R(ei,X)S)(ei,Y).

QED.

i

Take an orthonoemal basis el,...e2n in Tx(M) such that en+t = Jet

(t = 1,...,n) and an orthonormal basis vl,...,v2p for X(Mt such that vp+s = Jvs (s = 1,...,p), where we have put p = m-n. We calculate (V2S)(X,Y) for a Kaehlerian submanifold M of NF(c) in the following way. Since M is minimal, we obtain

27(R(ei,X)S)(ei,Y) = -2j[S(R(ei,X)ei,Y) + S(R(ei,X)Y,ei)

i

i

+ S(AB(X,el)ei,Y) - S(AB(ei,Y)X,ei) + S(AB(X,Y)ei,ei)].

Moreover, we have

-21[S(R(ei,X)ei,Y) + S(R(ei,X)Y,ei) = nc[S(X,Y) - Znrg(X,Y)]. i

the Ricci tensor S has the property S(JX,JY) = S(X,Y), and hence Lenin

1.1 implies that ES(AB(XY)ei,ei) = 0. We also have

-2I[S(AB(X,ei)ei,Y) - S(AB(ei,Y)X,ei)]

= -2 1 [g(Aaei,QY)g(AaX,ei) - g(AaX,Qei)g(AaY,ei)] i,a

= -21[g(QAaAaX,Y) - g(AQAaX,Y)], a where Q is the Ricci operator of M given by g(QX,Y) = S(X,Y). :onsequently, we have

184 22(R(ei,X)S)(ei,Y) = c[nS(%,Y) - irg(X,Y)] 1

- 22(g(QA0J,Y) - g( aQAAR,Y)]. a Therefore, lemma 1.2 implies the following (Bon [7])

THEOREM M. Let M be a complex n-dimensional Kaehlerian submanifold of a complex space form SP(c). If the scalar curvature r of M is

_

constant, then g(V2Q,Q) = c[nIQl2 - jr2] - GI[Q,Aa]I2.

a We give some applications of Theorem 1.1. First we obtain PROPOSITION 1.5. Let M be a complex space form of constant holomorphic sectional curvature c < 0, and let M be a Kaehlerian sutrnani-

fold of M. If the Ricci tensor S of M is parallel, then M is an Einstein manifold. Proof. From the assumption we have I Q 1 2 = r2/2n. This is equivalent

to the fact that M is Einstein.

QED.

PROPOSITION 1.6. Let M be a complex n-dimensional compact

Kaehlerian submanifold with constant scalar curvature of a complex space form Mm(e) (c > 0). If QAa = AaQ (a = 1,...,p), then M is an Einstein manifold.

Proof. If QAa = AaQ for a = 1,...,p, then we easily see that

QAa = AaQ for a = 1,...,2p. By the assumption and Theorem 1.1 we have

0 < JM1VQ12*1 = -JMg(O2Q,Q)*l = -cfM[nIQ12 - jr2]*1.

But we have always r2< 2nIQ12, hence we obtain aQ = 0. Therefore, we have r2 = 2nJQ12, which shows that M is an Einstein manifold.

W.

iL2 THEOREM 1.2,

Let M be a compact Kaehlerian hypersurface of a

complex space form Mn+1 (c) (c > 0). If the scalar curvature of M is constant, then M is an Einstein manifold.

Proof. Since the real codimension of M is 2, we can take an orthonormal basis v, Jv for Tx(M)`. Then we have

Q = #(n+1)cI - 2A2.

Therefore we obtain QAv = AvQ, and hence M is Einstein by Proposition

1.6.

QED.

We now compute the Laplacian for the square of the length of the second fundamental form of a complex n-dimensional Kaehlerian sutrnani-

fold M of a canplex space form Mm(c). From (3.10) of Chapter II, by a straightforward ccrnputation, we have

(1.6)

g(V2A,A) = j(n+2)cjAI2 -

E I[Aa,A ]I2 - E (TrAaAb)2. a,b a,b

LEMMA 1.3. Let M be a complex n-dimensional Kaehlerian submanifold of a canplex m-dimensional Kaehlerian manifold M. Then

(1.7)

!1AI4 < n

(1.8)

2p IAI4 <

I I[Aa,Ab]I2 a,b

IAI4,

E (TrAAb)2 < JIAI4, (p = m-n). a,b

If M is of constant holoaprphic sectional curvature c, then M is

Einstein if and only if EI[AaAb]I2 = IAI4/n. Proof. We put

Clearly A* is a symmetric, positive semi-definite operator. So we

have TrA* = I AI2. Fran la= 1.1 we see that

ISO

E I[Aa,Ab]I2 = 2 E ZrAaAb = 2Tr(A*)2. a,b a,b

We easily see that JA* = A*J. Thus, for a suitable basis, A* is

represented by a matrix form l

lx1

I

A* = ,

An+t = At,

at > 0 (t=l,...,n).

A2n

Then we have

2n Tr(A*)2=

2n )2 A2 _ [( a ajl 1 i=1 i=1 1 ij 1 2n n A.)2 - 4 E A.A. - Tr(A*)2, i=1 i#j

from which

JAI(1.9)

n

2Tr(A*)2 =

4 E A i X. < IAI4. J i#i

On the other hand, we obtain

2n

2n (1.10)

2Tr(A*)2 = 21 a? = i=1 1

a.)2 + (a. - a.)2 > 11AI4. ni>j 1 - n n i=1 1 (

Fran (1.9) and (1.10) we have (1.7). If 2 is of constant holy rphic sectional curvature c, then the Ricci operator Q of M is giben by Q = #(n+1)c - A*. If EI[Aa,Ab]I2 = IA14/n, 1i = aj for all i, j.

Thus Q is proportional to I and hence M is Einstein. The converse is also true.

We notice here that the following equation is satisfied:

(1.11)

II[Aa,Ab]I2 ° !IAI4 +

2IQI2 - nr2.

187

In the next place, we take a basis v1,.... v2p of X(M)` such that

a a E (TrAAb)2 = I(TrA2)2. a a,b Then we have

(1.12)

I(TrA2)2 =

a

JIAI4

p - 2

a

ab

(TrA2)(TrAb) < 1IAI4

a

p (1.13)

I (TrAa)2

= 2pIAI4 +

a

2

2nIAI42p

From (1.12) and (1.13) we have (1.8).

QED.

By Lemma 1.3 and (1.6) we have the following (Tanno [61)

THEOREM 13, Let M be a complex n-dimensional Kaehlerian submani-

fold of a corrgilex space form 1(c). Then -g(V2A,A) < 12[31A 12 - (n+2)c]IA12.

THEOREM 1.4, Let M be a ccq lex n-dimensional compact Kaehlerian

submanifold of a complex space form N(c). Then either M is totally geodesic, or IAI2 = (n+2)c/3, or at some point x of M, IAI2(x) > (n+2)c/3.

Proof. Since M is compact, we have

(1.14)

0 < fMIVAI2$1 = -fMg(O2A,A)*1 < JfM[3JAI2 - (n+2)c]IAI2*1.

Suppose IAI2 < (n+2)c/3 everywhere on M. Then VA = 0 and IAI2 = (n+2)c/3 or IAI2 = 0. Except for these possibilities, we get IAI2(x) > (n+2)c/3 at some point x of M.

PROPOSITION 1.7. Let M be a complex n-dimensional Kaehlerian sulmanifold of a complex space form En(c). If M is Einstein, then

0 < IVAI2 < #(n+2)[-11A12 - c]IAI2.

QED.

1ee Proof. Since M is Einstein, IAl2 is constant and hence -g(V2A,A) _ I VAI 2. Thus, from (1.6) and Lemma, 1.3, we have our inequality.

QED.

PROPOSITION 1.8, Under the same assumption as that of Proposition 1.7, either M is totally geodesic or I A12 > nc.

THEOREM 1.5, Let M be a complex n-dimensional Kaehlerian suYmanifold of a oonplex space form ff°(c) (c > 0). If IAI 2 = (n+2)c/3, then

M is an Einstein manifold of carplex dimension 1.

Proof. Since I A12 is constant, we have -g(V2A,A) = IVAI2. Fr an this and the assumption we see that the second fundamental form of M is parallel. Thus, from (1.9) and (1.12), we can assume that

ai = 0 (i = 2,...,n),

Aa = 0 (a = 2,...,P).

Therefore we see that A* = 2(A1)2. Since Q = !(n+1)cI - A*, we obtain QAa = AaQ for a = 1,...,p. Therefore, by Proposition 1.6, M is an Einstein manifold. Thus we have IA12 > nc, which implies that n = 1.

Thus we have our assertion.

QED.

THEOREM 1.6. Let M be a Kaehlerian hypersurface of a complex space form Mn+1(c). Then the following conditions are equivalent: (1) The Ricci tensor S of M is parallel; (2) The second fundamental form of M is parallel; (3) M is an Einstein manifold.

Proof. First of all we prove that (1) is equivalent to (2). If the second fundamental form of M is parallel, then the Ricci tensor S of M is obviousely parallel. We suppose that the Ricci tensor S of M is

parallel. We take a normal basis v, Jv of X(M)`. There exists a 1form s such that DXv = s(X)Jv for any vector field X tangent to M.

Thus we have, by (1.4),

(VXA)vAvY + s(X)AJvAvY + A (VXA )j + s(X)AvAJvY = 0.

Fran this and Lemma 1.1 we obtain

169

(VXA)VAVY + Av(VXA)vY = 0.

(1.15)

Let we take any two A, U of characteristic roots of Av at a point x of M. Now we define spaces by setting

TA = {X e Tx(M) : AvX = AX),

Tp = {X E Tx(M) : AvX = lX).

Then TA n Tu = {0} when A # p. Let Y be in TA. Then (1.15) irrplies

AA(VXA )j = -(VXA)vAvY = -1(VXA)vY

for any vector X. Thus, if Y E TX

,

then (VXA)vY E T

A

for any X. Let

X e TP, Y E TA and A # u. Then, using the Codazzi equation, we obtain

(VXA)vY ETA n T u, and hence (VXA)vY = 0. On the other hand, if A = p # 0, X, Y c TA, then we have

(VXA)vY E T_A and hence (VXA)v(VXA)vY e T. By the Codazzi equation we also have

(VXA)v(VXA )j = (V(VXA) YA)vX E T_A. v

Therefore, we have (VXA)v(VXA)vY = 0. Thus we obtain (VXA )j = 0. Let A = u = 0. We take X, Y e TO at x E M, and extend these to .local vector fields on M which are covariant constant with respect to

V at x. Then, by (1.15) we have

g((VXA)v(VXA)vY,Y) = 0,

from which (VXA)vY = 0.

From these considerations we have our assertion. Next we prove that (2) is equivalent to (3). If M is Einstein, then the Ricci tensor of M is parallel and hence the second fundamental form of M is parallel.

190 Conversely, we suppose that the second fundamental form of M is parallel. Then Theorem 1.1 implies

c[nIQ12 - ir2] = 0.

Thus, if c 0 0, then M is Einstein. Suppose that c = 0. Then (1.6)

shows that M is totally geodesic. Consequently, M is an Einstein

manifold.

QED.

We now prove the following theorem (Chern [1], Nomizu-3nyth [1],

Tsunero Takahashi [2]). THEOREM 1.7. Let M be a Kaehlerian hypersurface of a cciiplex space

form kn+1(c) with parallel Ricci tensor. If c < 0, then M is totally geodesic. If c > 0, then either M is totally geodesic, or an Einstein manifold with 1A122 = nc and hence r = n2c.

Proof. From Theorem 1.6 the second fundamental form of M is parallel. Thus, if c < 0, M is totally geodesic by (1.6). Let c > 0.

Then, by (1.6), Lemma 1.3 and Theorem 1.6, we have

(n+2)[c - nIA!2]IA12 = 0.

Therefore, I A12 = 0 or I A12 = nc. From these and (1.5) we have our

W.

assertion.

In the next place we give a global version of Theorem 1.7. For this purpose we prepare some lemmas. Let M be a Kaehlerian hypersurface of a complex space form Ium(c).

We take a local basis v, iv in the normal bundle for which the second fundamental tensors are given by AV, AJv = JAv. To simplify the notation we write Av by A. Moreover, we take a 1-form s such that

DXv = s(X)Jv. We put

D(X,Y) = AX A AY + JAX A JAY,

where X A Y denotes the skew-symmetric endomorphism which maps Z upon

191

g(Y,Z)X - g(X,Z)Y. Then we have

D(X,Y) = -R(X,Y) + R(X,Y) _ -*c[X n Y + JX A JY + 2g(X,JY)J] + R(X,Y).

LEMMA 1.4. At each point x of M we have

Ker A = {X a X(M) : D(X,Y) = 0 for all Y c T{(M))

= {X c X(M) : (R - R)(X,Y) = 0 for all Y c Tt(M)). Proof. Clearly Ker A is contained in the subspace defined by D.

On the other hand, if X

Ker A, then D(X,JX) = 2JAX A AX # 0, from

which we have the first equality. The second equality is clear by the definition of D.

QED.

The rank of A at x e M will be called the rank of N at x since Lemma 1.4 shows that it is intrinsic, that is, depends only on M.

Let f, f : M

> M be two Kaehlerian immersions. For each

x0 e M there is a neighborhood U(x0) of x0 in M on which we can choose of a unit normal vector field v (resp. v) for the immersion f (resp. f), there by giving rise to tensor fields A and s (resp. A and s) on U(x0). LEMMA 1.5. At each point x of U(x0) we have

(1) A = 0 if and only if A = 0;

(2) If A=A#OandVA=VA, then s

s.

Proof. (1) is immediate consequence from Lemma 1.4. We next

prove (2). From the Codazzi equation (1.2) we obtain

(VXA)Y - s(X)JAY - (VYA)X + s(Y)JAX = 0.

From the assumption we have

-s(X)JAY + s(Y)JAX = -s(X)JAY +'s(Y)JAX.

192

Thus we have

(s(X) - s(X))JAY = (s(Y) - s(Y))JAX.

Suppose that X

Ker A. Then, by setting Y = JX, we get

(s(X) - s(X))AX = (s(JX) - s(JX))JAY.

Since AX (# 0) and JAX are linearly independent, we conclude that

s(X) = s(X). If X c Ker A, then we choose Y

Ker A and get

(s(X) - s(X))JAY = 0, that is, s(X) = s(X).

Qom.

Left 1,6, Assuming R # R (that is, A # 0) at some point of M, let x0 be a point where the rank of M is maximal. There exists a neighborhood U(x0) of x0 on which we may choose unit normal vector fields v and v, with respect to the immersions f and f, respectively, such that A = A and s = s on U(x0).

Proof. On U(x0) on which the rank of M is constant and equal to k, say, we choose unit normal vector fields v and v, with respect to the immersions f and f, respectively. At each point x of U(x0) we can

choose an orthonormal basis el,...,en,Jel,...Jen of Tx(M) for which A is

Aei = aiei (i = 1,...,k),

Aei = 0 (j = k+l,...,n).

Then AJei = -aiJei (i = 1,...,k) and AJej = 0 (j = k+l,...,n). Since we have

(R - R)(ei,Jei) = -2Aei n JAei = -2Aei A JAei,

and the middle form of this identity being nonzero when i < k, we see that Ae i is a linear combination of Aeiand JAei, say, Aei = aiAei +

biJAei. Then ai + bi = 1. From

13

(R - R)(ei,ei) = Aei A Aej + JAei A JAej = Aei A Aej + JAei A JAej we easily see that ai = aj = a, say, and bi = bj = b, say, for i,j = 1,...,k. However, Ker A = Ker A, by Lemma 1.4, and therefore A = aA + bJA with a2 + b2 = 1 at each point of U(x0). From the assumption on the rank of M at x0 we can find a differentiable vector field X on U(x0) such that AX 0 0 and since a = g(AX,AX)/g(AX,AX), it follows that a (and hence b) is a differentiable function on U(x0) such that a = cosh, b = sinO. Then v' = cosOv + sinOJv is a unit normal vector field on U(x0) with respect to the iriiersion f and clearly A' = A. By

Lama 1.5, it follows that s' = s.

QED.

We now prove the rigidity theorem of Kaehlerian hypersurfaces (Nomizu-&nyth [1]).

THEOREM 1.8. A connected Kaehlerian hypersurface M of a simply connected complex space form Mn+1(c) is rigid in Mn+1(c). Proof. If R = R, then M is totally geodesic and hence M is rigid. If R # R at some point of M, let x0 be a point where the rank of M is maximal. Let-f, f : M

> M be two Kaehlerian immersions. From

Lemma 1.6, there exists a neighborhood U(x0) of x0 and suitably choosen unit normal vector fields v and v on U(x0) with respect to f and f respectively such that A = A and s = s on U(x0). We now resort to local coordinates to show that f =

on U(x0), 0 being a holomorphic motion

of M. In fact, since the group of holomorphic isanetries of M is transitive on the set of unitary frames, we may assume that without loss of generality that

f(x0) = f(x0),

f*(x0) = f*(x0),

v(x0) = v(x0)

and prove that f = f in a neighborhood of x0. Let (xl,...,x2n) be a 2n+2 system of local coordinates on U(x0) and let (u1,...,u ) be a system

194

of local coordinates on a neighborhood of f(x0) in &1 derived from a system of conplex coordinates. We use the following ranges for indices: i,j,k,l = 1,...,2n;

p,q,r,s = 1,...,2n+2.

Our notations (in the summation convention) will be afp

fp(x) = up(f(x)),

i

f (a ) = fP(a),

a

r yr = av

yr

ii

axi

iJ

ax

v = yr

etc.,

axiaxi

etc.,

axiaxi,

our

2 r 9V

=

a2fp

Jv = (Jv)r a

,

au

1 aup

k axi

i

fP. =

fp =

...

The corresponding notation for f is then self-explanatory. We also use

h.. i = g(A a., a ), k = g(Jt-a.,a).

i

ax1 axi

k4 = da ax1 taxi

,

ax1 axi

s( a ) = Si. 1 axi

Note that we have A = A and s = s so that we do not need the corresponding notation for f here. The Christoffel symbols are denoted by -vr+n+l rl for (x1) and by rp for (up). We note that (Jv)r = and qhr

(Jv)n+1 = vr. Then (I)

ri

_

i

Gauss and Weingarten formulas imply

fkrk + h.

r + k..(Jv)r,

We denote the corresponding equations for the iamersion f by (3) and (II). At x0 we have

fp(x0) = fp(x0),

fp(xO) = fp(x0),

(1.16)

(Jv)r(x0) = (Jv)r(x0).

vr(x0) = vr(xo),

995

We wish to show that f = 7 in a neighborhood of x0; since fp and fp are real analytic it sufficies to prove

(1.17)

fpj (x0) = fpj(x0),

fpj'k(x0),

(1.18)

and so on for all higher-order derivatives at x0. (1.17) follows from (I), (I), (1.16) and the equation A = A on U(x0), while

(1.19)

vi(x0) = vi(x0)

follows from (II), (II), (1.16) and the equations A = A ans s = s on U(x0). Now fib

and fink are obtained by differentiating (I) and (I)

and we deduce (1.18) from (1.16), (1.17), (1.19) and A = A on U(x0). In the same manner vii and vii are obtained by differentiating (II) and (II). Using the previous equations together with the equations A = A and s = s on U(x0), we infer

vr.(x0) = vii(x0).

We can then easily obtain

(x0)

The equations for higher-order derivatives are obtained in the same fashion. Thus f = f in a neighborhood of x0 and this completes the proof.

QED.

Moreover, we have the following (3myth [1]) THEOREM 1.9. If M1 and M2 are Kaehlerian hypersurfaces of Mr'+1(c)

which are simply connected complete Einstein manifolds with the same Ricci curvature, then they are holcmorphically isometric. Proof. Let fi denote the immersion of Mi into M (i = 1 or 2).

196

Take any xi u: Mi and let vi be any unit normal vector field of Mi with

respect to fi in a neighborhood U(xi) of xi. Let Ai be the second fundamental tensor for fi. We may choose an orthonormal basis ei,.:.,en,Jei..... Jen of x (hi) with respect to which i

AI n

0

0

-J1In

A. _ 1

> x2(M2)

We consider the holamrphic linear isometry F: T (M1) ,

given by F(ed) = e and F(Je]) = Jet, j = 1,...,n. Clearly On the other hand, from the assumption, the second fundamental form is parallel, and it follows from the Gauss equation that F maps the Riemannian curvature tensor of M1 at xl into the Rienannian curvature

tensor of M2 at x2. Since Mi is locally symmetric, the following lemm QED.

completes the proof.

LEMMA 1.7. Let Ml and M2 be arbitrary simply connected complete symmetric Kaehlerian manifolds. Any holomorphic linear isometry F: Tx1(M1)

> Tx2(M2) which preserves the Riemannian curvature

tensors extends uniquely to a holamorphic isometry of M1 onto M2.

Proof. F extends uniquely to an isanetry f: M1

> M2 (cf.

Kobayashi-Nomizu [ ;p.265]). Let yl be any point of M1 and set y2 = f(yl). Let c1 be any differentiable curve joining xl to y1, and set c2 =

Since parallel displacement along c2 corresponds, under f,

to parallel displacement along cl, and since the almost carplex structures of M1 and M2 are parallel, f maps the almost complex structure of M1 into that of M2. Thus f is holcmorphic.

W.

We now prove a global version of Theorem 1.7 (Namizu-Smyth [i]). THEOREM 1.10. (1) CPn and the complex quadric are the only complete Kaehlerian hypersurfaces in CPn+1 which have parallel Ricci tensors.

in

(2) Dn (resp. Cn) is the only complete Kaehlerian hypersurface Dn+1 (resp. Cn+1) which has parallel Ricci tensor. CPP+l

Proof. (1) Let f: M

>

be the Kaehlerian immersion.

197

If M has parallel Ricci tensor, then M is Einstein by Theorem 1.6. Let M be the universal covering manifold of M and let it be the covering

map. On M we take the Kaehlerian structure which makes n a holomorphic immersion. M is then simply connected and complete Einstein manifold. Moreover,

is a holamrphic isometric immersion of M in CPn+1: From

Theorems 1.7, 1.8 and 1.9 M immerses either onto a projective hyperplane or onto complex quadric in CPn+1. In either case

is a

is a covering map (see Theorem 4.6 in

simply connected and since

Kobayashi-Nomizu [11), it is one-to-one. Hence n is one-to-one and therefore M is holy orphically isometric either to CPn or to Qn. The same type of argument can be applied to (2).

QED.

From Theorem 1.2 and Theorem 1.10 we have (Kon [71) CPn+1.

THEOREM 1,11., Let M be a conpact Kaehlerian hypersurface of

If the scalar curvature of M is constant, then either M is CPn or Qn. We next consider a complex n-dimensional Ka.ehlerian submanifold

M with constant holcmorphic sectional curvature k immersed in a complex space form I

(c). First of all, the Gauss equation implies

(k-c)IA12 =

I

g(Abei,R(eitej)Abej-R(ei,ej)Abej) [g(Abei,Aaei)g(Aaej,Abej)

a,b,i,j

- g(Abei,Aa j)g(Abej,Aaei)] -

a,b

L(TrAaAb)2 - Tr(AaAb)2] _ - I (TrAaAb)2, a,b

from which

(1.20)

1)JA14.

E (TrAaAb)2 = (c-k)IAI2 = n(n+1

a,b From (1.8) and (1.20) we obtain

2p1AJ4

n(n+1)IAI4

Therefore we have (O'Neill [1])

(p = m-n).

198

THEOREM 1,12. Let M be a complex n-dimensional Kaehlerian suhmani-

fold of a complex space form

+p(c). If M is of constant holomorphic

sectional curvature k, and if_p < n(n+1)/2, then M is totally geodesic and hence c = k.

From (1.6), (1.20) and Lemma 1.3 we have (Ogiue [3], [4]) THEOREM 1.13. Let M be a complex n-dimensional Kaehlerian suUmani-

fold of a complex space form 9(c). If M is of constant holcmorphic sectional curvature k, then

IVA12 = n(n+l)(n+2)(c-k)(}c-k).

THEOREM 1.14. Let M be a complex n-dimensional Kaehlerian sutmanifold of a complex space form f"(c) (c > 0). If M is of constant holomorphic sectional curvature k, then either c = k, that is, M is totally geodesic, or c > 2k.

THEOREM 1,15. Under the same assumption as that of Theorem 1.13, 11 the second fundamental form of M is parallel, then c = k or c = 2k, the latter case arising only when c > 0.

999

2. ANTI-INVARIANT SUBMANIFOLDS OF KAEHLERIAN MANIFOLDS

Let M be a complex m,-dimensional (real 2n-dimensional) almost

Hennitian manifold with almost complex structure J and with Hennitian metric g. An n-dimensional Riemannian manifold M isometrically immersed in M is called an anti-invariant submanifold of M (or totally real

submanifold of M) if J X(M) C X(M f for each point x of M. Then we have m > n.

In this section we study an n-dimensional anti-invariant submanifold M of a complex m-dimensional Kaehlerian manifold M. We choose a local field of orthonormal frames

el, ... en'en+1'" ''em;el* Jet = Jen in M in such a way that, restricted

en* to M, el,...,en are tangent to M and hence the remaining vectors are Jen'e(n+l)*-Jen+1,...,e

normal to M. With respect to this frame field of M, let wl,...,wn n+1 m 1* n* (n+1)* m* W ,...,w ;w ,...,w ;w ,...,w be the field of dual frames. Unless otherwise stated, we use the conventions that the ranges of indices are respectively: i,j,k,l = 1,...,n;

a,b,c,d = n+l,...,m,1*,.... m*;

a,S,Y = n+l,...,m;

a,U,v

Then we have

Wi + Wi = 0,

w =

wa S + ws a = 0

wa =

w = wl*, wa*

i

wl + wa = 0,

a

S

S*p

Wa = wl

a

a*,

From wi = hijwJ we easily see that

(2.1)

hik = hk ,

B* ws a* = wa

i

wl = wa*

a

200

where we write h;k as

That is the second fundamental form A of M

satisfies

AMY = AjyX for any vector fields X and Y on M.

We assume that the ambient manifold M is of constant holomorphic sectional curvature c. Then the Gauss equation of M is given by

(2.2)

Rjkl = I°(didjl - dildjk) +

From (2.2) we see that the Ricci tensor Rij and the scalar curvature r of M are given respectively by

(2.3)

Rij = *(n-l)cdij +

(h

hij - hAhjk),

a,k (2.4)

r = $n(n-1)c +

I

a,i,j

ha hij (ha ha ii ii - ij

From (2.2), (2.3) and (2.4) we have PROPOSITION 2.1. Let M be an n-dimensional anti-invariant minimal submanifold of a complex space form Mm(c). Then M is totally geodesic if and only if M satisfies one of the following conditions: (1) M is of constant curvature }c; (2) S =i,(n-1)cg; (3) r = in(n-1)c.

Let M be an n-dimensional anti-invariant sutrnanifold of a complex m-dimensional Kaehlerian manifold M. Then we have the decomposition:

Tx(M)'' = JTX(M) ® Nx(M),

where Nx(M) is the orthogonal complement of JTX(M) in Tx(M). We see that the space Nx(M) is invariant under the action of J, that is,

J X(M) = NA(M). For any vector field V normal to M we put

JV=tV+fV, where tV is the tangential part of JV and fV the normal part of JV. We then have

tfV = 0,

f2V'= -V - JtV,

tJX = -X,

fJX = 0.

Therefore we have

f 3 + f = 0, which shows that, if f does not vanish, it defines an f-structure in the normal bundle (see Chapter VII). Using the Gauss and Weingarten formulas, we obtain

(VXf)V = -B(X,tV) - JAVX,

where we have put (VXf)V = DX(fV) - f(DXV). If VXf = 0 for all tangent vector field X, then the f-structure f in the normal bundle is said to be parallel.

LENMA 2.1. Let M be an n-dimensional anti-invariant sutxnanifold of a complex m,-dimensional Kaehlerian manifold M. If the f-structure

in the normal bundle is parallel, then

AV=0

forVcN{(M).

Proof. If V c N{(M), then tV = 0. Thus we have JAVX = 0 and hence

AV=0. Fran (3.9) of Chapter II and the fact that hick = h

QED.

iki

when the

ambient manifold is of constant holanorphic sectional curvature, we

have

202

LE u 2.2. Let M be an n-dimensional anti-invariant sutra nifold of a complex space form g°(c). Then

a,1,J

a a hph i i

i aa

h

=

a,i,J,k

+ }cj[TrA2 - (TrA )2] a a

a

+ jcj[TrAt - (TrAt)2] + aIb[Tr(AaAb - AbAa)2 - (TrAaAb)2 - TrAbTrAa2Ab],

where we have put At = At*. We now put

Sab = TrAaAb,

Sa = San,

S =

Sat

a so that Sab is a symmetric (2m-n,3n-n)-matrix and can be assumed to be diagonal for a suitable frame. S is the square of the length of the second fundamental form of M.

We prove the following theorem (Ludden-0kurrura-Yano [2)). THEOREM 2.1. Let M be an n-dimensional compact anti-invariant

minimal sulnanifold of a complex space form ho(c) (c > 0). If S < nqc/4(2q-1), where q = 2m-n, then M is totally geodesic. Proof. From Lemma. 2.2 and Lemma 5.1 of Chapter II we find

IVA12 - JAS = -

I Tr(A Ab - AbA )2 + jS2 - JncS - $cjTrA2 as t t a a a,b

[(2- 1/q)S - 4'nc]S - 1 E (S a a qa b

Sb)2 - ;cJTrAt. t

From the assumption we have

aIb(Sa -

Sb)2 = 0,

jTrA2 = 0.

Therefore we see that Sa = Sb for all a, b, and At = 0, which means that St = 0. Consequently, we obtain Sa = 0 for all a, and hence M is totally geodesic.

QED.

203 THEOREM 2.2. Let M be an n-dimensional compact anti-invariant

suhmanifold of a complex space form Mn(c). If M is minimal, then

0 < fMIVAl2*1 < JM[(2-1/n)S - i(n+1)c]*1.

Proof. From Lemma 5.1 of Chapter II we have

(2.5)

-

ITr(AtAs - AsAt)2 + JS2 - ;(n+1)cS t 's

t z - a(n+1)cS t#sts + ISt

< 2 E S S

(St - Ss)2.

_ [(2-1/n)S - 14(n+l)c1S - n

t>s

On the other hand, Lemma 2.2 implies

IVA12 -

AS = -tIs (AtAs - ASAt)2 + JS2 - a(n+1)cS.

Therefore, (2.5) gives our inequality.

QED.

Theorem 2.2 was given by Chen-Ogiue [2]. As an application of

Theorem 2.2 we have THEOREM 2.3. Let M be an n-dimensional compact anti-invariant minimal sutnanifold of a complex space form M"(c). If S satisfies S < n(n+1)c/4(2n-1), then M is totally geodesic. In Theorem 2.2 and 2.3 we study the case of m = n. When m > n,

by Lemma 2.1, if we assume that the f-structure in the normal bundle is parallel, then we have the similar results as those of Theorem 2.2

and Theorem 2.3 (see Yano-Kon [31). In the next place we consider the case of S = n(n+1)c/4(2n-1).

Without loss of generality, we assume that c = 4. First of all we

have

204

THEOREM 2.4. Let M be an n-dimensional (n > 1) anti invariant minimal submanifold of a complex space form Mn(4). If S = n(n+1)/(2n-1), then n = 2 and M is a flat surface of M2(4).-Moreover, with respect to

an adapted dual orthonormal frame field wl,w2,wl*,w2* in M2(4), the connection form (wB) of M2(4), restricted to M, is given by

0

0

-Aw2

-awl

0

0

-Awl

Aw2

Awl

0

0

-Aw2

0

0

Aw2 Awl

Proof. From the assumption the second fundamental form of M is

parallel and (2.5) implies

I (St - Ss) 2 = 0,

-Tr(AtA5 - ASAt) 2 = 2TrA2TrA2.

t,s

In view of Lenma 5.1 of Chapter II we may assume that At = 0 for t = 3,...,n, which means that St = 0 for t = 3,...,n. On the other hand, we have St = Ss for all t, s. Therefore we must have n = 2. Thus

Lemma 5.1 of Chapter II implies

0

1

AI=A

1

0

0

-1

A2=A 0

1

because of hit =A= hit. Since S = 2, we have 2A2= 1 and we may assume that A = (1/2)l/2. Then, from the Gauss equation, we see that M is flat. Moreover, we easily see the following:

W1*

1

= Aw2,

w2* = -Aw2, 2

These prove our assertion.

u2* = u1 = Awl,

wi = wi* = 0. QED.

2va Example 2.1. Let S5 be a 5-dimensional unit sphere with standard Sasakian structure (,,E,r1,g). The integral curves of the structure

vector field E are great circles S1 in S5 which are the fibres of the standard fibration n

:

S5

> CP2 onto complex projective space

CP2 of complex dimension 2 and of constant holomorphic sectional curvature 4. Let T = S1 x S1 be maximal torus which is imbedded in S5 as an

anti-invariant sutmanifold normal to the structure vector field F. The imbedding X : T

>

S5 is given by

X = 1 (cosul,sinul,cosu2,sinu2,cosu3,sinu3),

T

where U

3

= -ul - u2 in C3. We consider the following diagram:

We easily see that nIX(T) is one-to-one. Consequently T is imbedded in CP2 by

Thus S = 2. (For the detail, see §3 of Chapter VI.)

From this example and Theorem 2.4 we obtain (Ludden-0kumura-Yano

1:0) THEOREM 2.5. Let M be an n-dimensional (n > 1) compact anti-

invariant minimal submanifold of (pn. If S = n(n+l)/(2n-1), then M is S1 x S1 in Cp2.

LEMMA 2.3. Let M be an n-dimensional anti-invariant submanifold of a canplex n-dimensional Kaehlerian manifold M. Then M is flat if and only if the normal connection of M is flat. Proof. We have

3

i =dwij* + i * AW j* = dwj

which shows that

R

+

i n wjk = j

. Thus we have our assertion.

QED.

206

LEMMA 2,4, Let M be an n-dimensional anti-invariant submanifold of a carplex n-dimensional Kaehlerian manifold M. If the second funda-

mental forms of M a r e commutative, i.e., A P = A, Aa for all a and b, then we can choose an orthonormal basis {et} for which At satisfies Atet = atet, Ates = 0 (t # s) for each t = 1,...,n, that is, hij = 0 unless t = i = j. Proof. If AaAb = AbAa, we can choose an orthonormal basis el,...

en for which all A a are simultaneously diagonal, i.e., ha. = 0 when i # j, that is, hij = 0 when i # j. Thus we have hij = 0 unless t = i

= j by hij = ht j .

QED.

PROPOSITION 2.2. Let M be an n-dimensional (n > 1) anti-invariant,

totally umbilical submanifold of a complex n-dimensional-Kaehlerian manifold M. Then M is totally geodesic. =roof. From the assumption we have hij = 6ij(TrAt)/rr. Therefore

the second fundamental forms of M are commutative. Thus Lemma 2.4 implies that htj = 0 unless t = i = j. On the other hand, we have = At5ij/n.

Putting t # i = j, we find at = 0 and hence M is totally

hi3

geodesic.

QED.

LEM'1A 25, Let M be an n-dimensional anti-invariant submanifold of a complex space form Mn(c). Then M is of constant curvature ;c if and only if the second fundamental forms of M are commutative.

Proof. From (2.2) we have Rjkl = *c(6. 6 1 - dildjk) + t(hikhjl - hilhtjk)

4c(dikdji - dil$jk) + t(htkhtl

- h tl htk

which proves our assertion.

QED.

LEIT1A 2.6. Let M be an n-dimensional anti-invariant subma.nifold

of a carplex m-dimensional Kaehlerian manifold M. Then

ITrA2A2 = t 's

t Is(TrAtAs)2.

207 Proof.

AtAs

= t,srirjrkrlhklhlihijhjk

t r,s

ki(TrAkA2.

LEMMA 2.7. Let M be an n-dimensional anti-unvariant submanifold of constant curvature k of a complex space form Mn(c). Then

(2.6)

(c-k)j[TrA2 - (TrAt)2] = t£s[TrA2As - Tr(AtAs)2],

t

r

(n-1)(;c-k)IAI2 =

(2.7)

I [TrA22As - TrASTrA2As3.

t,s Proof. From the assurrption we have

(ac-k)(&Adjl - 6il6jk) _ J(hi

(2.8)

t

Multiplying the both sides of this equation by

summing up

with respect to i, j, k and 1 and using Lemma 2.6, we have (2.6). Putting i = k in (2.8) and summing up with respect to i, we have

(2.9)

(n-1)(jc-k)6ji =

ji

(hlhtj -

i

Multiplying the both sides of (2.9) by

and sunming up with

respect to j, k and 1 we obtain (2.7).

QED.

THEOREM 2.6. Let M be an n-dimensional anti-invariant submanifold with parallel mean curvature vector of a complex space form Mn(c). If M is of constant curvature k, then

IVA12 = -kl[(n+1)TrAt - 2(TrAt)2j. t

Proof. By the assumption we easily see that the square of the length of the second fundamental form of M is constant: Thus Lemma 2.2 implies

0 (2.10)

VA12 = -t[k(n+1)cTrAt -

(TrAt)2]

[Tr(At s - TrA5At)2- (TrAtAs)2 + Tr%TrAt2As].

t,s Substituting (2.6) and (2.7) into (2.10) and using Lemna. 2.6, we have our equation.

QED.

THEOREM 2,7, Let M be an n-dimensional (n > 1) anti-invariant subnanifold with parallel mean curvature vector of a complex space form Mn(c). If M is of constant curvature k, and if c > 4k, then k < 0 or M is totally geodesic and c = 4k.

Proof. From (2.9) we have

n(n-l)('c-k) = J[TrAt2 - (TrAt)2]. t

Since c > 4k, we have EtTrA2 > Et(TrAt)2. If k > 0, then Theorem 2.6 implies

0 = (n-1;JTrA2 + 21[TrAt - (TrAt)2],

from which we have I A 1 2 = 0 and hence M is totally geodesic. In this case c = 4k. Except for this possibility we have k < 0.

QED.

THEOREM 2.8. Let M be an n-dimensional (n > 1) anti-invariant submanifold of a complex space form Mn(c) and M be with parallel second fundamental form and of constant curvature k. If c > 4k, then either M is totally geodesic (c = 4k) or M is flat (k = 0). Theorems 2.6, 2.7 and 2.8 are given by Yano-Kon [2]. In case of dim M = m > n, see Yano-Kon [3].

We now give some examples of anti-invariant submanifolds of complex space forms. Excmrple 2.2. Let RPn be an n-dimensional real projective space.

By taking real and carplex hoar)geneous coordinates properly in RPn

2W and CIP, respectively, the imbedding of RPn into CPm is given as

follows. Let (x0,...,)J) and (z°,...,z) be the homogeneous coordinates of RPn and C2 4n, respectively. Our imbedding is given by natural imbe-

ding of Rn+1 into el, i.e., (x0,...,xn)

> (x0,...,xn,0,...,O).

In this case RIP is of constant curvature ic.

Example 2.3. Let S1(ri) _ {zi e C : Izil2 = ri},

i = 1,...,n, be

circles of radius ri. We consider

Mn = S1(rl) x ... x S1(rn)

in Cn, which is obviously flat. The position vector X of Mn in Cn has

components given by

X = (rIcosul,rlcosul,...,rncosun,rncosun).

Putting Xi = aiX = aX/aul, we have

Xi = ri(0,...,O,-sinul,cosu1,0,...,0),

i = 1,...,n.

On the other hand, we can take Vi = -(0,...,O,cosu1,sinul,O,...,o),

i = 1,...,n,

as orthonormal vectors normal to Mn. Then we obtain

JXi = r1V1,

i = 1,...,n.

Consequently Mn is a flat anti-invariant submanifold in Cn and it has parallel mean curvature vector and flat normal connection. Moreover, we see that Mn has parallel and commutative second fundamental forms. On the other hand, Cn is totally geodesic in C" (m > n). Thus Mn is an anti-invariant submanifold of a with parallel f-structure in the normal bundle.

210 Exwnple 2.4. As. an example similar to Example 2.3, we consider

Mn = S1(r1) x ... X S1( p) x Rn-p,

1 < p < n.

Obviously Rn-p is a totally geodesic anti-invariant submanifold of Cpn-p. Thus Mn is a flat anti-invariant submanifold of Cn. Moreover,

Mn is an anti-invariant submanifold with parallel and commutative second fundamental forms and with parallel f-structure of a (m > n). THEOREM 2.9. Let M be an n-dimensional (n > 1) complete antiinvariant submanifold of Cn and M be with parallel mean curvature

vector and with commutative second fundamental forms. Then either M is Rn, or a pythagorean product of the form

S1(r1) X ... X S1(rn)

or a pythagorean product of the form

S1(r1) x ... X

S1(r) x Rn-p,

1 < p < n.

Proof. By the assumption and Lemma. 2.5 we see that M is flat and

hence the normal connection of M is flat. Moreover, Theorem 2.6 implies that the second fundamental form of M is parallel. For a frame field, chosen as in Lemna. 2.4, we define the following distributions:

Tt(x) = {X C T,t(M) : AtX = AtX}

TO(x) = {X a Tx(M)

:

for

t # 0,

At = 0, 1 < t < n}.

Then each Tt is of dimension 1 and we have the decomposition

Tx(M) = T1(x) ® ..... 0 Tp(x) ® TO(M) (direct sum).

Moreover, since the normal connection of M is flat and the second fundamental form of M is parallel, we see that Tt is involutive and

211

totally geodesic in M. Thus we conclude that

M = Mlx... x p x MO

(Riemannian product),

where Mt and MO are the maximal integral suhmanifolds of Tt and TO respectively. Since M is complete, we have our assertion (see Theorem 3

of Yano-Ishibara (3]).

QED.

THEOREM 2.10, bet M be an n-dimensional (n > 1) cornnlete anti-

invariant submanifold of a simply connected complete complex space form Mn(c) and M be with parallel and commutative second fundamental form. If M is not totally geodesic, then M is a pythagorean product of

the form

S1(ri) x ... X S1(r)

in Cn

or a pythagorean product of the form Rn-p

S1(ri) x ... X S1(r) x

inCn,

1

Proof. From Lemma 2.4 and Theorem 2.6 we see that M is flat and hence c = 0 by Lemma 2.5. Thus Mn(c) is Ca. Therefore our theorem follows from Theorem 2.9.

QED.

In Theorem 2.6, if M is minimal, then

IVAI2 = -(n+l)kIA12.

In the following we prove that an n-dimensional anti-invariant minimal sulxnanifold M of constant curvature k of Mn(c) is totally geodesic or flat, which was proved by Ftijiri [1].

We need the following algebraic 1 emma.

212 Lam 2.8. Let T be a symmetric 3-linear map of re x R° x Rn into R such that n A[g(X,Z)g(Y,W) - g(X,W)g(Y,Z)] + E T(X,Z,fm)T(Y,W, m) nil n E T(X,W,fm)T(Y,Z,fm) = 0

and A > 0,

m=1 n E T(X,fm,fm) = 0,. M--1

where g is the Euclidean metric of Fin and f1,...,fn is an orthonormal

basis. If me choose an orthononmal basis e1,...,en, such that each ei is a maximum point of the cubic function T(X,X,X), restricted to {X E Rn :

I X I

= 1, and X is orthogonal to e1,...,ei-1}, then T has

the following expression:

T(ea,ea,ea) _ (n-a)[(n-a+l) + ... + (n-a+1)...n1

T(e

a

e.la,eja)

A

= -[(n-a+l) + .

1/2

A + (n-a+l)...n]aiaja'

where 1 < a < n and a < ia,ja < n unless i.a = ja = a.

THEOREM 2.11. Let M be an n-dimensional anti-invariant minimal sutmanifold of constant curvature k of a complex n-dimensional space form Mn(c). Then M is totally geodesic or flat (k = 0). Proof. We put T = -JB. Then T is a symmetric tensor field of

type (1,2) on M and satisfies

g(T(X,Y),Z) = g(T(X,Z),Y).

Moreover, equations of Gauss and Codaai are given respectively by

(c-k)[g(X,Z)g(Y,W) - g(X,W)g(Y,Z)] + g(T(X,Z),T(Y,W)) - g(T(X,W),T(Y,Z)) = 0,

213

(OXT)(Y,Z) = (DYT)(X,Z).

Since M is minimal, T satisfies the condition of Lemna 2.8 for A = c - k. We may assume that M is not totally geodesic, i.e., c 0 k and hence A 0 0. We easily obtain a local field of orthonormal frames e1,...,en such that the Lemma 2.8 holds. We denote by w

the Levi-

Civita connection with respect to e1,...,en. Using the Codazzi equation, we have

-(A/n)

n

1/2

i=1

n wl(e )e. a 1 i

i=1

n wI(e )T(e.,e ) a 1 1 l

n -(n-1)(A/n)1/21wi(ea)ei +

wI(e )T(e,e i=1

a

a

1

n 0,

for all a 0 1. Taking the innerproduct of it and e1, we obtain wa(e1) = 0. This, together with the innerproduct of the above and eb (b 0 1), implies wi(eb) = 0. As a result, e1 is a parallel vector field on M. Thus M is flat.

QED.

214

3. CR SUBMANIFOLDS OF KAEHLERIAN MANIFOLDS

Let M be a complex m-dimensional (real 2m-dimensional ) Kaehlerian manifold with almost complex structure J and with Kaehlerian metric g. Let M be a real n-dimensional Rienannian manifold isometrically immersed in M. We denote by the same g the Riemannian metric tensor field induced on M. from that of M. For any vector field X tangent to M, we put

(3.1)

JX = Px + FX,

where PX is the tangential part of JX and FX the normal part of JX. Then P is an endomorphism on the tangent bundle T(M) and F is a normal bundle valued 1-form on the tangent bundle T(M). For any vector field V normal to M, we put

(3.2)

JV = tV + fV,

where tV is the tangential part of JV and fV the normal part of JV. Clearly, P is skew-symmetric on T(M) and f is skew-symmetric on T(Mt .

From (3.1) and (3.2) we have

(3.3)

g(FX,V) + g(X,tV) = 0,

which gives the relation between F and t. We also have

(3.4)

p2 = -I - tF,

FP + fF = 0,

(3.5)

Pt + tf = 0,

f2 = -I - Ft.

We define the covariant derivative OXP of P by (VxP)Y = VX(PY) PVXY and the covariant derivative VXF of F by (VXF)Y = DX(FY) - FVXY.

215

Similarly, we define the covariant derivatives VXt of t and VXf of f by (V)t)V = Vx (tV) - tDXV and (VXf)V = DX(fV) - fDXV, respectively.

Then, from the Gauss and Weingarten formulas we have

tB(X,Y) = fB(X,Y) = (VXP)Y - AFyX + B(X,PY) + (VXF)Y.

Comparing the tangential and normal parts of the both sides of this equation, we find

(3.6)

(VXP)Y = AFyX + tB(X,Y),

(3.7)

(VXF)Y = -B(X,PY) + fB(X,Y).

Similarly, we have

-PAVX - FAUX = (VXt)V - AfVX + B(X,tV) + (Vxf)V,

from which

(3.8)

(V)t)V = AfVX - PAVX,

(3.9)

(VXf)V = -FAVX - B(X,tV).

Suppose now that the ambient manifold M is of constant holomorphic sectional curvature c. Then the curvature tensor R of M is given by

R(X,Y)Z = lc[g(Y,Z)X-g(X,Z)Y+g(JY,Z)JX-g(JX,Z)JY+2g(X,JY)JZ)

+ AB(Y,Z)X - AB(X,Z)Y + (VB)(X,Z) - (VXB)(Y,Z).

Comparing the tangential and normal parts of the both sides of this equation, we have the following equations of Gauss and Codazzi respectively:

216 (3.10)

R(X,Y)Z = jc[g(Y,Z)X-g(X,Z)Y+g(PY,Z)PX-g(PX,Z)PY+2g(X,PY)PZ]

+ B(YZ)X - AB(X,Z)Y, (3.11)

(VXB)(Y,Z) - (VYB)(X,Z) = *c[g(PY,Z)FX - g(PX,Z)FY + 2g(X,PY)FZ].

Moreover, equation of Ricci is given by

(3.12)

g(e(X,Y)U,V) + g([AV,l ]X,Y) _ tc[g(FY,U)g(FX,V)-g(FX,U)g(FY,V)+2g(X,PY)g(fU,V)].

Definition. Let M be a Kaehlerian manifold with almost complex structure J. A submanifold M of M is called a CR submanifold of M if there exists a differentiable distribution D : x

> Dx a 7x(M)

on M satisfying the following conditions: (1)

is invariant, i.e., JDx = Dx for each x c M, and

(2) the complementary orthogonal distribution

D': x

> DX e T X(M) is anti-invariant, i.e., JDX E T t(M' for each x n M. In the sequel, :

put dim M = 2m, dim M = n, dim D = h, dim D' = q

and codim M = 2m-n = p. If q = 0, then a CR submanifold M is a Kaehlerian submanifold of M, and if h = 0, then M is an anti-invariant submanifold of M. If p = q, then a CR submanifold M is called a generic submanifold of M. If h > 0 and q > 0, then a CR submanifold

M is said to be n^n-trivial (or proper). Remark. Sometimes, the definitions of CR sutmanifolds, generic subnanifolds and anti-invariant subaznifolds (totally real sutmanifolds) are respectively given as follows (cf. Wells [1], [2]). Let M be a real n-dimensional submanifold of a complex m-dimensional complex manifold M. We define

217

n J' X(M),

H {(td) =

as holomorphic tangent space to M at x. If dimCHx(M) is constant on M, then M is called a CR subnanifold. It is well known that

max(n-m,O) < dirncH(M) < in. If dimeX(M) = max(n-m,0) at each point of M, then M is called a generic subinanifold. Moreover, if dim X(M) = 0, then M is called a totally real sutmanifold.

Remark. We now state the result which justifies the name of CP. sutrnanifold. Let M be a differentiable manifold and T(M)C its ccmplexi-

fied tangent bundle. A CR structure on M is a complex subbundle H of T(M)C such that Hx A Hx = {0}

and H is involutive, i.e., for con lex

-vector fields X and Y in H, [X,Y] is in H. It is well known that on

a CR manifold there exists a (real) distribution D and a field of endomorphisn p : D

> D such that p2 = -ID. D is just Re(H®H) and

Hx = {X - VrlpX : X E Dx}. Blair-Chen [1] prove the following:

Let M be a CR sutmanifold of a Haermitian manifold M. If M is non-trivial, then M is a CR manifold.

We next give some characterizations of CR submanifolds of Kaehlerian manifolds. First of all, we prove the following (Yano-Kon [12]) THEOREM 3,1. In order for a sub manifold M of a Kaehlerian manifold M to be a CR sulmanifold, it is necessary and sufficient that FP = 0.

Proof. Suppose that M is a CR sutmanifold of M. We denote by Z and l

the projection operators on Dx and DX respectively. Then

Z+Z` =I,

Zz=Z,

Z`2=l,

Z11=Z1Z=0.

Fran (3.1) we have r X = 0, FZ = 0 and PZ = P, from which and the second equation of (3.4), we find

FP=0.

if = 0.

From (3.3) and (3.14) we obtain

(3.15)

tf = 0,

and hence, from the first equation of (3.5)

(3.16)

Pt = 0.

Thus, the first equation of (3.4) implies

p3+P=0.

(3.17)

From the second equation of (3.5) we also have

f 3 + f = 0.

(3.18)

Conversely, for a sukmanifold M of a Kaehlerian manifold M, assume that we have (3.13), that is, FP = 0. Then we have (3.14), (3.15), (3.16), (3.17) and (3.18). We put

11=l-Z.

Z=-P2, Then we see

Z+l`=I,

Z2=Z,

lye=l

,

111=1:11 =0,

which show that Z and Z' are complementary projection operators and

219 consequently define complementary orthogonal distributions D and Db respectively. Fran equation Z = -P2 we have P1 = P. This equation can

be written as PV = 0. But g(PX,Y) is skew-symmetric and g(l X,Y) is symmetric and consequently t P = 0. Thus we have Zl P1 = 0. Moreover,

by Z = P2 we have FZ = 0. These equations show that the distribution D is invariant and the distribution Dl is anti-invariant.

QED.

Fran (3.17) and (3.18) we have (Yano-Kon [12] ) THEOREM 3.2. Let M be a CR submanifold of a Kaehlerian manifold M. Then P is an f-structure in M and f is an f-structure in the normal bundle of M.

Let R be the curvature tensor of a canplex space form M(c). Then we have (Blair-Chen [1]) THEOREM 3.3. Let M be a suhmanifold of a complex space form M(c) with c # 0. Then M is a CR submanifold of M if and only if the maximal holamorphic subspace Dx = Tc(M) A JTx(M) defines a non-trivial differentiable distribution D on M such that

(3.19)

g(R(X,Y)Z,W) = 0

for all X, Y c D and Z, W E D`, where D` denoting the orthogonal complementary distribution of D in M.

Proof. If M is a CR submanifold of M(c), then we have R(X,Y)Z = zcg(X,JY)JZ. From this equation we have (3.19). Conversely, fran (3.19) we have g(R(JX,X)Z,W) = -Icg(X,X)g(JZ,W) _

0 for all X c D and Z, W c D. Frcm this we see that JDD is perpendicular to D . Since D is holcmorphic, JL is also perpendicular to

D. Therefore, JDx a X(Mt. This shows that M is a CR sub nn fold. QED. Let us now suppose that M is a Hermitian manifold and let St be the fundamental 2-form of M, i.e., S2(X,Y) = g(X,JY). M is a Kaehlerian

manifold if and only if cM = 0. However we consider a class of Hermitian manifolds slightly larger than that of Kaehlerian manifolds for

which cI

= 52 A w, w being a 1-form called the Lee form. When w is

closed we call these manifolds locally conformal symplectic manifolds. They include the-well known Bopf manifolds. We prove the following theorem (Blair-Chen [1]).

THEOREM 3,4, Let fd be a Hermitian manifold with dS2 = S2 A w. Then in order for M to be a (R sutmanifold of U it is necessary that Dl- be integrable.

Proof. Let X be vector field in D and Z, W vector fields in D . Then S2(X,Z) = 0 and S2(Z,W) = 0. Therefore S2 A w(X,Z,W) = 0 and hence 0 = 3dc2(X,Z,W) = -g([Z,W],JX),

but X and hence JX is arbitrary in D and [Z,W] is tangent to M, thereQED.

fore [Z,W] is in D'.

From Theorem 3.4 we see that the distribution DL of a CR submanifold of a Kaehlerian manifold is integrable. We next consider the condition that the distribution D is integrable. If the distribution is integrable and moreover if the almost cc Alex structure P induced on each integral submanifold of D is integrable, then we say that the f-structure P is partially integrable.

THEOREM 3.5, Let M be a CR suYmanifold of a Kaehlerian manifold M. Then the f-structure P is partially integrable if and only if

B(PX,Y) = B(X,PY)

for any vector fields X and Y in D.

Proof. Let X and Y be vector fields in D. Then (3.7) implies

F[X,Y] = FOXY - F yX = -(OXF)Y + (VyF)X = B(X,PY) - B(PX,Y).

Thus D is integrable if and only if B(X,PY) = B(PX,Y). In this case

0 the integral submanifold of D is invariant in M and hence it is also a Kaehlerian manifold. Thus the almost complex structure induced from

P on the integral submanifold of D is integrable.

QED.

In the next place we give some examples of generic sutmanifolds and CR submanifolds of complex space forms.

Ercvnple 3.1. Let Cm be the corrplex number space of complex dimension m. Let M be a product Rienannian manifold of the form Cp x Mq, where Mq is a real q-dimensional anti-invariant submanifold of Cq

Cam. Then m is a generic submanifold of Coq, and M is moreover a CR submanifold of Cm with m > p+q.

Example 3.2. Let e(r) be an m-dimensional sphere of radius r. We consider an immersion:

x ... x S"(rk)

>

C(n+k)/2,

where m1,...,mk are odd numbers. Then n+k is evev. We now consider (m.+l)/2

M.

S 1(ri) C C

(i = 1,...,k)

1

and

C(n+k)/2 = C(mj,.+1)/2

x .., x C

(rr+1)/2

(m.+1)/2 M. 1 Then each S '(ri) is a real hypersurface of C . We denote by (m.+l)/2 M. . Then Jv i is tangent to vi the unit normal of S '(ri) in C 1

m S 1(ri). Therefore S (r1) x ... x S of

CP(n+k)/2,

(rk) is a generic submanifold

and hence CR submanifold of Cm (2m > n+k) with parallel

mean curvature vector and flat normal connection. Similarly, we can consider an immersion:

S"l(r1) x ... x S

(rk) x RP

> C(n+k) /2

x Cp C G!,

where (n+k)+2p < 2m. Then the su nifold is a generic submanifold of C(n+k+2p)/2 and CR submanifold of do with parallel an curvature vector and flat normal connection. In the following we need some results of Riemannian fibre bundles,

which will be proved in Chapter IX.

Let S1 be a (2m+1)-dimensional unit sphere and &n be a complex m-dimensional projective space with constant holcmorphic sectional

curvature 4. Then there is a fibering

Let N be an (n+l)-dimensional submanifold immersed in S

1 and M be

an n-dimensional sulinanifold immersed in &n. We assume that N is tangent to the vertical vector field fibration n : N

of 52n+1 and there exists a

> M such that the diagram

N

i

t

> S2nt+1

n

n

M

1

> cr

commutes and the immersion i' is a diffecmorphism on the fibres. We denote by a the second fundamental form of the immersion i'. Then we have the following lemmas. LFNhIA 3,1. The second fundamental form a of N is parallel if and

only if the second fundamental form B of M satisfies

(3.20)

(VXB)(Y,Z) = g(X,PY)FZ + g(X,PZ)FY

and

(3.21)

fB(X,Y) = B(X,PY) + B(Y,PX).

LE1 A 3.2. The normal connection of N is flat if and only if M satisfies the conditions

Rl(X,Y)V = 2g(X,PY)fV

(3.22)

and

(3.23)

(Vxf )V = 0.

ExwTle 3.3. Let e(r) be an m-dimensional sphere with radius r. We consider the following immersion:

k

Sn+k

N

n+1 =

ml,...,mk

where m1,...,mk are odd numbers and ri +

Nmj,...,mk

=

S'1(r1)

x ... x

mi, i=1

+ rk = 1 and we have put

Smi(rk).

Then we have the following commutative diagram:

Nml,...,mk IT

i

CP(n+k-l)/2

ml,...,mk

where we have put M ml,...,mk

M

ml,...1,

= 7r (N

mt,...,mk

is a generic suhmanifold of

). Then we see that CP(n+k-1)/2

and is a CR

suhmanifold of CPFn (2m+1 > n+k). Since the normal connection of

Nml,...,mg

is flat, we see that the normal connection of M,, mll...+mk

satisfies R`(X,Y) = 2g(X,PY)f and Vf = 0 by Lemur. 3.2. Moreover, the

second fundamental form B of Mm

... mk satisfies (3.20) and (3.21)

because of Lemma 3.1. (See Example 4.1 of Chapter VI.) Excvrrple 3.4. In Example 3.3, if ri = (mi/(n+l))l/2 (i = 1,...,k),

then N is minimal and hence M mi,...,mk

ml......

k

is also minimal.

Let M be an n-dimensional CR sutmanifold of a complex projective space ?P(c). If the curvature tensor Rl of the normal bundle of M satisfies

R`(X,Y) = icg(X,PY)fV

(3.24)

for any vector fields X, Y tangent to M and any vector field V normal to M, then the normal connection of M is said to be semi-flat. The

justification of this definition is given by Lei 3.2. We notice that, if M is a generic submanifold of ho(c), then f vanishes identically and hence RL = 0. LEMMA 3,3, Let M be an n-dimensional CR sutma.nifold of Cpm with

semi-flat normal connection. Suppose that dim D = h > 4 and the f-structure in the normal bundle of M is parallel. If B satisfies (3.20), then the second fundamental form a of N of S The proof of this lemma. will be given in §

l is parallel.

of Chapter IX.

LE44A 3.4. Let M be an n-dimensional CR submanifold of CPm. Then we have

g(VB,VB) > 2hq,

where h = dim D and q = dim Dl, and the equality holds if and only if (3.20) holds.

Proof. We put

T(X,Y,Z) = (VXB)(Y,Z) + g(Y,PX)FZ + g(Z,PX)FY.

Then T = 0 if and only if (3.20) holds. Let {ei) be an orthonormal basis of Tx(M). Then

IT12 = IVBI2 + 2hq + 4

g((Ve B)(ei,Pei),Fej i,j

i

On the other hand, equation of Codazzi implies

B)(ei,Pei),Fej) - hq.

Since B is symmetric and P is skew-symmetric, the first term in the right hand side of the equation above vanishes. Consequently, we have

ITI2 = IVB12 - 2hq,

which proves our assertion.

QED.

LEMhA 3,5, Let M be a CR sutmanifold of a Kaehlerian manifold M. Then the f-structure in the normal bundle of M is parallel if and

only if AVtU = AUtV

(3.25)

for any vector fields U and V normal to M. Proof. From (3.9) we have

g((OXf)V,U) _ -g(FAVX,U) - g(B(X,tV),U) = g(AVtU,X) - g(AUtV,X),

W.

which proves our lemma. LEI4IA 3.6. Let M be a CR suthanifold of a complex space form

1F(c) with semi-flat normal connection. If the f-structure in the

normal bundle of M is parallel, then AfV = 0 V normal to M.

for any vector field

Proof. From equation of Ricci we find

g(EAV,AU]X,Y) = tcfg(FY,U)g(FX,V) - g(FX,U)g(FY,V)).

Thus we see that AfVAU = AUAfV for any vector fields U and V normal to M. On the other hand, (3.7) implies

o = g((VXf)fV,FY) = -g(f2V,(V F)Y)

= g(Af2 X,PY) - g(AfVX,Y), from which

A2

= A PAf2V.

Therefore we have

TrA

= -TrA PAf2V = TrAf2VPAfV = TrAfVAf2VP = TrAf2VAfVP = TIAfVPAf2V = -TrA2f.V.

Consequently, we have TrA

= 0 and hence Afv = 0.

QJED.

THEOREM 3.6. Let M be an n-dimensional canplete CR sutma.nifold of

CPm with semi-flat normal connection and h-> 4. If the f-structure f in the normal bundle of M is parallel and if IVB1 2 = 2hq, then M is totally geodesic invariant suhmanifold C1?nl2 of CPFn or M is a generic

sukmanifold of CP(n+q)/2 in CEP and is

tr(Sml(rl) x ... X Smk(rk)),

Emi = n+l,

Eri = 1,

where 2 < k < n-3 and ml,...,mk are odd numbers and q = k-1.

Proof. We assume that q = 0. Then M is a Kaehlerian sutmanifold of Cam. Fran equation of Ricci we have AvAu - A

= 0 for any vector

fields U and V normal to M. Thus Lenma. 1.1 inplies 0 = AjUA[j - ALTAN

2J 2 , and hence AU = 0. Therefore, M is totally geodesic in e n and

is C/2 Let us next assume that q > 1. From Lemtes 3.3 and 3.4 we see S2m+1 that the second fundamental form a of N in is parallel. From Lemma 3.6 we also have AfV = 0. If AFx = 0 for some vector field X tangent to M, (3.12) implies that c = 0 when q.1 2. This is a contradiction. When q = 1, if AFX = 0, then M is totally geodesic in CP° and

hence M is invariant or anti-invariant in C1. This is also a contradiction. Thus the first normal space of M is of dimension q. Moreover, the first normal space of M is parallel. Indeed, from (3.7) and (3.21) we have (VXF')Y = B(Y,PX), and hence g(DXFY,fV) = g((VXF)Y,f\) =

g(B(Y,PX),fV) = 0. Then we can see that the first normal space of N S2m+1 in is also parallel. Therefore, there is a totally geodesic S2m+1, (n+l+q)-dimensional suhmanifold Sn+l+q of and hence Cp(n+q)/2

of C' containing N and M respectively. Then, from the definition, M CP(n+q)/2. is a generic submanifold of Now, using Lemma 3.2, Example 3.3, Theorem 5.7 of Chapter II and Example 4.3 of Chapter II, we have our assertion.

QED.

THEOREM 3.7. Let M be an n-dimensional complete generic subnanifold of (PM with flat normal connection. If h > 4 and

I

2hp

(p = codim M), then M is

n(S1(r

X ... x

Smi(rk)

Fmi = n+1,

Zr? = 1,

where 2 < k << n-3, m = (n+k-1) /2 and ml,...,mk are odd numbers.

Theorem 3.6 was proved by Yano-Son [12] and Theorem 3.7 was

proved by Yano-Kon [11]. Let M be an n-dimensional CR suh-nifold of a complex m-dimensional Kaehlerian manifold M. Then we have the decomposition of the tan-

gent space X(M) at each point x of M:

X(M) = Dx 0 DX .

228 '

Similarly, we have

X(Mf = IVXL ® NX, where N; is the orthogonal complement of FDx in TX(Mt'. Thus JNX = f X

= N. We take an orthononnal frame {e1,...,e2m} of M such that, restricted to M, el,...,en are tangent to M. Then el,...,e. form an orthonormal frame of M. We can take {el,...,en} in such a way that e1,...,

en-q form an orthonor nal frame of DX and en-q+l, ... , en form an orthonormal frame of D' x, where q = dim Dx and n-q = h = dim D. Moreover, we can take {en+l,.... e2m } in such a way that en+1,...,en+q form an

orthonornal frame of F

x

and en+q+l,...,e2m form an orthonormal frame

of N. Unless otherwise stated, we use the conventions that the ranges of indices are respectively: i,j,k = l,...,n;

r,s,t = 1,...,n-q;

x,y,z = n+l,...,n+q;

a,b,c = n-q+1,...,n;

A,u,v = n+q+l,...,2m.

LE TIA 3,7, Let M be a CR sutirnnifold of CIO' with sari-flat normal connection and parallel f-structure f. If PAV = AvP for any vector field V normal to M, then

g(AUX,lY) = g(X,Y)g(tU,tV) - g(FX,U)g(FY,V) - g(AUtV,ei)g(AF'e.X,Y).

i

1

Proof. Fran the assumption we have g(AUPX,tV) = 0, from which

g((V A)UPX,tV) + g(AU(V P)X,tV) + g(AUPX,(V t)V) = 0.

Changing Y to PY in this equation, and using (3.6), (3.8) and Lemma. 3.6, we have

g((D.A)UPX,tV) + g(AUtB(PY,X),tV) - g(PAUX,P2AYY) = 0,

from which

g((v

.A)UP'X,tV) - Zg(AUtV,ei)g(AFe X,PY) = g(PALTX,AVY) = 0.

From this and the Codazzi equation (3.11) we have

g(PX,PY)g(tU,tV) -lg(AUtV,ei)g(AFe PX,Py) + g(P2AUX,AVY) = 0.

On the other hand, we have

g(PX,PY)g(tU,tV) = g(X,Y)g(tU,tV) - g(PX,FY)g(tU,tV),

- g(AUtV,ei)g(AFe PX,PY)

i

i.

Jg(AUtV,ei)g(A1e.X,Y) + g(AUtV,AFYX),

i

i

-g(AUX,AVY) - g(AUX,ATytV).

Moreover, from (3.12), we obtain

g(AUtV,AFyX) - g(AUX,AFYtV) = g(tU,tV)g(FX,FY) - g(FX,U)g(FY,V).

From these equations we have our assertion.

QED.

LEMMA 3.8. Let M be a CR submanifold of CPS with semi-flat normal connection and parallel f-structure f. If the mean curvature vector of M is parallel and if PAV = AvP for any vector field V normal to M, then the square of the length of the second fundamental form of M is

constant. Proof. From Lemmas 3.6 and 3.7 the square of the length of the second fundamental form of M is given by

1AI2 = jTrAX = (n-1)q +

x On the other

I g(Atetey)TrAy x,y

and, for any vector field V e FD, we have DXV E FD,

because of of = 0. Fran (3.7), we also have, for any V E N, DxV E N.

K Therefore, since R`(X,Y)V = 0 for anyV c Fb`, we can choose an orthonormal frame {e} of Fit such that Dex = 0 for all x (see Proposition 3.3 of Chapter II). Then we see that VX(tex) = -PAxX. Since the mean curvature vector of M is parallel and PAV = AVP, from the Codazzi

equation (3.11) and (3.25), we find

VXIAI2=x Ig((V xA)ytexX)TTA y

.

On the other hand, using PAV = AP and (3.6), we have

1g((VPe A)xPei,tV) = 0 and lg((Vpe A)xPei,PX) = 0.

i

i

i

i

Consequently, we have

J(VesA)xes = I(Vpi )xPei = 0

for each x. Since the mean curvature vector of M is parallel, using the Codazzi equation (3.11), we obtain

Axes

0 = J(Ve A)xei = J(Ves i

s

i

+ J(Ve A)xea a a

_ )(Ve A)xea = V(Vte A)xtey

a

a

y

y

for each x, and hence JA12 is Constant. t

QED.

3,9, Let M be an n-dimensional CR submanifold of C

° with

sari-flat normal connection, parallel f-structure f and parallel mean curvature vector. If PAV = AvP for any vector field V normal to M, then we have IVA12 = 2(n-q)q. Proof. From Lenn a 3.6 and Proposition 3.1 of Chapter II we have

g(V2A,A) =

E g((VeVeA)Rj' i

X, i,3

i

xej)

_ (n-3)jTrAx - J(TrAx)2 + 61[Tr(A?)2 - TrAx2P2]

x

x

x

RU + 3 1 [g( Xtey,Aktey) - g(Axtex,Aytey)] x,y -

g([ k,Aylei,[Ax,Ay]ei) +

E

x,y,i

E [3g(A{tex,tey)TrAy x,y

- (AA)2 + (TrA%)(TrAXAA)I, where we take {ex} such that Dex = 0 for all x and used the fact that

(VxA)vY = 0 for any V E N. On the other hand, the Ricci equation

(3.12) irlies E

g([Ax,Ay]ei,IAx,Ay]ei) = 2q(q-1),

x,y,i E

[g(Axt yAxtey) - g(Axtex,Aytey)] = q(q-1).

x,y,i Fran these equations we get

g(V2A,A) _ (n-3))TrA2 - J(TrAx)2 + 3II[p,A{]I2 + 2q(q-1)

x +

x

x

I [3g(Axtextey)T*rAy - (Tr XAy)2 + TrAyTrAXAA].

x,y Moreover, Lame 3.7 implies

I TrAyg(Axtextey) _ ITrA2 - (n-1)q, x,y x and Lemma 3.8 shows that IVAI2 = -g(V2A,A). Consequently, we have

(TAXAy)2 - njTrA2 + I(TrAX)2

VAI2 - 2(n-q)q =

x,y

x

x

x

E Tr ,TrAyAx + (n-1)q.

x,y On the other hand, lemma 3.7 implies

E (Tr X A)2._ (n-1)jTrA2 + E Tr ZTY XAyg(Axteytez x,y x x,y,z -

TrAxTrA2A = - I(TrA )2 + x Y x x x,y

y

x.Y

Y

Therefore, we nave

IVA1 2 - 2(n-q)q =

E TrAxg(Atey,tey) - JTrA2 + (n-1)q = 0.

x,y

x

Thus we obtain IVA12 = 2(n-q)q.

QED.

Here, we notice that IVBI2 = IVA12 in general. Moreover, if P_

= AVP, then we can prove Theorem 3.6 without the assumption h > 4.

Indeed, we see that

g(fB(X,Y),V) - g(B(X,PY),V) - g(B(Y,PX),V) _ -g(AVX,PY) - g(AVPX,Y) = 0.

Thus B satisfies (3.21). Therefore, Theorem 3.6 and Lemma 3.9 prove the following (Yano-Kon [13]) THEORE'l 3.8. Let M be an n-dimensional complete CR submanifold of

CPm with semi-flat normal connection, parallel f-structure f and parallel mean curvature vector. If PAV = AvP for any vector field V normal to M, then M is a totally geodesic Kaehlerian submanifold CPn12 of CPm or M is a generic submanifold of CP(n+q)/2 in CPm and is

Tr(S

(r1) x ... x Smi(rk)),

Emi = n+l,

Eri = 1,

where q = k-1 and m1, ...,mk are odd numbers.

From Theorem 3.8 we also have the following (Ki-Pak-Kim [1])

THEOREM 3.9. Let M be an n-dimensional complete generic submanifold of CPm with flat normal connection and parallel mean curvature vector. If PA. = AvP for any vector field V normal to M, then M is

n(Sml(rl) x ... x S7k (rk))I

Em1 = n+1,

Eri = 1,

where 2n-n = k-1 and m1,...,mk are odd numbers.

We give the meaning of the condition PAV = AvP in terms of the f-structure.

Let M be a CR submanifold of a Kaehlerian manifold M. The Nijenhuis tensor of the f-structure P on M is given by

NP(X,Y) = P2[X,Y] + [PX,PY] - P[PX,Y] - P[X,PY].

We put S(X,Y) = Np(X,Y) - t[(OXF)Y - (VYF)X].

The f-structure P is said to be normal if S vanishes identically. THEOREM 3.10. Let M be a CR sukmanifold of a Kaehlerian manifold M. Then the f-structure P on M is normal if and only if PAFX = AFXP

for any vector field X tangent to M. Proof. From the definition we have

S(X,Y) = (DOP)Y - (DpyP)X + P[(VYP)X - (VXP)Y] - t[(VXF)Y - (V F)X]

= (AF.YP - PAFY)X - (AW - PAFX)Y.

If

AW, then S = 0. Conversely, we suppose that S = 0. Then,

for any X eDi- and Y c D, we have AF 'Y = PAFXY. Let Z C D-. Then g(AFRPY,Z) = g(PAFXY,Z) = 0,

0 from Which ADZ a D' for Z E D. Thus we have AF,4PZ = PAZ for any Z E D'-. Consequently, we obtain PAS =ASP for any vector field X tangent to M. QED. Let M be an n-dimensional CR suhmanifold of G1n with flat normal

connection and with parallel f-structure f. Then the Ricci equation (3.12) implies AVAU = AUAv and AfV = 0 by Lemma 3.6. We suppose that

PAV = AVP. Then, by a method quite similar to that used in the proof of Lemma 3.7, we have

g(A,X,AVY)

Jg(AUtV,ei)g(A

X,Y).

From this we see that JA12 is constant when the mean curvature vector of M is parallel. We can also prove the following

LEA 3.10, Let M be a CR sulmanifold of Cm with flat normal connection and with parallel f-structure f. If the mean curvature vector of M is parallel and if PAV = AVP for any vector field V normal to M, then the second fundamental form of M is parallel.

From Lemma 3.10 and Theorem 5.5 of Chapter II we see that the sectional curvature of M is non-negative, and hence Theorem 4.1 of Chapter II implies the following (Yano-Kon [13])

THEOREM 3.11. Let M be an n-dimensional complete CR sutmanifold of Cm with flat normal connection and with parallel f-structure f. If the mean curvature vector of M is parallel and if PAV = AVP for any

vector field V normal to M, then M is a sphere Sn(r), a plane Rn, a Pythagorean product of the form

Spl(rl) x

.

.. x SPN(rN),

P1....,pN > 1, Epl = n, 1
or

SPl(r1) x ... x S

(rN) x Rp, p1,...1PN1p > 1, Epi+p=n, 1
0 From Theorem 3.11 we have (Ki-Pak [1))

THEOREM 3.12. Let M be a complete n-dimensional generic sutmanifold of C1D with flat normal connection and with parallel mean curvature

vector. If PAv = AvP for any vector field V normal to M, then we have the same conclusion as that of Theorem 3.11.

LE1MA 3,11, Let M be an n-dimensional CR sutmanifold of C

with

semi-flat normal connection and with parallel f-structure f. If U is a parallel section in the normal bundle of M, then

div(VtutU) = (n-1)g(tU,tU) + ITrA g(AxtU,tU) x - TrAU + JIL(tU)g12.

Proof. Fran equation of Ricci (3.12) and Lemma 3.6 we have fU = 0 and hence U e FD1. We also have OXtU = -FAUX, from which

div tU = Yg(Ve1tU,ei) _ -TrPAU = 0

because of AU is symmetric and P is skew-symmetric. Thus, from Theorem 4.3 of Chapter I, we find

div(VtutU) = S(tU,tU) + JIL(tU)gl2 - jOtUI2,

where S is the Ricci tensor of M. On the other hand, from (3.10) and

Lemma 3.6 we have S(tU,tU) = (n-1)g(tU,tU) + ITrAxg(AxtU,tU) - lg(A2tU,tU).

We also have

jOtU12 = TrAU - jg(AXtU,tU).

x From these equations we have our equation.

QED.

We notice here that

IL(tU)gl2 = I[P,AU]I2,

which means that tU is an infinitesimal iscc try of M if and only if

.PAU = AuP. We now prove the following (Yano-Kon [13]) THEOREM 3.13, Let M be an n-dimensional compact minimal CR sutmani_ fold of CPm with semi-flat normal connection and parallel f-structure f.

Then

f1[(n-1)q - IA12]*1 = -jfMMI[P, x]I2*1. x

Proof. We can take an orthonormal frame {ex} such that Dex = 0

for each x. From this and Lemma 3.11 we have our equation.

QED.

As an application of Theorem 3.13 we have THEOREM 3.14. Let M be an n-dimensional compact minimal CR submanifold of CPm with semi-flat normal connection and with parallel f-structure f. If IAI2 < (n-1)q, then M is CPn12, or M is a generic minimal submanifold of CP(n+q)/2 in Ci

n(SM, (rI) x ... x S

and is

(rk)),

ri = (mi/(n+1))1/2 (i=1,... ,k),

where £mi = n+l and m1,...,mk are odd numbers.

Proof. From Theorem 3.13 we obtain PA = AxP for all x. Therefore our theorem follows from Theorem 3.8.

QED.

THEOREM 3,15. Let M be an n-dimensional compact minimal generic submanifold of CPS with flat normal connection. If IAI2 < (n-1)q, then

M is n(SM,(rl) x ... x Smi(rk),

ri = (mi/(n+1))1/2 (i=1,...,k),

where Emi = n+l and m1,...,mk are odd numbers.

237

THEOREM 3.16. Let M be an n-dimensional complete CR submanifold of CPS with flat normal connection and parallel mean curvature vector. If PAV = AVP for any vector field V normal to M, then M is

7T

(S1(r1) x ... x S1(rn+1)),

Eri = 1

in CPn in CPm, or M is

x ... x Smi(rk)),

n(S 1(r

Emi = n+l,

Eri = 1,

where m1,...,nk are odd numbers and q = k-1, 2m = n+q. Proof. Fran the assumption and the Ricci equation (3.12) we have

'g([Afu,AU]PX,X) = 2g(PX,PX)g(fU,fU),

from which

TrA

AuP - TrAUAfUP = 2(n-q)g(fU,fU) = 0.

Thus we have n = q, that is, P = 0 and M is anti-invariant subrnanifold of CPm, or f = 0, that is, M is a generic submanifold of CP1°. Therefore our theorem follows from Theorems 3.8 and 3.9.

QED.

We next study the pinching problem of the sectional curvature of a generic minimal sulxnanifold M of CPP (Yano-Kon [14], [17]).

THEOREM 3.17. Let M be an n-dimensional compact generic minimal sulnenifold of CP(n+p)/2 with flat normal connection. If the minimum of the sectional curvature of M is (n-p)/n(n-1), then p = 1 and M is the geodesic minimal hypersphere

1T(SP(r

x

S1(r2)),

r1 = (n/(n+l)

)1/2 ,

r2 = (1/(n+1))1/2

or p = n and M is the flat anti-invariant submanifold n(S1(r1) x ... x SS(rn+1)), rl = ... = rn+1 =

(1/(n+1))112.

0 Proof. From Lemma 3.11 we have

div(jVJva Jva) _ (n-1)p - jTrA2a + JEI[P,Aa]I2,

a

a

a

where {va} denotes an orthonormal frame for T(M)y such that Dva = 0 for each a. On the other hand, using (3.1) of Chapter II, we obtain

g(v2A,A) =

I

g((R(ei,ei)AA)ej,Aaei)

a,i,3

+ ZfI[P,Aa]12 - 31TrA22 + 3(p-1)p,

a

a

where {ei} is an orthonormal frame of M. Since M is compact, these equations imply

0 < fM[IvAI2 - 2(n-p)p]*1 = fM[(n-p)p -

g((R(ej,ei)AA)ej,Aaei)]*l.

I

a,1,J

We can choose {ei} such that Aaei = ,aei (i = 1,...,n). Then

,Aaei) = j1IJ(X

- Xa)2Kij,

denotes the sectional curvature of M with respect to the Ki3 section spanned by ei, ej. By the assumption, we have K1 >

where

(n-p)/n(n-1) and hence

(n)1) I (X

- Xa)2 = n-1TrA2

1,3

Therefore, we have

0 < fM[IVAI2 - 2(n-p)p]*l <

n-1)p - ITrA2]*1 a a '

(n-)

P Aa]IZ*1. 2(n-1) IMaI[,

a

Fran this we have IVA12 = 2(n-p)p. We also have n = p or 1AI 2 = (n-1)p

and PAa = AaP for all a. If n = p, then M is an anti-invariant sutuenifold with flat normal connection and hence M is flat and obviously P = 0. Consequently, Theorem 3.9 implies that M is

n(S1(rl) x ... x Sl(

n+1))'

r1=. ..gin+1 (1/(n+1))1/2

or

7T(Sml(r1) x ... X Smi(rk).

ri = (mi/(n+1))1/2 (i=1,...,k).

But, by the assumption Kid > (n-p)/n(n-1) > 0 when n # p. Thus, we conclude that M is a geodesic hypersphere.

QED.

Any real hypersurface of a Kaehlerian manifold is obviously a generic sulmanifold. Thus Theorem 3.17 implies (Kon [14]) THEOREM 3.18. Let M be a compact real minimal hypersurf ace of CP(n+1)/2.

If the minimum of the sectional curvature of M is 1/n, then

M is the geodesic hypersphere

n(S&(rI) x S1(r2)),

r. =

(n/(n+l))1/2,

r2 =

(1/(n+1))1/2.

A CR sulmanifold M of a Kaehlerian manifold M is called a CR product if it is locally a Riemannian product of a holcmorphic suhmanifold MT and an anti-invariant suhmanifold M .

First we give a characterization of CR product (Chen [2]).

THEOREM 3,19. A CR sutmanifold M of a Kaehlerian manifold M is a CR product suhmanifold if and only if the f-structure P on M is parallel.

Proof. Suppose that P is parallel. Then (3.6) gives

A X = -tB(X,Y), from which tB(X,PY) = 0 for any vector fields X and Y tangent to M.

240

Let Y be in D. Then, for any vector fields. X, Z tangent to M, we have g(FVXY,FZ) = -g((VXF')Y,FZ) = g(B(X,PY),FZ) - g(fB(X,Y),FZ)

= -g(tB(X,PY),Z) + g(B(X,Y),fFZ) = 0.

Thus we have FVXY = 0 and consequently the distribution D is parallel. Let Y be in DL. Then we have PVXY = -(OXP)Y = 0 and hence DL is also parallel. Therefore, M is a CR product submanifold. Conversely, if M is a CH product submanifold, we easily see that

P is parallel.

QED.

Example 3.5. We define a mapping

fhp

:

CPh x Cep

> Cph+p+hp

by

(Zoo ...,Zh;wO,...,Wp)

> (ZOWO,...zlwj,...,ZhWp),

where (zO,...,zh) and (w O,..., P) are the homogeneous coordinates of CPh and CPP respectively. It is easy to see that fhp is a Kaehlerian

imbedding. Let My be a p-dimensional anti-invariant submanifold of CPh+p+hp. Cph x M' induces a natural CR product in

Cep. Then

A CR product submanifold M = MT x ML in CI is called a standard CR product if

(1) m = h+p+hp, and (2) MT is a totally geodesic Kaehlerian submanifold of &n, where h = dimCDx and p = dins"`. x

We shall prove that m = h+p+hp is in fact the smallest dimension

of C" for admitting a CR product. LEMMA 3,12, Let M be a CR product of CP. Then {B(XiX)}, i = 1,...,2h; x = 1,...,p, are orthonormal vectors in Nx, where {Xi} and {Zx} are orthonormal basis for Dx and DX respectively.

241

Proof. Since tB(X,PY) = 0, we obtain AFZPY = 0 for any vector fields X and Y tangent to M. From this, (3.7) and (3.11) we have

g(B(PX,Z),fB(X,Z)) = g(X,X)g(Z,Z)

for any X E Dx and Z E D X- On the other hand, (3.7) implies

0 = (VZF')X = -B(Z,PX) + fB(X,Z).

Thus we have

IB(X,Z)12 = g(X,X)g(Z,Z).

We suppose that JXJ = IZI = 1. Then IB(X,Z)l = 1. Therefore we obtain by linearlity that

g(B(Xi,Z),B(XX,Z)) = 0,

i # j.

Moreover, we see that B(X,Z) E N by AFZPY = 0. Hence, if dim Lx' = 1, the lesna is proved. If dim DX = p > 2, then

g(B(Xi, X),B( Xj,Zy)) + g(B(Xi,Zy),B(j, X)) = 0

for i # j, x # y. Since M is a CR product, we see that ) X,Zy) = 0 and hence, by (3.10) we obtain

g(R(Xi,

Xi

g(B(Xi,Zx)B(Xi, y)) - g(B(XiZy),B(j, X)) = 0.

Therefore we obtain g(B(Xi, X),B( Xiy)) = 0 and hence we have our

assertion.-

QED.

As an immediate consequence of Ira= 3.12 we have THEOREM 3.20. Let M be a CFi product submanifold of CPR. Then

m > h+p+hp.

242 THEOREM 3.21. Every CR product M of CL with m = h+p+hp is a standard CR product.

Proof. For any X, Y, Z in D and W cD; the Gauss equation (3.10) implies

g(B(X,W),B(Y,Z)) = g(B(X,X),B(Y,W)).

In particular, if Y = PX, then

g(B(X,X),B(PX,W)) = g(B(PX,Z),B(X,W)) = g(fB(X,Z),B(X,W)) _ -g(B(X,Z),BBPX,W)),

from which

g(B(X,Z),B(PX,W)) = g(B(X,PZ),B(X,W)) = 0.

Therefore we obtain

g(B(X,Z),B(X,W)) = 0

for any X, Z in D and W in Dl. Then by linearlity we have

g(B(X,Z),B(Y,W)) + g(B(Y,Z),B(X,W)) = 0.

Thus we have

g(B(X,Z),B(Y,W)) = 0.

On the other hand, Lemma. 3.12 implies that B(X,Z) lies in FD' for any

X, Z in D, because of m = h+p+hp. But we have AFXPY = 0, and hence we must have B(X,Z) a N. Consequently, we have B(X,Z) = 0 for any X, Z in D. Therefore, MT must be totally geodesic in CPm.

QED.

THEOREM 3,22, Let M be a CR product of CPS. Then we have

JA12 > 4hp.

If the equality holds, then MT and ML are both totally geodesic in

UP. Proof. First of all, we have JB(X,Z)I = 1 for any unit vectors X in D and Z in D`. Thus we have

2h

JAZZ = 4hp +

E

JB(X1.,X.)I2 +

i,j=1

I

X, y=1

IB(ZX Z

)12,

Y

where {Xi} and {ZX) are orthonornal basis of D and D'' respectively.

From this equation we have our assertion.

QED.

ExompZe 3.6. Let RPP be a real p-dimensional projective space. Then RPp is a totally geodesic anti-invariant submanifold of CPp. Then the composition of the immersions

CPh x RPP

>

> C,h+p+hp

Cpb x Cpp

gives a CR product in CPS with I A12 = 4hp. i

Theorems 3.20 and 3.21 are proved by Chen [2].

> CPm

244 EXERCISES

A. COMPLEX SPACE FORMS IMMERSED IN COMPLEX SPACE FORMS: Let Mn(k)

be a complex n-dimensional space form of constant holamrphic sectional curvature k immersed in a complex (n+p)-dimensional space form

Mn+p(c). Ogiue [4] proved the following THEOREM 1. If p = n(n+l)/2, then either c = k or c = 2k, the latter case arising only when c > 0. Furthermore, Nakagawa-0giue [1] proved the following theorems. THEOREM 2. If c > 0 and the immersion is full, then c = vk and n+p = (n+v) - 1 for some positive integer v. THEOREM 3. If c < 0, then c = k, that is, Mn(k) is totally geodesic in Mn+p(c).

These theorems are the local version of a classification theorem of Kaehlerian imbeddings of complete and simply connected complex space forms into complete and simply connected complex space forms (Calabi [1]).

In Theorem 1, the second fundamental form of M is parallel.

Nakagawa-Takagi [1] showed the classification theorem of complete Kaehlerian submanifolds imbedded in CPm with parallel second fundamental form.

B. KAEHLERIAN SUBMANIFOLDS WITH R(X,Y)S = 0 : Let M be a complex n-dimensional Kaehlerian submanifold of a complex space form bin+p(c). We denote by S the Ricci tensor of M and by R the Riemmannian curvature

tensor of M. We consider the condition

(*)

R(X,Y)S = 0

for any vector fields X and Y tangnet to M.

Then we have (Kon [5])

245

THEOREM 1. Iet M be a complex n-dimensional Kaehlerian subxrani-

fold of a complex space form MP(c) satisfying the condition (*). If c < 0, then M is totally geodesic.

When c > 0, Nakagawa-Takagi [2] proved the following THEOREM 2. Let M be a complex n-dimensional Kaehlerian submanifold of a complex space form Mn+p(c) (c > 0). If M satisfies the condition (*) and the codimension p is less than n-1, then M is Einstein.

When M is a Kaehlerian hypersurface, Ryan [2] proved THEOREM 3. The complete Kaehlerian manifolds with (*) which occur as hypersurfaces in a ccuplex space form Mn+1(c) (c 0 0) are (1) the complex projective space CPn and the complex quadric Q°, (2) the disk Dn of holomorphic sectional curvature c < 0.

When c = 0, Tsunero Takahashi [3] proved the following THEOREM 4. A complete complex hypersurface in Cn+1 satisfying the condition (*) is cylindrical.

C. POSITIVELY CURVED KAEHLERIAN SUB ANIFOLDS: Let M be a complex n-dimensional complete Kaehlerian submanifold in a complex projective

space C +p"with constant holomorphic sectional curvature 1. Then we have (Ogiue [2])

THEOREM 1, If every Ricci curvature of M is greater than n/2, then M is totally geodesic.

We denote by H the holomrphic sectional curvature of M. Then we have (Ogiue [1])

THEOREM 2. If H > 1 - (n+2)/2(n+2p), then M is totally geodesic. Moreover, we obtain the following theorem (Itoh [1]). THEOREM 3. If H > 3n/(4n+2), then M is totally geodesic. Thr positively curved Kaeblerian suhmanifolds, see Ogiue [5].

248

D. NORMAL CURVATURE: Let M be a ccaplex n-dimensional Kaehlerian

submanifold of a complex space form Mm(c) .. We denote by R. the curvature tensor in the normal bundle of M. We consider the following

condition on R' : (*)

Ri(%,Y) = fg(%,JY)J,

where X and Y are arbitrary vector fields tangent to M and f is a function on M. I. Ishihara [1] proved the following THEOREM. Let M be a complex n-dimensional (n > 2) Kaehlerian submanifold of a complex space form &P(c) and assume that M satisfies the condition (*). Then either M is totally geodesic or M is an Einstein Kaehlerian hypersurface of M1(c) with scalar curvature n2c.

The latter case occurs only when c > 0. Combining this theorem with Theorem 1.10, we can determine complete Kaehlerian sukmanifolds with (*) of simply connected complete

complex space fortes. E. KAEHLERIAN IMMERSIONS WITH VANISHING BOCHNER CURVATURE TENSOR: Kon [9] proved the following theorems:

THEOREM 1, Let M be a Kaehlerian manifold of complex dimension n+p with vanishing Bochner curvature tensor, and let M be a Kaehlerian suh:nanifold of M of complex dimension n with vanishing Bochner curva-

ture tensor. If p < (n+l)(n+2)/(4n+2), then M is totally geodesic in M.

THEOREM 2. Under the same assumption as in Theorem 1, if p = 1 and n > 2, then M is totally geodesic in M.

F. ANTI-INVARIANT SUBMANIFOLDS WITH FLAT NORMAL CONNECTION: Let M be an n-dimensional anti-invariant submanifold of a complex m-dimensional Kaehlerian manifold M. If n = in, by Lemma 2.3, we see that M is flat if and only if the normal crnnection of M is flat. When n < m, Yano-Kon-Ishihara [1] proved the following theorem.

247

THEOREM, Let M be an n-dimensional (n > 3) anti-invariant submanifold of a complex space form TP+p(c) (c 0 0) with parallel mean curvature vector. If the normal connection of M is flat, then M is a flat anti-invariant submanifold of some Mn(c) in Mn+p(c), where Mn(c) is a totally geodesic Kaehlerian submanifold of Mn+p(c) of complex dimension n. G. PARALLEL SECOND FUNDAMENTAL FORM: For an anti-invariant

minimal subrmnifold of a complex space form with parallel second fundamental form we have (Kon [6])

THEOREM. Let M be an n-dimensional anti-invariant minimal submanifold with parallel second fundamental form of a complex space form Mn(c). Then either M is totally geodesic or M has non-negative scalar curvature r > 0. Moreover, if r = 0, then M is flat. H. TOTALLY UMBILICAL ANTI-INVARIANT SUBMANIFOLDS: Yano [7] proved the following theorems. THEOREM 1. Let M be an n-dimensional (n > 3) totally umbilical,

anti-invariant submanifold of a complex m-dimensional Kaehlerian manifold with vanishing Bochner curvature tensor. Then M is conformally flat.

THEOREM 2. Let M be an n-dimensional (n > 4) anti-invariant

submanifold of a complex n-dimensional Kaehlerian manifold M with vanishing Bochner curvature tensor. If the second fundamental forms of M are commutative, then M is conformally flat. 1. CONFORMALLY FLAT ANTI-INVARIANT SUBMANIFOLDS: Jet M be an

n-dimensional anti-invariant submanifold of a complex projective space CPn with constant holomorphic sectional curvature c > 0. We denote by K and r the sectional curvature and the scalar curvature of M respectively. Verheyen-Verstraelen [1] proved the following theorems.

THEOREM 1. Let M be compact, conformally flat and of dimension n > 4. Then r > ((n-1)3(n+2)/4(n2+n-4))c implies that M is totally geodesic.

218 THEOREM 2. Let M be complete, conformally flat and of dimension

n > 4. Then K > ((n i)2/4n(n2fa-4))c

implies that M is totally geodesic.

J. TOTALLY UMBILICAL CR SUBMANIFOLDS: Bejancu 13] proved the

following theorem. THEOREM. Let M be a totally umbilical non-trivial CR submanifold

of a

M. If dim D+ > 1, then M is totally geodesic

in M.

K. REAL HYPERSURFACES: Let M be a real (2n-1)-dimensional real hypersurface of a Kaehlerian manifold M of complex dimension n (real

dimension 2n). Then M is obviously a generic submanifold of M. We give examples of real hypersurfaces in a complex projective space CPn with constant holomorphic sectional curvature 4 (see R. Takagi [2]). Let Cn+l be the space of (n+l)-tuples of complex numbers (z1,....zn+1). Put

-

= ((z1,...,z) n+1 E Cn+1

S2n+1

:

n+l + Iz.12 = 1}. j=1 J

For a positive number r we denote by MO(2n,r) a hypersurface of S2n+1 defined by n Iz.I2 = rizn+112,

I

j=l

n+1 I Iz.I2 = 1. j=1 J

For an integer m (2 < m <'n-1) and a positive number s, a hypersurface S2n+1

M'(2n,m,s) of

is defined by

m

n+l

J -1 IZ.12

= sJ

JI

I+1

IZj12,

n+1 J 11 IZ.12 = 1.

For a number t (0 < t < 1) we denote by M'(2n,t) a hypersurface of S2n+1 defined by n+l

n+1 2212

E

j=1 J

= t,

g

j=1

12.12

J

= 1.

249

Let

be the natural projection of 52n+1 onto CPn. Then MO(2n-l,r) =

n(%(2n,r)) is a connected compact real hypersurface of (

with two

constant principal curvatures. We call MO(2n-l,r) a geodesic hyper-

sphere of (P. Moreover, M(2n-l,m,s) = n(M'(2n,m,s)) (n > 3) and M(2n-l,t) = 7(M'(2n,t)) (n > 2) are connected compact real hypersurfares of CPn with three constant principal curvatures. R. Takagi [1] [2] proved the following theorems.

THEOREM 1. If M is a connected complete real hypersurface in CPn (n > 2) with two constant principal curvatures, then M is a geodesic hypersphere.

THEOREM 2, If M is a connected complete real hypersurface in CPn (n > 3) with three constant principal curvatures, then M is congruent to some M(2n-l,m,s) or M(2n-l,t).

We denote by C a unit normal of a real hypersurface of a Kaehlerian manifold M. We put JC = -U. Then U is a unit vector field on M. We define a 1-form u by u(X) = g(X,U). If the Ricci tensor S of M is of the form

S(X,Y) = ag(X,Y) + bu(X)u(Y)

for some constants a and b, then M is called a pseudo-Einstein real hypersurface of M. We have (Kon [13])

THEOREM 3. If M is a connected complete pseudo-Einstein real hypersurface of (

(n > 3), then M is congruent to some geodesic

hypersphere MO(2n-l,r) or M(2n-1,m,(m-1)/(n-m)) or M(2n-1,1/(n-1)). When a and b are functions, see Cecil-Ryan [1]. L. GENERIC MINIMAL SUBMANIFOLDS: Let M be a compact n-dimensional generic minimal submanifold of a real (n+p)-dimensional complex projective space CP(n+p)/2 with constant holomorphic sectional curvature 4.

Then we have (Kon [15])

250 THEOREM. If the Ricci tensor S of M satisfies S(X,X) > (n-1)g(X,X) + 2g(PX,PX),

then M is a real projective space RPn (p = n), or M is the pseudoEinstein real hypersurface tr(Sm(r) x Sm(r)) (m = (n+l)/2, r = (1/2)1)

of CP(n+l)/2 (p = 1). M. SUBMANIFOLDS OF A QUATERNION KAEHLERIAN MANIFOLD: Let 0 be a 4n-dimensional quaternion Kaehlerian manifold with structure (F,G,H). Let M be a Riemannian manifold of dimension m (m < n) immersed in R by an isometric immersion f. We call M a totally real submanifold of R if Tx(M) J FTx(M), 7x(M) J GTx(M), TA(M) J H X(M) for any point x of M. Then Funabashi [1] proved the following THEOREM. Let HPn be a quaternion projective space of dimension 4n and M a connected and complete submanifold of dimension n immersed by f : M

> HPn. Assume M is a compact, totally real and minimal

submanifold satisfying the inequality !A12 <(n+l)/2(3n-1) for the square of the length of the second fundamental form A of M. Then M is an n-dimensional real projective space RPn, and the immersion f being congruent to the standard immersion i : RPn

> HPn or, M is the

unit sphere Sn, f being congruent to the standard immersion > HPn, where n : Sn

> RPn is the natural projection.

Sn

251

CHAPTER V

CONTACT MANIFOLDS In this chapter, we study the various almost contact manifolds and contact manifolds. In §1, we give the definitions almost contact

manifolds and almost contact metric manifolds. §2 is devoted to the study of contact manifolds. We give some fundamental properties and examples of contact manifolds and contact metric manifolds. In §3,

we define the torsion tensor field of a almost contact manifold and study the normality of the manifolds. Moreover, we define a K-contact

Riemannian manifold and give some conditions for the manifold to be K-contact. In §4, we consider the contact distribution on a contact

manifold. §5 is devoted to the study of Sasakian manifolds. We define

a Sasakian manifold and give sane examples of Sasakian manifolds. Moreover, we define the q-sectional curvature of a Sasakian manifold and give the typical examples of Sasakian space forms, that is, Sasakian manifolds of constant 4-sectional curvature. In §6, we

discuss regular contact manifolds. We prove theorems of Boothby-Wang [1], and consider a principal fibre bundle, which is called the

Boothby-Wang fibration. Furthermore, we consider the relation of the Boothby-Wang fibration and Sasakian structures and prove a theorem of Hatakeyama [1]. In the last §7, we consider the Brieskorn manifold.

We give a contact structure on a Brieskorn manifold. Moreover, we show that there exists a Sasakian structure on a Brieskorn manifold

by using the deformation theorey of the standard Sasakian structure on a unit sphere.

For contact manifolds, we refered to Blair [1] and Sasaki [2].

252

1. ALMOST OONTACT MANIFOLDS

Let M be a (2n+1)-dmensional manifold and 0, g, n be a tensor field of type (1,1), a vector field, a 1-form on M respectively. If

E and n satisfy the conditions

n(E) = 1,

(1.1)

(1.2)

c 2X = X + n(X)E

for any vector field X on M, then M is said to have an almost contact structure (4,E,n) and is called an almost contact manifold.

By the definition every almost contact manifold must have a nonsingular vector field E over M. However, the Euler characteristic of any compact manifold is equal to zero and then there exists at least one non-singular vector field over the manifold. Therefore, the condition for an almost contact structure and that for an almost complex structure may be considered to impose almost the same degree of restrictions for odd and even dimensional manifolds respectively.

PROPOSITION 1.1. For almost contact structure (c,E,n) we have

(1.3)

$E = 0,

(1.4)

nW) = 0,

(1.5)

ranks = 2n.

Proof. From (1.1) and (1.2) we have 0ZE = 0, and hence OE = 0 or 4E is a non-trivial eigenvector of ¢ corresponding to the eigenvalue 0. Suppose that OE # 0. Then we have 0 = 02($E) _ -g + 71(0E)E, from which OE =

and hence

0. But we have 0 = 0ZE _

n(cE)c. This contradicts to the fact that 4E # 0 and n(oE) 0 0.

0 Therefore, we have F = 0. From this and (1.2) we easily see that 0.

In the next place, since cF = 0, rank¢ < 2n. If X is another vector of M such that cX = 0, then (1.2) implies that X = n(X)F, that is, X is proportional, to F. Therefore, we have ranlj = 2n.

QED.

We now prove that every almost contact manifold admits a Riemannian metric tensor field which plays an anologous role to an almost Hermitian metric tensor field. We first prove the following lemma.

LEMA 1.1, Every almost contact manifold M admits a Riemannian metric tensor field h such that

h(X,F) = n(X)

for any vector field X on if.

Proof. Since M admits a Riemannian metric tensor field f (which exists provided M is pa.racorrpact), we obtain h by settibg

h(X,Y) = f(X - n(X)F,Y - n(Y)F) + n(X)n(Y).

QED.

PROPOSITION 1,2. Every almost contact manifold ,lf admits a Riema-

nnian metric tensor field g such that

n(X) = g(X,E),

(1.6)

g(4X,cY) = g(X,Y) - n(X)n(Y).

(1.7)

Proof. We put

g(X,Y) = j(h(X,Y) +

n(X)n(Y)).

Then we can easily verify that this satisfies (1.6) and (1.7).

From (1.2), (1.6) and (1.7) we have

QED.

254

g(9,Y) + g(X,OY) = 0.

(1.8)

This means that 0 is a skew-symmetric tensor field with respect to g. We call the metric tensor field g, appearing in proposition 1.2, an associated Riemannian metric tensor field to the given almost contact structure

If M admits tensor field

g being an asso-

ciated Riemannian metric tensor field of an almost contact structure then M is said to have an almost contact metric structure and is called an almost contact metric manifold.

PROPOSITION 1.3. Let M be a (an+1)-dimensional manifold with Then the structure group of its

almost contact structure

tangent bundle reduces to U(n) x 1. The converse is also true.

Proof. First of all, we can choose 2n+1 mutually orthogonal unit

vectors e1,...,en0ej,...,0en,C, which form an orthononnal frame of M, and is called an adapted frame. Then with respect to this frame, we see

I In (1.9)

g =

0

01

In

0

-In

0

0

0

0

0J

0 1

0

In

0

0

0

1J

1

ng with

Now take another adapted frame

respect to which g and 0 have the same canponents as (1.9) an3 put

ei = rei,

=

0ei = r4ei,

then we can easily see that the orthogonal matrix

r = I Cn 0 must have the form

Dn

0

0

1

2'5

r=

Thus the structure group of the tangent bundle of M can be reduced to U(n) x 1.

Conversely, if the structure group of the tangent bundle of M can be reduced to U(n) x 1, then we can define g and

as tensors having

(1.9) as canponents with respect to the adapted frames. We can also give a 1-form n and a vector field & by (0,...,0,1) and t(0'...,O,1) respectively. They satisfy the desired properties.

QED.

Since the structure group of the tangent bundle of an almost contact manifold M reduces to U(n) x 1 and the determinant of every element of U(n) x 1 is positive,

have the following

PROPOSITION 1.4. Every almost contact manifold is orientable.

2. CONTACT MANIFOLDS

A (2n+1)-dimensional manifold M is said to have a contact structure and is called a contact manifold if it carries a global 1-form rj such that

(2.1)

n A (dn )n 0 0

everywhere on M, where the exponent denotes the nth exterior power. We call n a contact form of M.

A quadratic form 6 of the Grassuan algebra AV*, V* being dual to a vector space V, is said to have rank 2r if the exterior product 6r 0 0 and 6r+' = 0. Equivalently, ranks = dimV - dimV0, where V 0 {X : X e V, 6(X,V) = 01. It follows that on a contact manifold M the condition (2.1) implies that the quadratic form dry in the Grassman

algebra

has rank 2n. We then have that VO = {X : X E X(M)

256

do (X,T x(M)) = 0} is a subspace of dimension 1 on which n # 0, and which is thus complementary to the 2n-dimensional subspace on which n = 0. Let Ex be the element of VO on which n has value 1, then E is a vector field, which we call an associated vector field to n, defined over M by n, and which is never zero since n(g) = I.

THEOREM 2.1, Let M be a (2n+1)-dimensional manifold with contact structure n. Then there exists an almost contact metric structure

such that

g(X,4Y) = dn(X,Y).

Proof. For the contact form n there exists a vector field

such

1 and dn(C,Tc(M)) = 0 at every point x of M. We define a

that

skew-symmetric tensor field ¢ of type (1,1) as follows. First of all we can prove that there exists a Riemnannian metric

tensor field h such that n(X) = h(X,E). On the other hand, do is a syirplectic form on the orthogonal complement of E and hence that there

exists a metric g' and an endrxorphism 0 on the orthogonal complement of C such that g'(X,OY) = dn(X,Y) and 02 = -I. Extending g' to a metric g agreeing with h in the direction E and extending 0 so that 4E = 0, we have an almost contact structure (O,C,n,g).

For an almost contact metric structure

(2.2)

QED.

on M we put

7(X,Y) = g(X,cY).

We call

the fundamental 2-form of the almost contact metric structure.

Since 0 has rank 2n, we have n A @n # 0.

An almost contact metric structure constructed from a contact form n, appearing in Theorem 2.1, is called a contact metric structure

associated to n and a manifold with such a structure is called a contact metric manifold.

An almost contact metric structure with @ = do is a contact metric structure.

In the next place, we give a definition of contact structure due

257 to Spencer [1], which is called a contact structure in the wider sense. First of all, we notice that the following theorem of Darboux was obtained (see Cartan [1], Sternberg [1]).

THEOREM 2.2. Let w be a 1-form on an n-dimensional manifold M and

suppose that w A (du)p # 0 and (cb)P+1 = 0 on M. Then about every point there exists a coordinate system (x 1,...,xp,y1,...,yn-p) such that

w=

dy1 -

i=1

yidxi .

From this we see that for every point of a (2n+1)-dimensional contact manifold M, there exists coordinates (xl,yl,z), i = 1,...,n, such that

n yldxl.

n = dz i=1

n,z) be cartesian coordinates in (2n+1)-

Let (xl,...,x dimensional Euclidean space

R2n+1,

and let no be the 1-form on R2n+1

defined by

n0 = dz -

(2.3)

n I Yldxl. i

Then we can easily verify that

(2.4)

no A (dn0)n # 0.

R2n+1.

no is called a contact form on

A diffearorphisn f : U

> U' , where U and U' are open subsets

of R2n+1 is called a contact transformation if and only if f*n0 = Tno,

where T is a non-zero, real valued function on U. We denote by r the set of all contact transformations. r is a pseudo-group in the following sense:

(i) if f : U

> U' and g : V

mations and U' n V # 46, then

a contact transformation,

> V' are contact transfor-

f 1(U' n V)

g(U'rl V) is also

259 (ii) by the carposition in (i), r is associative and (iii) each element of r admits its inverse in r.

A contact transformation f e r such that

f*nO = n0

(2.5)

is called a strict contact transformation. The set r0 of all such transformations is a sub-pseudo-group of r.

A (2n+1)-dimensional manifold M will be called a contact manifold in the wider sense if there exists an open covering {ti} of M with homeonorphisn fi : Ui

> Vi c-

R2n+1 such that fij = fi-fi

for all pairs (i,j) such that fij is defined. Two such coordinate

systems {U ,fi} and {UJ,fl} will be called equivalent if

e r

whenever defined. An equivalence class will be called a contact structure in the wider sense on M.

From the definition we see that there exists a non-zero function

Pij on fj(Uitl i) such that

(fi-f3-)* n0 = PijnO.

Therefore we have

fIn0 =

If we define the 1-form ni on every Ui by setting

ni = fin0,

then we have

ni = on non-empty Ui n U3. Since no satisfies (2.4), we obtain

ni A (drai)n 0 0. Let D be the subbundle of the tangent bundle T(M) whose fibre D is given by

D = {X a T{(M): ni(X) = 0)

for x e U1. Recall that a vector bundle over a manifold with standard fibre lip is said to be orientable if the structure group of its asso-

ciated principal fibre bundle with group GL(p,R) can be reduced to GL+(p,R), which is a subgroup of GL(p,R) consisting of matrices with

We put ni = Tijnj on Ui n U3. Then we have

positive determinants.

ni A (drai)n = Tn+1 (nj A (dnj)n

and

is just the Jacobian of the coordinate transformation. Thus,

if M is orientable and n is even, Tij must be positive and hence vector bundle D is orientable.

We now prove the following theorem (G. W. Gray [1]). THEOREM 2.3. Let M be a (2n+1)-dimensional orientable contact manifold in the wider sense. If n is even, then M is a contact manifold.

Proof. From the assumption we see that T(M) and D are orientable, and hence the quotient bundle T(M)/D admits a global cross section S without zeros. On the other hand, ni defines a local cross section Si

over U1 by the equation ni(Si) = 1, and hence Si = hiS, where the Ws are non-vanishing functions of the same sign. We define n by n = h n i i on U1. Then we obtain a global 1-form n such that n A (dn)n # 0. QED. We now give some examples of contact manifolds. Excanple 2.1. Let

Then

R2a+1

be a (2n+1)-dimensional Euclidean space.

260

n dz - E yldxi i=1 is a contact form on R2n+1,

(xi,yi,z)

being cartesian coordinates.

Then the vector field & is given by a/az. Excmpte 2.2. Let M be a (2n+1)-dimensional regular hypersurface of R2n+2 (i.e., C with a unique tangent plane at every point). In R2n+2 (x1,...,)2n+2), we consider a 1-form with cartesian coordinates

defined by

a = xldx2 - x2dxl + ... +

x2n+1dx2n+2 - x2n+2&2n+1.

Then we have

dx2n+2),

da = 2(dxl A dx2 + ,,, + dx2n+1 A

from which

2n+1 a A (da)n =

2n-1n![

(-1)1-lxidxl

A dx2

A ...

i=1 ... A

dxi-1

A dxi+l A ... A dx2n+2].

We denote by v1,...,v2n+l 2n+1 linearly independent vectors which span the tangent space of M at x0 = (x.,...,x0 +2). We put

wi = *dxd(vl,...,v.+1),

where * denotes the Hodge star operator of the Euclidean metric on R2n+2.

Then a vector w with components w3 is normal to the hypersurface

spanned by v1,...,v2n+1. We also have

(a A (da)n)(vl,...,v2n+l) = xO.w,

where (

2n+2

) denotes the ordinary scalar product in R.

269

On the other hand, the equation of the tangent space of M at

is

x0 given by w (x-x0) = 0. Therefore, the tangent space of M at x passes

through the origin if and only if

0, that is, a A (da)

= 0 at

x0. Moreover, we see that n = i*a, i being an immersion of M into

R2n+2,

satisfies

n n (dn)n = i*(a A (da)m).

Therefore, we see that n A (dr1)n vanishes at x0 on M if and only if

the tangent space of M at x0 passes through the origin of R2n+2. Consequently, we have the following theorem (G. W. Gray [1]). THEOREM 2.4. Let M be a smooth hypersurafce immersed in R2n+2 If the tangent space of M does not pass through the origin of R2n+2 then M has a contact structure.

As a special case of Theorem 2.4, we see that an odd-dimensional R2n+2 carries a conatact structure. Furthermore, since (xl,...,x2n+2) > (-x1,..,-x2n+2), a is invariant under substitution sphere Stn+1

a also induces a contact structure on the real projective space RP2n+1 Exa.npZe 2.3. In On we put

S=

n

x6 n+i .

i=1 Let Ri be the subspace of R2n defined by xl = 0, i = 1,...,n and Rn the subspace of R2n defined by xJ = 0, j = n+1,...,2n. Then (3 induces

a contact form on a hypersurafce M of dimension 2n-1 immersed in R2n if and only if M n Ri = 5, dim(M (1 R2) = n-1 and no tangent space to

M q in R2 containes the origin in E.

Excmrple 2.4. Let M be an (n+l)-dimensional Riemrannian manifold and T(M)* its cotangent bundle. We denote by (x1,.. .,)cp+l) local coordinates on U and (p1,...,p +1) fibre coordinates over U defined with

respect to dxIs. If n : T(M)* then (pl,gl =

> M is the natural projection,

are local coordinates on T(M)*. We put

262

n+1

i pd4

i=1

on a coordinate neighborhood. We denote by T1(M)* the bundle of unit cotangent vectors. Then T1(M)* has empty intersection with the zero section of T(M)*. Moreover, the intersection with any fibre of T(M)* is an n-dimensional sphere and no tangent space to this intersection contains the origin of the fibre. Therefore, from Example 2.3, we see that S induces a contact structure on the hypersurface T1(M)*. Similarly, we obtain a contact structure a on the bundle T1(M) of

unit tangent vectors. We denote by g.. the carponents of the metric with respect to the J1 coordinates (xl,...,xn+1) and by (v1...... vn+1) the fibre coordinates

on T(M). We define a 1-form S locally by

S =

g1.vdgl,

where we have put ql = xi.7r. From this we have our assertion. Ezcvrrple 2.5. We now give an example of a contact manifold in the

wider sense.

Let M = 10+1 x RPn, where RP denotes n-dimensional real projective space. Let ()0,...,xn) be coordinates in Rn+1 and (t0,...,tn)

homogeneous coordinates in PPP. The subsets

{Ui},

i = 1,...,n defined

by ti # 0, form an open covering of M. In Ui we define a 1-form ni by

ni =

n

ti j=0E t.dxJ. J

Then we have ni ^ (drai)n # 0 and ni = (tj/ti)r1j. Thus, M has a contact

structure in the wider sense, but for n even, M is non-orientable and hence cannot carry a global contact form.

0 3. TORSION TENSOR OF ALMOST CONTACT MANIFOLDS

let M be a (2n+l)-dimensional almost contact manifold with almost contact structure (4),C,n). We consider a product manifold M x R, where

R denotes a real line. Then a vector field on M x R is given by (X,f(d/dt)), where X is a vector field tangent to M, t the coordinate of R and f a function on M x R. We define a linear map J on the tangent space of M x R by

(3.1)

J(X,fat) =

Then we have J2 = -I and hence J is an almost ccrrplex structure on

M x R. The almost carplex structure J is said to be integrable if its Nijenhuis torsion NJ vanishes, where

NJ(X,Y) = J2[X,Y] + [JX,JY] - J[JX,Y] - J[X,JY].

If the almost complex structure J on M x R is integrable, we say that the almost contact structure (4),E,n) is normal.

In the following we seek to express the condition of normality in term of the Nienhuis torsion N0 of ¢, which is defined by

N0(X,Y) = $2[X,Y] + [4X,4)Y] - $[4)X,Y] - [X,oY].

Since NJ is a tensor field of type (1,2), it sufficies to compute NJ((X,0),(Y,0)) and NJ((X,0),(O,d/dt))'for any vector fields X and Y on M. From (3.1) we have

NJ((X.0),(Y,0)) = -([X,Y],O) +

-(+[X,4)Y]-(Xn(Y))f,n([X,4)Y]}dt)

234

=

e)Y-(L,yn)R

NJ((R,0),(O,d/dt)) = (-[$X,E],

),

d )

=

dt).

Here we define four tensors N(1), N(2), N(3) and N(4) respectively by

N(1)(X,Y) = NN(X,Y) + N(2)(X,Y)

N(3)(X)

= (LLXn)Y - (LLYn)X,

= (L Ox,

N(4)(X) = (LLn)X.

It is clear that the almost contact structure (O,C,n) is nprmal if and only if these four tensors vanish. Larva 3.1, If N(1) = 0, then N(2) = N(3) = N(4) = 0.

Proof. If N(l) = 0, then we have

(3.2)

[C,X] + 4[E,0] - (Cn(X))C = 0.

Thus we have

n([C,X]) - sn(x) = 0,

which shows that N(4) = 0. From this equation we also have n([E,4Xl) = 0. On the other hand, applying ¢ to (3.2), we see that

0 = OLCX - LE¢X +

from which (LEO)X = 0 and hence N(3) = 0. Finally, from N(l) = 0 we

have

2S 0 = N1,(4X,Y) + 2dn(4X,Y)

= -[OX,Y] - [X,4Y] - (cYn(X))E - n(X)[OY,E] c[OX,$Y] + (cXr(Y))E.

Applying n to this equation and using n([E,4X]) = 0, we get

$y-X(Y) - n([4X,Y]) - 4Yn(X) + n([cY,X]) = 0.

Thus we have N(2) = 0.

QED.

In view of Team 3.1 we have PROPOSITION 3.1. The almost contact structure (c

,n) of M is

nonn3l if and only if

N+2dn®E=0. LEMMA 3.2. let M be a contact metric manifold with contact metric structure

Then N(2) and N(4) vanish. Moreover, N(3) vanishes

if and only if E is a Killing vector field with respect to g. Proof. We have

dn(cX,,Y) = 0(4X,4Y) =

-g(X,43Y) =

dn(X,Y),

from which

dn($X,Y) + dn(X,4Y) = 0.

This is equivalent to N(2) = 0. On the other hand, we have 0 = g(X,OE) = dn(X,E) = #(Xn(E) - En(X) - n([X,E])).

288

Thus we obtain

En(x) - n([E,X]) = o.

Therefore, we have LEn = 0 and hence N(4) = 0. Moreover, we see that

(LEg)(X,E) = Ei(X) - n([E,X]) = (I n)X = 0.

On the other hand, we easily see that LEdn = 0 and consequently

(LLdn)(X,Y) = (LE(D)(X,Y) = 0,

from which

0 = Eg(X,4Y) - g([E,X],$Y) - g(X,4[E,Y])

.

= (LEg)(X,,Y) + g(X,(LLc)Y) = (LEg)(X,cY) + g(X,N(3)(Y)).

Thus E is a Killing vector field if and only if N(3) = 0.

QED.

L DM 3,3, For an almost contact metric structure (4,E,n,g) of M

we have

2g((V4)Y,Z) = 3dO(X,bY,4Z) - 3d4(X,Y,Z) + g(N(1)(Y,Z),4X)

+ N(2)(Y,Z)n(X) + 2dn(cY,X)n(Z) - 2dn(4Z,X)n(Y).

Proof. The Riemannian connection V with respect to g is given by

2g(VXY,Z) = Xg(Y,Z) + Yg(X,Z) - Zg(X,Y) + g([X,Y],Z) + g([Z,X],Y) - g([Y,Z],X).

On the other hand, d$ is given by

297

3dt(X,Y,Z) = 74(Y,Z) +

74 (X,Y)

- 4'([X,Y],Z) - 0([Z,X],Y) - 4)([Y,Z],X)

Fran these equations and (2.2) we have our equation.

QED.

LEMMA 3.4. For a contact metric structure ($,E,r,g) of M with 0 = do and N(2) = 0, we have

2dn(4Y,X)n(Z) - 2dn(OZ,X)n(Y).

2g((V3&)Y,Z) =

Especially we have V0 = 0. Proof. The first equation is trivial by the assumption. We prove

that V = 0. Fran N(2) = 0 we have dn(X,&) = 0. Thus the first equation implies that V 0 = 0.

QED.

E

In the case of Lemma 3.4 it is also easy to see that the integral curves of F are geodesics, that is, V&& = 0.

Let M be a (2n+1)-dimensional contact metric manifold with contact metric structure

If the structure vector field E is a Kill-

ing vector field with respect to g, then the contact structure on M is called a K-contact structure and M is called a K-contact manifold.

Fan Lemma 3.2 we have the following PROPOSITION 3,2, Let M be a contact metric manifold. Then M is a K-contact manifold if and only if N(3) vanishes. Since we have

dr)(X,Y) = #(g(VXE,Y) - g(V,X)) = g(V,Y), for a K-contact structure, we obtain VX _ -4X. Conversely, if y _ as $ is skew-synnEtric, l; is a Killing vector field. Thus we have

268

PROPOSITION 3.3. Let M be a contact metric manifold. Then M is

a K-contact manifold if and only if VXE _ -4X.

We now give a geometric characterization of K-contact manifolds.

THEOREM 3.1, In order that a (2n+1)-dimensional Riemannian manifold M is K-contact, it is necessary and sufficient that the following two conditions are satisfied: (1) M admits a unit Killing vector field E;

(2) The sectional curvature for plane sections containing E are equal to 1 at every point of M. Proof. Let M be a K-contact manifold. Then

g(R(X,E)E,X) = g(-Q2X,X) = g(X,X) = 1,

where X is a unit vector field orthogonal to E.

Conversely, we suppose that M satisfies the conditions (1) and (2). Since E is a Killing vector field, we have

(3.3)

R(X,E)Y = VXVYE - VVXYE.

We put n(X) = g(X,E) and OX = -V. Then we easily see that 4E = 0. Fran (3.3) we also have

1 = g(R(X,E)E,X) = -g(c2X,X),

where X is a unit vector field on M orthogonal to E. Therefore, we obtain ¢2X = -X for every vector field X of M orthogonal to E and hence

02Y = -Y + n(Y)E

2" for any vector field Y of M. Moreover, we see that

drl(X,Y) = j(g(V

Consequently,

,Y) - g(VYE,X)) = -g(V 1 ,X) = g(X,cY).

is a K-contact structure on M.

QED.

From (3.3) we also have

R(X,E) _ZX = X - n(X)C.

(3.4)

We easily see that a (2n+l)-dimensional Riemannian manifold M admitting a unit Killing vector field C which satisfies (3.4) is a K-contact manifold.

4. CONTACT DISTRIBUTION

Let M be a (2n+1)-dimensional contact manifold with contact form n. Then n = 0 defines a 2n-dimensional distribution D of the tangent bundle. The distribution D is called the contact distribution which is as far from being integrable as possible from the fact that n A (dn)n

00. In the following we consider an integral submanifold of the distribution D (Sasaki [1]).

THEOREM 4.1. Let M be a (2n+1)-dimensional contact manifold with contact form n. Then there exist integral sub nnifolds of the contact distribution D of dimension n but no higher dimension. Proof. We can choose local coordinates (xl,yl,z) such that n = dz - Eyldxl on the coordinate neighborhood. Then for a point x with coordinates 4,y0',z(,) in the coordinate neighborhood, xl = xp, z = z0 defines an n-dimensional integral sukmanifold and a

maximal

integral sutmanifold containing this coordinate slice is an integral suhmanifold of D in M.

Let N be an r-dimensional integral sulmanifold of D. We suppose that r > n. We denote by e1, ...,er r linearly independent local vector

270

fields tangent to N and extend these to a basis er+1,...,eW2n+1 of M. Then we have

n(ei) = 0,

dn(ei.e? = 0,

i,j = 1,...,r.

Thus, since r > n, we see that (n A (dn)n)(e1,...,e2n+1) = 0, which is a contradiction.

QED.

PROPOSITION 4.1. Let N be an r-dimensional submanifold immersed in a (2n+1)-dimensional contact manifold M. Then N is an integral submanifold of D if and only if n and do vanish on N. Let (0,E,n,g) be an associated almost contact metric structure. Then N is an integral sukmanifold of D if and only if every tangent vector X of N belongs to D and X is normal to N in M. Procf. For any vector fields X and Y tangent to N we see that n(X) = n(Y) = 0 and hence dn(X,Y) = 0. Conversely, if n and do vanish

on N, we have

0 = dn(X,Y) = - ln([X,YI)

for any vector fields X and Y tangent to N. Thus N is an integral

suYmanifold of D. The second statement follows immediately fran the fact that dn(X,Y) = g(X,cY).

QED.

LB$'A 4.1. Let (xl,yl,z) be local coordinates for x = (xx,Yo, ZO)

such that n = dz - Eyldxl on the coordinate neighborhood. In order that

r linearly independent vectors Xt, t = 1,...,r < n, at x with canponents (a1,b1,ct) be tangent to an r-dimensional integral submanifold it is necessary and sufficient that n(Xt) = 0 and dn(Xt,Xs) = 0, that is,

ii

ct = YOat,

lait bi s

=

lasbit.

Proof. Since the necessity is clear, we prove the sufficiency.

0 We put c

= Ealbl and choose a sufficiently small neighborhood V of

the origin of R with coordinates (u 1 ...,u ) such that

xl

1

_

z = zo +

Call t t

ICtut

+

t

O y1 + tbtut,

Y1

+ tt

ctsutuS

t,s

define a mapping i of V into M. Then

axl/au = at,

ayl/aut = bt,

az/aut = Ct + 1ctsus = i.yI(axl/au ) + I (axu/aut)(ayu/auS)uS is s i

= 51(axl/au ) i

and hence the napping i defines an integral submanifold tangent to QED.

X1,...,X, at X. THEOREM 4.2.

be a vector at a point x of M belonging to D.

Then there exists an r-dimensional integral sub manifold N (1 < r < n) of D through x such that X is tangent to N.

Proof. Let'(bcl) be the conponents of X with respect to the local coordinates (xl,yl,z). Since X E D we have cl = Eiyoal. If not all *the ai's vanish, choose a2,...,al such that rank(al) = r and define

C2,...,cr by ct = Eiyoat, t = 2,...,r. We now define b2,...,b. inductively as follows. We suppose that b1,s are given. We take bs+1 (1 < s < r-1) as a set of solutions of

lfl = jal bl I

S+111

.... '

jalfl = jal bl 8+1 s s

which exists as rank (at) = r. Then {(t,bt,ct)} satisfy the conditions of the previous Lema 4.1 and hence we have an integral subnanifold N with X tangent as desired. If on the other hand, all of the si's vanish, then c1 also vanishes, so choosing b2, ...,bb such that rank(bt) = r we again have an

0 r-dimensional integral submanifold with X tangent to by Lemma 4.1. QED.

A diffeomorphism f: M

> M is called a contact transformation

if f*D = Tn for some non-vanishing function r on M. If moreover t = 1,

then f is called a strict contact transformation. The following lemma is trivial. LEMMA 4.2, A diffeomorphisn f on a contact manifold M is a contact transformation if and only if f4X belongs to D for every X in D. THEOREM 4.3. A diffeomnrphism f on a contact manifold M is a contact transformation if and only if f maps n-dimensional integral submanifolds of D onto n-dimensional integral sutmanifolds of D. Proof. Let f be a contact transformation and 11 be an n-dimensional

integral submanifold. Then we have f*(T{(N)) in D. Thus f(N) is an integral submanifold. Conversely, for any vector X at x in D, we have seen that there exists an integral sutmanifold N through x with X as a tangent vector. Since f(N) is also an integral sulmanifold, f *X is in

D and f is a contact transformation by Lemma 4.2.

QED.

5. SASAKIAN MANIFOLDS

Let M be a (2n+1)-dimensional contact metric manifold with contact

metric structure ($,C,n,g) If the contact metric structure of M is normal, then M is said to have a Sasakian structure (or normal contact metric structure) and M is called a Sasakian manifold (or normal contact metric manifold). We denote by V the operator of covariant differentiation with

respect to g. Then we have THEOREM 5.1, An almost contact metric structure is a Sasakian structure if and only if

(5.1)

(Y)Y = g(X,Y)l; - n(Y)X.

on M

Proof. If the structure is normal, we have (P = do and N(1) = N(2)

= 0. Thus lama 3.3 implies (5.1). Conversely, we suppose that the structure satisfies (5.1). Putting

Y =

in (5.1), we have --OV

this, we obtain V

= n(X)E - X, and hence, applying 0 to

= -X. Since 0 is skew-symmetric, we see that C is

a Killing vector field. Moreover, we obtain

dr,(X,Y) = #((VXrj)Y - (VYn)X) = g(X,4)Y) = 4)(X,Y).

Thus the structure is a contact metric structure. Furthermore, by a straightforward computation we have N4) + 2dn 0

= 0. Therefore, the

structure is Sasakian.

QED.

If M is a Sasakian manifold, from (5.1) we have

R(X,Y) = n(Y)X - n(X)Y,

(5.2)

where R denotes the Rie annian curvature tensor of M. From (5.2) we also have

(5.3)

R(X,E)Y = -g(X,Y)C + n(Y)X.

'THEOREM 5,2. let M be a (2n+1)-dimensional Riemannian manifold

admitting a unit Killing vector field C. Then M is a Sasakian manifold if and only if (5.3) holds.

Proof. From Theorem 3.1 we see that M has a K-contact structure Then we have

R(X,t)Y = VXVYE - VVgy, _ -(VX4))Y.

Thus Theorem 5.1 proves our assertion. We now give some examples of Sasakian manifolds.

QED.

274

Example 5.1. Let M be a real (2n+1)-dimensional hypersurface of a Kaehlerian manifold M of ccixplex dimension n+1. We denote by J the

almost complex structure tensor field of M and p the operator of covariant differentiation in M. The operator of covariant differentiation with respect to the induced connection on M will be denoted by V. We denote by C a unit normal of M in

and by A the second fundamental

tensor of M. We put JC = -C. Then

is a unit vector field on M. For

any vector field X tangent to M we put

JX = Ox + n(X)C,

where n is a 1-form dual to E. Then we have

42X = -x + n(X)C,

4

= 0,

n(4X) = 0,

n(X) = g(E,X)

Thus (O,C,n,g) defines an almost contact metric structure on M.

bbreover, the Gauss and Weingarten formulas for M are respectively given by

OXY = VXY + g(AX,Y)C

and

OXC = -AX.

Fran these equations we have

(Vx4)Y = n(Y)AX -

V

= 4,AX.

We suppose that AX = -X + Sn(X),, where a is a function on M. Then

we have

g(V,Y) + g(V,X) =

g(¢Y,X) = 0.

Therefore, E is a Killing vector field on M. Furthermore, we have

(Vxo)Y = g(X,Y)E - n(Y)X.

275 Thus Theorem 5.1 shows that M is a Sasakian manifold. Conversely, if M is a Sasakian manifold, we obtain

AX = -x + n(AX+x)E,

from which

AX = -X + On(x)E,

where we have put S = n(4) + 1. Consequently, a real hypersurface M of a Kaehlerian manifold M is a Sasakian manifold if and only if its second fundamental form A satisfies A = -I + an ® E. We notice that such a real hypersurface is an n-umbilical hypersurface. For a real hypersurface M of a Kaehlerian manifold 22 we see that

N0(X,Y) + 2dn(X,Y) = n(X)(bA-A$)Y - n(Y)(8A-Ac)X.

From this we easily see that 4A = A¢ if and only if N0(X,Y)+2dn(X,Y)F = 0. Therefore, the almost contact metric structure (0,&,n,g) on M is normal if and only if i; is a Killing vector field on M.

Excnnpie 5.2. Let Cn+1 be a caiplex (n+l)-dimensional number space. Stn+1 be a (2n+1)-

Then Cn+l admits a Kaehlerian structure J. Let dimensional unit sphere, i.e.,

Stn+1 = {Z E Cn+1

:

IZI = 1}.

Then 52n+1 is a real hypersurface of Cn+l and the second fundamental Cn+1 is given by A = -I. Therefore, from Example form A of g2n+1 in 1.1, Stn+1 admits a Sasakian structure (0,E,n,g) which is called a Stn+1 natural Sasakian structure on Excmrpte 5.3. It R2a+2 be a (2n+1)-dimensional number space. We

put

R2 n n = i(dz -

yidxi), i=1

(xl,yl,z) being cartesian coordinates. Then the structure vector field

E is given by E = -2a/az and the Riemannian metric tensor field g is given by

n ((dxl)2 + (dyl)z).

g = 4(n ®n +

i=1 This gives a contact metric structure on R2n+1 as follows. First of all we have

16iJ+ylyJ

0

0

6ij

0

-yl

0

1

g = 4

j

We give a tensor field c of type (1,1) by a matrix form

0

-6.. 0

6ij

0

0

0

yj

0

The vector fields Xi = 2a/ayl, Xn+i = 2(a/axl + yla/az),

form a

-basis for the contact metric structure. On the other hand, we can

see that N0 + 2dn ® = 0 and hence the contact metric structure is normal.

Let M be a (2n+1)-dimensional Sasakian manifold with Sasakian structure

From (5.1) we easily see

277 (5.4)

R(X,Y)4>Z = 4>R(X,Y)Z + g(4>X,Z)Y - g(Y,Z)4>X

+ g(X,Z)4>Y - g(tY,Z)X.

Erom (5.4) we also have the following equations:

(5.5)

g(Y,Z)X - g(X,Z)Y

R(X,Y)Z =

- g(QY,Z)4>X + g($X,Z)4>Y,

(5.6)

g(R($X,cY)4>Z,4>W) = g(R(X,Y)Z,W) - n(Y)n(Z)g(X,W)

- n(X)n(W)g(Y,Z) + n(Y)n(W)g(X,Z) + n(X)n(Z)g(Y,W).

A plane section in X(M) is called a 4>-section if there exists a

unit vector X in X(M) orthogonal to E such that {X,¢X) is an orthonormal basis of the plane section. Then the sectional curvature K(X,4>X) = g(R(X,4>X)4>X,X) is called a 4>-sectional curvature, which will

be denoted by H(X). We shall show that on a Sasakian manifold the 4>-sectional curvatures determine the curvature completely.

In the following we prepare some lemmas. We put

P(X,Y;Z,W) = g(Y,Z)g($X,W) - g(4>X,Z)g(Y,W)

+ g(4>Y,Z)g(X,W) - g(X,Z)g(4>Y,W).

Then we have

P(X,Y;Z,W) = -P(Z,W;X,Y).

If {X,Y} is an orthonormal pair orthogonal to E and if we put g(X,4>Y) = cos9, 0 < e < n, then

P(X,Y;X, Y) = -sin2O.

278

We now put B(X,Y) = g(R(X,Y)Y,X)

for any vectors X and Y and D(X) =

for any vector X orthogonal to E. Then we have LEMMA 5,1, For any vectors X and Y orthogonal to E we obtain

B(X,Y) = 32(D(X+4Y) + 3D(X-cY) - D(X+Y) - D(X-Y) - 4D(X) - 4D(Y) -

Proof. First of all we have

D(X+Y) + D(X-Y) = 2[D(X) + D(Y) + 2B(X, Y) + 2g(R(X,OX)OY,Y) + 2g(R(X,OY)$X,Y)].

Putting OY instead of Y in this equation we obtain

D(X+qY) + D(X-4Y) = 2[D(X) + D(cY) + 2B(X,Y) + 2g(R(X,¢X)cY,Y) + 2g(R(X,Y)4Y,4X)].

Since D(cY) = D(Y), we find

D(X+Y) - D(X-Y) - 4D(X) - 4D(Y)

= 12B(X,Y) - 4B(X,¢Y) + 8g(R(X,4X)OY,Y) +

On the other hand, from (5.4) and the Bianchi's identity, we have

0 8g(R(X,$X)$Y,Y) = 8[B(X,Y) + B(X,$Y) + 2P(X,Y;X,$Y)].

We also have the following equations:

12g(R(X,Y)$Y,$X) = 12[B(X,Y) + P(X,Y;X,$Y)],

4g(R(X,$Y)Y,$X) = 4[-B(X,tY) + P(X,$Y;X,Y)].

From these equations we have our assertion.

QED.

We notice that D(X) = H(X) if and only if X is a unit vector and B(X,Y) = K(X,Y) if and only if {X,Y} is an orthonormal pair.

LEMMA 5.2. Let {X,Y} be an orthonoru l pair of the tangent space

of a Sasakian manifold M orthogonal to . If we put g(X,$Y) = cose (0 < 6 < n), then

3(1-cos6)aH( X_$Y )

K(X,Y) = 8[3(l+cose)2H(

- H( X

YX+Y

) - H( X Y) - H(X) - H(Y) + 6sin2e ] .

Proof. Since D(Z) = IZ14H(Z/IZI) for any Z, we see that

D(X+$Y) = JX+$Y14H( X+$Y

4(1+cose)2H( X+$Y ).

Similarly, we obtain

D(X-$Y) = 4(1-cose)2H(

IX-9yj

D(X+Y) = 4E( X+Y ),

D(X-Y) = 4H( X Y ).

),

From these equations and Lemma. 5.1 we have our equation.

QED.

THEOREM 5.3. The 0-sectional curvatures determine the curvature of

a Sasakian manifold.

280

Proof. Since the sectional curvatures of a Rienannian manifold determine the curvature, it sufficies to show that the sectional curvatures are determined by the 0--sectional curvatures uniquely.

Let {X,Y} be an orthonormal pair. We put

X = n(X)g + aZ,

Y = n(Y)g + bW,

where a = (1 - 3(X)2)1/2, b = (1 -

n(Y)2)1/2,

Z, W being orthogonal

to E. Then Z and W are unit vectors. By a simple computation we find

K(X,Y) = a2n(Y)2 - 2abn(X)n(Y)g(Z,W) + b2n(X)2 + a2b2g(R(Z,W)W,Z).

Noticing that

g(Z,W)

isnWn(y),

g(R(Z,W)W,Z) = (1-g(Z,W)2]K(Z,W) = [1 - b2 a (X)2n(Y)2]K(Z,W),

we obtain

K(X,Y) = 3(X)2 + n(Y)2 + [1-3(X)2-n(Y)2]K(Z,W).

On the other hand, by Lemma 5.2, K(Z,W) is determined by 0-sectional curvatures. This ccmpletes the proof.

QED.

THEOREM 5.4, If the Q-sectional curvature at any point of a Sasakian manifold of dimension > 5 is independent of the choice of 0-section at the point, then it is constant on the manifold and the curvature tensor is given by

R(X,Y)Z = }(c+3)[g(Y,Z)-g(X,Z)Y] + *(c-1)[n(X)n(Z)Y-n(Y)n(Z)X +g(X,Z)n(Y)C-g(Y,Z)n(X)C+g(oy,Z)o% g(ox,Z)OY+2g(X,OY)OZ],

289

where c is the constant ¢-sectional curvature.

Proof. We notice that R(X,E)X = - and R(U,X)E _ -X for any vector X orthogonal to E. Thus we have actually prove that any vector field of type (5.3) on a Sasakian manifold satisfying the symmetries of the curvature tensor, the Bianchi's identity, (5.4) and which coincides with the values of the 4>-sectional curvatures must be the curvature tensor. From this we see that R(X,Y)Z is of the given form.

Conversely, we easily see that c is constant when dimension > 5. QED.

A Sasakian manifold M is called a Sasakian space form if M has constant ¢-sectional curvature c, and will be denoted by M(c).

Example 5.4. Let 52n+1 be a unit sphere with natural Sasakian structure (4>,F,n,g). We consider the deformed structure:

n* = an,

F* = a 1E,

g* = Cg + (a2 - a)n 0 n,

where a is a positive constant. We call this deformation D-homothetic deformation. Then

is a Sasakian structure on Stn+1 Stn+l(c)

with constant 4-sectional curvature c = 4/a - 3. We denote by the Sasakian manifold with this structure.

Example 5.5. Let R2n+1 be a (2n+1)-dimensional number space with Sasakian structure as in Example 5.3. It is checked that R2n+1 is of constant q-sectional curvature -3 and we denote it by R2n+1(-3). Example 5.6. let CDn be a simply connected bounded complex danai.n

in Cn with constant holomorphic sectional curvature c < 0. We denote by (J,G) a Kaehlerian structure on CEP. Since the fundamental 2-form S2 of the Saehlerian structure is closed, S2 = dw for some real analytic 1-

form w. Let t denote the coordinate on R and put Ti = 2w + dt on a product space R x CUP. If we consider R as an additive group, then n is an infinitesimal connection form on the trivial line bundle (R,CD"). We have F = 3/at and g = ,m*G + n 0 Ti, where w : (R,&')

> CEP is

282

the projection. T1 is also written as r = 2mr*w+ dt, and we have drt = 2n'. Therefore, these tensors define a Sasaan structure on (R,CD") with constant 0-sectional curvature k = c-3 < -3. We denote. this by

(R,&)(k). We show that three types of model spaces in these examples 5.4, 5.5 and 5.6 are unique up to isomorphism's, where an isomorphism means

a diffecmorphism which maps the structure tensors into the corresponding structure tensors (Tanno [41). THEOREM 5.5. Let M be a (2n+1) -dimensional complete simply connec-

ted Sasakian manifold with constant 0-sectional curvature c. (1) If c > -3, then M is isamorphic to S

+1(c) or M is D-harothe-

tic to S2n+1

(2) If c = -3, then M is isomorphic to

R2n+1(-3);

(3) If c < -3, then M is isomorphic to (R,C')(c). Prcof. From the assumption M admits local 0-holomorphic free mobility and hence M admits global 0-holarorphic free mobility because of M is complete and simply connected. Thus M admits an autarorphism group Aut(M) such that, for any point x and y, any 0-section at x carried to any other 0-section at y by some element of Aut(M). Aut(M) is of (n+1)2-dimension and M is diffeomorphic to a homogeneous space Aut(M)/(isotoropy group). Thus we can assume that M is real analytic, and also that g is real analytic. We denote by M* one of the model spaces corresponding to c > -3, = 3 or < -3 and by

the

structure tensors of M*. For any point x of M and x* of M*, let

(el,...en0el,...,0en,E) and

i,...,en,#

be orthonormal

0-basis at x :uld x* respectively. We define a linear isomorphism F: TY(M)

> TY*(M*) by Fei = e1 4l, Foei = 0*et (i = 1,...,n) and

FE = F*. Then we have F0 = 0*F and F is isanetric at x, that is, F is isomorphic at x. Since both 0- and 0*-sectional curvatures are equal to c, F maps R into R*, F being considered as a map of tensor algebra. The covariant derivatives of 0 and E are written in terms of and hence the covariant derivative of R is expressed by

$, E, g

0, E and g,

that is, F maps (VR) x into (V*R*)x*. Likewise, we see that F maps

283

(vkR)x into (v*kR*)x* for every positive integer k. Then we have an

isanetry f of M onto M* such that f(x) = x* and the differential of f at x is F (cf. Kobayashi-Nomizu [1; p.259-261]). We then have that (vE)x is mapped to (V*E*)x*. Thus we have

(V*(fl;))x* =

f is also a Killing vector field. By (f&)x*

f

_ CX* and (V*(fE))x* = (V*C*)x*, we get f _

and n*) are determined by g and

*. Because 0 and n (4*

(g* and *), f is an isomorphism

between M and M*.

QED.

In the following we study the properties of the Ricci tensor of a Sasakian manifold. First, let M be a (2n+1)-dimensional K-contact mani-

We denote by S and Q the Ricci

fold with stricture tensors

tensor and the Ricci operator of M respectively. We prove PROPOSITION 5.1. If the Ricci tensor S of a K-contact manifold M is parallel, then M is an Einstein manifold. Proof. Since

X - n(X)E, we obtain S(C,C) = 2n. Thus we

have (VXS)(C, ) = 2S(cX,E) = 0, and hence S(X,F) = 2nn(X). From this we see that S(X,gY) = 2ng(X,OY). Therefore, we obtain S(X,Y) = 2ng(X,Y) for any vector fields X and Y, which means that M is an Einstein mani-

fold.

QED.

PROPOSITION 5.2, Let M be a K-contact manifold. If M is locally symmetric, then M is a Sasakian manifold with constant curvature 1. Proof. By the assumption we have

-R(X,4Y) -

g(cY,X)E + n(X)¢Y.

Replacing Y by cY in this equation, we see that

R(X,Y)l; + R(X,E)Y = 2n(Y)X - n(X)F -

From this we find

284 R(X,Y)Z + R(X,Z)Y = 2g(Z,Y)X - g(X,Y)Z - g(Z,X)Y.

Thus, for any orthonormal pair {X,Y}, we obtain

K(X,Y) = g(R(X,Y)Y,X) = 1,

which shows that the sectional curvature of M is 1 and hence M is a Sasakian manifold.

QED.

From Propositions 5.1 and 5.2 we see that-the notion of parallel Ricci tensor and locally symmetric are not so essential on a K-contact manifold and consequently on a Sasakian manifold.

From (5.6) and the Bianchi's identity we have LEW 53, The Ricci tensor S of a (2n+1) -dimensional Sasakian manifold M is given by

2n+1 g(gR(X,AY)ei,ei) + (2n-1)g(X,Y) + n(X)n(Y).

S(X,Y)

i=1

Moreover, by (5.6), we easily see that LE('MA 5,4, The Ricci tensor S of a (2n+1)-dimensional Sasakian

manifold M satisfies the following equations:

S(X,E) = 2nn(X),

S(¢X,cY) = S(X,Y) - 2nn(X)n(Y).

From Lemma 5.3 and the Bianchi's identity we also have LE74MA 5.5. The Ricci tensor S of a (2n+1)-dimensional Sasakian

manifold M satisfies

(VzS)(X,Y) = (VxS)(Y,Z) +

n(X)S(4Y,Z)

- 24(Y)S(4X,Z) + 2nn(X)g(4Y,Z) + 4nn(Y)g(gX,Z).

0 If the Ricci tensor S of a K-contact manifold M is of the form

S(X,Y) = ag(X,Y) + bn(X)n(Y),

a and b being constant, then M is called an n-Einstein manifold. Obviously, we have PROPOSITION 5.3. If a Sasakian manifold M is of constant +-sectional curvature, then M is an n-Einstein manifold.

PROPOSITION 5.4. Let M be a (2n+1)-dimensional Sasakian manifold. If the Ricci tensor S of M satisfies S(X,Y) = ag(X,Y) + bn(X)n(Y), then

a and b are constant. Proof. From the assunuption on the Ricci tensor S we have a+b = 2n

and r = (2n+l)a + b, where r denotes the scalar curvature of M. Then we have Za = -Zb'and Zr = (2n+l)Za + Zb = -2nZb. On the other hand,

Team 5.5 implies

Zr = 2Za + 2(N71(Z) = -2Zb +

Therefore, we obtain

(n-1)(Zb) = -(Cb)n(Z).

Putting Z =

in this equation, we find b = 0 and hence Zb = 0, which

shows that b is constant. Then a is also constant.

QED.

298 6. REGULAR CONTACT MANIFOLDS

Let M be a (2n+1)-dimensional contact manifold with a contact form n. We denote by

the associated vector field to n. We say that

a contact structure n is regular if each point has a regular coordinate neighborhood , i.e., a cubical coordinate neighborhood U such that

the integral curvaes of

passing through U pass through the neighbor-

hood only once. Then the vector field E; is- said to be regular.

Hereafter we will assume that the manifold M to be compact. Then, if E is regular, & is a closed vector field, that is, the orbit of through an arbitrary point is a closed curve. Let B be the set of orbits of C. Then we have the natural projection

IT

: M

> B,

making correspond every point p of M to the orbit through p. Then we see that B with the quotient topology is a 2n-dimensional differentiable manifold and n is a differentiable map (see Palais [1]). We put

11E(p) = inf( t : t > 0,

0t(p) = p).

Clearly, VC is constant on every orbit. We call }r(p) a period functions of E. If & is regular, we see that µ,(p) is constant on M.

We now give the theorems of Boothby-Wang [1]. THEOREM 6.1, Let n be a regular contact form on a compact manifold M of dimension 2n+1. Then there exists a gauge transformation n' = on such that the vector field E' associated to n' has the following

properties: (1) The group of diffeomorphisms of M which is generated by E'

is a 1-dimensional ccepact Lie group; (2) Each element of this group except the identity does not leave

any point of M invariant.

0 Proof. We denote the period by y and put n' = (1/11)n. Then we

have &' = p&. Thus, F' has the same set of orbits as E. We may easily verify that p'(p) = pC,(p) = 1 for any p c M. Hence, the group of diffeomorphisms of M generated by &' depends only equivalence classes of the real variable t mod 1. Consequently, it is a 1-dimensional compact Lie group. We call this group the circle group and denote it by S1. The assertion (2) is evident.

QED.

If a 2n-dimensional manifold admits a closed exterior 2-form with maximal rank c, i.e., a 2-form 52 such that d0 = 0, (S2n) 0 0,

then we call the manifold a symplectic manifold with fundamental form 9 and the structure given by n the symplectic structure.

THEOREM 6.2. Let M be a (2n+1)-dimensional compact contact manifold with a regular contact form n. Then we have the following: (1) M is a principal circle bundle over B; (2) n defines a connection in this bundle;

(3) The base manifold B is a 2n-dimensional symplectic manifold whose fundamental form 0 is the curvature form of n, i.e., dQ = 7010 is the equation of structure of the connection; (4) S2 determines an integral cocycle on B.

Proof.-By Theorem 6.1 we can modify n so that E generates the circle group S1. (1) Since

is regular we can choose an open covering {U1) of (x1,...,x2n+1)

M such that on Ui we have coordinates

with the integral

curves of C being given by xl = const......x2n = const. Then {n(Ui)) is an open covering of B. We define local cross sections si : n(Ui) si(xl,...,x2n) _ (xl,...,x2n,c) for some l > M withasi = id. by constant c, x ,...,x

being regarded as the coordinates on rr(Ui).

We define the maps fi -: n(Ui) X S1

> M by fi(p,t) = ¢tsi(p)

p c M, 0t being elements of the I -parameter group of diffeomorphisms

generated by E. Then fi are coordinate functions for the bundle. (2) Since we have LEn = O and Ldn = 0, rj and do are invariant

under the action of Si. Now, the Lie algebra 6 of the group S1 is 1-dimensional vector space. So, we identifyrel with the line R, and

283 take A = d/dt as a basis of Cg

Then n = nA is an invariant form on

M which has its values in d1. For the sake of simplicity we write n instead of n again. To prove that in defines a connection we must show that the following two conditions: (a)

n(A*) = A,

where At is the fundamental vector field corresponds to A; (b)

Rtn = (ad t -')n,

where Rt is the right translation by t e S1, Rt is the dual map of Rt on the space of 1-forms on M.

First we have n(A*) = n(A*)A = nQ)A = A. This proves (a). Next, as Rt(p) = 4t(p) and n is invariant under the group S1, we have Rtn = n. Since S1 is abelian, we see that ad(t 1) = 1. Thus we have (b). (3) We denote by S2 the curvature form of the connection n. Then

dn(X,Y) _ - j[n(X),n(Y)] + Q(X,Y).

However, as the group S1 is abelian [n(X),n(Y)] = 0 and hence

do = S2. On the other hand, do is invariant and

0. Therefore there exists a 2-form Q on M such that do = n*St. Now, n*dit = dn*52 = den= 0, so that dSt = 0 and n*(Stn) _ (n*S2)n = (dn)n 0 O'giving Stn 0 0. Therefore M is symplectic. (4) Finally, as the transition functions fib : Ui

Uj

S1

are real (mod 1) valued, one can check that [S2] a H2(M,Z) (see

Kobayashi [1] for details).

QED.

Conversely we now prove the following THEOREM 6.3, bet B be a 2n-dimensional symplectic manifold such that its fundamental form Q determines an integral cohcmology class. Then there is a principal circle bundle it : M

> B and a connec-

tion n on M such that (1) n is a contact form on a (2n+1)--dimensional manifold M, and

(2) its associated vector field E generates the right translations of the structure group S' of the bundle.

Proof. We take A = d/dt as the basis of the Lie algebra of S1.

0 According to a theorem of Kobayashi [1], there exists a principal circle bundle M over B and a connection n on M such that n defined by n = nA satisfies do =Tr*r. Since Tr* is an isomorphism, we have (dn)n = (,1*Q)n = ,ff*O # 0. Therefore, if we denote the fundamental

vector field corresponding to A by A* and 2n linearly independent horizontal vectors by X1,.... X2n, then we have (n n (dn)n)(A*,X1,...,

X2n) 0 0. Thus, n is a contact form of M. Moreover, since n(A*) = A, we have n(A*) = 1. On the other hand, for arbitrary vector X of M we have dn(A*,X) = Tw*Q(A*,X) = Q(TiA*,TrX). Hence, we get d (A*,X) = 0.

Therefore, A* is the vector field associated to n, i.e., A* = C. Hence,

generates the group S1 of right translations of the bundle

W.

M. Let Tr : M

> B be a Boothby-Wang fibration (Theorem 6.2).

Since B carries a global symplectic form Q, there exists a Riemannian metric h and a Tensor field J of type (1,1) such that (J,h) is an almost Kaehlerian structure on M with Q as its fundamental 2-form. Let n be the contact form on M with do = Tr*Q and

vector field. We define a tensor field

OX = (JTr*X)*,

its associated

of type (1,1) on M by

X C Tx(M),

where * denotes the horizontal lift with respect to n. Then

02X = (JTr*(JT*X)*)* _ (J2Tr,'X)* = -(,T*X)* = -X + n(XX.

that is,

2 =-I+n0 C. Therefore,

is an almost contact structure on M. We define a

Riemannian metric g of X by g(X,Y) = h(n*X,TT*Y) + n(X)n(Y),

X,Y E x(M),

290 that is,

g=n*h+n0n Clearly E is a unit Killing vector field with respect to g. Moreover

we have

g(X,cY) = h(n*X,n*(JTr*Y)*) = h(Tr*X,JTr*Y) = S2(n, X,n*Y) = n*S2(X,Y) = dn(X,Y) and similarly

g(gX,OY) = h(n*X,n*Y) = g(X,Y) - n(X)n(Y).

Thus we have (Hatakeyama [1]) THEOREM 6.4, Compact regular contact manifold carries a K-contact structure.

Let N0 and NJ be the Nijenhuis torsions of 0 and J respectively. Since we have LEA = 0, we obtain

N(E,X) + 2dn(E,X)E = 42[E,X] - O[E,OX] = 0.

On the other hand, for projectable horizontal vector fields X and Y, we have

N0(X,Y) + 2dn(X,Y)E _ (J27r*[X,Y])* + [(Jn,X)*,(Jn*Y)*] - (Jn*[(J1T*X)*,Y])*

- (Jn*[X,(Jn*Y)*])* + 2dn(X,Y)E

_ (JZ[n*X,n*Y])* + [Jn+X,JTr*Y]* + n([(Jn*X)*,(Jn*Y)*l) - (J[Jn*X,n*Y])* - (J[Tr,,X,Jn*Y])* + 2dn(X,Y)E

291

= (NJ(n, C,n*Y))* - 2(0(Jn

,Jn*Y))* - (SZ(w*X,n*Y))*)

= (NJ(n*X,Tf*Y))*.

Thus we see that K-contact structure (c,C,n,g) is Sasakian if and only if M is Kaehlerian.

A Hodge manifold is by definition a compact Kaehlerian manifold such that the fundamental 2-form 0 determines an integral cocycle over the manifold. Thus, by the previous considerations, we have (Hatakeyama [1])

THEOREM 6.5, In order that a (2n+1)-dimensional manifold M with a regular contact structure admit a Sasakian structure is that the base manifold B of the Boothby-Wang fibration of M is a Hodge manifold.

7. BRIESKORN MANIFOLDS

Let Cn+1 be the complex vector space of (n+l)-tuples of complex numbers (z 0,zl,...,zn). A Brieskorn manifold is by definition a (2n-1)-

dimensional submanifold

BBn-1

(aO,al,...,an) in Cn+1 defined by equa-

tions

z0 +za,+...+zan=0

(7.1)

and

zOzO + zlzl + ... + znzn = 1,

(7.2)

where a0,al,...,an are positive integers. We denote by B2n

B2n-l(a0,al'".,an)

for simplicity. We also denote by B2n the complex hypersurface

in Cn+l - {0) defined by (7.1). The Brieskorn manifold 52n+1.

intersection of B2n with the unit sphere Let us consider the C-action on Cn+1 defined by

B2n-1

is the

292

mw/atz

j,

z =e

(7.3)

where m is the least carmon multiple of integers a0,al,...pan and w

is a complex variable. We can easily see that the Faction fixes the origin 0 and transforms 92n onto itself. Therefore, restricting w to at s = 0, we see that

its real part s and differentiating

ze Oa

ul = (a z3),

is a tangent vector of B2n at z. Similarly, restricting w to its purely

imaginary part it (t E R), we see that

U2 = iu1 = (a izj),

z E

B2n

J

is a tangent vector of On at z orthogonal to ul. When we restrict w to it, (7.3) gives a S1-action on Cn+1 and the S1-action leaves T3 2n, S2n+1 and so B2n-1. Therefore, if z E B2n-1, the orbit of z under this B2n1 and so u2 is a tangent vector of B2n-1 action lies on

We denote by dz the differential at a point z on On. Then we have

1Z-dz=0 j , J where f(z0,zl,...,zn) is the polynomial on the left hand side of (7.1). This is equivalent to = 0, where the bracket means the inner product of two vectors 3f/3z (the canplex conjugate of of/az) and dz

in C. Thus we have Re = 0

and

Re = 0.

Therefore we see that

a -l v1 = (a-) _ (ajZjJ

a.-1 ),

v2 = (izj) _ (iajzjJ

) = ivl

29 3

are normal vectors of B2n at the point z. We easily see that u1, u2, v1 and v2 are mutually orthogonal.

We now restricted the point z to the one on

B2n-1.

Then the unit

normal vector n of S2n+1 has z. as its caxponents. We see that v1, v2 -1 and n are normals to B2n in Cn+1 and they are linearly independent. We put

v=n+avl+uv2, where a. A = -(Re(Eai zj3)/),

a.

u = Im(Eajzj3)/.

Then v1, v2 and v are normal vectors of B2n-1 in Cn+1 orthogonal with B2n-1 which each other, which shows that v is a normal vector of tangent to B2n at each point z c

B2n-1.

B2n inherits the ccinplex structure from that of Cn+l. If we denote

by << , >> the Kaehlerian inner product, then

<> = Re.

Then we have

n

<> = jiJI (zjdzj - zjdzi-). We define a real 1-form tt on Bin-1 by

Ti = jij(zi dzj - zi dzj).

We notice that n = <> = <>.

Sasaki-Hsu [1] proved that the form n is a contact form on B2n-1. THEOREM 7.1. Every Brieskorn manifold is a contact manifold.

Proof. We shall show that the 1-form n on 4-1 is a contact form.

294

Since

n

do = i j! dz4 A dz j we have

(7.4)

n A

(dn)n-1

= din{I J (zjdzj - -Zjd )} A

dZk A dzk) n-1

n _ i(n-1)!in[{J) (zJ.d =:O

J

- dz.)}

A {E (dz0 A dz0) A ... A (dzj Aid-zj -)

j
where roofs mean factors which should be omitted.

We may first restrict ourselves on the domain Dn on

B2n-1

where

zn # 0. On Dn we have n-1 dzn = - E fpdzp,

A

(7.5)

a .-l fp = tp/tn,

tj = ajziJ

We denote the equation ccriplex conjugate to (7.5) by (7.5). On the

other hand, we have, by (7.2), n I (zjdzj + zjdzj) = 0. j

From this and (7.5) we have n-1 (7.6)

I (m dz +

P=0

where we have put

0,

216

Mp = p - znfp, Since the functions m0,ml,.:.,m

p = zp - zn'p. 1 defined on Dn can not vanish

simultaneously at any point of Dn, we may consider the subdarrin

Dn,n-1 in n such that mn_1 # 0. Then n-1 n-2 d n-1 = -m( I pdzp + Imkdyk)

(7.7)

on Dn

n-1'

where we have put m = 1i

n-1'

Now, if we pay attention to the domain D

n ,n-1

on B

2n-1

(7.4) '

can be written as

(dn)n-1

(7.8)

TI A

= i(n-1)!in(A + B + C),

where A, B and C are (2n-1)-forms defined as follows: A: the sin of monomials each of which contains zkdzk - zkdzk (k = 0,1,...,n-2) as its factor,

B; the sum of monomials each of which contains zn-ldzn-1

zn-ldz n-1 as its factor, and C: the sum of monomials each of which contains zdz

n

n

- z dz

n

n

as

its factor.

We shall compute A, B and C on D

n , n-1.

To simplify the notation,

we put wi = dzi n dzi. (i) Calculation of A: If we fix the value of k, any non-zero monomial in (7.4) which contains zkdzk - zkdzk does not contain wk as its factor. So A can be written as

A=Al+A2+ A3, where A1, A2 and A3 are (2n-1)-forms with the following additional

properties:

296

A1: the star of monomials each of which contains

factor, but does not contain

n as

n-1 as its

its factor,

A2: the scan of monomials each of which contains

n as

its

factor, but does not contain wn-1 as its factor, A3: the stan of monomials each of which contains both wn-1 and

n as

its, factor.

Then we see that

n-2 Al = k-0w0 A ... A wk-1 A (71

_

d Zk-k) A wk+1 n ... A n-1.

Substituting (7.7) into the equation above, we find

n-2

Al= -mI (zkmk+kn )"' k=O

where we have put

S2 = W0 A W1 A ... A wn-2 A dzn-1' Next, we have

n-2

A2 = kI0W0 A ... A wk-1 A (zk k zkdzk) n wk+1 A ... A %-2 A wn. Substituting (7.5) and (7.5) into the equation above, we obtain

n-2 A2

kI (Zkfkfn-lwo A ... A wn-2 n din-1 +

zkfn-lfn-iw0 n ... A wk-1

+

zkfn-lfn-lw0 A ...

A dZk n wk+1 n ... A wn-1

A wn-2 A dzn-1

zkfn-lf n-lw0 n ... AWk 1 n dzk A wk+1 n ... A wn-1). Using (7.7), A2 reduces to

n-2

A2 = mI{zkfn-l(fkmn-1 - fn-1mk) +

fn-1km-1 n(f

zk

-

fn-lmk ))S2,

We next ccmpute A3. We set

A3

A3 + A3,

where we have put

n-2 n-2 A3

Jo j=k+l O n ... A Wk-l A (zk A Ik+l A ... A^WA ...

2k

)

wn-2

wn-1,\ wn,

n-2 k-1

A3

kI hI O A ... n wh n

... A wk-1

A(zkdzk zkdzk) A Wk+l A ... A wn-2

A n-1 A n'

Substituting (7.5) and (7.5) into A3 and using (7.7), we find

n-2 n-2 A3

3-k+l{zkfj(fk j-fjmk) + zkfj(fkmj-tnkfj))S2. mk--0

Similarly, we have

n-2 n-2

_ +l{zkfh(fkmh-fhmk) + zkfk(fkmh-fhmk))S2.

= mho k Changing indices h and k to k and j respectively, we have

n-2 n-2 A3 = in 3 k+i{zjk(fjn)

fkmj) + zjfk(fjmk-fkmj))S2.

k=O

consequently, we have

n-2 n-2 A3 = m E

E

k=O j=k+l

{(fk fjmk)(zIjj

Jfk)

+ (fk j-mkfj)(zkf j-zjfk))S2.

298 (ii) Calculation of B: Clearly B can be written as

B=B1+B2, where B1 and B2 are (2n-1)-forms with the following additional properties:

B1: the monomial which contains zn-1dzn-1

-

zn-idzn-1 as its

factor, but does not contain wn as its factor, B2: the sum of monomials each of which contains both of zn-ldzn-1 zn-ldzn-1 and n as its factors.

-

Substituting (7.7) into the equation

B1 = w0 n WI n ... n

wn-2 A (zn-ldzn-1 Zn-idzn-1

we obtain

B1 = -m(zn-1mn-1 +

zn-lmn-1)Q.

Moreover, we have n-2

B2 =

I W0 n ... A k A ... A n-2 A (zn-1 dzn-1 in-idzn-1) A

n.

Thus (7.5), (7.5) and (7.7) imply n-2

B2 = mI{zn-1fk(fn-lc-fkmn-1) + zn-lfk(fn-lmk-fkmn-1)}SZ.

(iii) Calculation of C: C can be written as

C=C1+C2, where C1 and C2 are (2n-1)-forms with the following additional properties:

C1: the monomial which contains zndzn - zdzn as its factor, but does not contain

n-1'

299 C2: the star of monanials each of which contains both of zndzn

ndzn and n-1 as its factors. First, we see that

C1 = WO A ... A wn-2 A (zndzn zndzn).

Then, by (7.5), (7.5) and (7.7), we find

C1 = m(znfn-1

n-1 +

znfn-lmn-1)Q.

Similarly, fran the equation

n-2 C2 =

1 i 0 A ... A Wk 4 ... A wn-2 A n-1 A (zndzn zndzn)

k

we obtain n-2 C2 = m E (znfkmk + znfkmk) k=O

We now define a function F on D ,n-1 by

n

(dn

)n-1

=

(n-1) ! inFil .

Then we have

Fat=A+B+C=(A1+B1)+ (C1+C2)+{A3+(A2+B2)}. To show n

(dn)n-1 # 0, it is sufficient to show that F # 0.

We have the following equations

Al+B1

n-1 P=O

(pp+ PP)Q,

n-1

C1 + C2 = m(zn P=O

Moreover, we have

n-1 _ Page + zn I fPrtP)S2 P=O

300 n-2 A2 + B2 = mk-0{(:'On-1

a-lmk)(zkfn-1 n-lfk) fn-lmk)(kfn-1 zn-1fk)}0'

from which

n-2 n-1 A 3 + (A2 + H2) = m

k=OJ11

{(fk j-fjmk)(zkfj-zjfk)

+ (f1&nkfj)(zkfj-zjfk)}S2. Consequently, we obtain

n-1

n-1

n-1

n-1

iF = p=zpzp + p=zpfpzn +zpfpzn - p=fpfpznzn P=O n-2 n-1 (zkfj-zjfk)(kfj-zjfk).

k=O j=+1 Thus we have n-1 n-1 n-1 itntnF = -pI Itnzp12 ItpznI2 + 2 ) Re((tnp)(tpzn))

n-2 n-1 Izktj-zjk12

k j=+1 E

n-1

_ -I{Re(tn p)-Re(tpzn)}2 -

n-1 pI{Im(tnzp)+II(tpzn)}2

P=I

n-2

n-1

- I

k=O j=k+1

nit.-z.tk J 12.

Thus, we have F < 0 on Dn,n-1' If F = 0, then

Re(tnp) = Re(tpzn),

for p = 0,1,...,n-1, and

Zktj = zjfk

Im(tnp) = -Im(tpzn)

3M for k = 0,1,...,n-2 and j = k+1,...,n-1. Then we have a contradiction. (dn)n-1 #

0 on D

Therefore, F < 0 and hence n A

,n-1'

Quite similar argument can be performed for other domains Dn (dn)n-1

(k = 0,1,...,n-2) similarly defined as Dn,n-1. Thus n A (dn)n-1 #

on D . In the same way, we can prove that n A

k

00

0 for domains

DO,D1,..., n-1. Consequently, n is a contact form on B2n-1.

QED.

In the sequel, following Takao Takahashi [1], we shall show that the Brieskorn manifold has a Sasakian structure. First of all, we prove

the following THEOREM 7.2. Let M be a Sasakian manifold with structure tensors (q,E,n,g) and u be a vector field on M which satisfies the following three conditions: (a) Lug = 0,

(b) [u,F] = 0,

Define new structure tensors

(c) 1 + n(u) > 0.

n,g) by

n = (1 + n(u))-l

ix = (x - n(x)D,

,

Z =

+ u,

g(X,Y) = (1 + n(u)) lg(X-n(X)LY-n(Y)D + n(X)n(Y) for any vector fields X and Y on M. Then

is a Sasakian

structure on M.

Proof. Let D be the distribution defined by n = 0. By the definition of n we see that D is also defined by n. From the definition of structure tensors we have

1 and g(,Z) = 1. For any vector field

X of D we have n(X) = 0,

0 and 42X = -X, hence

2X =

(4X) = O(Ox n(OX)Z) = 42X = -X.

Because f-C = 0 and the equation above holds for any vector field of

D, we conclude that

i2 =-I+n® 1.

302

Let X and Y be vector fields which belong to D. Then we see that

OX = X, Y = jY and n(X) = n(Y) = 0. Thus we have

N(X,Y) + 2dn(X,Y) = $2[x,y] + [cX,4Y] - $[OX,Y] - [X,cY] -

([X,Y])&

= Nc(X,Y) + q(n([ X,Y])& + n([X, Y])&)

= (1+n(u))-1(n([4X,Y]) + (1+n(u))

1N(2)(Y'X)$E = 0,

by L m a 3.1. Moreover, we easily see that

0.

Consequently, we obtain

N + 2dn 0

0.

Next, we prove that dn(X,Y) =

If X is a vector field in D,

then

0.

If X and Y are vector fields in D, then

2dn(X,Y) = -n([X,Y]) _ -(1+n(u))n([X,Y])

= -2(1+n(u))g(OX,Y) = 29(X,$Y).

Therefore, we obtain dn(X,Y) = a Sasakian structure on M.

Consequently,

ng) is QED.

Example 7.1. Let Stn+1 be the unit sphere with standard Sasakian 1 xn+1 n+1 ) in R2n+2 with coordinates (xl n

structure ($

At a point p on+1 we put = nIl(x3By3-Y-ax3),

nIl(x3dy3-y3dx3),

j=l

j=1

where vie have put ax3 = a/ax3, ayj = a/ay3.

is defined to be the

restriction of the almost complex structure J of R2n+2 which is the orthogonal complement of in the tangent space of Stn+1 and 0 on The Riemannian metric g on Stn+1 is induced from that of

R2n+2.

We put

U =

I rj(xJayl-yJaxJ), j=1

where (rj) is a (n+l)-tuple of real numbers such that

1 +

n+1 I r.((xJ)2 + (yJ)2) > 0 j=1

52n+1.

on

Then u satisfies the conditions of Theorem 7.2. The new

trajectriy of Z with the initial condition p is given by

x3(t) = x3cos(1+rj)t - yisin(1+r)t,

yy(t) = x3sin(1+rj)t + yJcos(1+r)t

for j = 1,2,...,n+1.

THEORE14 7,3, Every Brieskorn manifold admits many Sasakian

structures. Proof. We define a mapping F of Cn+1 onto C by

F(z0,zl,...,zn) = (7O)2 + (z1)2 + ... + (zn)2.

Then the Brieskorn manifold

B2n-1

is given by On-" = Stn+1 A Fr-1(0) 52n+1.

and B2n-1 is a submanifold in

Let xO,yO,...,xn,yn be the real

coordinates of Cn+1 such that zj = xj + iyj (j = 0,1,...,n). We define a real vector field Z on C°

by

304

n = JI AJ(xJ-2yJ

is

where A. = (a ) 1A for a positive constant A. (j = 0,1,...,n). +1.

tangent to

We put .t =

- E. Then N satisfies the conditions

of Theorem 7.2 because of 1 + n(u) = we have a Sasakian structure

0 (see ale 7.1). Then on Stn+1 with respect to V. The

explicit formula of n is given by

n

J E

where K = Aj((xj)2+(yj)2). Let P and Q be real valued functions on Cn+1 defined by

P(x0,Y0,...,xn,Yn) = the real part of F(z0,...,zn), Q(x0,Y0,...,xn,Yn) = the imaginary part of F(z0,...,zn).

Then it is easy to see that P = -AQ and Q = AP. These show that the restriction of of

0,

to

B2n-l

is tangent to B2n-1. Since the restrictions

and J to D are the same mapping and F 1(0) - {0} is a coaplex

submanifold of Cn+1, for any vector X tangent to B2n-1 which belongs to D on

Stn+l,

we get

(JX)(F) = iX(F) = 0

because of the Cauchy-Riemann equations for the ccnplex analytic function F. Since we have the orthogonal decomposition, with respect B2n-1 to the induced metric on

Tp(B2n-1) = (Dp + <E >) ' Tp(B2n-1) = (Dp it Tp(B2n-1))+< fp where <EP> is the 1-dimensional space generated by C at p. Since

¢ = 0, we see that

maps Tp(B2n-1) into itself. Therefore,

B2n-1

is

MG an invariant suhmanifold of Stn+1 with respect to

Any

invariant suLmanifold of a Sasakian manifold is also a Sasakian manifold (see §1 of Chapter VI). Consequently, B2n-1 admits a Sasakian

structure.

QED.

306

EXERCISES

A. K-CONTACT RIEMANNIAN MANIFOLDS: Let M be a K-contact Riemannian

manifold with structure tensors

We denote by R and S the

Riemannian curvature tensor and the Ricci tensor of M respectively. Then we have (Tanno [1]) THEOREM. Let M be a K-contact Riemannian manifold. (1) If

0 for any X, then M is an Einstein manifold;

(2) If R(X,F)R = 0 for any X, then M is of constant curvature. B. IMMERSED K-CONTACT RIEMANNIAN MANIFOLDS: Takahashi-Tanno [1] studied K-contact Riemannian manifolds immersed in a space form as hypersurfaces and proved the following theorems. THEOREM 1. If an n-dimensional K-contact Riemannian manifold M is isometrically immersed in an (n+1)-dimensional space form M, then M is a Sasakian manifold.

This theorem gives a sufficient condition for a K-contact Riema-

nnian manifold to be Sasakian. THEOREM 2. Let M be an n-dimensional K-contact Riemannian manifold isometrically immersed in an (n+l)-dimensional space form M of constant curvature 1. Then (1) the type number t < 2, and (2) M is of constant curvature 1 if and only if the scalar curvature r = n(n-1).

THEOREM 3. Let M be an n-dimensional K-contact Riemannian manifold isometrically immersed in an (n+l)-dimensional space form M of constant curvature c # 1. Then c < 1 and M is of constant curvature 1. C. PRODUCT OF ALMOST CONTACT MANIFOLDS: bet M and M be manifolds of dimension 2n+1 and 2m+1 and let E =

and E =

be

almost contact structures on M and M, respectively. We define J by

J(X,X) _ (% Then, it is easily seen that Jz = -I, which shows that J is an almost complex structure on M x M. We call J the induced almost complex struc-

ture onMxMbyEandE. If the induced almost complex structure J on M x M by E is integ-

rable (i.e., complex structure), we call E is integrable. Morimoto [1] proved the following theorems.

THEOREM 1. The induced almost complex structure on M x M is integrable if and only if z and E are both integrable.

THEOREM 2. An almost contact structure E is integrable if and only if E is normal.

Since an odd dimensional sphere is normal, we obtain (CalabiEckmann [1]) THEOREM 3. The direct product of two spheres of odd dimension

has a complex structure. D. TRIPLE OF KILLING VECTORS: Assume that a Riemannian manifold M admits three unit Killing vectors F, n and S which are mutually orthogonal and satisfy

F;

=

[n10 ,

n =

[;,F],

; = #R,n]

We call such a set {F,ri,l} a triple of Killing vectors. A triple {F,n,c} of Killing vectors is called a K-contact 3-structure, if F,

n and r are K-contact structures. A K-contact 3-structure is called a Sasakian 3-structure if all of F, n and c are Sasakian structures.

If, for a K-contact 3-structure

any two of C. n and L are

Sasakian structures, then {F,r,r} is necessarily a Sasakian 3-structure (see Kuo [1], Kuo-Tachibana [1], Tachibana-Yu [1]). Kashiwada [1]

proved

0 THEOREM. A Riemannian manifold with Sasakian 3-structure is an Einstein manifold. E. SASAKIAN +-SVhMETRIC MANIFOLDS: In view Proposition 5.2 a symmetric manifold condition is too strong for a Sasakian manifold.

Toshio Takahashi [1] introduced the notion of Sasakian 0-symmetric manifolds. Let M be a Sasakian manifold with structure tensors (0,g,7j,g).

We denote by R the Riemannian curvature tensor of M. A Sasakian manifold M is said to be a locally 4-symmetric manifold if

2[(OVR)(X,Y)Z] = 0

for-any horizontal vectors X, Y, Z and V, where a horizontal vector means that it is horizontal with respect to the connection form n of the local fibering; namely a horizontal vector is nothing but a vector which is orthogonal to C. Takahashi [1] proved the following

THEOREM 1, A Sasakian manifold is a locally -symmetric manifold if and only if each Kaehlerian manifold, which is a base space of local fibering, is a Hermitian locally symmetric manifold.

A complete and simply connected Sasakian locally 0-symmetric manifold is called a globally 4-symmetric manifold.

THEOREM 2. A Sasakian globally 4-symmetric manifold is a principal G1-bundle over a Hermitian globally symmetric space, where G1 is a

1-dimensional Lie group which is isomorphic to the 1-parameter group of global transformations generated by E.

F. CONTACT BOCHNER CURVATURE TENSOR: Let M be a (2n+1)-dimensional Sasakian manifold with structure tensors

We denote by R, Q

and r the Riemannian curvature tensor, the Ricci operator and the scalr curvature of M respectively. Then the contact Bochner curvature

tensor B of M is defined by (Matsumoto-C ann [1])

309 B(X,Y) = R(X,Y) + t1 {QY A X - QX A Y + QTY A OX - Q¢X A ¢Y

+ 2g(QQX,Y)4 + 2g(4X,Y)% + n(Y)gX A F + n(X)& A 4Y)

AY A X

- m+4 Y A OX + m+4 n(Y)FA X + n(X)YA F),

where we have put m = 2n, k = (r+m)/(m+2) and (X A Y)Z = g(Y,Z)X g(X,Z)Y. Then we have

[1])

THEOREM 1. Let M be a Sasakian manifold with vanishing contact Bochner curvature tensor. If M is an n-Einstein manifold, then M is of constant 4-sectional curvature. We define the contact Ricci tensor L of M by setting

L(X,Y) = g(QX,Y) + 2g(X,Y) - 2(n+1)n(X)n(Y).

Then we have (Ikawa-Kon [1]) THEOREM 2. Let M be a (2n+1)-dimensional Sasakian manifold with constant scalar curvature and vanishing Bochner curvature tensor. If the contact Ricci tensor of M is positive semi-definite or negative semi-definite, then M is of constant $-sectional curvature. G. FIBRATION OF COMPACT NORMAL. ALMOST CONTACT MANIFOLDS: Similar

to the Boothby Wang fibration of compact regular contact manifolds, we can construct a fibration of compact normal almost contact manifolds with F regular (see Hatakeyama [1], Morimoto [21). Conversely we have the following (Morimoto [2]) THEOREM. Let M be a (2n+1)-dimensional compact normal almost contact manifold with structure tensors

and suppose that r; is

a regular vector field. Then M is the bundle space of a principal

circle bundle n : M

> B over a real 2n-dimensional almost complex

manifold B. Moreover n is a connection form and 2-form '' on M such that do = n*'Y is of bidegree (1,1).

310 H. BRIESKORN MANIFOLDS: Let B2n-1 be the Brieskorn manifold with contact form n = jiE(z.dz.-z.dz.). We denote by g the metric tensor J J J J B2n-1 field of induced from the natural metric of Then n is Cn+l.

given by n(X) = g(Jv,X), X e X(B2-1), where v = n + av1 + uv2, and Cn+1.

J is an almost complex structure of

We put N = v/Ivi and

n' = n/IvI. Kashiwada [2] proved the following THEOREM. For a Brieskorn manifold B2n-1(a0,a1,...,an) the following conditions are equivalent:

(1) The induced structure (2)

B2n-1

of B2n-1 is Sasakian;

is n-umbilical in the Kaehlerian manifold

On;

(3)a0=a1=...=an; (4) B2n-1 is umbilical in the Kaehlerian manifold B2n.

311

CHAPTER VI

SUBMANIFOLDS OF SASAKIAN MANIFOLDS

In this chapter, we study submanifolds of Sasakian manifolds, especially those of Sasakian space forms. In §1, we study invariant submanifolds of Sasakian manifolds. We compute the Laplacian for the second fundamental form and the Ricci tensor of an invariant submanifold of a Sasakian space form, and give some integral formulas. As applications of these integral formulas we prove the classification theorem of invariant submanifolds of codimension 2 of Sasakian space forms with n-parallel Ricci tensor. We also study invariant submanifolds of codimension 2 with constant scalar curvature. In §2, we discuss anti-invariant submanifolds, tangent to the structure vector field, of Sasakian manifolds. We give sane examples of anti-invariant suhnanifolds tangent to the structure vector field of Sasakian space forms and prove some theorems which characterize these examples. In §3, we study submanifolds of Sasakian manifolds which are normal to the structure vector field. Then the submanifolds are anti-invariant submanifolds. §4 is devoted to the study of contact CR submanifolds of Sasakian manifolds. We construct some examples of contact CR submanifolds of a unit sphere, which have parallel mean curvature vector and flat normal connection. We also prove some theorems concerning these examples. In the last §5, we discuss the structure induced on

sutmanifolds.

312

1. INVARIANT SUBMANIFOLDS OF SASAKIAN MANIFOLDS

Let M be a (2m+1)-dimensional Sasakian manifold with structure tensors ($,E,n,g). A (2n+1)-dimensional silhnanifold M of I is said to

be invariant if the structure vector field E is tangent to M everywhere

on M and g is tangent to M for any vector X tangent to M a every point of M, that is, gTx(M) c Tx(M) for all x e M. We easily see that any invariant submanifold M with induced structure tensors, which will

as M, is also a Sasakian mani-

be denoted by the same letters fold.

Let V (resp. V) be the operator of covariant differentiation with respect to the Levi-Civita connection of rd (resp. M). Since

is tan-

gent to M, for any vector field X tangent to M, we have -$X = OX _ 0

+ B(X,E). Thus we obtain B(X,E) = 0. For any vector fields X and Y

tangent to M we obtain OX,Y = V Y + B(X,W = (OXO)Y + JOXY + B(X,OY) = g(X,Y)E - n(Y)X + OVXY + B(X,$Y),

vx¢Y = (OX4)Y + ¢OXY = g(X,Y) - n(Y)X + BOXY + OB(X,Y).

From these equations we have ¢B(X,Y) = B(X,4Y). Since B is symmetric,

we also have B(X,Y) = B(OX,Y). Consequently, we have the following lemma for the second fundamental form B of M, or equivalently, for the associated second fundamental form A of M.

LEJT A 1,1, Let M be an invariant submanifold of a Sasakian manifold M. Then A

(1.1)

B(X,C) = 0,

(1.2)

B(X,cY) = B(cX,Y) = OB(X,Y),

(1.3)

cAOX = -AOX = AO X.

= 0,

313

PROPOSITION 1.1. Any invariant sutmanifold M of a Sasakian manifold M is a minimal sutmanifold. Proof.

Let el,...en4el,...,0en,E be a -basis of Tt(M). By (1.1)

and (1.2) we see that

I[B(ei,ei) + B(gei,¢ei)] +

0

because of B(Oei, ei) = 02B(ei,ei) _ -B(ei,ei). This means that M is a minimal sutmanifold of M.

QED.

PROPOSITION 1.2. If the second fundamental form of an invariant

suimanifold M of a Sasakian manifold M is parallel, then M is totally geodesic.

Proof. By the assumption on B we have

0 = (OXB)(Y,E) = B(Y,cX) _ OB(X,Y),

from which

B(X,Y) = -¢2B(X,Y) - n(B(X,Y))C = 0.

Therefore, M is totally geodesic.

QED.

Let R (resp. R) be the Riemannian curvature tensor of M (resp. M). From equation of Gauss and (1.2) we have

(1.4)

g(R(X,¢X)QX,X) =

2g(B(X,X),B(X,X)).

Thus we have the following PROPOSITION 1.3, bet M be an invariant submanifold of a Sasakian space form M(c). Then M is totally geodesic if and only if M is of

constant -sectional curvature c. We now suppose that the ambient manifold M is a Sasakian space form with constant 4-sectional curvature c. Then we have

314 (1.5)

R(X,Y)Z = *(c+3)[g(Y,Z)X-g(X,Z)Y] + j(c-1)[n(X)n(Z)Y

-n(Y)n(Z)X+g(X,Z)n(Y)g-g(Y,Z)n(X)t+g(oY,Z)+X

AB(Y,Z)X - AB(X,Z)Y. (1.6)

(VXB)(Y,Z) - (V B)(X,Z) = 0.

We denote by S and r the Rioci tensor and the scalar curvature of M respectively. Then, from (1.5) we have

(1.7)

S(X,Y) = i(n(c+3)+(c-1))g(X,Y) - l(n+l)(c-l)n(X)n(Y) - lg(B(X,ei),B(Y,ei)), i

(1.8)

r = n2(c+3)+n(c+1) -

E g(B(eiej ),B(ei,ei)),

i,J where {ei} is an orthonormal basis of M. In view of Proposition 1.2, the notion of parallel second funda-

mental form of an invariant submanifold of a Sasakian manifold is not essential. So, we need the following. Definition. Let M be an invariant sutmanifold of a Sasakian manifold M. The second fundamental form B of M is said to be n-pcrallel if

0 for all vector fields X, Y and Z tangent to M. It will be proved that the notion of n-parallel second fundamental

form corresponds to that of parallel second fundamental form for invari-

ant submanifolds of Kaehlerian manifolds. We easily see that the second fundamental form B is n-parallel if and only if

(1.9)

(OXB)(Y,Z) = n(X)gB(Y,Z) + n(Y)¢B(X,Z) + n(Z)¢B(X,Y).

315

mw 1,2, Let M be an invariant sulmanifold of a Sasakian manifold M. Then the second fundamental form of M is n-parallel if and only

if IVAI2 = 31A12. Proof. For the proof of this we take a cu-basis e1,...,e2n+1

(en+t = Oet, e2n+1 = E) We also take a basis vl,...,v2p of the normal space of M, where 2p = codim M. Then we have

A)aC) = g(Aaei,Aaei),

g((De i

i

g((oe1A)a4ej,(VelA)aOe.)

= g(ei,Aaej)2 + g(O(oelA)aejO(VelA)aej),

where A. = Av

(a = 1,...,2p). These equations imply

a

2n+1 E

g((o A)e.,(D A)ej)

a=1 i,j=1 2n

I a=1 I a=1

I

i

g((Ve A)a$ej(Ve

i,j=1 2n I

i

21A12 i

i

g(O(De A)aejO(Ve A)aej

i,j=1

i

+ 31A12.

i

Therefore we have 2n (1.10)

IVA12 - 31A12

=

I

a=1 i,3=1

g(O(0 A)a eO(o A) ae) > 0, ei ei

which proves our asertion.

QED.

LE41A 1.3. If the second fundamental form of an invariant sutrmnifold M of a Sasakian manifold M is n-parallel, then the square of the length of the second fundamental form of M is constant, that is, IA 12

is constant. Proof. Let {ei} be an orthonoimal basis of M. Then we have the following equation.

316 VXJAl 2 = vX[iIJg(B(ei,ee),B(ei,ee))] = 2i1 g((v33)(0i,eJ),B(ei,eJ))

=2

[n(X)g( B(ei,ei 1, +n(e.)g(4B(X,ei),13(t.i,t.J)) )

= 2Z[g(gB(X,ei),B(E,ei)) + g(pB(X,ei),B(,ei))] = 0. 1

Therefore, JA12 is a constant.

QED.

In the sequel we give an integral formula, so-called the Simons' type formula, for an invariant sulznanifold of a Sasakian space form

by computing the Laplacian of the square of the length of the second fundamental form.

Let M be a (2n+1)-dimensional invariant sukmanifold of a Sasakian space form M¢1(c). Hereafter we put p = m-n. We take air orthonormal basis el,...,e2n+1 of M such that en+t = cet (t = 1,...,n), e2n+1 = ' We take also an orthonormal basis v1,...,v2p of the normal space of M such that vp+s = Ovs (s = 1,...,p). In view of Proposition 3.1 of Chapter II, by a straightforward

computation, we have

(1.11)

-g(V2A,A) - 31A12 =

(TrAaAb)2 +

I[a,A,]12 a,b=l

a,b=1

- J(n+2)(c+3)JA12.

LE" 1.4, Let M be a (2n+1)-dimensional invariant submanifold of a (2m+1)-dimensional Sasakian manifold M. Then we have

(1.12)

1JAI4

I

<

(Tr Ab)2 < JJA14,

a, b=1

(1.13)

2

'JAl4

I[Aa,a]I2 < JAI4

<

a,b=l

317

If M is a Sasakian space form, then M is TI-Einstein if and only if EI[Aa,Ab]1 2 = IAl4/n.

2p

I A. Then we have

Proof. We put A* _

a=1 2p l l[Aa,Ab]12 = 2Tr(A*)2.

a

Since A* is synmetric, positive semi-definite and ¢A* = A*¢, choosing a suitable basis, A* is represented by a matrix form

x1

0

A* =

An+t = at,

,

at > 0.

0 A2n J Then we have n 2Tr(A*)2 = JAI 4 - 4 F a a t s t#s 2n n (ai - A )2. 2Tr(A*)2 = nIA14 +

n

i> j

From these equations we obtain (1.13). If EI[Aa,Ab]12 = IA14/n, then we have Ai = aj for all i, j. Fran this and (1.7) we easily see that M is n-Einstein if and only if the equality above holds. In the-next place, choosing a suitable basis {va} we have

I

(TtA Ab)2 =

a,b=1

(TrA2)2.

a

a=1

a

From this we obtain

I

(TSrAaAb) 2= j JA J4- I TrA$TYAb,

a,b=1

aft (TrA2)2 >

(TIA Ab) 2 =

a,b=1

a

a=1

a

-

-(

TrA2) = 1 IAI4. 2P a

a=1,

318

Therefore we have (1.12).

QED.

From (1.11) and lemma 1.4 we have

(1.14)

-g(v2A,A) - 31A12 < 1[31AI2 - (n+2)(c+3)]IA12.

Thus we have the following theorems (Kon [4], (81). THEOREM 1.1, Let M be a (2n+1)-dimensional compact invariant

sutmanifold of a Sasakian space form M21(c). Then

0 < fM(IvAl2 - 3IAI2)*1 < fM[31A12 - (n+2)(c+3)]IAI2#1.

THEOREM 1.2, Let M be a (2n+1)-dimensional compact invariant

submanifold of a Sasakian space form bl(c). Then either M is totally geodesic, or IAI2 = (n+2)(c+3)/3, or at some point x of Mt IAI2(x) > (n+2)(c+3)/3.

THEOREM 1.3. Let M be a (2n+1)-dimensional n-Einstein invariant submanifold of a Sasakian space form NI

1(c). Then

0 < IVAI2 - 31A12 < j(n+2)[!lAl2 - (c+3)]IAI2.

n

Proof. If M is an n-Einstein manifold, then the scalar curvature r of M is constant. Thus, by (1.8), IAI2 is also constant and hence -g(V2A,A) = IVAI2. Therefore, (1.11) and Lemma 1.4 imply our inequality. QED.

THEOREM 1,4, Let M be a (2n+1)-dimensional n-Einstein invariant

submanifold of a Sasakian space form h1'(c). Then either M is totally geodesic, or IAI2 > n(c+3). If an invariant submanifold M is of constant 4-sectional curvature k, by using the Gauss equation, we have

(1.15)

n(n+1

1 (TrAaAb)2 = (c-k)IAI2 = a,b

l)

IAI4.

319

Therefore we have THEOREM 1.5. Let M be a (2n+1)-dimensional invariant submanifold 22m+1 (c). If M is of constant 4>-sectional of a Sasakian space form curvature k, then

0 < IVA12 - 31A12 = n(n+l)(n+2)(c-k)(>(c+3)-(k+3)).

THEOREM 1.6. Let M be a (2n+1)-dimensional invariant submanifold of a Sasakian space form M

1(c). If M is of constant 4>-sectional

curvature k, and if c > -3, then either M is totally geodesic, that is, c = k, or (c+3) > 2(k+3).

THEOREM 1.7. Let M be a (2n+1)-dimensional invariant submanifold 22m+1(c) with constant 4>-sectional curvature

of a Sasakian space form

k. If the second fundamental form of M is n-parallel, then M is totally geodesic, or (c+3) = 2(k+3), the latter case arising only when c > -3. From (1.12) and (1.15) we have

THEOREM 1.8. Let M be a (2n+1)-dimensional invariant submanifold

of a Sasakian space form M1(c) with constant 4>-sectional curvature k. If p < n(n.+l)/2, then M is totally geodesic, where p = m--n.

In the next place, we consider a (2n+1)-dimensional invariant

subn=ifold M of a Sasakian space form i(c) with respect to the Ricci tensor S of M. First of all we ccinpute the Laplacian for the Ricci operator Q of M (Kon [8]). We put

T=Q-al-bn0E, a, b being constant and a+b = 2n, (2n+l)a+b = r. We notice that T = 0 if and only if M is n-Einstein.

LEMMA 1.5. Let M be a (2n+1)-dimensional invariant submanifold of a Sasakian space form M2m+1(c). If the scalar curvature r of M is

constant, then

320

-g(V2Q,Q) - 21QJ2 - 16n3 - 8n2 + 8nr

_ -n(c+3)ITI2 + I I[Q,Aa]I2. a=1 Proof. Fran Lemur. 5.5 of Chapter V we obtain

2n+1 (V2S)(X,Y) _

(VeV S)(X,Y) i=1

i

i

= 1[(R(ei,X)S)(ei,Y) + (R(ei,$Y)S)(ei,$X)] - 4S(X,Y) + 8ng(X,Y) + (3r-12n2-6n)n(X)n(Y).

On the other hand, by equation of Gauss, we have

l(R(ei,X)S)(ei,Y) = -F[S(R(ei,X)ei,Y) + S(ei,R(ei,X)Y)

+ S(AB(X,ei)ei,Y) - S(AB(ei,Y)X,ei) +

S(AB(X,Y)ei,ei)].

Moreover, we have 2n+1

-IS(AB(X,Y)ei,ei) _

[g(AaAB(X,Y)Aaei,ei) a=1 i=1

+

i,ei)] = 0,

[S(AB(ei,Y)X,ei) - S(AB(X,eei,Y)]

_ l[g(AQAa,Y) - g(QAaAaX,Y)] a

From these equations and Theorem5.4 of Chapter V we obtain

(R(ei,X)S)(ei,Y) = 1(n(c+3)+2)S(X,Y) - *(c+3)rg(X,Y) + jn(c-i)g(X,Y) - Jn(2n+1)(c-1)n(X)n(Y) + *(c-1)ri(X)n(Y)

+ )[g(AaQAaX,Y) - g(QAAaX,Y)]. a

321

Similarly we have

i(n(c+3)+2)S(X,Y) - *(c+3)rg(X,Y) + Jn(c-l)g(X,Y) - n(n(c+3)+2)n(X)n(Y) + *(c+3)rn(X)n(Y) - jn(c-l)n(X)n(Y) + J[g(AaQAaX,Y) - g(QAaAa ,Y)]. a Fran these equations we have our assertion.

QED.

If the Ricci tensor S of a Sasakian manifold M satisfies (VXS)(gY,gZ) = 0 for all vector fields X, Y and Z on M, then we say that the"Ricci tensor S of M is n-parallel.

We now prepare the following lemras.

LEMMA 1,6. Let M be a Sasakian manifold with n-parallel Ricci tensor. Then we have the following: (1) The scalar curvature r of -M is constant;

(2) The square of the length of the Ricci operator Q of M is

constant, that is, IQ12 = constant. Proof. The Ricci tensor S of M is n-parallel if and only if

(VXS)(Y,Z) = 2n[g(X,Y)n(Z) + g(¢X,Z)n(Y)]

+ n(Y)S(X4Z) +

Thus we have VXr = E(VX.S)(ei,ei) = 0, which shows that r is constant.

Moreover, since g(QX,Y) = S(X,Y), we obtain VXIQ12 = 2Eg((VXQ)ei'Qei)

= 0 and hence IQ12 is constant.

QED.

LEMMA 1.7, The Ricci tensor S of a (2n+1)-dimensional Sasakian manifold M is n-parallel if and only if the following equation is satisfied:

IVQ12 - 21Q12 - 16n3 - 8n2 + 8nr = 0.

Proof. By using a 4-basis el,...e2,+1

(en+t = 4et, e2n+1

we have

2n+1 IVQI2 =

g((V Q)e.,(V Q)e.)

E ei J i,j=1 J ei 2n+1 2n 2n+1 £ g((Ve Q)eJ.,(V Q)e) + I E g((Ve Q)F,(Ve Q)F) J i=1 j=1 i i i=1 ei i 2n+1 2n+1 2n JI g((Ve Q)F,(e1Q)E) =

2n+1 2n = 2IQI2 + 16n3 + 8n2 - Bar +

I g($(0 Q)e,¢(V Q)e.).

E

j=1

i=1

ei

J

ei

J

On the other hand, we can see easily that the Ricci tensor S of M is n-parallel if and only if ui

Q)ej,4(De Q)ej) = 0. Thus we have 1

our assertion.

1

QED.

Here we notice that

ITI2 = IQI2 - r2/2n + 2r - 4n2 - 2n.

Fran this we also have

fI[Aa,Ab]I2 = 2Tr(A*)2 =

nIAI4

+ 2ITI2.

PROPOSITION 1.4. Let M be a (2n+1)-dimensional invariant su mani22m+1(c) (c < -3). If the Ricci tensor

fold of a Sasakian space form

S of M is n-parallel, then M is ri-Einstein.

Proof. From Lemma 1.6, IQI2 is constant. Thus we have -g(V2Q,Q)

IVQI2 Therefore Lemmas 1.5 and 1.7 imply

-n (c+3) ITI2 + 11 [Q,Aa] 12 = 0. Since c < -3, we have ITI2 = 0 and hence T = 0, which shows that M is n-Einstein.

QED.

PROPOSITION 1.5. Let M be a (2n+1)-dimensional compact invariant submanifold of a Sasakian space form

Vm+1

(c > -3) with constant

scalar curvature. If QAa = AaQ (a = 1,...,p), then M is n-Einstein. Proof. By the assumption and lenma. 1.5 we easily see that 1T12= 0.

Therefore M is n-Einstein.

hp

QED.

an invariant submanifold of codimension 2 of be a Sasakian space form Vn+3(c). Then the Ricci tensor S of M is 1,8. let M

n-parallel if and only if the second fundamental form of M is n-parallel.

Proof. Suppose that the second fundamental form of M is n-parallel. Then it is easy to see that S is n-parallel. Conversely, we assurne that S is n-parallel. We shall prove that

B is n-parallel. To prove this it sufficies to show that, at each point x of M,

O(VoXA)OvoY = 0

for any X, Y e TT(M), where {v,cv} is an orthonormal basis of the

normal space of M. There exists a 1-form s such that DXv = s(X)O.

From (1.7) we have ¢(VXAv)Z$Y = c(VXA)vAv4Y + s(X)02AA0 + 4Av(VXA)v4Y +

0.

Since 4Av = -Avd, the above equation reduces to

(1.16)

¢(VXA)vAvlY + (Av(VXA)voY = 0.

We take any two A, p of characteristic roots of Av at a point x of M.

We naw define spaces by setting

TX = {X c TT(M): AAX = AX},

T- = {X a TA(M): AvX = lpX}.

334

Then TA A 71 = {0} when A # u. If Y E 71, then $Y e T u. let Y e Tu.

Then (1.16) implies

v$(VXA)v$Y = -U$(VXA)vvY,

which shows that c(VXA)vcY e T T. let X E TX, Y c TV and A # tit. Then,

from equation of Codazzi, we obtain

¢(VOXA)v$Y E T_X (% T_4.

Thus we have (VOXA)vOY = 0. On the other hand, if X E T, then

anda=u#0. Then we find

02XETX. Let XcTX , Y

C)2(VOXA)v$Y c TX.

From this and equation of Codazzi we have

c TX,, T_X,

0. From this we have

which means that 0.

Let A = µ = 0. We take X, Y e TO at x E M, and extend these to local vector fields on M which are covariant constant with respect to V at x. Here we notice that if X e T0, then OX c T0. From (1.16), we obtain, at x e M,

gWVOXA)v(VOXA)v$Y,Y) = 0,

and hence

0. Consequently we have

cases. On the other hand, if

(V

0, then

-g(cX,AviY)F,

0 for any

325

0. This completes the proof of our assertion.

from which

QED.

LENS 1.9. Let M be an invariant submanifold of codimension 2 of a Sasakian space form on+3(c). If the Ricci tensor S of M is rI-parallel, then M is an 71-Einstein manifold.

Proof. Fran Lemmas 1.5, 1.6 and 1.7 we have 2

n(c+3)ITI2 =

I I[Q,Aa]12. a=1

Since M is of codimension 2, we can take a basis {v,qv) for T(M)L. Then (1.7) implies

S(AVX,Y) = i(n(c+3)+(c-1))g(AvX,Y) - 2g(AV3X,Y) = S(X,AvY),

which shows that QAv = AvQ and hence n(c+3)IT12 = 0. Thus if c # -3, then M is n-Einstein. Let c = -3. Then (1.11) and Lemma 1.8 show that

M is totally geodesic and hence M is n-Einstein.

QED.

From these lemmas we have (Kon [83) THEOREM 1.9. Let M be an invariant sutmanifold of codimension 2 of a Sasakian space form

jP+3(c).

Then the following conditions are

equivalent: (1) The Ricci tensor of M is n-parallel; (2) The second fundamental form of M is n-parallel; (3) M is n-Einstein.

THEOREM 1.10, Let M be an invariant subnanifold of codimension 2 K?n+3 (C). If

with n-parallel Ricci tensor of a Sasakian space form

c < -3, then M is totally geodesic. If c > -3, then either M is totally geodesic, or an n-Einstein manifold with JAl2 = n(c+3) and hence r = n(n(c+3)-2).

Proof. From Theorem 1.9, M is n-Einstein. Then (1.11) and

Lemma, 1.4 imply

326

j(n+2)11IAI2 - (c+3)]IAIZ = 0.

This proves our theorem.

QED.

THEOREM L.U. bet M be a compact invariant suhrmnifold of codimension 2 of a Sasakian space form I1+3(c) (c > -3). If the scalar curvature r of M is constant, then either M is totally geodesic, or an n-Einstein manifold with the scalar curvature r = n(n(c+3)-2). Proof. In this case we have already seen that QAv = AAQ. Then Proposition 1.5 states that m is an n-Einstein manifold. Thus our theorem follows from Theorem 1.10.

QED.

Here we consider the case that the second fundamental form of M satisfies IAI2 = (n+2)(c+3)/3. We prove

THEOREM 1.12. Let M be a (2n+1)-dimensional invariant submanifold 0r'+1 of a Sasakian space form (c) (c > -3). If IAI2 = (n+a)(c+3)/3, then M is an n-Einstein manifold of dimension 3 and has the scalar curvature r = (c+l).

Proof. Since IAI2 is constant, we have -g(V2A,A) = IVAI2. Then (1.11) implies

0 < IDAI2 - 31A12 = I(TrAaAb)2 + 1l(Aa,Ab]I2 - J(n+2)(c+3)IAI2 = ',-[3IAI2-(n+2)(c+3)]IAI2

n - 4 1 a a TrA2TrA2. b a i s t#s aft

From the assumption we have

n I A X = 0 i s t#s

and

TrA2T-rAb = 0.

aft

a

Thus we may assume that at = 0 for t = 2,...,n and 'Na = 0 for a = 2,..

.,p. Therefore we have

327

QX = i(n(c+3)+(c-1))X - 2(n+1)(c-l)n(X)E - 2A1X.

From this we see that QAa = AaQ for all a. From this and Lemma 1.5, M is n-Einstein because of the second fundamental form of M is n-parallel and hence Q is n-parallel. Then Theorem 1.4 implies

3(n+2)(c+3) = IA12 > n(c+3),

from which n = 1 and hence M is of dimension 3. Thus we have r = (c+l) by (1.8).

QED.

We now assume that M is a regular Sasakian manifold. Let M/C denote the set of orbits of E. Then M/t is a real 2n-dimensional Kaehlerian manifold. Then there is a fibering 7r : M

> M/F.

Henceforth X*, Y* and Z* on M will be horizontal lifts of X, Y and Z over M/C respectively with respect to the connection n. Then we have

(S'(Y,Z))* = S(Y*,Z*) + 2g(Y*,Z*),

where S' denotes the Ricci tensor of M/&. From this we have

((VXS')(Y,Z))* = (VX*S)(Y*,Z*),

where V' denotes the operator of covariant differentiation in M/E. therefore we see that the Ricci tensor S' of M/E is parallel if and only if (VX*S)(Y*,Z*) = 0 which is equivalent to

0

for any U, V, W c T{(M) because of the horizontal space is spanned by {4U: U c TX(M)). On the other hand, (V,US)(cV,¢W) = 0 implies that (V02US)(¢V,¢W) = -(VUS)(gV,¢W) = 0 and the converse is also true. Therefore the Ricci tensor S' is parallel if and only if (VTS)(4V,4W) = 0, which states the meaning of the definition of n-parallel Ricci tensor.

0 THEOREM 1,13, Let M be a regular Sasakian manifold. Then the Ricci tensor S of M is ri-parallel if and only if the Ricci tensor S' of M/g is parallel. Excurrple 1.1. Let R2n+3 be a (2n+3)-dimensional Euclidean space

with standard Sasakian structure of constant q-sectional curvature c = -3. A (2n+3)-dimensional unit sphere S2n+3 has the standard Sasakian structure of constant 4-sectional curvature c > -3. By Can+l, R and (R,CDn+1) we denote the open uni ball in a complex (n+l)-dimensioCan+1 nal Euclidean space Cn+l, a real line and the product bundle R x

Then (R,CDn+1) also has a Sasakian structure with constant -sectional curvature_c < -3. Let (P be an n-dimensional complex quadric in a complex projectivae space C

+1. We denote by (S,Qn) a circle bundle

over Qn. Then (S,Qn) define a Sasakian structure which is n-Einstein S2n+3. R2n+1, (R,CDn) and (S,Qn) is an invariant submanifold of and Stn+1 are totally geodesic invariant submanifolds of (R,CDn+1) R2n+3,

and S2n+3 respectively. We prove the following (Kon 18])

THEOREM 1.14. (1)

S2n+l

and (S,(p) are the only connected complete

invariant submanifolds of codin ension 2 in S2n+3 which have n-parallel Ricci tensor;

(2) (R,C) (resp. R2n+1) is the only connected complete invariant R2n+3) sukmanifold of codimension 2 in (R,CDn+l) (resp. which has n-parallel Ricci tensor. S2n+3

Proof. Let M be one of the

and (R,CDn+l) and B be

Cn+l (if M = R2n+3), CPn+1 (if i = S2n+3) and (fin+1 (if M =

(R,a)n+1)).

Then M is a principal G1-bundle over B, where G1 is a circle or a line. Then an invariant sub manifold M of codimension 2 of M is also regular. We can consider the following commutative diagram:

M

>M

7Tl

mg

lTr

>

M By Theorem 1.13, the Ricci tensor S' of M/l; is parallel. Since M/F is

an invariant submanifold of B of codimension 2, by Theorem 1.6 of

Chapter IV, WE is Einstein. From Theorem 1.10 of Chapter IV our theorem reduces to the following lemma.

LEMMA 1.10, Let M be a non-totally geodesic connected complete n-Einstein invariant submanifold of codimension 2 of S2n+3. Then there is an automorphisn 6 of S2n+3 such that 6M = (S,QW). Proof. By Theorem 1.10 of Chapter IV, B = nM is holomorphically isemetric to Qn. Frcm Theorem 1.8 of Chapter IV, there is a holomorphic iscmetry 6 of CPn+l such that eB = Qn. Let x c B, Ox = y and T(t), 0 < t < 1, be a curve joining x and y. Then we have a continuous family of J-basis (T(t),ei(t),Jei(t)), i = 1,...,n+1, on T(t) such that ei(1) = 6*(ei(0)). Thus a is contained in the connected component e+1.

of the automorphism group of

Bence there are finite numbers of

infinitesimal autonorphismis X1,X2,...,Xs of CPn+1 such that 6 =

expts s...expt1X1. By Lemma 5.1 of Tanno [4] there are infinitesimal automorphisms

Y1..... Ys of S2n+3 such that nYk = Xk, n(exptkYk)(u)

= exptkXk(mi), u c M, k = 1,...,s. Putting 8 = expts s...exptlY1, we have

9M = Qn. Since SM and (S,(;") have the same fibre, we have QED.

9M = (S,(P) By Theorem 1.11 and Theorem 1.14 we have (Kon [8]) THEOREM 1.15.

S2n+1 and (S,(P) are the only compact invariant

sutmanifolds of codimension 2 in S2n+3 which have constant scalar curvature.

2. ANTI-INVARIANT SIIRMANIFOLDS TANGENT TO THE STRUCTURE

VECTOR FIELD OF SASAKIAN MANIFOLDS

let rd be a (2m+1)-dimensional almost contact metric manifold with structure tensors (4),E,n,g). An n-dimensional sutmanifold M immersed in ti is said to be anti-invariant in 2 if OTx(M) c T{(M)' for each x c M.

Then we see that, 0 being of rank 2m, n < m+1. When n = m+l we have

330

PROPOSITION 2.1. let M be an almost contact metric manifold of

dimension 2n+1 and let M be an anti-invariant si

nifold of M of

dimension n+l. Then the structure vector field E is tangent to M. Proof. By the assumption we have ¢T{(M) = T{(M)- at each point x of M. For any vector field X tangent to M we have gQ,$X) = -g(c

,X)

= 0, which shows that the structure vector field E is tangent to M. QED.

PROPOSITION 2.2. Let M be a submanifold tangent to the structure vector field E of a Sasakian manifold M. Then E is parallel with respect

to the induced connection on M, i.e., OE = 0, if and only if M is an anti-invariant submanifold of M.

Proof. Since g is tangent to M, we have X = VV - B(X,g). Thus 9 is parallel if and only if X =

and hence X is normal to M. QED.

In this section we sail study anti-invariant su

n folds tangent

to the structure vector field of a Sasakian manifold. Therefore, in this section, we mean by an anti-invariant submanifold U of a Sasakian manifold M an anti-invariant submanifold M tangent to the structure vector field

of M.

PROPOSITION 2.3. let M be an (n+l)-dimensional anti-invariant subnanifold of a (2n+1)-dimensional Sasakian manifold M. If n

1,

then M is not totally umbilical.

Proof. If M is totally umbilical, then B(X,Y) = g(X,Y)u, u being the mean curvature vector of M. Then B(E,C) = 0 implies u = 0 and hence M is totally geodesic. This contradicts to the fact that OX = -B(X,E)

# 0.

QED.

Here we choose a local field of orthonormal frames e0 E,el,...,en; en+1,...,em;el*= ell...,en* en;e(n+l)* en+l,.... em* em in M such

that, restricted to M, e0,el,...,en are tangent to M. With respect to r ,wl,...own;wn+l,...,(d,3 ...,ten* this frame field of, let

W

(n+l)*

,...,

u

be the dual frames. Unless otherwise stated, we use

the conventions that the ranges of indices are respectively:

331

A,B,C,D = i,j,k,l,s,t = 0,1,...,n,

x,y,z,v,w = 1,...,n,

a,b,c,d = X,u,v = n+l.... m,(n+l)*,...,m*, a,a y = n+l,.... m.

Then we have the following equations:

* Wy =

Wy*,

Wa = Wu s

* y* Wy = Wx

,

a = WS

,

U)

s*'

x _ x*

Wa - Wa*,

s

a

x* _

a*

W x= -W0 ,

W x

= WO,

Wa = -WO o

Wa

= Wa 0,

Wa - Wx

We restrict these forms to M and then we have Wa = 0. Thus we have

WO = WO* = wo = 0. Therefore (2.13) of Chapter II implies

x_ y

z

hyz - hxz = hxy,

x

a

h00 = 0,

h0i - 0,

x h i = -'xi'

where we use h-ij in place of hid to simplify the notation. For each a,

the second fundamental form Aa is represented by a symmetric (n+l,n+l)matrix Aa = (ha.). Then we have

x

0 0

-1

0

10 hx

for all x,

yz

for all A.

AA =

0

332 Hereafter we put

a =

n7), iihich is a symmetric (n,n)-matrix. We

notice that

IAi 2 = IH12 + 2n, where I H12 = ETrH2a. We also have a

Tra -Tra for all a, and hence M is minimal if and only if Tr a = 0 for all a. Since 07{(M) c T{(M)- at each point x of M, we have the following decanposition:

x(M),

T X(Mt =

where NA(M) is the orthogonal conplement of $X(M) in X(M)y. If V e a(M), then 4V a NS(M). For any vector field V normal to M we put

¢V = tV + fV,

where tV is the tangential part of cV and fV the normal part of 4V. Then t is a tangent bundle valued 1-form on the normal bundle T(M)'

and f is an endomorphisn on T(M). Then we have

tfV = 0,

f2V = -V - 4tV,

tqX = -X + n(X)

foX = 0.

From these we also have

f3+f=0. Since f is of constant rank (see Stong [1]), if f does not vanish,

it defines an f-structure in the normal bundle T(M)r' (see Chapter VII).

By the Gauss and Weingarten formulas we have

(Vxf)V = -B(X,tV) - lAVX.

If (OXf)V = 0 for all X and V, then the f-structure f in the normal bundle of M is said to be paraZZel. LE14 A 2.1. It M be an (n+l)-dimensional anti-invariant sutrnani-

fold of a (2m+l)-dimensional Sasakian manifold M. If the f-structure f in the normal bundle of M is parallel, then AV = 0 for V E NX(M), or equivalently, AA = 0.

Proof. If V E X(M), then tV = 0. Thus we have (AVX = 0 and hence ¢2AVX = -AVX + n(AVX)E = 0. On the other hand, we see that n(AVX) = g(B(X,E),V) = -g(gX,V) = 0. Thus we have AV = 0.

QED.

We can easily see that if Ha = 0 for all a, then f is parallel. We denote by R and R the Riemannian curvature tensors of M and M respectively. Then we have

(2.1)

y*ij = Ryij R0

iXjy - iyjX(2.2)

i

= ROjk = 0.

Suppose that the ambient manifold M is of constant ¢-sectional curvature c. Then the Gauss equation is given by

(2.3)

Rjkl = *(c+3)(6 ik6jl - 6il6 jk) +

+ nirlldjk - ninkdjl) +

(c-1)(njnkdil - njnldik

a

fran which we find the Ricci tensor Rij of M is given by-

(2.4)

Rij = *[n(c+3)-(c-1)]dij - j(n-1)(c-1)ninj

+

I (hkkhij a,k

334

On the other hand, we have

a,k(

h x

hil -

fix)

- lij .+ nine.

a,x

Thus (2.4) reduces to

(2.5)

R.. = i(n-1)(c+3)6.. - a[(n-1)c+(n+3)]ninj

+

a,x

(ha ha -

Since we have

(h7 )2 =

(hXy)2 + n,

a,x,y

a,i,x

the scalar curvature r of M is given by

r = $n(n-1)(c+3) +

(2.6)

F

a,x,y

(ha hay

-

ha hay

From (2.5) and (2.6) we have PROPOSITION 2.4. Let M be an (n+l)-dimensional anti-invariant submanifold of a Sasakian space form M

1(c). If M is minimal, then

the Ricci tensor S and the scalar curvature r of M satisfiy (1) S - J(n-1)(c+3)g +

;[(n-1)c-(n+3)]n O n is negative semi-

definite;

(2) r < jn(n-1)(c+3).

PROPOSITION 2,5. Let M be an (n+l)-dimensional anti-invariant suirnanifold of a (2n+1)-dimensional Sasakian manifold M. Then M is

flat if and only if the normal connection of M is flat.

Proof. First of all we have

g Y#i]

L(hixhjy - hix hiy)

Ryij

Yi.] + z

because of hix = h' and (2.1). This combined with (2.2) proves our

assertion.

QED.

In the following we compute the Laplacian of the square of the length of the second fundamental form of M. Since hljk - hikj = Of (3.1) of Chapter II reduces to

(v2B)(X,Y) = j(R(ei,X)B)(ei,Y) + DXDy(TrB), i

from which

ha.6ha. _ i3

CC

a,l,j

1J

a,i,j,k

a

+ ha

a

(hai3 kki_i

ij

+ hhtiRkjk ij

t

ijk

- h

ijNiRbjk).

Therefore, (2.16) and (2.18) of Chapter II imply

Rijkhijhkt + Rkjkhijhti

£ a,i, j hi.jnhij a,i,,7,k - jkhljhid) -

[(h hkj -

£

t

a,b,i,j,k,t -

hijhbjhtkhbk - hijhithtihkk3.

Consequently we obtain

(2.7)

jha

hij = lhijh

j + Q(n+1)(c+3)lTrAa - 4(c-1)jTrA2

- 12(c+1)1(Tr X)2 - 14(c+3)1(TrA )2 - jn(n+l)(c-1)

+ J[Tr(AaAb - AbAa)2 - (TrAaAb)2 + TrAbTrAa2Ab].

PROPOSITION 2,6. Let M be an (n+l)-dimensional anti-invariant suboanifold of a Sasakian space form M

iAIA12 - J(ba

)2 = jha h

1(c). Then

j + a(c+3)1[nTrHa - (TrHa)2]

+ *(c+3)1[TrHX - (TrHX)2] + J[Tr(H Hb - HbHa)2 - (Tr aHb)2 + TrHbTrHaHb].

338 Proof. First of all, we have

TrAX = TrH + 2,

Trq =

TrAa = WE.-

We also have the following equations:

JTr(AaAb - AbAa)2 = JTr(HaHb - HbHa)2 - 4E(Tr X)2 - 41Tr a + 81TrHX - 2n(n-1),

J(TrAaAb)2 = I(TrAa)2 = J(TrHa)2 + 41TrH2 + 4n,

ITrAbTrAa2Ab = JTrHbTrHaHb + 21(TrHX)2 + 1(mra)2.

Moreover, we obtain

le-Ah'J = JAIA12 - Z(haJk)2

(2.8)

= JAIAI2 - E(hayZ)2 - 31TrH2,

(2.9)

- I(TrH,)2.

j = IhX]h

jhijh

J

Substituting these equations into (2.7), we have our equation.

QED.

FYcm (2.8) and (2.9) we also have PROPOSITION 2.7. Let M be an (n+l)-dimensional anti-invariant suhnanifold of a (2m+1)-dimensional Sasakian manifold M. (1) If the second fundamental form of M is parallel, then HA = 0 for all A;

(2) If the mean curvature vector of M is parallel, then TrHx = 0 for all A.

We now put

Tab =

hayhb xi y

,

Ta =T an,

T = ITa = IH12.

a

THEOREM 2.1, Let M be an (n+1) -dimensional compact anti-invariant

minimal sutmanifold of a Sasakian space form

FP+1(c).

Then

0 < fMIVAI2*1 < fM[(2 - n)T - <(n+l)(c+3)]T*l.

Proof. By the assumption on the dimensions, f vanishes identically. Thus Proposition 2.6 implies

jAIA1 2 - IVAI2 = a(n+l)(c+3)T - JT2 + jTr(HXHy - HYHx)2

because of E(ha )2 = E(hijk)2 by (2.8). Applying Lemma 5.1 of Chapter xyz II, we obtain

-jTr(

xy

- HYH{)2 + JT2 - }(n+l)(c+3)T

< 2 yTxTy + JT2 - }(n+l) (c+3)T

-

(T

[(2 - n)T - I(n+l)(c+3)]T - n

x>y

Ty)2.

QED.

Therefore, we have our assertion.

THEOREM 2.2. Let M be an (n+l)-dimensional compact anti-invariant minimal sulmanifold of a Sasakian space form M2n+1(c). Then, either T = 0, or T = n(n+l)(c+3)/4(2n-1), or at some point x of M, T(x) > n(n+l)(c+3)/4(2n-1). THEOREM 2.3. Let M be an (n+l)-dimensional anti-invariant minimal

suhmanifold of a Sasakian space form Mn+1(1). If 1A12 = (5n2-n)/(2n-1), then n = 2 and M is flat. With respect to an adapted dual orthonormal

frame field wO,wlw2,wl*,w2*, the connection form (wB) of M5(1), restricted to M, is given by

0

0

0

0

0

0

0

0

0

w1

--OO+aw

awl

0

0

w2

w0+Aw1

Aw2

0

0

-wl up-Aw2

-awl

-w2

-Aw1 _w0-Aw2

, A=

338

goof. Since 1 ALI 2 is constant, we have 0 < IVA12 < [(2 - n)T - (n+1)]T.

Thus, by the assumption and T = IAl2 - 2n, we see that the second fundamental form of M is parallel. Moreover, we have, from Lemma. 5.1 of Chapter II,

T")2 = 0,

E (T X>y

- Tr(HXHy - HyHX) 2 = 2TrHXTrH2,

and hence Tx = Ty for all x, y and we may assume that x = 0 for x = 3,...,n. Therefore, we mist have n = 2 and we obtain

0

1

1

0

1

0

0

-1

H1=X by putting h12 = h1

X. Then we have

=

0

-1

0

Al = -1

0

X

0

X

0

A2

,

0

0

0

X

-1

0

-1

0. -X

On the other hand, we find

x

x

d

xd

yx

dhab = hadwb + hdbwa - habwy.

Putting x = 1, a = 1 and b = 0, we see that dX = wo = 0, which shows that X is a constant. Since T = 2, we get 4X2 = 2. Thus we may assume that X = 1/v!E. Moreover, we have

w0 = -wl* = 0 1 w2* = Xw l

1

w0 - -w2* = 2 w1* _ -w0 + Xw2, 1

0,'

0

1

01*

= w ,

0 2 w2*=w, 2

W2 2

_ -w0 - Xw

w1 2 - 0.

KK On the other hand, from equation of Gauss,

easily see that M is

flat and hence the normal connection of M also flat. These prove our theorem.

QED.

Example 2.1. Let J = (ats) (t,s = 1,...,6) be the almost complex structure of.C3 such that a2i,2i-1

= li a2i-1,2i = -1 (i = 1,2,3) and the other components being zero. Let S (1//) _ {z E C: I ZI2 = 1/3), a plane circle of radius

M3 =

Sl(1//)

We consider

X

Sl(1//)

x

S1(1/15)

in 55 in C3,which is obviously flat. The position vector X of M3 in S5 in C3 has components given by

X = (1//)(cosul,sinul,cosu2,sinu2,cosu3,sinu3),

ul, u2, u3 being parameters on each S1(l//). Putting Xi = aix/aui,

we have

X1 = (1//)(-sinul,cosu1,0,0,0,0), X2 = (1/,/3)(0,0,-sinu2,cosu2,0,0),

X3 = (1//)(0,0,0,0,-sinu3,cosu3).

The structure vector field & on S5 is given by

t = JX = (1/v)9)(sinul,-cosu1,sinu2'-cosu ,sinu3,-cosu3).

Since C = -(X1 + X2 + X3), C is tangent to M3. On the other hand, the structure tensors (c,E,n) of S5 satisfy

4Xi = JXi - n(Xi)X,

i = 1,2,3,

which shows that 4%j is normal to M3 for all i. Therefore, M3 is an

anti-invariant suhinifold of S5. Moreover, M3 is a minimal suhmanifold

340

of S5 with I Al2 = 6 and the normal connection of M3 is flat. Since the connection form (4AA) of S5, restricted to M3, coincides

with that in Theorem 3.2, we have THEOREM 2.4. Let M be an (n+1)-dimensional compact anti-invariant

minimal submanifold of S +1. If 1A12 = (5n2-n)/(2n-1), then M is

S1(1/,/)

x

S1(1/,/) x S1(1/T)

in S5.

THEOREM 2.5. Let M be an (n+l)-dimensional compact anti-invariant

minimal such nifold of S S. If the second fundamental form of M is parallel and if JA12 = (5n2-n)/(2n-1), then M is

S1(1/T) x S1(1/f)

x

S1(1/T)

in an S5 in S

1.

Proof. Since the second fundamental form of M is parallel, by Proposition 2.7, AA = 0 for all X. Thus we have the same inequality as in the proof of Theorem 2.1 and then we have n = 2. We also have T1 = T2 0 0. Thus the first normal space of M is spanned by el*, e2*, that is, the first normal space of M is 4Tx(M). For any vector fields

X and Y tangent to M and any vector field V in N(M) we have g(DXpY,V) =

g((OX4)Y,V) + g(fv XY,V)

= g(OVXY,V) + g(OB(X,Y),V) = -g(AoX,Y) = 0,

which shows that the first normal space of M is parallel. Thus M is S2m+l in an S5 in and M is anti-invariant in S5. Therefore our theorem follows from Theorem 2.4.

QED.

Example 2.2. Let S1(ri) _ (zi a C: Izil2 = ri}, i = 1,...,n+1. We consider

Mn+1 = S1(rl) x ... x S1(rn+l)

'S1

in Cn+1 such that ri + ... + rn+1 = 1. Then

0+1 is a flat subm nifold

of S2n+1 with parallel mean curvature vector and with flat normal connection. The position vector X of Mn+1 in Cn+1 has components given

by

+1,rn+1sinun+1 ).

X = (r1cosu1,rlsinu ,...,rn+lcosu

Then X is an outward unit normal vector of

Stn+1

in C. Putting

Xi = aiX = ax/au l, we have

X1 = r1(-sinul,cosu ,O,...,O), .......................... Xn+1 = rn+1(0,.... 0,-siniin+1 ,,,,n+l).

The structure vector field C on S2n+1 is given by its carponents

C = -JX =

(rlsinu1,-rlcosul,...,rn+lsinun+l,-rn+lcosu +1 ).

Therefore we see that

= -(X1 + ... +

Xn+1),

which means that the

vector field E is tangent to M. So the structure tensors (4,E,n,g) Stn+1 of satisfy

cXi = JXi - n(X1)X,

i = 1,...,n+1.

Thus M is normal to M for all i. Therefore Mn+1 is anti-invariant in S2n+}.

THEOREM 2.6. It M be an (n+l)-dimensional compact anti-invariant sulmanifold of S2n+1 with parallel mean curvature vector. If the normal connection of M is flat, then M is

S1(r1) X ... X S1(rn+l)'

Erl = 1.

Proof. Since the normal connection of M is flat, Proposition 2.5 implies that M is flat. On the other hand, as the mean curvature vector

342

of M is parallel, IAl2 is constant. From these and Proposition 4.4 of Chapter II we see that the second fundamental form of M is parallel. Therefore our theorem follows from Theorem 4.4 of Chapter II.

QED.

THEOREM 2.7. Let M be an (n+l)-dimensional ccapact anti-invariant

sutmanifold with flat normal connection of 521. If the second fundamental form of M is parallel, then M is

S1(rl) x ... x Sl(rn+l)

in an S2,,+1 in

S3n+1 .

where Zr? = 1.

Proof. From the assumption we have AA = 0 for all A and we see that M is flat. We shall show that the first normal space of M is of dimension n and parallel with respect to the connection induced in the normal bundle. Since M is flat, by equation of Gauss, we have

0 = (dikdjl - 8il8jk) + a

If the dimension of the first normal space N1(M) of M is less than n,

then for sane x, Ax = 0. Thus we have

J(hixhjl -

a

y _ I(hiyhS Y

Therefore we have 0 = (6 ix6jl

-

- hilhx

= 0.

6il6jx), which is a contradiction.

We also see that N1(M) is parallel by the similar method used in the proof of Theorem 2.5.

QED.

THEOREM 2.8. Let M be an (n+l)-dimensional compact anti-invariant

minimal submanifold of a Sasakian space form M2

1(c) (c > -3). If

T < }nq(c+3)/(2q-1) (q = 2m-n), then T = 0. Proof. From Proposition 2.6 and Lemma. 5.1 of Chapter II we have

343

J(hXyz)2 - JAIA12 = -ITT(HaHb

1T2

HbHa)2 +

EL

- }n(c+3)T - }(c+3)ITrHX [(2 - q)T - }n(c+3)]T -

1 (Ta - Tb)2 - *(c+3)

Therefore, by the assertion, we obtain

1(Ta - Tb)2 = 0,

ITx = 0.

Thus we see that x = 0 for all x and Ta = Tb for all a and b. Hence we have Ta = 0 for all a. Consequently, we obtain T = 0.

QED.

We give examples of anti-invariant submanifolds with T = 0.

Example 2.3. Let Stn+1 be a unit sphere of dimension 2n+1 with standard Sasakian structure and let CPn be a complex projective space of real dimension 2n with constant holarorphic sectional curvature 4.

A real projective space RPn of dimension n with constant curvature 1 is imbedded in CPn as an anti-invariant and totally geodesic submanifold (see Abe [1]). We consider the following cormutative diagram:

(S,RP1)

>

S2n+1

V

where (S,RPn) denotes a circle bundle over RPn. Then (S,RPn) is an

anti-invariant subnanifold with T = 0 of

52n+1.

Example 2.4. Let R2n+1 be an Euclidean space with cartesian coordinates ()l,...,xn,yl,.... yn,z). As in Example 5.5 of Chapter V

we derive the standard Sasakian structure in R2n+1 with constant -sectional curvature -3. We consider the following natural imbedding of Rn+1 into R2n+1:

344

Then we easily see that Rn+1 is an anti-invariant submanifold of R2n+1 which has T = 0.

3. ANTI-INVARIANT SUF1ANIFOLDS NORMAL TO THE STRUCTURE VECTOR FIELD OF SASAKIAN MANIFOLDS

From the consideration of §4 of Chapter V it is interesting to study the following suhmanifolds of contact manifolds, especially

those of Sasakian manifolds. First of all, we prove THEOREM 3,1. Let M be an n-dimensional sutrnanifold of a (2n+l)-

dimensional K-contact manifold M. If the structure vector field E is normal to M, then M is an anti-invariant submanifold of M, and n < m. Proof. From the Weingarten formula we have

g(¢X,Y) = -g(V,Y) =

for any vector fields X and Y tangent to M. Since AC is symmetric and

is skew-symmetric, we have AE = 0 and X is normal to M. Thus M is an anti-invariant sub manifold of M. We also easily see that n < m. QED.

Throughout in this section we mean by an anti-invariant submanifold M of an almost contact metric manifold M, a sutmanifold M of M normal to the structure vector field E of M. Especially we consider a submanifold M of a Sasakian manifold M normal to E. We state some fundamental properties of the second fundamental

form of an anti-invariant submanifold Mof a Sasakian manifold M. We already have the following

(3.1)

AE = 0.

345

Moreover, we easily see

(3.2)

for any vector fields X and Y tangent to M.

LEhMA 3.1, Let M be an n-dimensional anti-invariant sutmanifold of a (2m+l)-dimensional Sasakian manifold M. If the second fundamental form of M is parallel, then

AqX = 0

for any vector field X tangent to M. If moreover n = m, then M is totally geodesic in M.

Proof. Prom the assumption on the second fundamental form of M we obtain

0 = g((OXB)(Y,Z),0 = g(B(Y,Z),OX) = g(AOXY,Z),

from which

0. If n = m, the normal space Tx(M)L is equal to

cT{(M) 9 {F}..Therefore M is totally geodesic in M.

QED.

Since 4Tx(M) C TA(M)'' at each point x of M, we have the decompo-

sition of TX(M)` into the direct sum

Tx(M)` = $TT(M) ® N(M),

N(M) being the orthogonal complement of X(M) in Tx(M)L. We have

N(M) = ON(M) 0 W. For any vector field V normal to M we put

4V=tV+fV, where tV is the tangential part of V and fV the normal part of 4V. Then t is a tangent bundle valued 1-form on the normal bundle of M and f is an endomorphisn of the normal bundle of M. We find

346 f2V = V - $tV + n(V)g,

tfV = 0,

t9 = X,

f$X = 0.

Moreover, we have

f3 + f = 0, which means that f defines an f-structure in the normal bundle of M.

LE" 3,2. If the f-structure f in the normal bundle of M is parallel, then

AV = 0

for V E NA(M).

Proof. First of all, we have (VXf)V = -B(X,tV) - cAVX. Since

Vf = 0 and tV = 0, we obtain AVX = 0. Thus we have

o2AVX = -AVX + n(AVX)C = -AVX = 0,

which proves our equation.

QED.

Let u be the mean curvature vector of M. Then, from (3.1), we have

g(DRu,E) = g(u,cX).

From this we have the following

LEMM 3,3, Let M be an n-dimensional anti-invariant sutmanifold of a (2m+l)-dimensional Sasakian manifold A. If the mean curvature

vector u of M is parallel, then u E Nx(M) at each point x of M. Moreover, if n = m, then M is minimal.

From Lamas 3.1, 3.2 and 3.3 we have

347

PROPOSITION 3.1. let M be an n-dimensional anti-invariant su)manifold of a (2m+1)-dimensional Sasakian manifold M with parallel f-structure f in the normal bundle of M.

(1) If the second fundamental form of M is parallel, then M is totally geodesic;

(2) If the mean curvature vector of M is parallel, then M is minimal.

We choose a local field of orthonormal frames el,...,en;en+1,"

.,

em*4em in M in such a

n;e(n+i)* n+l1...1 l,...,en* em;eO* E,el* way that, restricted to M, el,...,en are tangent to M. Unless otherwise stated, we use the conventions that the ranges of indices are respectively:

i,j,k,l,t,s = 1..... n,

a,b,c,d = p,q,r = n+l,...,m,l*,.... m*.

In the following we put h0j =

h0

= g(Aei,ej) and hlj = hid _

g(e ei,ej) to simplify the notation. t* We now assume that the ambient manifold M is a Sasakian space

form M

(3.3)

l(c). Then the Gauss equation of M is given by

Rjkl = }(c+3)(dikdjl

- dilajk) +

J(hikhjl -

hik hjk)

Rij = *(n-1)(c+3)bij +

a,k r = in(n-1)(c+3) +

E

a,i,j

ij ij

(ha ha- - ha-ha.). ii ii

If M is minimal, we have Z1hii = 0 for all a, and hence we have

the following

348

PROPOSITION 3.2. Let M be an n-dimensional anti-invariant submanifold of a Sasakian space form M

l(c). Then M is totally geodesic

if and only if M satisfies one of the following conditions: (1) M is of constant curvature 4(c+3); (2) S = 4(n-1)(c+3)g;

(3) r = 4n(n-1)(c+3),

where S and r be the Ricci tensor and scalar curvature of M respectively.

Let M be an n-dimensional anti-invariant sutmanifold of a Sasakian space form M

l(c). Then we have (see §2)

haAha ij ij a.i.j

=

+ 4(c+3)F[nTrA2 - (TrA )2] ha. a i,7hkkij a a a a.i.j.k

+ 4(c-l)t[TrAt2 - (TrAt)2] + aF[Tr(AaAb - AbAa)2 b

- (TrAaAb)2 + TrAbTrAaA.].

On the other hand, we have

E

hij&i

= JAIA12 -

a,i,j

(hljk)2 F a,i,j,k

= iAIA12 -

p.i.j.k(

i.jk)

2 _ TEA2 . t

t

From these equations we obtain

PROPOSITION 3.3. Let M be an n-dimensional anti-invariant submanifold of a Sasakian space form 1faD+1(c). Then

#AIA12 - F(hpjk)2 = Fhijhkkij + 4n(c+3)IA12 - F(TrAa)2

- 4(c+3)1(TrAa)2 + 4(c+3)1TrA2 - 4(c-l)F(TrAt)2

+ F[T-(AaA. - AbAa)2 + TrAbTrAa2Ab].

349

THEOREM 3.2. Let M be an n-dimensional caiipact anti-invariant sutmaniiffold with parallel mean curvature vector of a Sasakian space 22n+1 form (c). Then

0 < JM (hp.k)2*1 < 1M[(2 - i)IA12 - *(n+1)(c+3)]IA12*l. I n p,i,j,k Proof. From Lemma 3.3 we see that M is minimal. Thus Proposition 3.3 implies

oIAI2 - 1(hpjk)z = *(n+1)(c+3)IAI2 - F(TrAt)2+ 1Tr(AtAS AsAt)2.

On the other hand, we obtain

-ITr(AtAs - ASAt)2 + I(TrA2)2 - *(n+l)(c+3)IAI2

.S2

_

I

TrA2A2 + 1(TrA2)2 - }(n+l)(c+3)IAI2

n)IAI2 - *(n+l)(c+3)]JAI2 - n (TrA2 - TrA2)2. t>s

Thus we have our inequality.

QED.

THEOREM 3.3. Let M be an n-dimensional compact anti-invariant submanifold with parallel mean curvature vector of a Sasakian space form M2n+1(c). Then either M is totally geodesic, or IAI2 = n(n+l)(c+ 3)/(4(2n-1), or at some point x of M, IAI2(x) > n(n+l)(c+3)/4(2n-1). We now take a unit sphere S2n+1 as an ambient manifold. Then we

have THEOREM 3.4. Let M be an n-dimensional anti-invariant submanifold with parallel mean curvature vector of 52n+1 (n > 1). If IAI2 = n(n+l)/(2n-1), then n = 2 and M is a flat surface of S5. Proof. Fran the assumption we have

j(TrAA - TrAs) = 0,

350

Tr(AtAS - ASAt) = 2TrAATrAs.

Therefore, TrAt = TrAs for all t, s. E Qn Iemma 5.1 of Chapter II we m a y a s s um e

that At = 0 for t = 3,...,n. Then we must have n = 2 and we

can put

A0=0,

0

1

1

0

1

0

0

-1

A2=A

Al = X

Since I AI2 = 2, we obtain 2A2 = 1. Thus we may assus

that k = 1/a.

Moreover, the Gauss equation (3.3) shows that M is flat. Ermr:e 3.1. Let C:n+1 be a complex (n+l)-dimensional number space with almost complex structure J and let S2n+1 be a (2n+1)-d n ensional unit sphere in G'n+1 with standard Sasakian structure (4,,E,-n,g). Let S1

be a circle of radius 1. Let us consider

Si.

T n = S1 x ... x

Then we can constract an isometric minimal immersion of Tn into

S2n+1

which is anti-invariant in the following way. Let 1: Tn

X=

1

>

n+l( cosu

1

S2n+1 be a minimal immersion represented by

sinu

1 ,

n+1 n+1 ...,cosun,sinun,cosu sinu

where we have put un+1 = -(ul + ... + un). We may regard X as a posiCn+1. tion vector of S2n+1 in The structure vector field E of S2n+1' restricted to Tn, is then given by

=

-JX

n1(sinus,-cosul,...,sinun+l'-'cosun+1).

Putting X1 = aX/aul, we have

0,-sinus,cosui,0,...,0,sinun+l,-cosun+l).

Xi =

n+l(0,....

51

i = 1,...,n. Thus X1,...,)cn are linearly independent and rI(Xi) = 0

for i = 1,...,n. Therefore the immersion X is anti-invariant. Moreover, the immersion X is a minimal immersion with 1A12 = n(n-1). From these considerations we have (Yano-Kon [4]) THEOREM 3.5, Let M be an n-dimensional compact anti-invariant

sutmanifold with parallel men curvature vector of S2n+1 (n > 1). If IA12 = n(n+1)/(2n-1), then M is S1 x Si.

4. CONTACT CR SU3IMIFOLDS

Let M be a (2m+l)-dimensional Sasakian manifold with structure tensors (p,E,n,g). We consider a Riemannian manifold M isometrically immersed in M with induced metric tensor field g.

Throughout in this section, we assume that the sutmanifold M is tangent to the structure vector field C of a Sasakian manifold M. Fbr any vector field X tangent to M, we put

qX = PX + FX,

(4.1)

where PX is the tangential part of OX and FX the normal part of OX. Similarly, for any vector field V normal to M, we put

V = tV + fV,

(4.2)

where tV is the tangential part of OV and fV the normal part of 4V. We easily see that P and f are skew-symmetr;.c. Fran (4.1) and (4.2)

we have

g(FXtV) + g(X,tV) = 0.

Moreover, we obtain

P2=-I-tF+n0 C,

FP+fF=O,

352

Pt + tf = 0,

f2 = -I - Ft.

Since we have 4F = PP + FE = 0, we find PP = 0 and FE = 0. For any vector field X tangent to M, we have 0X4 = -4X = 0

+ B(X,&). Thus,

we have the following

0

(4.3)

= -PX,

B(X,C) = -FX,

Especially, we have

= tV.

A.

0. Let X and Y be vector fields tangent

to M. Then we obtain

(4.4)_:-

(VxP)Y = AF X + tB(X,Y) + g(X,Y)E - n(Y)C,

(7XF)Y = -B(X,PY) + fB(X,Y).

(4.5)

For any vector field X tangent to M and any vector field V normal to

M, we also have

(4.6)

(TXt)V = AfVX - PA.X,

(4.7)

(7xf)V = -FAUX - B(X,tV).

Leett+M be an (n+l)-dimensional sub manifold of a Sasakian space

form M' 1(c). Then we have the following equations of Gauss and Codazzi respectively:

(4.8)

R(X,Y)Z =i,(c+3)[g(Y,Z)X-g(X,Z)Y] + 4'(c-1)1n(X)n(Z)Y

+2g(X,PY)PZ] +

(4.9)

AB(Y,Z)X

AB(X,Z)

Y,

(VXB)(Y,Z)-(VYB)(X,Z) = }(c-1)[g(PY,Z)FX-g(PX,Z)PY+2g(X,PY)FZ].

Moreover, we have equation of Ricci

(4.10)

g(Rl(X,Y)U,V) + g([AV,A)X,Y) = l(c-1)[g(FY,U)g(FX,V)-g(FX,U)g(FY,V)+2g(X,PY)g(fU,V)).

Definition. Let M be a submanifold tangent to the structure vector field ; isanetrically inmersed in a Sasakian manifold M. Then M is called a contact CR submanifold of M if there exists a differentiable distribution D : x

> Dx c Tx(M) on M satisfying the following

conditions:

(1) D is invariant with respect to 0, i.e., 4Dx c Dx for each

xcM, and (2) the canplementary orthogonal distribution D`: x

DX C_

X(M) is anti-invariant with respect to 0, i.e., 4DX c X(M)` for each x c M.

In the sequel, we put

dim M = 2n+1,

dim .M = 2nt1,

dimD`=p,

dim D = h,

codimM=2n-n=q.

If p = 0, then a contact CR suhmanifold is an invariant sut=nanifold

of M, and if h = 0, then M is an anti-invariant submanifold of M tangent to the structure vector field E. If q = p, then a contact CR submanifold M is called a generic submanifoZd of M. In this case we

have OX(M)l c T{(M) for every point x of M. If h > 0 and p > 0, then a contact CR sukmanifold M is said to be non-trivial (proper).

Let M be a contact CR snh nifold of a Sasakian manifold M. We denote by Z and l

the projection operators on D and DL- respectively.

Then we have

Z+l =I,

Z2=Z,

Z`2=Z`,

Since we have 4ZX = PZX + FIX, we obtain

ZZI =l l=0.

354

l'pl =O,

FL =0.

From $ZtX = WX + Fl-X we have PV- = 0, and hence PZ = P. Moreover we have

(4.11)

FP=O,

fF=0,

(4.12)

tf = 0,

Pt = 0.

Thus we find

(4.13)

P3 +P=01

(4.14)

f3 + f = 0.

These equations show that P is an f-structure in M and f is an fstructure in the normal bundle. Conversely, for a sutra nifold M, tangent to the structure vector

field F, of a Sasakian manifold M, we assume that we have FP = 0. Then we have IF = 0, (4.12), (4.13) and (4.14). We put

Z=-Pz+n0E,

l =I - Z.

Then we can easily verify that

Z+Z`=I,

Z2=Z,

Z`2=Z1,

ll`=ll=0,

which mean that Z and Z- are complementary projection operators and consequently define orthogonal distributions D and D` respectively. From Z = -P2 + n 0 E, we have PI = P because of p3 = -P and PE = 0.

This equation can also be written as Pt' = 0. But g(PX,Y) is skewsymmetric and g(V-X;Y) is symmetric and consequently Z''P = 0. Thus

we have Z'Pl = 0. On the other hand, we obtain FZ = 0 because of FP = 0

and

= 0. Consequently, the distribution D is invariant and DLis

anti-invariant with respect to 0. Moreover, we have Z&

1-

= 0

and consequently the distribution D contains E.

On the other hand, putting

V. = I+P2, we still see that Z and Zt define canplementary orthogonal distributions D and D1 respectively. We also have P1 = P, l P = 0, FZ = 0

and PZ'' = 0. Thus we see that b is invariant and D1 is anti-invariant with respect to 0 and also that ZE = 0, r = E, which mean that Dcontains F.

From these considerations we have (Yano-Kon [16])

THEOREM 41. In order for a sutmanifold M, tangent to the structure vector field F, of a Sasakian manifold M to be a contact (fit sub-

manifold, it is necessary and sufficient that FP = 0.

THEOREM 4.2. Let M be a contact B sutmanifold of a Sasakian manifold M. Then P is an f-structure in M and f is an f-structure in the normal bundle. LEM'1A 4.1. Let M be a contact CR suhnanifold of M. Then

AMY - A X = n(Y)X - n(X)Y

for X, Y E Dl.

Proof. Let X, Y be in D. Then PX = PY = 0, and hence

g((OZP)X,Y) = g(OZPX,Y) - g(PVZX,Y) = 0

for any vector field Z tangent to M. Frcm this and (4.4) we find

g(AZ,Y) + g(tB(Z,X),Y) = n(Y)g(Z,X) - n(X)g(Z,Y)

Thus we have our equation.

QED.

356

THEOREM 4.3. Let M be an (n+l)-dimensional contact CR sutmanifold of a (2m+1)-dimensional Sasakian manifold 2. The distribution D` is completely integrable and its maximal integral submanifold is a qdimensional anti-invariant submanifold of M normal to F or a (q+l)dimensional anti-invariant submanifold of M tangent to E. Proof. For any vector fields X and Y in D` we have

[X,Y] = P[X,Y] + F[X,Y] = -(VXP)Y + (VYP)X + F[X,Y]

= AMY - AX - n(Y)X + n(X)Y + F[X,Y] = F[X,Y].

Thus we have ¢[X,Y] E T(M), and consequently [X,Y] E D.

QED.

THEOREM 4.4, Let M be an (n+l)-dimensional contact CR submanifold of a (2m+1)-dimensional Sasakian manifold M. Then the distribution D is completely integrable if and only if

B(X,PY) = B(Y,PX)

for any vector fields X, Y e D, and then F E D. bbreover, the maximal integral suhnanifold of D is an (n+l-q)-dimensional invariant submanifold of M.

Proof. Let X, Y E D. Then (4.5) implies

q[X,YJ = P[X,Y] + F[X,Y] = P[X,Y] + (VYF)X - (VXF)Y = P[X.Y] + B(X,PY) - B(Y,PX).

Thus we see that [X,Y] E D if and only if B(X,PY) = B(Y,PX) for every X, Y e D. If D is normal to F, then g([X,Y],F) = 2g(PX,Y) for X,Y E D.

Thus, if D is completely integrable, we have g(PX,Y) = 0, which shows that dim D = 0. This proves our assertion.

QED.

We now give examples of contact CR sutmanifolds and generic submanifolds of a Sasakian manifold.

357 Exanple 4.1. Let S

l be a (2m+1)-dimensional unit sphere with

standard Sasakian structure (0,C,rl,g). We denote by Sm(r) an m-dimen-

sional sphere with radius r. We consider the following immersion:

S'(rl) x ... x

k

Smi(rk)

>

n+1 =

Sn+k,

m. i=1 1

where ml,...,mk are odd numbers and ri + ... + rk = 1. Here n+k is m. m.+1 (m +1)/2 m. 1 1 also odd. Let v. be a point of S 1(r.) in R = C . S 1(r 1 1 (m1+1)/2 -1 1 is a real hypersurface of C with unit normal ri vi. Thus v

= C(n+k+l)/2. We restrict (mi+l)/2 the almost caoplex structure of C (n+k+l)/2 to C . Then each _ (v1,...,vk) is a unit vector in

Jv

1

Rn+k+l

is tangent to S 1(r). Thus Jv is tangent to M 1 -n1,...,mk

x ... x Smk(rk). We then consider the normal space of

%

= S(r ) "1 1

,...,mk

in

Sn+k which is the orthogonal complement of the 1-dimensional space spanned by v in the space spanned by v1,...,vk. That is,

)`' $ T x(M ml P... 1Mk Let w1,...,wk

1

= ). Then wi

be an orthonormal frame for Tx(M

is given by a linear combination od v1,...,vk. Thus Jwi 1s tangent to

M

, and hence we see that

¢wi=Jwi-rl(wi)v=Jwi. Therefore

i is tangent to Mml,...,mk for T (

x ml,.,mk )A.

Consequently,

1,...,mk

c T (M

all i = 1,...,k-1. Thus

x ml,...,mk

).

is a generic submanifold of Sn+k. M ml,...,mk

has parallel second fundamental form and flat normal connection. Flu thermorer

r...r

is a contact CR submranifold of S1

K

(2m+1 > n+k) with parallel second fundamental form and flat normal connection. Example 4.2. In Example 4.1, if ri = (m./(n+l))1/2 (i = 1,...,k),

then M

n]., ... ,mlc

is a generic minimal submanifold of Sn+k and hence

minimal contact CR submanifold of S2"+1 (2m+1 > n+k). Then the square of the length of the second fundamental form of M is given by I Al 2 = (n+l)(k-1).

We notice that in Examples 4.1 and 4.2, if we put mi = 1 for all

i = 1,...,k (k = n+l), then M

is an anti-invariant submanifold

m1,...mk

of S2n+1

In the following we consider a contact CR submanifold with flat

normal connection. Let M be an (n+l)-dimensional contact (R submanifold of a (2m+1)-dimensional Sasakian manifold M. Then we have the following decomposition of the tangent space Tx(M) at each point x of M:

Tx(M) = H{(M) ®

Nx(M),

where Hx(M) = OHt(M) and Nx(M) is the orthogonal complement of

Hx(M) ®

in Tx(M). Then 4Nx(M) = F x(M) C Tx(M)1. Similarly, we

have

Tx(Mf = %(M) ® X(M)l, where Nx(M)- is the orthogonal complement of FN (M) in Tx(Mf . Then X

ONx(Mf = f x(M)1 = N (M)''. We now take an orthonormal basis el,...,e2m+1 of M such that,

restricted to M, el,...en+1 are tangent to M. Then el,...en+1 form an orthonormal basis of M. We can take el,...'en+l such that el,...ep form an orthonormal basis of N{(M) and ep+1,...,en form an orthonormal

basis of H{(M) and en+1 = F, where p = dim X(M). Moreover, we can of an orthonormal basis of Tx(MY such that 2m+1 form an orthonormal basis of FN (M) and en+2+p,..., en+2"** en+l+p x form an orthonormal basis of NX(MY. In case of need, we can e2m+1 take en+2' ...,e

take en+2"* ., en+l+p such that en+2 Fe V .. " en+l+p Fep .

35

Unless otherwise stated, we use the conventions that the ranges of indices are respectively:

i,j,k = 1,...,n+l;

x,y,z = 1,...,p;

a,b,c = p+1,...,n;

a,s,Y = n+2,...,n+l+p.

Here, we take S2n+1 as an ambient manifold M. Then we have

LDM 4.2. If the normal connection of M is flat, then

AfV=0

for any vector field V normal to M.

Proof. Since R` = 0, (4.10) implies that AVAU = AUAV. Thus, from (4.3), we obtain AVtU = AUtV. From this and tf = 0, we see that AfVtU = 0 and AfVA = 0. Moreover, from (4.7), we obtain (VXf)fV = 0. Thus, from (4.5) and (4.14), we have

g((VXf)fV,FY) = -g(f2V,(VXF)Y) _ -g(AfVX,Y) + g(Af2VX,PY) = 0.

From this and the fact that AfVAf2V = Af2VAfV, we have

TrAzfV = TrAfVPAfV = -TrAtVPAf2V = -TrAf2VAfVP = -TrAfVAf2VP = -TrAf2VPAfV = -TrA}.V:

Consequently, we have TrAf2.V = 0 and hence AfV = 0.

QED.

LEM V1 4.3. Let M be an (n+l)-dimensional contact CR submanifold of

S

1 with flat normal connection. If PAV = AVP for any vector field V

normal to M, then

g(AOX,AVY) = g(X,Y)g(tU,tV) -

Jg(AUtV,ei)g(A.Fe

i

X,Y).

i

Proof. From the assumption we have g(AUPX,tV) = 0, which implies

EE g((V A)UPX,tV) + g(AU(V P)X,tV) + g(AUPX,(VYt)V) = 0.

Thus, from (4.4) and (4.6), wee have

g((V A) PX,tV) - g(X,Y)g(AUg,tV) + n(X)g(AUY,tV)

+ g(AUAY,tV) + g(AUtB(Y,X),tV) + g(AUPX,AY) - g(AUPX,PAVY) = 0,

from which and Lemma 4.2, we find

g((4.,A)UPX,tV) + g(X,PY)g(tU,tV) + g(AUtV,tB(PY,X)) - g(AUPX,PAVPY) = 0.

On the other hand, we have

g(AUtV,tB(PY,X) = -lg(AUtV,ei)g(AFe X,PY),

-g(AUPX,PAVPY) = g(A PX,AVY).

From these equations we obtain

g((V

A)UPX,tV) + g(X,PY)g(tU,tV) - lg(AUtV,ei)g(AFe1X,PY) + g(AUPX,AVY) = 0.

Therefore, the Codazzi equation implies

g(X,PY)g(tU,tV) - lg(AUtV,ei)g(AF,e X,PY) + g(AUPX,AVY) = 0,

i

from which

i

369

(4.15)

g(PX,PY)g(tU,tV) - g(AUtV,ei)g(AFe PX,PY) + g(AUP2X,AVY) = 0.

On the other hand, we have

g(PX,PY)g(tU,tV) = g(X,Y)g(tU,tV) - n(X)n(Y)g(tU,tV) - g(FX,FY)g(tU,tV),

-lg(AUtV,ei)g(AFe.PX,PY) = -lg(AUtV,ei)g(AFe.X,Y) + n(Y)g(AUtV,X) + n(X)n(Y)g(tU,tV) - Jg(AUtV,ei)g(AFe X,tFY),

i

i

g(AUP2X,AVY) = -g(AUx,AVY) - n(Y)g(AUtv,x) - g(A.X A tFY).

Substituting these equations into (4.15), we find

g(X,Y)g(tU,tV) - lg(AUtV,ei)g(AFeX,Y) - g(AUX,AVY)

i

i

- g(FX,FY)g(tU,tV) - lg(AUtV,ei)g(AFe X,tFY) - g(AUX,AVtFY) = 0.

i

i

Moreover, we obtain

-lg(AUtV,ei)g(Ape X,tFY) = g(AUtV,AFYX) + g(FX,FY)g(tU,tV) - g(AUX,AVtFY)

g(AUtV,AFYX).

Fran these, we obtain our equation.

QED.

LEA 4,4, Let M be an (n+l)-dimensional contact CR sutmanifold of S

l with flat normal connection. If the mean curvature vector

of M is parallel, and if PAV = AVP for any vector field V normal to M, then the square of the length of -the second fundamental form of M is constant. Proof. Since AfV = 0, we have IA 12 = aTrAa2. On the other hand,

3'82

Le= 4.3 gives JA12 = (n+l)p +

E g(A(% te(I ,te0)TrA8.

a,$

Since the normal connection of M is flat, we can take {ea} such that Dea = 0 for each a, because, for any V F- FN(M) we have DXV a FN(M)

by (4.7) and AVtU = AUtV. Then we have

VXIAI2 =

E g((VXA)atea,tea)TrAa = I g((Vte A)atea,X)TrA3 a,s a,a a

by using VX(tea) _ (VXT)ea = Afe X - PAaX and Pt = 0. On the other

hand, using PAV = AVP, we have a

g((VPe A)aPei,X) = )[g((VPe P)A(X ei,X) + g(P(VPe A)aei,X) 1 1 1

i

- g( a(VPe P)ei,X)]. 1

Since Aa is symmetric and P is skew-symmetric, using (4.4), we see that

jg(Aa(VPe P)ei,X) = 0. 1

Jg((VPe P)Aaei,X) = 0, 1

i

i

Therefore, we have

lg((Vpe A)aPei,X) _ Ig(P(VPe A)oei,X) 1 1

i

Jg((Vpe A)aei,PX)

Ig((VpXA)aPei,ei) = 0,

i

i

1

where we have used the Codazzi equation and the fact that (VPXA)a is symmetric and P is skew-symmetric. Since we have E(VeaA)aea = E(Vpe A)aPei, the equation above implies 1

a (VeA)aea = 0.

Moreover, we see that

(VA)

= 0.

Since the mean curvature vector of M is parallel, we have

0 = E(Ve A)aei = j(Ve A)aea + (V&A) + J(Ve A)aex i x x i a a A)ate8. = }(Ve A)aex = Y(Vte

x

x

B

B

Therefore, the square of the length of the second fundamental form of M is constant.

QED.

Frcm Lemmas 4.2, 4.4 and Proposition 3.1 of Chapter II we have LF1l4A 4.5. Let M be an (n+1)-dimensional contact CR submanifold

of S1 with flat normal connection. If the mean curvature vector of M is parallel, and if PAV = AvP for any vector field V normal to M, then

IVA12 = -(n+1)ITTA22 + J(Tr a)2 a a

+

TrABTrAaAB. F (TrAaAB)2 a,B a,B

LEIT1A 4.6. Under the same assumptions as those of Lemma 4.5, the second fundamental form of M is parallel.

Proof. From Lemma 4.3 we obtain TrA2AB = TrAag(ea,eB) + jTr(AYAa)g(AYteateB),

Y TrAaAB = (n+l)g(ea,e8) + :TrAyg(AyteateB).

Y Fran these equations we have

(TrAaAB)2 = (n+1)jTrAa +

a,B

a

E TrAaABTrA a,B,Y

TrASTrA2A0 = -I(TYA a)2 a a,0

F

TrA A0TrA

Yg(Y

a,Y

a,te8)

Substituting these equations into the equation of Iemm 4.5, we find IVAI2 = 0, which shows that the second fundamental form of M is paraQED.

llel.

We prove the following theorems (Yano-Kon [16]). THEOREM 4.5. Let M be an (n+l)-dimensional ccnplete contact CR

submanifold of S1 with flat normal connection. If the mean curvature vector of M is parallel, and if PAV

AVP for any vector field V normal

to M, then M is Sn+l or

S"(r1) x ... X Smk(rk),

n+l = Imi,

1,

2 < k < n+l,

i

in some

Sn+1+P,

where m1,...,mk are odd numbers.

Proof. We first assume that F = 0, thst is, M is an invariant

submanifold of S

1. Then we have PAV + AVP = 0. Thus we have P.AV = 0

and hence AV = 0. Consequently, M is totally geodesic in S1 and M is Sn+1.

We next assume that F # 0. Since the second fundamental form of M is parallel and the normal connection of M is falt, by Theorem 5.5 of Chapter II, the sectional curvature of M is non-negative. On the other hand, from Lemma. 4.3, we see that AV # 0 for any V E EN (M). Thus

X Lemma 4.2 shows that the first normal space of M is of dimension p. Therefore, by Theorem 4.3 of Chapter II and Example 4.1, we have our QED.

assertion.

THEOREM 4.6. Let M be an (n+l)-dimensional complete generic

submanifold of S

1 with flat normal connection. If the mean curvature

vector of M is parallel, and if PAV = AvP for any vector field V normal

to M, then M is

Sml(rl) X ... X Smi(rk),

n+l = Imi, i

ri = 1, 1

2 < k < n+l,

0 where m1,...,mk are odd numbers. Let M be an (n+l)-dimensional contact CR suhmanifold of

S2m+1

with flat normal connection. Let V be a parallel vector field normal

to M. Then we have VXtV = PAvX. Hence we have div tV = -TrPAV = 0. Therefore, from Theorem 4.3 of Chapter I, we find

div(Vt,7tV) = S(tV,tV) + JIL(tv)gl2 - IvtVl2.

In the following we suppose that M is minimal. Then the Ricci tensor S of M is given by

S(X,Y) = ng(X,Y) - lg(A22X,Y). a

On the other hand, we have

IvtV12 = TrAA - g(tV,tV) - jg(AZatV,tV). a

From these equations we obtain

div(vtvtV) = (n+l)g(tV,tV) - TrA + JIL(tV)gl2.

Since the normal connection of M is flat, we can take {e such that De

a

of FN(M)

= 0 for each a. Thus we have

div(Pte tea) = (n+l)p - IA12 + JEIL(tea)gl2. a a a THEOREM 4.7. Let M be an (n+1)-dimensional contact minimal contact S2m+l -CR submanifold of with flat normal connection. Then

0 < J fMf IL(te(x )g12*1= IM[ IAI2 - (n+l)p]*1. a As an application of Theorem 4.7, we have

366

THEOREM 4.8. Let M be an (n+l)-dimensional ccmpabt minimal contact S2m+1 CR submanifold of with flat normal connection. If I Al 2 = (n+1)p, then M is

S71(rl) x ... X Smi(rk),

ri = (mi/(n+1))1/2

Sn+l+p,

n+l = Flni, Err = 1, 2 < i < n+1, in some

(i = 1,...,k),

where ml,...,nk are

odd numbers.

Proof. From the asswption ae have JL(tea)gj = 0 for each a. Thus we have

0 = (L(tea)g)(X,Y) = g(OXtea,Y) + g(VYtea,X) = g((AaP - PAa)X,Y)

for each a. Consequently, PAV = AvP for any vector field V normal to M. Therefore, our assertion follows from Theorem 4.5.

QED.

5. INDUCED STRUCTURES ON SUWANIFOLDS

Let M be an n-dimensional manifold. We assume that there exist on M a tensor field f of type (1,1), vector fields U and V, 1-forms u and v, and a function A satisfying the conditions:

(5.1)

f2X = -X + u(X)U + v(X)V,

(5.2)

u(fX) = Av(X),

(5.3)

u(U) = v(V) = 1 - A2,

v(fX) = -Au(X),

fU = -AV,

fV = AU,

u(V) = v(U) = 0,

for any vector field X on If. In this case, we say that the manifold M has an (f,U,V,u,v,A)-structure. Then we have (Yano-Okiunura (1])

367

THEOREM 5.1. A manifold M with (f,U,V,u,v,X)-structure is of even dimensional. Proof. Let x be a point of x at which A2 # 1. Then we see that U # 0 and V # 0 at x. Two vectors U and V are linearly independent.

Fbr, if there are two numbers a and b such that aU + bV = 0, then u(aU + bV) = au(U) = a(l - A2) = 0, v(aU + bV) = bv(V) = b(1 - A2) = 0.

Thus we have a = b = 0. Thus U and V being linearly independent at x, we can choose n linearly independent vectors X1 = U, X2 = V, X3,...,

a which span the tangent space Tx(M) and such that u(Xa) = 0, v(Xa) _ 0, for a = 3,...,n. Consequently, we have f2Xa = -Xa for all a, which shows that f is an almost complex structure in the subspace Ex of TT(M)

at x spanned by X3,...,Xn and that Ex is even dimensional. Thus Tx(M) is also even dimensional.

Next, let x be a point of M at which A2 = 1. In this case, we see that u(U) = u(V) = v(U) = v(V) = 0. We also see that if u # 0,

then v # 0, and if u = 0, then v = 0. We first consider the case in which u # 0, v # 0. Then u and v are linearly independent. Because, if there are two numbers a and b such that au + by = 0, then (au + bv) (fX) = A(bu - av)(X) = 0 and hence bu - av = 0, A being different from zero. Thus we have (a2 + b2)u = 0, from which a = 0, b = 0. Thus, u and v being linearly independent at x, we can choose n linearly independent covectors wl = u, w2 = v, w3,...,w1 which span the cotangent space X(M)* of M at x. We denote the dual basis by X11X21 ..., a. If U and V are linearly independent at x, we can assume that

n-1 = U,

Xn = V. Then we have

f2Xa = -Xa + u(Xa)U + v(a)V = -Xa,

a = 3,...,n

which shows that f is an almost complex structure in the subspace Ex of TA(M) at x spanned by X3,...,Xn and that Ex is even dimensional and consiquently T{(M) is also even dimensional.

If U and V are linearly dependent, there exist two numbers a and b such that aU + bV = 0 and a2 + b2 # 0. Applying f to this equation, we find A(-aV + bU) = 0, and hence bU - aV = 0. Thus, we must have

360

U = V = 0. Consequently, we have f2X = X for any vector X in X(M) and Tx(M) is even dimensional.

If u = 0, v = 0, we also have fzX = X for any vector X in TX(M) and consequently TA(M) is even dimensional. Thus we have our assertion.

W. The structure (f,U,V,u,v,A) is said to be normal if

S(X,Y) = N(X,Y) + du(X,Y)U + dv(X,Y)V = 0

for any vector fields X and Y on M, N being the Nijenhuis torsion of f.

We consider a product manifold M x R2, where R2 is a 2-dimensional Euclidean space. Then, (f,U,V,u,v,A)-structure gives rise to an almost carplex structure J on M x Rz:

J =

as we can easily check using (5.1), (5.2) and (5.3).

Computing the Nijenhuis torsion of J, we can easily prove PROPOSITION 5.1. If J is integrable, then (f,U,V,u,v,A)-structure is normal.

We assume that, in M with (f,U,V,u,v,A)-structure, there exists a positive definite Riemannian metric g such that

v(X) = g(V,X),

(5.4)

u(X) = g(U,X),

(5.5)

g(fX,fY) = g(X,Y) - u(X)u(Y) - v(X)v(Y).

We call such a structure a metric (f,U,V,u,v,X)-structure and denote it by (f,g,u,v,A).

Example 5.1. Let M be a (2n+1)-dimensional almost contact metric manifold with structure tensors (q,C,n,g). Then the structure tensors

0 satisfy

¢2=-I+n®&, n(X) = gQ,X),

OE=0,

n(4X) = 0,

n(;) = 1,

g(gX,¢Y) = g(X,Y) - n(X)n(Y)

for any vector field X on M.

Let M be a 2n-dimensional hypersurface of M. We denote by the same g the induced metric tensor field on M. The unit normal of M in M will be denoted by C.

Fbr any vector field X tangent to M we put

0 = fX + u(X)C,

v(X) = n(X),

C = V + AC,

4C = -U,

A = n(C) = g(c,C),

where f is a tensor field of type (1,1), u, v 1-forms, U, V vector fields and A a scalar function on M. Then they satisfy

f2X = -X + u(X)U + v(X)V,

fU = -AV,

fV = AU,

u(U) = 1 - A2,

u(fX) = Av(X),

u(V) = 0,

v(fX) = -Xu(X),

v(U) = 0,

v(V) = 1 - A2.

Moreover, we have

g(U,X) = u(X),

g(V,X) = v(X),

g(fX,Y) = -g(X,fY),

g(fX,fY) = g(X,Y) - u(X)u(Y) - v(X)v(Y).

Therefore, the hypersurface M admits (f,g,u,v,A)-structure. Example 5.2. Let M be a real (2n+2)-dimensional almost Hermitian manifold with structure tensors (J,g). Then J2 = -I and g(JX,JY) =

370

g(X,Y) for any vector fields X and Y on R. Let M be a si+hnifold of cod.imension 2 of M with orthonormal frame C, D for X(M -. We put

JX = fX + u(X)C + v(X)D,

JC = -U + AD,

JD = V - AC.

Then the induced structure (f,g,u,v,A) on M satisfies (5.1) - (5.5). Therefore, the sutmanifold M admits an (f,g,u,v,A)-structure. ExcvnpZe 5.3. Let M be a (2n+1)-dimensional Sasakian manifold and

let M be a hypersurface of M. We have the Gauss and Weingarten formulas

OXY = VXY + g(AX,Y)C,

VXC = -AX.

Then we have

(VXf)Y = -v(Y)X + u(Y)AX + g(X,Y)V - g(AX,Y)U,

VXU = AX + fAX,

VXV = -fX + AAX.

On the other hand, we obtain

S(X,Y) = (VfXF)Y - (Vfy f)X + f(VYf)X - f(VXf)Y

+ g(VXU,Y)U - g(VYU,X)U + g(VXV,Y)V - g(VYV,X)V = u(Y)(AfX - fAX) - u(X)(AfY - fAY).

Thus, if Af = fA, then the hypersurface M is normal. We shall prove the converse. Let S(X,Y) = 0 for all X and Y and put PX = (Af-fA)X. Then

u(U)PX = u(X)PU.

Also, it can be shown that g(PX,Y) = g(X,PY) so that

31 u(X)g(PU,Y) = u(Y)g(PU,X),

that is to say,

g(PU,Y) = au(Y)

for some a. Thus we have

u(U)g(PX,Y) = u(X)g(PU,Y) = au(X)u(Y),

but since the trace of P is zero, we have a = 0, i.e., P = 0, which means that Af = fA. Consequently, M is normal if and only if Af = fA.

372

EXERCISES

A. K-CONTACT SUBMANIFOLDS: Let Si be a (2n+1)-dimensional K-contact

Riemannian manifold with structure tensors

A sutmanifold M

of M is said to be invariant in Si if OX is tangent to M for any tangent

vector field X to M, and

is always tangent to H. Any invariant sub-

manifold M of a K-contact Riemannian manifold M is also a K-contact Riemannian manifold with respect to the induced structure on M. Then we have (Endo [1]) THEOREM 1, Any invariant submanifold M of a K-contact Riemannian manifold Si is a minimal submanifold.

On the other hand, we obtain (cf. Kon [1]) THEOREM 2. Let M be an invariant submanifold of a K-contact

Riemannian manifold M. Then M is totally geodesic if and only if the second fundamental form of M is parallel.

B. COSYMPLECTIC MANIFOLDS: Let M be a normal almost contact metric manifold such that the fundamental 2-form 0 is closed and do = 0. Then M is called a cosymplectic manifold. The cosymplectic structure is characterized by a

= 0 and OXn = 0. Let M be an

invariant submanifold of a cosyrrplectic manifold M. Then M is also a cosymplectic manifold with respect to the induced structure on M.

Ludden [1] proved the following THEOREM. If M is a cosymplectic manifold of constant 0-sectional curvature and M is an invariant submanifold of codimension 2 of M which is n-Einstein, then M is locally symmetric. C. FLAT NORMAL CONNECTION OF INVARIANT SU3 ANIFOLDS: Let M be

a Sasakian space form of constant -sectional curvature k and M be an invariant submanifold of Si. Then we have

THEOREM. The following conditions are equivalent: (a) The normal connection of M is flat, i.e., R` = 0;

373

(b) k = 1 and M is totally geodesic in M.

In the case of codimension 2, Kenmotsu [2] proved the above theorem. The theorem above was proved by Kon [3] when the codimension of M is greater than 2.

D. AXIOM OF 4-HOLOMORPHIC PLANES: Let M be a (2n+1)-dimensional Sasakian manifold with structure tensors

We say that m

admits the axiom of 4-holcooorphic (2r+1)-planes if, for each point x

of M and any (2r+1)-dimensional ¢-holamrphic subspace S of Tx(M), 1 < r < n, there exists a (2r+1)-dimensional totally geodesic suhnanifold N passing through x and satisfying T{(N) = s, where we mean a 0-holomorphic subspace S by a subspace of Tx(M) satisfying OS C S. I. Ishihara [2] proved the following

THEOREM. A Sasakian manifold is of constant -sectional curvature if and only if the manifold satisfies the axiom of ct-holomorphic (2r+1)-planes.

E. REDUCTION THEOREMS OF ANTI-INVARIANT SUBMANIFOLDS: I. Ishihara [2] studied the reduction theorems of codimension of anti-invariant sukmanifolds of Sasakian space forms. Let M be an (n+l)-dimensional

anti-invariant sutmanifold, tangent to the structure vector field of a (2m+1)-dimensional Sasakian manifold M. If Eiha

,

= 0 for all

indices a and x, then the mean curvature vector of M is said to be pseudo-parallel. (For the ranges of indices, see §2.) The normal

connection of M is said to be pseudo-flat if R&y = 0 for all indices. Then THEOREM 1. Let M be an (n+l)-dimensional (n > 3) anti-invariant sutmanifold, tangent to the structure vector field F, of a Sasakian

space form M1(c) (c # -3) with pseudo-parallel mean curvature vector. If the normal connection of M is pseudo-flat, then there is

in Ml(c) a totally geodesic and invariant sutmanifold M2n+1(c) of dimension 2n+1 in such a way that M is immersed in M2n+1(c) as a flat

anti-invariant sutmanifold. Let M be an n-dimensional anti-invariant sutrranifold, normal to

the structure vector field F, of a (2m+1)-dimensional Sasakian

374 manifold M. If Ekh

d

= 0 for all indices i and p, then the mean

curvature vector of M is said to be n paraZZeZ. (Fbr the ranges of indices, see §3.) For the normal curvature tensor It we consider the condition

(*)

g(E' (X,Y)U,V) = g(4Y,V)g(4X,U) - g(0X,V)g(¢Y,U)

for any vector fields X, Y tangent to M and any vector fields normal

to M. Then we have (I. Ishihara [3]) THEOREM 2. Let M be an n-dimensional (n > 3) anti-invariant sukmanifold, normal to the structure vector field E, of a Sasakian space.-. form

M2n+1(c) (c # -3) with n -parallel mean curvature vector.

If it satisfies (*), then there is in M2n+l(c) a totally geodesic and invariant submanifold M2n+1(c) of dimension 2n+1 in such a way M2n+l(c) that M is immersed in as a flat anti-invariant minimal suhmanifold.

F. CONFORMALLY FLAT ANTI-INVARIANT SUBMANIFOLDS: Let M be an (n+l)-dimensional anti-invariant suhmanifold, tangent to the structure vector field E, of a (2m+l)-di.nensional Sasakian manifold M. If the

second fundamental form B of M satisfies

B(X,Y) = [g(X,Y)-n(X)n(Y)]a + n(X)B(X,E) + n(Y)B(X,E)

for any vector fields X, Y tangent to M, where a denotes a normal vector field to M, then M is said to be a contact totaZZy wnbilical.

We have (Kon [10]) THEOREM 1, Let M be an (n+l)-dimensional (n > 3) contact totally

umbilical, anti-invariant submmifold, tangent to the structure vector field E, of a (2m+l)-dimensional Sasakian manifold M with vanishing contact Bochner curvature tensor. Then M is locally a product of a confonnally flat Riemannian manifold Mn and a 1-dimensional space M1.

Yano [8] proved the following

375

THEOREM 2, Let M be an n-dimensional (n > 3) totally umbilical anti-invariant sutxmanifold, normal to the structure vector field E,

of a (2m+1)-dimensional Sasakian manifold M with vanishing contact Bochner curvature tensor. Then M is conformally flat.

G. GENERIC SUBMANIFOLDS: Yano-Kon [la] proved the following THEOREM, Let M be an (n+l)-dimensional complete generic minimal

submanifold of

t+1

with parallel second fundamental form. If M is

Einstein, then M is

Sq(r) x ... x Sq(r) (N-times),

r = (q/(n+1))1/2,

where q is an odd number and 2m-n = N-1, Nq = n+1. H. PSEUDO-UMBILICAL HYPERSURFACE: Let M be a hypersurface of a Sasakian manifold M, tangent to the structure vector field. If the second fundamental form A of M is of the form

AX = a[X-n(X)E] + bu(X)U + n(X)U + u(X)C

for any vector field X tangent to M, a and b being functions, then M is called a pseudo-umbilical hypersurface of M, where a vector field U and a 1-form u are defined to be U = -4C, u(X) = g(U,X) respectively for a unit normal C of M. The notion of pseudo-umbilical hypersurfaces of Sasakian manifolds corresponds to that of n-umbilical real hypersurfaces of Kaehlerian manifolds. If M is a pseudo-umbilical hypersurface of

52n+1 (n

> 2), then

M has two constant principal curvatures with multiplicities 2n-1 and 1 respectively. Then we have (Yano-Kon [101) THEOREM 1. Let M be a compact pseudo.-umbilical hypersurface of

52n+1 (n > 2). Then M is

S2n-t(rl) x S1(r2),

r1 + r2 = 1.

376

A Sasakian manifold M of dimension 2n+1 is said to satisfy the P-axiom if for each x e M and each 2n-dimensional subspace S of Tx(M),

a S, there exists a pseudo-umbilical hypersurface N such that Tc(N) = S, x e N and g(AU,U) = a+b = constant. Yano-Kon [10] proved the following

THEOREM 2. If a (2n+1)-dimensional Sasakian manifold M (n > 2) satisfies the P-axiom,

is a Sasakian space form.

To prove the theorem above, we need the following theorem of Tanno [7].

THEOREM 3. A (2n+1)-dimensional (n > 2) Sasakian manifold M is a Sasakian space form if and only if R(X,gX)X is proportional to OX for any vector field X of M such that n(X) = 0. 1. PSEUDO-EINSTEIN HYPERSURFACES: Let M be a 2n-dimensional

hypersurface of 52+1 tangent to the structure vector field. If the Ricci tensor S of M is of the form

S(X,Y) = a[g(X,Y)-n(X)n(Y)] + bu(X)u(Y) + n(X)S(l;,Y) + n(Y)S(i;,X) - n(X)n(Y)SQ,E),

a and b being constant, then M is called a pseudo-Einstein hypersurface S2n+1.

of

Yano-Kon [10] proved the following theorems (see also Kon

[12]).

THEOREM 1. Let M be a pseudo-Einstein hypersurface of

S2n+l

(n > 3). Then M has two constant principal curvatures or four constant principal curvatures. 5"n+1.

We give examples of pseudo-Einstein hypersurfaces of

Let

Cn+l be the space of (n+l)-tuples of complex numbers

Put S2n+1 = {(zl,...,zn+l) E 0+1 : EIz.I2 = 1}. For a positive number r we denote by MO(?rn,r) a hypersurface of 52n+1 defined by n+1

n F.

j=1

Iz.I2 = 1.

rizn+ll2,

j=1

J

377

For an integer m (2 < m < n-1) and a positive number s a hypersurface M(2n,m,s) of S2n+1 is defined by

m

n+1 IZjl2 = S

n+1 j1llzjl2 = 1.

Izj12,

jI

For a number t (0 < t < 1) we denote by M(2n,t) a hypersurface of S2n+1 defined by

n+1 I

n(+'1

ZJIZ = t,

L

j=1

IZjl2 = 1.

j=1

MO(2n,r) and M(2n,m,s) have two constant principal curvatures and M(2n,t) has four constant principal curvatures (Nomizu [ ], R. Takagi [ ]). MO(2n,r) is always pseudo-Einstein. M(2n,m,s) is pseudo-

Einstein if s = (m-l)/(n-m) and M(2n,t) is pseudo-Einstein if t = 1/(n-1).

THEOREM 2. If M is a canplete pseudo-Einstein hypersurface in S2n+1 (n > 3), then M is congruent to some MO(2n,r) or to some M(2n,m,(m-1)/(n-m)) or to M(2n,l/(n-1)).

J. HYPERSURFACES WITH (f,g,u,v,A)-STRUCTURE: Let M be a 2n-dimensional hypersurface of

S2n+1.

Then M admits an (f,g,u,v,A)-structure.

Nakagawa-Yokote [1] proved the following THEOREM 1. Let M be a compact hypersurface of S2n+1 satisfying Af = fA = 2af, a being function. If n > 3, then one of the following two assertions (a) and (b) is true: (a) M is isometric to one of the following spaces: (1) the great sphere S2n; (2) the small sphere S2n(c), where c = 1+a2;

(3) the product manifold S2n-1(c1) x S1(c2), where c1 = 1+a2, c2 = 1+1/a2;

(4) the product manifold Sn(c1) x Sn(c ), where c1 = 2(1+x2 +a(1+a2)1/2), c2 = 2(1+x2-a(1+a.2)1/2);

(b) M has exactly four distinct constant principal curvatures a±(1+a2)1/2, (-1±(1+a2)1/2 )/a with maltiplicities n-1, n-1, 1 and 1, respectively.

378

In the case that (b) holds, the hypersurface M is congruent to

some M(2n,t) in Exercise (I) (see R. Takagi. (3]). When the second fundamental tensor A com sites to f, Bon [12]

proved THEOREM 2, Let M be a coaplete hypersurface of Stn+1 (n > 2). If Af = fA, then M is congruent to S2n(a2+1), a = v(AV)/(1-X2), or to S2p+1(r1) X S2q+1(r2). p+q = n-1.

379

CHAPTER VII

f-STRUCTURES

In this chapter, we study a manifold which admits an f-structure. In §1, we define an f-structure on a manifold and give a necessary and sufficient condition for a manifold to admit an f-structure (Yano [41). We also give integrability conditions of an f-structure (Ishihara-Yano [1].

§2 is devoted to the study of the normality of an

f-structure (S. Ishihara [1]). In §3, we consider a manifold with a globally framed f-structure (Goldberg-Yano [2]). In the last §4, we discuss hypersurfaces of framed manifolds and give some theorems (Goldberg [2], Goldberg-Yano [1]).

1. f-STRUCTURE ON MANIFOLDS

A structure on an n-dimensional manifold M given by a non-null tensor field f satisfying

f3 + f = 0 is called an f-structure (Yano [4]). Then the rank of f is a constant, say r (Stong [1]). If n = r, then an f-structure gives an almost complex structure of the manifold M and n = r is necessary even. If M is orientable and n-1 = r, then an f-structure gives an almost contact structure of the manifold M and n is necessary odd.

We put

Z=-f2,

m=f2+I,

380 I denoting the identity operator, then we have

Z+m=I,

12=Z,

ft = If = f,

m2=m,

mf = mf = 0.

These equations show that the operators Z and m applied to the tangent space at each point of the manifold M are complementary projection operators. Then there exist in M two distributions L and N corresponding to the projection operators Z and m respectively. When the rank of f is r, L is r-dimensional and N (n-r)-dimensional. We now introduce in M a local coordinate system and denote by fi, 1i, Ti the local components of the tensors f, Z, m respectively. We also introduce a positive definite Riemannian metric in M and take r mutually orthogonal unit vectors ua (a,b,c,... = 1,2,...,r) in L and n-r rutually orthogonal unit vectors uA (A,B,C,... = r+l,...,n)

in M. We then have Ziub

ub h i h miUB = UB.

miub = 0,

From fm = 0, that is, fim = 0, we find, contracting with uB and taking account of the last equation of (1.1),

fiuB = 0.

If we denote by (vi,vi) the matrix inverse to (ub,uB), then vi and

vi are both components of linearly independent covariant vectors and satisfy

ai

a

viuB = 0,

viub = 0,

viub = 6B

v iu = b (1.2)

301

and (1.3)

iu+viA=bi.

Now from (1.1) and (1.3), we find

(Zhva)

h a

i = day

x hub hvA

i

(Zivh)uB = 0

b

(mivh)uB = dB,

= 0,

which show that

Zhlvha

mivh = vi,

= vi,

from which

(1.4)

mivh = vA.

mivh = 0,

From mf = 0, that is, fimh = 0, we find, contracting with vA and taking account of the last equation of (1.4),

ua, we find

On the other hand, from

Zjiua

via'

Zj(di

that is,

(1.5)

Zi

i a

by (1.1) and (1.3). Similarly, we get

- viA)

Vial

382

h= viuA. Ah

(1.6)

mi

If we change ub and uB into ub and uB respectively by orthogonal transformations

ub

cb a'

uB

A'

where

c

= d cb, cc*b

cCCB = 'CB,

then v a and vi are change into va and vA respectively by the rules

-a V.

bb

= cavi,

-A BB vi = cAvi,

and we have vivi = vivi,

vivi = vivi.

Consequently, if we put

v

(1.7)

v i +

a globally defined positive definite Riemannian metric

with respect to which (ub,uB) form an orthogonal frame such that

(1.8)

i a v = aj iua,

A

as we can easily verify it. If we put

Zji =

we find, from (1.5) and (1.6),

i

vj = aj iuA,

mji =

383

mji =

Zji =

because (1.8). Consequently

Zji + mji = aji,

that is, Zji and mji are both symmetric and their scan is equal to aji.

We can easily verify the following relations:

Zji,

0,

mji.

If we put

gji =

(aji +

then gji is again a globally defined positive definite Riemannian metric satisfying

j =

9J-ii

mji = mjgti-

Thus the distributions L and N which were orthogonal with respect to aji are still orthogonal with respect to gji and uA which were mutually orthogonal unit vectors with respect to aji are still mutually orthogonal unit vectors with respect to gji. We can easily verify that gji satisfies

If we put figst = fit-

we have

384

mji = gji

and

mji = -gji.

Hence we have

ft(fit + fti) = 0. The rank of f being r and n-r linearly independent solutions of 0 being given by vt, these equation give

fit + fti = vAW

for certain WA, from which

= 0, or

fit+fti=0 by fituB = fivi = 0 and ftiuB = (ftuB)gji = 0. Thus fji is skewsymmetric tensor of rank r, hence r must be even. Gathering the results,

we have (Yano [4]) THEOREM 1,1, Let M be an n-dimensional manifold with f-structure f of rank r. Then there exist complementary distributions L of dimen-

sion r and N of dimension n-r and a positive definite Riemannian metric g with respect to which L and N are orthogonal and such that

fjfigts + mjgti = gji' fji = -fij,

fji = fjgti

Thus the rank r of f must be even. Take a vector uh in the distribution L, then the vector fhui is also in L and orthogonal to uh, and moreover has the same length as uh with respect to gji. Consequently, we can choose in L r = 2m mutually orthogonal unit vectors ub such that

395

+l -

iul ' um +2 -

i2....... u2m = f um.

Then with respect to the orthogonal frame (ub,uB), the tensors gji and fii have components:

(1.9)

f=

,

g =

IM denoting (m,m)-identity matrix. We call such a frame an adapted frame of the structure f.

Now take another adapted frame (ub,uB) with respect to which 9ji

and fii have the same components as (1.9) and put

h =

nb

-h- Ah

ah

rbua

uB = rBuA,

then vie-can easily see that the orthogonal matrix

m0

A I

r

mm 0

0

1

0

n-2m

must have the form

An

r=

Bm

0

m Am

0

0

0n-2rn

0

396

Thus the group of the tangent bundle of M can be reduced to U(m) x O(n-2n). Conversely, if the group of the tangent bundle of M can be reduced to U(m) x 0(n-2n), then we can define a positive definite Riemannian metric g and a tensor f of type (1,1) and of rank 2m as tensors having (1.9) as corponents with respect to the adapted frames. Tben we have

f3 + f = 0. Thus we have (Yano [4] ) THEOREM 1.2, A necessary and sufficient condition for an n-dimensional manifold M admit an f-structure f of rank r is that r is even (r = 2m) and the group of tangent bundle of M be reduced to the group U(m) x 0(n-2n). In the next place we consider the integrability conditions of the distributions L and N.

The Nijenhuis tensor Nf of f is given by

Nf(X,Y) = [fX,fY] - f[fX,Y] - f[X,fY] - Z[X,Y].

Then we have the following identities:

Nf(mX,mY) = ZNf(mX,mY) = -Z[mX,mY],

mNf(X,Y) = m[fX,fY],

mNf(ZX,ZY) = m[fX,fY],

f(P) = m[ZX,ZY]. Since Z = -f2, If = f, if we have Nf(ZX,ZY) = 0 for all vector fields X and Y, then Nf(fX,fY) = 0 and conversely. We also see that the following three conditions are equivalent:

(i) mNf(X,Y) = 0, (ii) mNf(ZX,ZY) = 0, (iii) mNf(fX,fY) = 0

for any vector fields X and Y. The Lie derivative Lyf is, by definition, given by

387

(LYf)X = f[X,Y] - [fX,Y].

We also have

Nf(ZX,mY) = f(LYf)ZX = f[Z(Lyf)ZX],

from which

fNf(ZX,mY) _ -Z(Lmyf)ZX.

The distribution N is integrable if and only if Z[mX,mY] = 0 for any vector fields X and Y. Thus we have

PROPOSITION 1,1. A necessary and sufficient condition for N to be integrable is that

Nf(mX,mY) = 0

or equivalently

ZNf(mX,mY) = 0

for any vector fields X and Y.

The distribution L is integrable if and only if m[ZX,ZY] = 0 for any vector fields X and Y. Thus we have

PROPOSITION 1.2. A necessary and sufficient condition for the distribution L to be integrable is that one of the conditions (i), (ii) or (iii) be satisfied.

Because of Z + m = I, Nf(X,Y) can be written in the form

Nf(X,Y) = ZNf(ZX,ZY) + mNf(ZX,ZY) + Nf(ZX,mY) + Nf(mX,ZY) + Nf(mX,mY).

Thus we have

388

PROPOSITION 1,3. A necessary and sufficient condition for both of two distributions L and N to be integrable is that

Nf(X,Y) = INf(ZX,ZY) + Nf(ZX,mY) + Nf(nX,ZY)

for any vector fields X and Y. Suppose that the distribution L is integrable and take an arbitrary vector field X' in an integral manifold of L. We define an operator f' by f'X' = fX'. Then f' leaves invariant tangent spaces of every

integral manifolds of L, and f' is an almost complex structure on each integral manifold of L. For any vector fields X' and Y' tangent to integral manifold of L, we denote by N'(X',Y') the vector-valued two form corresponding to the Nijenhuis tensor of the almost complex structure induced on each integral manifold of L from the structure f. Then

we have

Nf(ZX,ZY) = N'(ZX,ZY)

for any vector fields X and Y on M.

If the distribution L is integrable and moreover if the almost

complex structure f' induced from f on each integral manifold of L is integrable, then we say that the f-structure is partially integrable. We have the following theorem.

THEOREM 1.3. A necessary and sufficient condition for an f-structure to be partially integrable is that one of the following equivalent conditions be satisfied:

Nf(ZX,ZY) = 0,

or

Nf(fX,fY) = 0

for any vector fields X and Y on M.

Next, from Propositions 1.1, 1.3 and Theorem 1.3, we have

389

PROPOSITION 1.4. In order that the distribution N is integrable and the f-structure be partially integrable, it is necessary and sufficient that

Nf(X,Y) = Nf(ZX,mY) + Nf(mX,ZY)

for any vector fields X and Y.

PROPOSITION 1.5. The tensor field Z(Lnyf)Z vanishes identically for any vector field Y if and only if

Nf(ZX,mY) = 0

for any vector fields X and Y.

When two distributions L and N are both integrable, we can choose a local coordinate system in such a way that L's are represented by putting n-r local coordinates constnat and N's by putting the other r coordinates constant. Then the projection operators Z and m can be supposed to have the ccagonents

0

0

respectively. Since f satisfies fl = if = Z and fm = mf = 0, f has the components

f =

0

0

where fr is an (r,r)-n atria. Thus, for a vector field mY in M, the

Lie derivative LWf has components

390

LmYf =

Hence, if we assume that the tensor field Z(LmYf)Z vanishes identically for any vector field Y, then we have L Yf = 0, which means that the

m

components of f are independent of the coordinates which are constant along the integral manifold of L in an adapted coordinate system. Conversely, if the components of f are independent of these coordinates,

then we easily see that Z(LmYf)Z vanishes. Thus we have PROPOSITION 1.6. Suppose that the two distributions L and N are both integrable and that an adapted coordinate system has been chosen. A necessary and sufficient condition for the local components of the f-structure to be functions independent of the coordinates which are constant along the integral manifold of L is that

Nf(ZX,mY) = 0

for any vector fields X and Y.

From Propositions 1.3 and 1.6 we have PROPOSITION 1.7. Suppose that L and N are both integrable and that an adapted coordinate system has been choosen. The carponents of f are independent of the coordinates which are constant along the

integral manifold of L if and only if

Nf(X,Y) = INf(ZX,ZY)

for any vector fields X and Y. We now assume the following: (a) f is partially integrable, that is, Nf(X,Y) = Nf(ZX,mY) + Nf(mX,ZY);

391

(b) The distribution N is integrable, that is,

Nf(mX,mY) = 0; (c) The carponents of f are independent of the coordinates which

are constant along the integral manifolds of L in an adapted coordinate system.

In this case we say that the structure f is integrable. Combining Propositions 1.1, 1.7 and Theorem 1.3 we have (Ishihara-Yano [1]) THEOREM 1,4, A necessary and sufficient condition for the structure f to be integrable is that

Nf(X,Y) = 0

for any vector fields X and Y on M.

When the structure f is integrable, the components of f have the

f

foam

01

Or 0

in an adapted coordinate system and fr is an (r,r)-

matrix whose elements are functions independent of the coordinates which are constant along the integral manifolds of L. Since fr defines a complex structure on an integral manifold of L, we can effect a change of adapted coordinate system in such a way that fr becomes 0

fr =

-Im I

M

0

where r = an. The converse being evident, we have (Ishihara-Yano [1]) THEOREM 1.5. A necessary and sufficient condition for an f-structure to be integrable is that there exist a coordinate system in which f has the constant components

0

f=

Im 0

r = an being rank of f.

m 0

0

0

0

-I

0

,

392

2, NORMAL f-STRUCi1RE

Let M be an n-dimensional manifold with an f-structure f. We denote by L and N the distributions corresponding to the projection operators Z and m respectively. We put r = rank f. Then L is r-dimensional and N (n-r)-dimensional. Let U be an arbitrary coordinate neighborhood of M. If we take in M arbitrary an adapted set {fx} of n-r vector fields fx spanning the distribution N at each point, then there exists uniquely in U an ordered set {fy} of n-r 1-forms fy such that 2n-r +l Efx

0 fx

= m,

fx(fy)

= Y

where the indices x,y,z,... run over the range {n+1,n+2,...,2n-r}.

We then have fy(fX) = 0,

ffx = 0

for any vector X at each point of M. We call such an ordered set {fx} an (n-r)-frame and the ordered set {fy} an (n-r)-coframe being dual to {fx}.

If a 1-form 0, global or local, satisfies OX = 0 for any X E L, 4>

is said to be transversal to L. It is expressed uniquely by 0 = 0yfy.

Similarly, any vector field v in N is expressed uniquely by v = v fx. Denoting by fb, fy, fb respectively the components of f, fy, fx with respect to local coordinates (na) in U, where the indices a,b,c,... run over the range {1,2,...,n}. Then we have

a c a_= fbfy y a - db,

fbfc

fafx = O,

ca

fyfc = 0,

fyfx = dy.

393

The set of all tangent vectors belonging to the distribution N > M over M, which is a subbundle

forms a vector bundle p : N(M)

of the tangent bundle T(M) of M. Let N(M)* be the vector bundle dual belonging to the fibre N* of N(M)*

to N(M). If we take an element

at x e M, then there exists at x uniquely a 1-form

of M, which is

transversal to L, such that qv = Ov for any element v belonging to the fibre x of N(M) at x. Conversely, for any 4> transversal to L at x, there exists uniquely

of N* such that cv = Ov for any v c N.

Thus N(M)* can be identified naturally with the set of all 1-forms transversal to L, and hence the bundle N*(M) can be regarded as a subbundle of the cotangent bundle T(M)* of M. In a coordinate neighborhood U, {fx} is a basis of N{ and {fy} is a basis of N*. Let v be a vector field of N and 4, be a transversal 1-form to L. Then we have v = vxfx, 4> = 0f3' in U with functions vx and 0

defined in U. We call

(vx) and (0y) the components of v and p respectively with respect to f

x Let U and U' be twv coordinate neighborhoods of M such that

U a U' # 6. If Ifx) and {fx,} be (n-r)-frames in U and U' respectively, then we have

fy, - AY, y fy

in U A U', where the matrix (Ay,) is a function in U A U'. Taking v mr N and a 1-form 0 transversal to L, we have Oy'fyo

V=vfx,

0 _ 4yfY,

v = vx'fx "

0 =

respectively in U and in U', and

V = AX,v in U A U', where (AX') =

, ,

4y = Ayo0y,

X,)-1

let there be given a connection w* in N(M). Then w* has n(n-r)2 components

y with respect to local coordinates (na) of M and an

c

394

(n-r)-frame { X} in U. Denotong by ray and ra.x0y, the carponents of the given connection w* respectively with respect to {na,fx} and

to

we find in U A U' rc,xvy,

i AX'(r

_

X AY, + 3 Ay.),

an

where ac = mane. Taking a vector X and a 1-form p at a point x of M, we consider an element Tx(X,p) of the fibre FR of the vector bundle

N(M)* 0 N(M) at x and suppose for VX,p) to be bilinear with respect to its arguments X and p. The correspondence s : (X,p)

> x(X,p)

is called an Fr_valued tensor of type (1,1) at x. If there is given

a correspondence T : x

> T{, it is called an N(M)* 0 N(M)-valued

tensor field of type (1,1) and its differentiability is naturally defined. Let v and ¢ be respectively a vector field and a 1-form and T an N(M)* 0 N(M)-valued tensor field of type (1,1). Denote by T(V,c) a cross-section of N(M)* 0 N(M) such that its value at x is given by

Tx(vxOx), where vx and ¢x are respectively the values of v and 0 at x E M. Then T(av,T¢) = aTT(v,q) for any functions a and T. T(X,p) is locally expressed by

2nr T(X,p) _

n

7Ccp Tb,

b,c=1

b

c

Tb =

c

bxfy y 0 f x

x,y n+lTc

with functions Tcbyx defined in U and Xa, ph are respectively canponents

of X and p with respect to (rya). Fbr any vector field v = vxfx E N any transversal 1-form

yfy,

if we put Vcvx = acvx + rcxyvy,

V 0 = achy - rcxyoxI c y

then it is easily verified that (Vcvx)fx and

are globally

defined covariant vector fields in M which take their values respectively in N(M) and,in N(M)*. In this sense, we call

Vcv = Vcvxfx

and

Vc¢ = Vc¢yf

395

or simply Vcvx and Vcy, respectively the covariant derivatives of v and 0 with respect to the connection w*.

Let there be given a linear connection w in M and denote by rc

a b

its components with respect'to local coordinates (r1a) in U. If we

consider now an N(M) 0 N(M)*-valued vector field Ta, then we can put

Ta = Tayxfx 0 fy

in U with components Tayx with respect to {na,fx}. On putting in U

VcTayx

z 7A x ya = acr x + rcabT"y + rc z1 yz - rc y z .

then the tensor field

VcTa = (VcTayx)fx 0 fy

defined in each neighborhood U determines globally in M a tensor field of type (1,1) which takes values in N(M) 0 N(M)*. In this sense, we call VcTa, or VCTayx, the covariant derivative of Ta with respect to connections w and w*.

In the same way, we can define the covariant derivatives of tensor field of any mixed type. Sunning up, if there are given connections w and w* respectively in M and N(M), we can introduce the covariant differentiation Vc operating on tensor fields of any mixed type. In general case, two connections w and w* may be given independently. However, if there is given a linear connection w in M, then there exists in N(M) a connection w* defined by components

rcxy = (2cfy + re dfy)

e,

where rc b are the components of the given linear connection w. By identifying each tangent space of the fibre of N(M) with the fibre itself, the tangent space TQ(N(M)) of N(M) at a e N(M) is

expressible as a direct sum by

396

To(N(M)) = TT(M) ® x = Lk ® Nx ®

x,

x being the point p(a) of M. There exists naturally an identification

Nx

> Fx.

Let there be given a connection w* in the vector bundle N(M). Taking a tangent vector X of the base space M at x, we denote by X*

the horizontal lift of X at each point a of the fibre p1(x) with respect to the connection w*. We define a linear operator a applied to the tangent space a(N(M)) of the manifold N(M) at a point x by

JQ(X*) = (fx)*,

JQ(Y*) = j(Y),

Ja(Z) = -0 -1(Z))*,

where X, Y and Z belong respectively to Lx, N and

x being the

point p(a). It is easily verified that the operator Ja defined in each tangent space Ta(N(M)) determine an almost coaplex structure J in the manifold N(M), i.e., J2 = -I. We shall now give the tensor representation Ji of the almost corrplex structure J. The fibre of N(M) being (n-r)-dimensional vector

space Rn-r, the collection P-1 (U) = U x Rn-r of local product of N(M) over U's forms an open covering of N(M). In p-1 (U), any element v of N(M) such that p(v) c U is expressed by (na,vx), where (na) are coordinates of p(v) and v = vxfx. Any tangent vector of the bundle space N(M) is expressed by

if the tangent space of Rn-r are identified with Rn-r itself. That

is to say, (navx) are local coordinates defined in each p-1 (U) of N(M).

Let there be given a linear connection w* iq N(M) and rcxy its components with respect to local coordinates (na) and (n-r)-frame {fx} in U of M. Then, in the tangent space of N(M) at any point (na,vx) of p 1(U) x Rn-r , the horizontal plane is defined by a linear

39!

equation

Vx + ra`va

(ra' = ra yvy),

and the vertical plane is defined by a linear equation

Va = 0.

If, in each tangent space of N(M), we consider a frame consisting of 2n-r vectors V(i) with carponents V(i) such that

V(b)

cv(b))

(V(y)

V(y)

1V(b)

then V(b) are horizontal and V(y) are vertical, where the indices h,i,j,... run over the range {l,2,...,n,n+l,...,2n-r}. We now define in each tangent space of N(M) a linear operator J by 2n-r

n

J(V(b))

+1

b

(x)'

n faV(a).

J(V(Y)) aE=1

We see that J has the components

(db -rb

0

fb

-fYl

fb

0

a

0 l-1

f db

x b

-r

y

,

-faY fZ'ex x

x yre

398 We can easily verify that

- a 2 0

ab

0

0

ax y

Consequently, we obtain J2 = -I. Hence J is an almost canplex structure in the bundle space N(M). Stunning up, we have

THEOREM 2,1, If a manifold M admits an f-structure f of rank r, then there exist almost complex structures in the bundle space of the (n-r)-dimensional vector bundle N(M) over M. Given a connection with

components Icxy in N(M), then an almost canplex structure J = (J) is determined by (*).

In the next place, we define a tensor field Scba of type (1,2) by

Scba = Ncba + (2cfb -

-

-

abfc)fa

(fcrb u

fur

zc

u)fa,

where Ncba is the Nijenhuis tensor of f, i.e.,

eba = fc2efb - fb2efe - (2cfb - abfc)fe.

We put

Scbx = fe(a e fb

- fb(2ef - 2c e)

2b e)

b

Scat' = fy2efc - fc2efy

+

Scyx = fy(2efC

- 2c e)

Saxy = fx2efy - fy2efx -

fe2cfy

+

- (fefb -

fcfzl e Zy,

+ fcfyrexz - Tcxy'

(fxI'ezy - fylezx)fz

Then we have

Scba + Sbca = 0,

Scbx + Sbcx = 0,

Saxy + Sayx = 0.

exz

In term of covariant differentiation Oc these are expressed respectively as follows:

Scba = Ncba + (V fb - Vbf,)fa,

Scbx = fe

b - Ob

e) - fb(o efc - O c fe

Sa c y = fy e fea - fc e fya + fe cfy , Scyx = fe(oefx - Oce), a

Saxy = fxe Oefy -

fy efx

We denote by Hjih the Nijenhuis tensor of the almost ccrrplex

structure J of N(M), which is given by

Hjih =

J181J - (aji -

with respect to local coordinates (rla,vx) in P 1(U) = U X Rn-r, where nx

we have put

= vx and 2i = Vans. We find

Hcba = Scba - (Tcz3baz - TbzS. z) + - (fe R c

ebz - fb eC z)fa

z

+ (Tczfb ebu

cbx = Scb - (Tczczx - TbzScz ) -

a

Tczrbus

- Tb feR

Scberex

cu)fu,

+ (TczSbez

- TbzScez)Tex - TczTb SzyeTex + (Rcbx - fCb ed )

zfeF UfdR x + (Tczfefd z c edx - T z b - T bzfefd)R

c zb ued

+

(feR

- fbbRecz

z

d-

ze a a a cy = Sc Y + TczSzy + ce fy z

(Tczfz eb -

e

)f;rd

400

Hcy

a

= Sc

a

y

+ T czSzya + R

ce

zfyfa ,

Hcyx = Scyx - Scey"ex + rczSyzerex +

+

fcfy

edx

zfdrd

x + feeR

ed

c z

y eC

,

Hzya=Sazy, Hx zy =

zy e Serx

-

efd - fz yRcd

cbyxVY and Rebyx is the curvature tensor of the linear byx being defined by

where Rcbx

connection rcy, R Rcby

x

x

x

= 8crb y - abrc y + rc

x

z

x

HlihJ

= 0,

z

r - rb zrc y. z b y

Noticing that

Hj.lhJi + HiikJk = 0,

Hi.lhJi -

we obtain

Scb

x

ae

xfey

= Sce fbfa - ey

c b'

Shay = Sdeafbfy - Se zyfbfe, x

= Sdef cy f

Sa

= -S

zy

e

x d e

Scy

f + S zycfxe,

afcfb.

cb z Y

These identities show that

Scba = 0 implies Scbx = 0, Scay = 0, Scyx = 0, Sazy = 0

(cf. Nakagawa [1], Ishihara-Yano [1]).

401

When the almost complex structure J is conplex in the bundle space N(M), we say that the given f-structure f is normal with respect to a connection w* given in the vector bundle N(M). f is normal if and only if Hjih vanishes identically. If Hjih = 0, then

(2.1)

Scba = 0, Scb = 0, Scya = 0, Scyx = 0, S

= 0.

Therefore we have Rcbx = 0, and hence Rcbyx

(2.2)

= 0.

Conversely, if we assume that (2.1) and (2.2), then Hjih = 0.

Therefore, we obtain PROPOSITION 2.1. A necessary and sufficient condition for an f-structure f to be normal in M with respect to a connection w* given in the vector bundle N(M) is that the tensor fields Scba' Scbx' Scay' Scyx' SaZY

vanish identically and the connection w* is of zero curvature. If Scba vanishes, then Scbx, Scay, Scyx, SaZ, are equal identically to zero: Thus we have THEOREM 2.2. A necessary and sufficient condition for an f-structure f to be normal in M with respect to a connection w* given in the vector bundle N(M) is that the tensor field Scba vanishes identically and the connection w* is of zero curvature.

If an f-structure in M is normal with respect to a connection given in N(M), then the connection has zero curvature by means of

Theorem 2.2. Thus, M assumed to be simply connected, the vector bundle N(M) is trivial, that is, it is a product bundle. Therefore, we have PROPOSITION 2.2. If an f-structure in M is normal with respect to a connection given in the vector bundle N(M), then the vector

bundle N(M) induced from N(M) by the covering projection n : M -> M is trivial,- where M is the universal covering space of M.

402

If the vector bundle N(M) is trivial, then it is naturally identi-

fied with the product space M x Rn-r. Then there exists naturally a

connection m of zero curvature in N(M). In this case, a necessary and sufficient condition for f to be normal with respect to WS is

Scba = cba + (2cfb - 8bfe)fa = 0 (cf. Nakagawa [1]).

3. FRAMED f-STRUCTURE

Let M be an n-dimensional manifold with an f-structure f of rank r. If there are n-r vector fields Ea spanning the distribution N at

each point of M, and if there exist n-r differential form na satisfying

a(Eb) = db,

where a,b,c,... = 1,...,n-r, and if

f2=-I+na0Ea, then M is said to have a globally framed f-structure or, simply, a framed structure. We call such a manifold M a framed manifold, and denote it by M(f,Ea,na). We see that

fEa = 0,

0,

a = 1,...,n-r.

A framed manifold M(f,Ea,,a) is called a framed metric manifold if a Riemannian metric g on M is distinguished such that (i) na = a = 1,...,n-r, and (ii) f is skew-symmetric with respect to g. It can be shown that a framed manifold carries a metric with these properties.

We put F(X,Y) = g(fX,Y) and call it fundamental 2-form of the framed

manifold. A framed metric manifold M(f,n,g) a is said to be covariant

constant if the covariant derivatives with respect to g of its structure tensors are zero. For a framed manifold M, if we put

f = f + n2i

n2i-1

® Eli-1 -

0 E

i = 1,...,[(n-r)/2],

,

an almost contact structure f is defined on M if dim M = n = 2m, and an almost contact structure

1) if n = 2m-l. If M is

a framed metric manifold, then an almost complex structure f is defined on M with n = 2m in terms of which the metric g is Hermitian. Setting

F(X,Y) = g(fX,Y), we obtain

F = F +

21n2i

n2i-1

A

i

If the fundamental 2-form F and the na are closed forms, the almost Hermitian structure on M is almost Kaehlerian. It is Kaehlerian if f has vanishing covariant derivative with respect to g, that is, if the structure tensors f and Ea are covariant constant with respect to the metric g.

THEOREM 3.1. A covariant constant even dimensional framed manifold M carries a Kaehlerian structure. M(f,na,g)

In the odd dimensional case the framed metric structure gives rise to the almost contact metric structure M(f,1 - 2m -r-1, g). Then

g(fX,fY) = g(X,Y) -

n2n-r-1(%)n2m-r-1(Y).

We put

_ =

f,

E = Z2m-r-1'

71 = n

2m_r--1

then 2i-1 = F + 21n2i A 'n

,

4(X,Y) = g(fX,Y),

404

If the fundamental 2-form @ and the 1-form n are closed, the almost contact structure on M is almost cosymplectic. It is cosymplectic if and only if the almost contact structure is normal.

THEOREM 3.2. A covariant constant odd dimensional framed manifold M carries a cosymplectic structure.

Proof. Since f and na have vanishing covariant derivatives with respect to the Riemannian metric g, so does 0. Thus we can easily verify that the torsion of 0 vanishes. Therefore, M(q,E,n) is normal. QED.

In the next place vie consider the relation of the normality of

the framed structure and the integrability (normality) of the induced almost complex (almost contact) structure. First of all, we have

LEMMA 3.1. Let M(f,Ea,na) be a framed manifold. Then

X(na(Y)) _ (LXna)(Y) + na([X.Y]).

dna(Eb,X) _ (LEbna)(X),

dna(fX,Y) _ (LfXna)(Y).

Since the f-structure is framed, there exists a connection of

zero curvature in N(M). Then a framed f-structure f is normal if

S=Nf+dna'0Ea vanishes. LEMMA 3.2. Let M(f,Ea,na) be a normal framed manifold. Then

LEnb=0, a LE'`f = 0,

[Ea,Eb] = 0,

dna(fX,Y) + dna(X,fY) = 0

a for any vector fields X and Y and a,b = 1,...,n-r.

405 Proof. Since the structure of M is normal, we have

Nf(X,Y) + dna(X,Y)Ea = 0,

from which

-f[fX,Eb] + f2[X,Eb] + da(XEb)Ea = 0,

that is,

f(

f)X + da(XEb)EzL = 0.

Taking the interior product of the both sides of the equation above nb = 0.

by roc, we obtain dnc(X,Eb) = 0 which is equivalent to I

a

Moreover, we obtain

f2[Ea,Eb] + dnc(Ea,Eb)Ec = f2[Ea,Eb] = 0,

and hence f[Ea,Eb] = 0. Thus, [Ea,Eb] =

for some function dab

on M. Then we get nc([Ea,Eb]) = acab = 0.

We also have fi.E f = 0, so (I M = ua(X)Eb for some function

a

a

)i a(X). Consequently, we get r1b((I

0 = (I,E a

n f))X = ((I,Ear1b)f)X + fb(,Eaf)X), f = 0 for all a.

the ya(X) vanish, that is, l

Next, we find

f)X) = ua(X). Thus, since

a

na([fX.fY]) + dna(X,Y) = 0,

so that

0 = na([fX,f2Y]) + dna(X,fy)

_ -r1a([fX,Y]) + fX(na(Y)) + drla(X, fY) .

0 On the other hand,

have

.

fX(na(Y)) - Y(na(fX)) - na([fX,YI) = dna(fX,Y).

This proves the last equation.

QED.

THEOREM 3.3. Let M(f,Ea,na) be a an-dimensional normal framed manifold of rank r, a = 1,...,2m-r. Then the induced almost complex n2i-1 structure f = f + n2i ® E2i-1 ® E2i on M is integrable. Proof. Let Nf be the torsion tensor of f. By a straightforward

computation, vve find dn21(X,fY))E21 Nf(X,Y) = Nf(X,Y) + dna(X,Y)Ea + (dn2i(fX,Y) + (dn2i-1(fX,Y)

+ dn2i-1(X,fY))E21 + n2i(X)(LE f)Y 2i-1 - n 2i(Y)(LE2i-1 n2i-1 f)X 2i -

2i

E

2i-1].

Our theorem is now a consequence of Lemma 3.2.

QED.

We state the following converse of Theorem 3.3.

THEOREM 3.4. Let M(f,Ea,na) be an even dimensional framed manifold of rank r, a = 1,...,n-r, where induced almost complex structure f = n2i n2i-1 ® Eli-1 ® E2i is integrable. Then, if (a) the dna are of bidegree (1,1) with respect to f, i.e., dna(fX,Y) + dna(X,fY) = 0, (b) the vector fields Ea are holomnrphic and (c) [E2i-1,E2i] = 0, the f-structure is normal. If M(f,Ea,na) is an even dimensional normal framed manifold, dna then the are of bidegree (1,1) with respect to the induced almost

complex structure f. We have

dna(fX,Y) = dna(fX,Y) - r

(X)dna(E2i-1,Y)

+ n2i-1(X)dna(E2i,Y).

407

But, by Lemma 3.2,

dna(Eb,Y) = Eb(na(Y)) - na([Eb,Y])

= %(rja(y))

- a(V )

= Eb(na(Y)) - LEb(na(Y)) = 0.

THEOREM 3,5, Let M(f,Ea,na) be a (2m-1)-dimensional normal framed manifold of rank r, a = 1,...,2m-r-1. Then, the induced almost contact - n2i-1 ® structure f = f + n2i 0 Eli on M is normal. EZi-1 Proof. Let S be the torsion tensor of f. Then

S(X,Y) = [fX,fY] - f[fX,Y] - f[X,fY] - [X,Y]

+ dn2m-r-1(X,Y)EZm-r-1 +

[fX,fY] - f[fX,Y] - f[X,fY] - [X,Y] + (na([X,Y]) + dna(XY))% - (,,r-1([X,Y]) + dn2m-r-1(X,Y))E2r-1 +

(n2m-r-1([X,Y]

+ dn2m-r-1(X,Y))E2m-r-1

= Nf(X,Y) + dna(X,Y)Ea = 0.

Thus,

(f,E2m-r-1'n2m-r-1)

is normal.

QED.

We also have the following converse.

THEOREM 3.6. Let M(f,E,na) be an odd dimensional framed manifold a n2i ® E2 of rank r where induced almost contact structure f = f + ;_1 nai dna 1 0 E2i is normal. Then, if (a) the are of bidegree (1,1) with respect to f, (b) LE f vanishes and (c) [E2i-i,E21] 0, the

=

f-structure is normal.

a

408 4. WYPERSURFACES of FRMIED MANIFOLDS

Let M(?,Ea,na) be a framed manifold of dimension n and rank r, a = 1,...,n-r. We consider an (n-1)-dimensional hypersurface M immersed in d such that: Fbr each x e M the vectors Ea, a = 1,...,n-r-1 belong to the tangent space TX(M) and :!En_r 4 Tx(M). The vector field E

En-r is then an affine normal to M, so we may write

Ix =fX+B(%)E,

TE =0,

where f and B are tensor fields on M of type (1,1) and (0,1), respectively. If B = 0, the submanifold is an invariant hypersurface of M. On the other hand, if 0 # 0, it provides a measure of the deviation of M from this property. Such a hypersurface will be called noninvariant or a normal variation of M. A hypersurface may, of course, be neither invariant or noninvariant. However, in the sequel, unless otherwise specified, M will be a noninvariant hypersurface of the framed manifold M.

First of all, we have -x + na'(X)Ea = f2X + B(fX)E.

Since there are vector fields E on M such that

Ex = Ex,

x = 1,...,n-r-1,

we obtain

f2 = -I + nx 0 Ex,

cB=n,

nx =

nx,

n=n-n-r

cB is the 1-form on M defined by c8(X) = B(fX).

x = 1,...,n-r-1,

409 Moreover, we obtain

x(Ey = nx(Ey) = 6y ,

T1

f X = 0,

e(Ex) = 0,

fEx = fEx + e(Ex)E,

x = 1,...,n-r-1.

THEOREM 4,1. A noninvariant hypersurf ace of a framed manifold

admits a framed structure of the same rank as the ambient manifold. Moreover, it admits a 1-form a determining an (n-r-1)-dimensional distribution complementary to the distribution determined by the Pfaffian system nx = 0, x = 1,...,n-r-1.

If M is integrable, we obtain THEOREM 4.2. A noninvariant hypersurface of a normal framed

manifold M(,Ea,na) is a normal framed manifold of the same rank r carring a 1-form whose differential has bidegree (1,1) with respect to the induced f-structure.

Proof. Given a symmetric affine connection 0 on M, an affine connection V is defined on M with respect to the affine normal E by the Gauss formula

pXY = VXY + h(X,Y)E,

where h is the second fundamental tensor of the immersion with respect to E. We express ST in the form

Sf(XX,Y) = (O

f)Y - (V T)X + 1(DY7f )X - 1(OXf)Y

+ [(OXna)(Y) - (DYna)(X)]Ea

for any vector fields X and Y on M. For any vector fields X and Y on M, by a straightforward computation, we obtain

Sf(X,Y) = Sf(X,Y) + [de(fX,Y) + de(X,fY)JE.

410 From the equation above we have our assertion.

QED.

Let M(f,ja,g) be an n-dimensional framed metric manifold with rank r. Let F be the fundamental 2-form of M defined by F(X,Y) = g(i!,Y). Let F be the fundamental 2-form of the induced f-structure

on M. We then have

F(X,Y) = g(fX,Y) = g(fX,Y) + e(X)j(Y) = F(X,Y) + (e A c6)(X,Y),

that is,

F=F+6Ac6. Since f is not of maximal rank, the tensor field

G=g-c6®c6 is not a Riemannian metric. However, if n = 2m+1 and r = 2m, f is of maximal rank, so G defines a positive definite metric. In this case, it is easily checked that G is Hermitian with respect to f. In fact,

if F is closed, G is an almost Kaehlerian metric and F + 0 A CO is the fundamental 2-form of the almost Kaehlerian manifold M(f,G). If the structure on M is integrable, then M is Kaehlerian.

THEOREM 4.3. In addition to the canonical framed metric structure

(f,rlx,g) the noninvariant hypersurface M(f,Exrix) admits the framed metric structure (f,nx,g*), where g* = g + 6 ® 6.

Proof. We have

g(fX,Y) + 6(X)n(Y) = -g(X,fY) - 6(Y)n(X).

Hence we have g(fX,Y) + 6(X)ce(Y) = -g(X,fY) - e(Y)c6(X),

411

that is,

(g + 6 0 e)(fX,Y) = -(g + 6 0 e)(X,fY).

Moreover, we obtain r1X(X) = g(X,EX) = g*(X,£X) - e(X)e(EX) = g*(X,EX)

W.

412 EXERCISES

A. AUTOMORPHISMS: Let M(f,Ea,rla) and M'(f',Ea,n'a) be framed

manifolds of the same rank. A diffeemorphism p of M onto M' is called an isomorphism of M onto M' if

f' u#

µ*Ea = Ea.

and

If M' = M and f' = f, Ea = .,

,a

=

a, a = 1,...,n-r, then p is

said to be an automorphism of M. Then we have (Goldberg-Yano [2]) THEOREM. The group of automorphisms of a compact framed structure is a Lie group.

B. PRODUCT FRAMED MANIFOLDS: Let M(f,Ea,na), a = 1,...,n-r, and M(f,Ex,nx), x = i.....n-r, be framed manifolds. An f-structure may be defined on the product manifold M x M canonically as follows: For any X E Tp(M) and X e Tp--(M), we put

f(X,%) = (fX,fX).

Then

f2 = -(I,I) + (na e Ea,O) + (O,nx ® Ex),

where 0 is the zero vector and I, I the identity tensors of M, M respectively. Clearly, f3 + f = 0 and f has rank r + I'. We put

Ex)'

Ea = (E a.10),

En-r+x = (0,

na

= (na'0)'

Then

fEA = 0,

A(EB) = dB'

n-r+x

= (0,nx).

413 where A,B =

Thus M x M carries a framed structure

(f,EA,nA) of rank r+r. Then we have (Goldberg [3)) THEOREM. The direct product of two normal framed manifolds is a normal framed manifold.

For the product framed manifolds see also Millman [1] and Nakagawa [1].

C. QUASI-SYMPLECTIC MANIFOLDS: An even dimensional framed metric manifold M(f,Ea,g) of rank r is called quasi-symplectic if the fundamental 2-form F is closed and parallel along the integral curves of

the vector fields Ea. It is symplectic if dim M = 2n and r = 2n. A quasi-symplectic manifold with zero torsion will be called an integral quasi-symplectic manifold. Goldberg [4] proved

THEOREM, The betti number b2q(M) of a conpact integral quasisymplectic manifold M is different from zero for q = 0,1,...,r/2. For the quasi-symplectic manifolds see also Goldberg-Yano [3].

D. HYPERSURFACES: Let M be a hypersurface of a framed manifold M(f,Ea;na). We suppose that M is covariant constant, that is, Of = 0 and ona = 0 for all a. If for any vector field X on M, OXE is propor-

tional to E, then M is said to be totally flat, where E is the affine normal of M. Then we have (Goldberg [2]) THEOREM, A noninvariant hypersurface of a covariant constant

framed manifold is a totally flat covariant constant framed manifold and the connection in the affine normal bundle is trivial. If the hypersurface is invariant, it is also totally geodesic.

414

CHAPTER VIII

PRODUCT MANIFOLDS

In this chapter we give the fundamental results concerning geometry of product manifolds. We also prove some theorems of submanifolds of product manifolds. In §1, we discuss locally product manifolds. §2 is devoted to the study of locally decomposable Riemannian manifolds. We give some properties of Riemannian curvature tensors of locally decomposable

Riemannian manifolds. Moreover, we define an almost product Riemannian manifold. In §3, we discuss st

n folds of almost product Riemannian

manifolds which are invariant with respect to an almost product structure. In the last §4, we consider Ka.ehlerian product manifolds

and its sulinanifolds. For the theory of product manifolds we refer to Tachibana [2]

and Yano [5]. 1. LOCALLY PRODUCT MANIFOLDS

Let us consider an n-dimensional manifold M which is covered by such a system of coordinate neighborhoods (Eh) that in any intersec.

tion of two coordinate neighborhoods

al = Eas(E;a), with

and (E

Cx, _

) we have

e, (E,),

415

IaxEx'I # 0,

Iaaa"I # 0,

where the indices a,b,c,... run over the range 1,2,...,p and the indices x,y,z,... run over p+1,...,p+q=n. Then we say that. the manifold

M admits a locally product structure defined by the existence of such a system of coordinate neighborhoods called separating coordinate system. We call locally product manifold a manifold which admits a locally product structure.

Let vh = (va,vx) are components of a contravariant vector, then

(Va,0) and (0,v) are also components of contravariant vectors. Similarly, if wi = (wb,wy) are components of covariant vectors and also if, for example,

b

Ty

IX

Y

are components of a tensor, then f74b

0

0

10

0

to

Ty

0 x)

are all components of tensors of the same type as T. For vh = (va,v we see that (v ,-vx) are also components of a contravariant vector. This process may be represented by vh

Then we have

F-OFi =

Ihj,

> Five, where

416 where I denotes the, identity tensor, I = 6

.

If we put 0

db

0

0

0

6x

}y,,

_

,

0

Qh =

0

in a separating coordinate system, then Ph is an operator which

projects any vector

vh = (va,vx)

into

Phvl = (va,0)

and Qh is an operator which projects

into

vh =
Qhv1 = (0,vx).

These operators may be expressed in terms of Ii and I

P h =

( I h +

) .

Qh = i(Ih -

as follows

),

from which

Ih=Pi+Qh,=Ph-Qh. If a tensor Th has components of the type

I

b

0

I

Th = i

10

yj

ve say that Th is pure, and if it has coq)onents of the type

b

0

417

we say that Ti is hybrid. Similarly if a tensor field Tai has components of the type

0

we say that Tai is pure, and if it has components of the type

Tji =

we say that Tji is hybrod.

We denote by BF the system of subspaces defined by

Cx

= const.

and by Mq the system of subspaces defined by Ea = const. Then the manifold M is locally the product Mp x Mq of two manifolds. Now, is given in the manifold M.

suppose that a linear connection

The linear connection r has the following transformation law: rh'

E

_

J'i'

(a

3Ci

a

.-y-,h aEi

Ji

+ a1 aEi

).

We can see the quantities

a rcy'

a rzb'

a rzy'

I,x cb'

rxzb'

rx cy

are all carponents of tensors. We can prove the following propositions.

PROPOSITION M. An arbitrary contravariamt vector tangent to IF, displaced parallelly along 0, is still tangent to MP, if and only if 17b = 0.

If this condition is satisfied, we say that MP is totally geodesic in M.

418 PROPOSITION 1,2. An arbitrary contravariant vector tangent to MP, displaced parallelly along Mq, is still tangent to MP, if and only

ifrb=0. PROPOSITION 1.3. An arbitrary contravariant vector tangent to MP, displaced parallelly in any direction, is still tangent to Mp if and only if rcb = 0 and rZb = 0.

PROPOSITION 1.4. An arbitrary contravariant vector tangent to Mq, displaced parallelly along MP, is still tangent to M4, if and only if rCY = 0.

PROPOSITION 1.5. An arbitrary contravariant vector tangent to Mq, displaced parallelly along Mq, is still tangent to Mq, if and only

if rary= 0. In this case Mq is totally geodesic in M.

PROPOSITION 1.6. An arbitrary contravariant vector tangent to Mq, displaced parallelly in any direction, is still tangent to Mq, if and only if ray = 0 and ray = 0. In this case, Mq is parallel in M.

2. LOCALLY DECOMPOSABLE RIEMANNIAN MANIFOLDS

Suppose that, in an n-dimensional locally product manifold M, there is given a Riemannian metric

ds2 =

which satisfies

tj FFi gts = gji Then we see that, g 1 is pure, and consequently that

ds2 = gcb(VdcdEb + 9Zy(E)dE;Zdy.

419

or equivalently that the manifolds LIP and Mq are orthogonal. We call

such a manifold a locally product Riemannian manifold.

It is

easily seen that gih is also pure. If we put Ftgj = Fji, then Fji = Fij and in fact

F..l = J

the tensor Fji is pure. If we put

-g

0

pji = pjgti' Qji = Qjgti'

then we

have also

0

0 Qji =

0

0

and

ji = pji - Qji' Suppose that the matrix has the form

ds2

that is,

b + gZy(

-

are functions of

d ',

)d

a only, gcy = 0, and gZy(t;) are

functions of F only, then we call the manifold a locally decomposable Riemannian manifold.

For a locally decomposable Riemmnnian manifold M, we have

(2.1)

r c b - 0,

r

= 0,

= 0,'

r

= 0.

cy

Conversely, if (2.1) are satisfied in a locally product Riemaannian

manifold, we can easily prove that gc(E) are functions of Ea only, gcy = 0 and g2,(F) are functions of

only. Thus we have

THEOREM 2.1. A necessary and sufficient condition for a locally product Riemannian manifold M to be a locally decomposable Rienannian manifold is that (2.1) are satisfied in a separating coordinate system. Combining the propositions in the previous section and this theorem, we have THEOREM 2.2. A necessary and sufficient condition for a locally product Riesnnnian manifold MP x Mq to be a locally decomposable Riemannian manifold is that MP and Mq are both totally geodesic and are parallel.

On the other hand, we obtain

V Fb = 0,

VJFF =

VJFX = 0,

7jFb =

(2.2)

21'

.

Thus, from Theorem 2.1 and (2.2), we have THEOREM 2.3. A necessary and sufficient condition for a locally product Riemannian manifold M to be a locally deconposable Riemannian manifold is that

vjr

= o,

or equivalently

vjFih = 0.

Since, in a locally decomposable Riemnnnian manifold M, the

Christoffel symbols rji are all zero except rob and Zy, and besides, rob are functions of a only and I'xZy are functions of Ex only, the

curvature tensor, (2.3)

=

R jih

akrji

ajrki

+ ktrji

rjtrki

421

is pure in all its indices and also the successive covariant deriva-

tives of R ih, V1Rkjih' VmV1Rkjih' .. are all pure in its indices. Naw suppose that gcb and ggy are both Einstein, then

(2.4)

cy = u gzy

Rcb = 'gcb'

for certain constants A and u provided that p,q > 2, where Rji is the

Ricci tensor of M. The equations (2.4) may also written in the form

Rji = +j(A+.i)gji + 4(A-u)Fji.

Conversely suppose that the Ricci tensor of a locally decomposable

Riemannian manifold has the form

Ri=agji+bF

ji.

Then we have

RZy = (a-b)gZy.

cb = (a+b)gcb'

Thus, if p,q . 2, then gcb and gzy are both metrics of Einstein manifolds, and a+b = A, a-b = p are both constants. Thus we have THEOREM 2.4. In a locally decomposable Riemannian manifold Mp x Mq (p,q > 2), a necessary and sufficient condition that the two components are both Einstein is that the Ricci tensor of the manifold has the form

Rji=agji+bFjl, a and b being necessarily constant.

We next suppose that the two components are both of constant curvature. Then we have

Rdcba = A(gdagcb - geagdb)'

cyxw = u(gzwgyx - gywgzx)

for certain constants X and u. The equations above may also written in the form

Rkjih = *(X+u)[(gkhgji - gjhgki) + (FkhFji - FjhFki)1

+ *U-11)[(F

gji - Fjhgki) + (gkhFji - gjhFki)1.

Conversely suppose that the curvature tensor of a locally decomposable Riemannian manifold has the form

Rkjih = a[(gkhgji - gjhgki) +

(F

Fji

- FjhFki)]

+ b[(Fkhgji - Fjhgki) + (gkhFji - gjhFki)].

Then we have

Rdcba = 2 (a+b) (gd-.gcb - gcagdb) Raw = 2(a-b)(gzwgyx - gywgzx). Thus if p,q > 2, then gcb and gzy are both metrics of manifolds of constant curvature, and 2(a+b) = X. 2(a-b) = u are both constants. Thus we have THEOREM 2,5. In a locally decomposable Riemannian manifold

MP x 0 (p,q > 2), a necessary and sufficient condition that two components are both of constant curvature is that the curvature

tensor of the manifold has the form

Rkjih = a[(gkhgji - gjhgki) + (FkhFji - FjhFki)] + b[(Fkhgji - Fjhgki) + (gkhFji - gjhgki)]'

a and b being necessarily constant. In the following we define an almost product manifold. Let M be an n-dimensional manifold with a tensor field F of type (1,1) such

that

F2 = I. Then we say that M is an almost product manifold with almost product structure F. We put

Q = j(I - F).

p = }(I + F),

Then

P+Q= I,

P2 = P,

Q2 = Q,

PQ=QP=0,

F = P - Q. Thus P and Q define two complementary distributions P and Q globally. We easily see that the eigenvalues of F are +1 or -1. An eigenvector corresponding to the eigenvalue +1 is in P and an eigenvector corresponding to -1 is in Q. Thus, if F has eigenvalue +1 of multiplicity p and eigenvalue -1 of multiplicity q, then the dimension of P is p and that of Q is q.

Conversely, if there exist in M two globally caiplementary distributions P and Q of dimension p and q respectively, where p + q = n and p,q > 1. Then we can define an almost product structure

F on M by F = P - Q. If an almost product manifold M admits a Riemannian metric g such that

g(FX,FY) = g(X,Y)

for any vector fields X and Y on M, then M is called an almost product Riemannian manifold.

424

3. SUBMANIFOLDS OF PRODUCT MANIFOLDS

Let 2 be an m-dimensional almost product Riemannian manifold with structure tensors (F, g). Let M be an n-dimensional Riemannian

manifold isometrically immersed in M. For any vector field X tangent

to Mwe put F% = fX + hX,

where fX is the tangential part of FX and hX the normal part of FX.

For any vector field V normal to M we put

FV = tV + sV,

Where tV is the tangential part of FV and sV the normal part of FV.

We then have

f2X = X - thX,

hfX + shX = 0,

s2V = V - htV,

ftV + tsV = 0.

We easily see that

g(fX,Y) = g(X,fY),

g(fX,fY) = g(X,Y) - g(hX,hY).

If Frx(M) C X(M) for each x E M, then M is said to be Finvariant in M. Then h vanishes identically, and hence f2 = I and g(fX,fY) = g(X,Y). Therefore, (f,g) is an almost product Riemannian structure on M. Conversely, if (f,g) is an almost product Riemannian structure on M, then h = 0 and M is F-invariant in M. Consequently,

we obtain (Adati [1])

43 THEOREM 3,1, Let M be a sutmanifold of an almost product Riemannian manifold M. A necessary and sufficient condition for M to be Finvariant is that the induced structure (f,g) of M is an almost product Riemannian structure.

Let M be a locally decomposable Riemannian manifold, that is, VXF = 0, where V denotes the operator of covariant differentiation in M. We denote by V the operator of covariant differentiation in M with respect to the induced connection on M. If M is F-invariant in

M, then we easily see that F X(MY'c Tx(M)` for each x e M. Then we have

VXFY = F9XY = FDXY + FB(X,Y) = fVXY + sB(X,Y),

VXFY = VXfY = (VXf)Y + fVXY + B(X,fY).

Canparing the tangential and normal parts of these equations, we obtain (Vxf)Y = 0 and sB(X,Y) = B(X,fY). Thus M is locally decomposable.

THEOREM 3.2. Let M be an F-invariant submanifold of a locally decomposable Riemannian manifold M1 x T d-1. Then M is a locally decan-

posable Riemannian manifold M1 x M 1, where M1 is a submanifold of M1 and M-1 is a sutmanifold of R-1, M1 and M 1 being both totally geodesic in M.

Proof. We put

T1(x) = {X c Tx(M) : fX = X},

T 1(x) = {X a T;(M) : fX = -X}.

Then the correspondence of x c M to T1(x) and that to T 1(x) define two distributions T1 and T 1 in M respectively. Let Y e T1. Then for any vector field X tangent to M, we have fVXY = VXfY = VXY and hence VXY E T1. This shows that the distribution T1 is parallel. Similarly, we see that T 1 is also parallel. consequently, the integral manifolds of T1 and T-1 are both totally geodesic in M. We denote than by M1

426

and M 1 respectively. We now stew that M1 is in M1. Let X e T1. Then

QX = J(IX - FX) = J(X - fX) =0. Thus X belongs to the tangent space Tx(M1). Therefore M1 is a sr

M

1

nifold of M1. Similarly we see that

is a submanifold of M 1.

QED.

A sutmanifold M of an almost product Riemannian manifold M is

said to be F-anti-invariant if 7x(M)

X(Mf for each x e M.

THEOREM 3,3. Let M be a submanifold of a locally decomposable Riemannian manifold M. If M is anti-invariant with respect to F,'then AhXY = 0. Moreover, if 2dim M = dim M, then M is totally geodesic. Proof. Let X and Y be vector fields tangent to M. Then

VXFY = OXhY = -Ahy,X + DXhY, OXFY = F7XY = hVXY + tB(X,Y) + sB(X,Y).

Fran these equations we obtain

-g(AhyX,Z) = g(tB(X,Y),Z) = g(B(X,Y),hZ).

Since B is symmetric, we have A. Y = AhYX. Thus we obtain

-g(& ,X,Z) = g(AhZR,Y) = g(AhYX,Z).

Thus we have AhXY = 0. If dim M = 2dim M, the normal space Tx(M)` is spanned by {hX : X E Tx(M)}. Therefore AhgY = 0 means that M is totally geodesic. QED.

Let Sn be an n-dimensional sphere of radius 1, and consider Sn X Sn as an ambient manifold M. We denote by P and Q the projection operators of the tangent space of M to each components SP respectively.

The almost product structure F of M satisfies TrF = 0, where TrF is the trace of F. The Riemannian curvature tensor of M is given by

W (3.1)

R(X,Y)Z = i[g(Y,Z)X-g(X,Z)Y+g(FY,Z)FX-g(AX,Z)AY]

for any vector fields X, Y and Z on M. We can easily see that M is an Einstein manifold.

Let M be a hypersurface of M. We denote by N the unit normal of M in M. We can put

FN=U+XN.

FX=fX+u(X)N,

Then f, u, U and A define a symmetric linear transformation of the tangent bundle of M, a 1-form, a vector field and a function on M respectively. Moreover, we easily see that

g(U,X) = u(X).

The Gauss and Weingarten formulas of M are given by

VXY = OXY + g(AX,Y)N,

VXN = -AX,

where A is the second fundamental tensor of M with respect to N.

The Gauss and Codazzi equations of M are given respectively by

(3.2)

R(X,Y)Z = i[g(Y,Z)X-g(X,Z)Y+g(fY,Z)fX-g(fX,Z)fY] + g(AY,Z)AX - g(AX,Z)AY,

and

(3.3)

(VXA)Y - (VyA)X = J[u(X)fY - u(Y)fX].

Moreover, we obtain

f2X = X - u(X)U, fU = -AU,

u(fX) = -Xu(X),

Trf = -A,

XX = -2u(AX),

u(U) = 1 - A2, VXU = -fAX + AAX.

428

We also have

(VXf)Y = g(AX,Y)U + u(Y)AX,

(VXu)(Y) = Xg(AX,Y) - g(AfX,Y).

We now asstmm that the hypersurface M has constant mean curvature. Then, by a straightforward computation, we have

(3.4)

JAIA12 = -XTrfA2 + Tr(fA)2 + A(TrA)g(AU,U) - (TrAf)2 + 2a(TrA)(TrfA) + g(AU,AU) - 2(TrA)2 - IA12(IA12-(n-1)) + (TrA)(TrA3) + IVA12.

On the other hand, we obtain

(3.5)

div((TrfA)U - fAU) = g(AU,AU) - (TrfA)2 + X(TrA)(TrfA) - (TrA)g(AU,U) + Tr(fA)2 - XTrfA2 + (n-1)(i-X2),

(3.6)

div((TrA)U) = -(TrA)(TrfA) + X(TrA)2.

From these equations we obtain (Ludden-Okumura [1]) THEOREM 3.4. Let M be a hypersurface of SP x Sn. If the mean curvature vector of M is constant, then

2AIA12 - div((TrfA)U - fAU) - Idiv((TrA)U)

= 2(TrA)g(AU,U) - 2(X-1)(TrA)(TrfA) - 2(1+X)(TrA)2 - IA12(1AI2-(n-1)) + (TrA)(TrA3) - (n-1)(1-X2) + JVAI2.

40 In the following we shall give some applications of Theorem 3.4

(Ludden-0kumira [1]). THEOREM 3.5. A compact minimal hypersurface M of SP x SP (n > 1)

satisfying fM[`AI2 - (n-l)]IA12*1 > JMIVAI2*1

is an F-invariant hypersurface. Proof. From the assumption TrA = 0 and hence

fM[IA12(n-1) - 1AI4 - (n-1)(1-A2) + IVAJ2]*1 = 0.

Fran this we see that 12 = 1. Therefore, M is an F-invariant hyper-

surface of SP x e.

QED.

In view of Theorem 3.2 we can prove LE T1A 3.1, A complete F-invariant hypersurf ace M of SP x Sn is a

Riemannian product manifold U' x Sn, where M' is a hypersurf ace of Sn.

Since M' is totally geodesic in M' x SP, the second fundamental form of M' in.Sn has the quite similar properties to A. From this and Theorem 5.8 of Chapter II, we have THEOREM 3.6. The n((m/(n-1))1/2) x Snl(((n-m-1)/(n-1))1/2)xSn in Sn x SP are the only compact F-invariant minimal hypersurf aces of Sn x SP satisfying JAI2 = n-1.

4. SUBMANIFOLDS OF KAEHL.ERIAN PRODUCT MANIFOLDS

In this section we study sibmanifolds of Kaehlerian product manifolds (see Yano-Kon [7]). Let 1m be a Kaehlerian manifold of complex dimension m (of real dimension 2m) and if be a Kaehlerian manifold of caiplex dimension n (of real dimension 2n). We denote by Jm and Jn almost complex structures of SP and Mn respectively. We consider the Kaehlerian product

430

M=K°xMn and put

ix =

JnP7C + JJQX

for any vector field X on M, where P and Q denote the projection operators. Then vm see that

JnPP=PJ, J2 = -I,

Jn@=QJ,

FJ=JF,

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

V$J = 0,

F being an almost product structure on M. Thus J is a Kaehlerian structure on M. If Mm is of constant holocmrphic sectional curvature c1 and Mn is of constant holarnrphic sectional curvature c2, then the Riemannian curvature tensor R of M is given by

(4.1)

R(X,Y)Z =

16(cl+c2)[g(Y,Z)X-g(X,Z)Y+g(JY,Z)JX-g(JX,Z)JY

+2g(X,JY)JZ+2g(FY,Z)FX-g(FX,Z)FY+g(FJY,Z)FJX -g(FJX,Z)FJY+2g(FX,JY)FJZ] + 16(c1 c2)[g(FY,Z)X-g(FX,Z)Y+g(Y,Z)FX-g(X,Z)FY

+ g(FJY,Z)JX-g(FJX,Z)JY+g(JY,Z)FJX-g(JX,Z)FJY +2g(FX,JY)JZ+2g(X,JY)JFZ]

for any vector fields X, Y and Z on M.

We now consider an F-invariant submanifold M of a Kaehlerian product manifold M = On x Mn. Suppose that M is a Kaehlerian suth nifold of M. Since M is F-invariant, M is a Riemannian product manifold

Mp x Mq and MP is a sub mnifold of ff° and Mq is a sutmanifold of Mn. We now show that MP is a Kaehlerian subrmnifold of Mm and Mq is that of Mn. Let X E TX(M#). Then

431

JX = JmfiX + JnQX = Jm C X(

) n T{(M) = TX(Mp).

Therefore MP is a Kaehlerian submanifold of SP. Similarly, Mq is a Kaehlerian submanifold of Mn. Thus we have

THEOREM 4,1, Let M be an F-invariant Kaehlerian sutrrsnifold of a Kaehlerian product manifold M = PP x SP, Then M is a Kaehlerian product manifold MP x Mq, where IF is a Kaehlerian submanifold of Mr0 and Mq is a Kaehlerian su}manifold of Mn.

We next assume that M is an F-invariant, anti-invariant submanifold of M. Then M is the product MP x Mq. Let X E Tx(Mp). Then we have

JX= JmPX+ JnQX= J 11 c TX(M)''.

Since QJX = JnX = 0, we see that QJm = 0. This means that JmX is in Tr(f'). Thus IF is anti-invariant in X . Similarly, Mq is antiinvariant in Mn. THEOREM 4.2. Let M be an F-invariant submanifold of a Kaehlerian product manifold M = Mm x Mn. If M is anti-invariant in M, then M is a Riemannian product manifold MP x Mq, where IF is an anti-invariant

submanifold of M
(4.2)

R(X,Y)Z = 16(c1+c2)[g(Y,Z)X-g(X,Z)Y+g(FY,Z)FX-g(FX,Z)FY]

+

ci-c2)Ig(FY,Z)X-g(FR,Z)Y+g(Y,Z)FX-g(X,Z)FY]

+ AB(Y,Z)X - AB(X,Z)Y

(4.3)

(VXB)(Y,Z) = (V,gB)(X,Z).

432

From (4.2) we see that the Ricci tensor S of M is given by

(4.4)

S(X,Y) = 1 c1+c2)[(p+9-2)g(X,Y)+TrFg(FX,Y)l

+

16(cx-c2)[(p+q-2)g(FX,Y)+TYFg(X,Y)]

+ l[g(B(ei,ei),B(X,Y))-g(B(X,ei),B(Y,ei))], i

where {ei} is an orthonormal frame of M and hence the scalar curvature

r of M is given by

(4.5)

r = 16(cl+c2)[(p+q)(p+q-2) + (TrF)2] + 16(cl c2)[(p+q-2)TrF + (p+q)TrF]

+

[g(B(ei,ei),B(ej,ej)) - g(B(ei,ej),B(ei,ej))].

From (4.2) we have

PROPOSITION 4.1. Let M be a (p+q)-dimensional F-invariant, antiinvariant submanifold of a Kaehlerian product manifold 11°(c1) x Mn(c2).

If M is totally geodesic, then M = Mp(}c1) x Mq(tc2), where Mp(jc1) is a real space form of constant curvature ic1 and Mq(lc2) is a real

space form of constant curvature Ic2. If M is minimal, then EB(ei,ei) = 0. Then (4.2), (4.4) and (4.5)

imply the following PROPOSITION 4.2. Let M be a real 2n-dimensional F-invariant, anti-invariant minimal sulmanifold of Mn(c) x Mn(c). Then M is

totally geodesic if and only if M satisfies one of the following conditions:

(a) M is a Riemannian product manifold of two Lanac); (b) S = *(n-1)cg; (c) r =

n(n-1)c.

0 If, for any vectors X, Y and Z tangent to M, A(X,Y)Z is also

tangent to M, i.e., A(X,Y)Tt(M)e T;(M) for each x E M, then M is said to be curvature invariant. THEOREM 4,3. Let M be a real (p+q) -dimensional F-invariant sub-

manifold of a Kaehlerian product manifold Sp(c) x Mn(c) (c # 0). If M is curvature invariant, then M is a Riemannian product manifold IF X Mq such that

(a) IF and Mq are invariant in f(c) and M (c) respectively; (b) MP and Mq are anti-invariant in SP(c) and Mn(c) respectively; (c) MP is invariant in °(c) and Mq is anti-invariant in Mn(c);

(d) MP is anti-invariant in '(c) and Mq is invariant in KNO. Proof. Since c # 0, (4.1) implies that

g(JY,Z)JX-g(XX,Z)JY+2g(X,JY)JZ+g(FJY,Z)FJX-g(FJX,Z)FJY +2g(F%,JY)FJZ E Tx(M)

for any vector fields X, Y and Z tangent to M. Putting Y = Z in the equation above, we obtain

g(X,JY)JY + g(X,JFY)JFY c Tx(M).

On the other hand, M is of the form MP x Mq. If X e Tx(MP), then FX = X and if X E TG(IF), then FX = -X. Now let Y e TX(MP). Then g(X,JY)JY E Tx(M) for each point x of M. Therefore JY E TX(M) or g(X,JY) = 0. If JY E TT(M), then JY = JmY E Tx(MP) and hence MP is invariant in FP(c). If g(X,JY) = 0 for X, Y c

R{MP),

then MP is

anti-invariant in itn(c). Similarly we see that Mq is invariant or

anti-invariant in Mn(c). In Theorem 4.3, if M is totally geodesic, M is obviously curva-

ture invariant. Thus we have

QED.

434

THEOREM 4.4. Let M be a real r-dimensional complete F-invariant

sutmanifold of CEP x &. If M is totally geodesic, then M is Cpp x Cpq (2p+2q=r), or RPp x Rpq (p+q=r), or Cpp x Rpq (2p+gZ) or RPP x Cpq (p+2q:--r).

We now study invariant sutmnnifold (Kaehlerian sutmanifold) M of a Kaehlerian product manifold f(c) x Mn(c) (c # 0). We now assume that M is curvature invariant. Then, for any vectors X, Y and

Z tangent to M, (4.1) implies

(4.6)

g(FY,Z)FX-g(FX,Z)FY+g(FJY,Z)FJX -g(FJX,Z)FJY+2g(FX,JY)FJZ E X(M).

Putting Y = Z in (4.6), we find

(4.7)

g(FY,Y)FX-g(FX,Y)FY+3g(FX,JY)FJY a Tx(M).

Replacing Y by JY in (4.7), we see that

(4.8)

g(FY,Y)FX-g(FX,JY)FJY+3g(FX,Y)FY e TX(M).

From (4.7) and (4.8) we have

(4.9)

g(FY,Y)FX+g(FX,Y)FY+g(FX,JY)FJY a TX(M).

Therefore (4.7) and (4.9) imply

(4.10)

g(FY,Y)FX + 2g(FX,JY)FJY a X(M),

fran which

(4.11)

g(FY,Y)F) + 2g(FX,Y)FY c Tx(M).

Putting X = Y in (4.11), we get

4z g(FY,Y)FY a Tx(M).

Thus we see that FY c TX(M) or g(FY,Y) = 0. If g(FY,Y) = 0, then (4.11) implies that g(FX,Y)FY c TX(M) and hence FY c TX(M) or

Oc)nsequently we see that for any Y c TX(M), FY a TT(M) or FY a X(M). Thus we have F X(M) C TX(M) or FTX(M) C X(M7`. FY E

This means that M is F-invariant or F-anti-invariant. Therefore we

have THEOREM 4,5, Let M be a Kaehlerian submanifold of a Kaehlerian product manifold

(c) x Mn(c) (c # 0). If M is curvature invariant,

then M is F-invariant or F-anti-invariant. THEOREM 4.6. Iet M be a Kaehlerian sutrnanifold of a Kaehlerian

product manifold M"(c) x EP(c) (c 0 0). If M is totally geodesic, then

M is F-invariant or F-anti-invariant.

436 EXERCISES

A. DECOMPOSABLE VECTORS: Consider a contravariant vector field Eb

v1Q) in a locally product manifold M. When va are functions of

only, and v functions of Ey only in a separating coordinate system, we call

v a contravariant decomposable vector field in M. Then we

have the following theorems (see Tachibana [1], Yano [5]).

THEOREM 1. A necessary and sufficient condition for a contravariant vector field vh in a locally product manifold to be decomposable is that the derivative of Fh with respect to vh vanishes.

THEOREM 2. A Killing vector field in a compact locally decomposable Riemannian manifold is deccnposable.

THEOREM 3, A conformal Killing vector in a compact locally decomposable Riemannian manifold is a Killing vector and is decomposable.

B. INTEGRABILITY CONDITIONS: Let M be an almost product manifold with almost product structure F. Then the integrability condition of the distributions P and Q are stated as follows:

THEOREM 1. In order that the distribution P (resp. Q) be carpletely integrable, it is necessary and sufficient that

Njih -

NjiV

= 0

(resp. Njih + NjikFh = 0).

Consequently, in order that both of the distributions P and 0 be completely integrable, it is necessary and sufficient that

Njih = 0,

where Njih is the Nijenhuis tensor formed with Fi, i.e.,

Njih =

l(alFi - airi) - F

(a1F

- ajFi

437

We also have THEOREM 2. In order that an almost product manifold be integrable,

it is necessary and sufficient that it is possible to introduce a symmetric affine connection with respect to which the structure tensor F is covariantly constant. (See Fukami [1], Lichnerowicz [1], [2), [3], de Rham [1], Walker [1], Willmore [1], Yano [3].)

C. HYPERSURFACES OF Sn x Sn: Matsuyama [1] proved the following theorems.

THEOREM 1. A compact minimal hypersurface with non-negative sectional curvature of Sn x Sn (n > 1) satisfying

!M[IA14 - (n-1)IAl2]*1 > 0

is an F-invariant hypersurface. THEOREM 2.

e-1(r) x Sn(l) and Sn(s) X

Sn-m-1((1-s2)V2) x Sn(l)

are the only ccmpact F-invariant hypersurfaces of Sn x Sn with constant mean curvature and non-negative sectional curvature. D. SUl31

NIFOIAS OF CPn x CPn: Let M be a Kaehlerian hypersurface

of ('.P4 x c?. We denote by v, Jv an orthonormal basis of the normal

space of M in CPn x CP4, and by H the second fundamental tensor with respect to v. Then the second fundamental form with respect to Jv is given by JH. We put S = 2TrH2, that is, the square of the length of the second fundamental form of M. Then we have (Matsuyama [2]) THEOREM 1. A compact Kaehlerian hypersurface of C P

satisfying

fM[2n+iE2 - (n+l)S]*1 > 4JMIVA12*1

is an F-invariant hypersurface.

x CV3

438 THEOREM 2. CPn-1 x C P" and

Q°-1

x CP are the only Cixpact F_

invariant Kaehlerian hypersurfaces of CPn x (PP' with constant scalar

curvature, where Qn-1 is the complex quadric.

For anti-invariant such nifolds we have (Yano-Eon [7] ) THEOREM 3. Let M be a real (m+n)-dimensional (m > n) coapact

F-invariant, anti-invariant minimal suhmanifold of CPn x Cam. If JAI2 = n(n+1)/(2m-1), then m = n = 2 and M is of the form

S1

x

S1 x Rp2 or Rp2 x S1 x S1.

439

CHAPTER IX SUBMERSIONS

In this chapter we study the submersions and its applications to the theory of suh anifolds.

In §1, following O'Neill [2] we give the fundamental equations and some examples of submersions. §2 is devoted to the study of

almost Hermitian submersions (Watson [1]). In the last §3, we discuss the relation between Sasakian manifolds and Kaehlerian manifolds which is a special case of the submersion with totally geodesic fibres. We also consider the relation between submanifolds of Sasakian manifolds and submanifolds of Kaehlerian manifolds. Then the various notions in sulmanifolds of Sasakian manifolds which correspond to that of Kaehlerian manifolds will be clear.

1. FUNDAMENTAL EQUATIONS OF SUBMERSIONS

Let M and B be Riemannian manifolds. A surjective mapping 1r

: M

> B is called a Riemannian submersion (O'Neill [2]) if: (Si) n has maximal rank;

that is, each derivative map n* of n is onto; hence, for each x E B, n 1(x) is a suimanifold of M of dimension dim M - dim B. The sutmani-

folds 7T(x) are called fibres, and a vector field on M is vertical

if it is always tangent to the fibres, horizontal if always orthogonal to fibres; (S2) n* preserves lengths of horizontal vectors.

440

For a submersion n : M

B, let H and V denote the projec-

tions of the tangent spaces of M onto the subspaces of horizontal and vertical vectors, respectively. We denote by V the operator of covariant differentiation of M. Following O'Neill [2] we define a tensor field T of type (1,2) for arbitrary vector fields E and F on M by

TEF = EVVE(VF) + WVE(HF).

We shall make frequent use of the following three properties of T: (a) TE is a skew-symmetric operator on the tangent space of M, and it reverses the horizontal and vertical subspaces; (b) T is vertical, that is, TE = TVE;

(c) For vertical vector fields, T has the symmetric property, that is, for vertical vector fields V and W, TVW = TWV.

This last fact, well known for second fundamental forms, follows immediately from the integrability of the vertical distribution.

Next we define a tensor field A of type (1,2) on M by

AEF

= WHE(HF) +

HVHE(VF).

A has the following properties: (a') AE is a skew-symmetric operator on the tangent space of M, and it reverses the horizontal and vertical subspaces; (b') A is horizontal, that is, AE

= AHE; (c') For horizontal vector fields X, Y, A has the alternation

property AXY = -.X. The last property (c') will be proved in the proof of Lemma 1.2. A basic vector field on M is a horizontal vector field X which is n-related to a vector field XX on B, that is, n, Cp = X*,m(p) for

all p E M. Every vector field X* on B has a unique horizontal lift X to M, and X is basic. Thus X <

> X* is a one-to-one correspondence

between basic vector fields on M and arbitrary vector fields on M.

In the following we prepare same lemmas which give the basic

formulas for submersions.

441

LEWIA 1,1, Let X and Y be vector fields on M. Then (1) G(X,Y) = g(X*,Y*).r, where G is the metric tensor field on M and g the metric tensor field on B; (2) H[X,Y] is the basic vector field corresponding to [X*,Y*]; (3) HVXY is the basic vector field corresponding to V* Y*9 X* where v* is the operator of covariant differentiation of B.

Proof. The first assertion (1) follows from (S2), the second from the identity r*[X,Y] = [X*,Y*]. We shall prove (3). For any basic vector fields X, Y and Z on M we have

2G(VXY,Z) = XG(Y,Z) + YG(X,Z) - ZG(X,Y)

- G(X,[Y,Z]) - G(Y,[X,Z]) + G(Z,[X,Y])

Thus we

But, for example, XG(Y,Z) = have

G(VXY,Z).

g(V*X*

Therefore, VXY is r-related to V*X Y*. Thus we have (3).

QED.

LEMMA 1.2. If X and Y are horizontal vector fields on M, then

AXY = JV[X,Y].

Proof. First of all we have

V[X,Y] = VVXY - VVYX = AXY - AyX.

Thus it sufficies to prove the alternation property (c'), or equiva-

lently, to show that AX = 0. We may assume that X is basic, hence that 0 = VG(X,X) = 2G(VVX,X) for any vertical vector field V. Since V is r-related to the zero vector field, we see that [V,X] = VVX-VXV is vertical. Hence

442

G(Vvx,X) = G(Vxv,x) = -G(V,Vxx) = -G(v,AXx).

Since AxX is vertival, the result follows.

QED.

We denote by p the Riemannian connection along a fibre with respect to the induced metric. We see that pVW = vvVW for any vertical

vector fields V and W. We easily see the following LEhMA 1,3, Let X and Y be horizontal vector fields, and V and W

vertical vector fields on M. Then (1) VVW = TvW + VVW,

(2) VVX = HVVX + TVX,

(3) VXV = AxV + IVVXV,

(4) V Y = HVXY + AxY,

(5) if X is basic, HVVX = AxV.

L64 M 1. 4, Let X, Y be horizontal vector fields and V, W be

vertical vector fields on M. Then. (1) (VVA)W = -ATVW,

(VXA)W = -A W,

(2) (VxT )y = -TAXY,

(VVT)y = -TT

Y.

V

Proof. We will only prove (1). Let E be an arbitrary vector field on M. Then

(VVA)WE = VV(AWE) - AVVWE - AW(VVE).

Since A is horizontal, we see that AW = 0. On the other hand, we have

AVVWE = AHVVWE = ATVWE.

Thus we obtain (11.

QED.

443

Furthermore, we prepare the following lemmas. LEMMA 1.5. If X is a horizontal vector field and U, V, W are vertical vector fields, then

G((VUA)XV,W) = G(TUV,AXW) - G(TUW,AXV).

LEMMA 1.6. Let X and Y be horizontal vector fields and V and W be vertical vector fields. Then we have (1) G((VEA)XY,V) is alternate in X and Y; (2) G((VET)VW,X) is symmetric in V and W.

LEMMA 1.7. Let X, Y and Z be horizontal vector fields and V be a vertical vector field. Then we have

G G((VZA)XY,V) = GG(AXY,TVZ),

where G denotes the cyclic scan of over the horizontal vector fields

X, Y and Z. Proof: Since this is a tensor equation, we can assume that X, Y and Z are basix, and even that all three brackets [X,Y],.. are vertical. Thus, Lemma. 1.2 implies that j[X,YJ = AxY. Hence

#G([[X,Y],z],V) = G([AXY,Z],V) = G(VAXYZ,V) - G(VZ(AXY),V).

On the other hand, we have 7,V) = G(TAXYZ,V) = -G(Z,TAxYV) G(VISF

= -G(Z,TV(AXY)) = G(TVZ,AXY).

Thus, using the Jacobi's identity, we find G G(VZ(AXY),V) = SG(TVZ,AXY).

444

Thus it remains only to show that SG(VZ(AXY),V) = GG((VZA)XY,V). But

G(VZ(AXY),V) - G((VZA)XY,V) = G(AVZXY,V) + G(AX(VZY),V).

The first term on the right hand side of this equation equals to

-G(Ay(HVg),V),

and since we assume that [X,Z] = 0, this becomes -G(Ay(HVXZ),V), from which the projection H may now be deleted. From this we have our assertion. For a submersion it : M

QED.

> B we now derive the equations

analogous to the Gauss and Codazzi equations of an immersion.

Since we can consider the fibres as sutmanifolds of M, we have the following theorem. Let R be the Riemannian curvature tensor of M and R the Riemannian curvature tensor of the fibre. Then THEOREM 1.1. If U, V, W, F are vertical vector fields and X is

horizontal, then G(R(U,V)W,F) = G(R(U,V)W,F) + G(TUW,TVF) - G(TVW,TUF), G(R(U,V)W,X) = G((VUT)VW,X) - G((VVT)UW,X).

Let R* be the Riemannian curvature tensor of B. Since there is no danger of ambiguity, we denote the horizontal lift of R* by R* as well. Explicitely, if X, Y, Z and W are horizontal vectors of M, we set

G(R*(X,Y)Z,W) = g(R*(n*X,n*Y)r*Z,n*W).

THEOREM 1.2. If X, Y, Z, H are horizontal vector fields and V is vertical, then-,

G(R(X,Y)Z,H) = G(R*(X,Y)Z,H) + 2G(AXY,AZH) - G(AyZ,AXH) + G(AXZ,AyH),

G(R(X,Y)Z,V) = -G((VZA)XY,V) - G(AXY,TVZ) + G(AyZ,TVX) - G(AXZ,TVY).

Proof. Since the two equations are tensor equation, we can assume that X, Y and Z are basic vector fields whose brakets are vertical. Then [X,Y] = 2AxY. We write the basic vector field AVyZ as V*yZ. Then VYZ = V*YZ + A.Z. From IR ma 1.3 we obtain

VXVyZ = V*XV*yZ + AXV*yZ + A

Z + VVX

Z,

V[X,y]Z = 2AZAXY + 2T zZ.

Therefore we have

R(X,Y)Z = VXVyZ - VYVXZ - V[X,Y]Z

= V*XV*YZ - V*yV*XZ + AXAYZ - AyAXZ - 2AZAXY - 2TpyZ + VVXAYZ - VVyAXZ + AXV*yZ - Ayv*XZ

= R*(X,Y)Z + AXAyZ - AyAXZ - 2AZAXY - Zr yZ + V(VXAyZ - VyAXZ) + AXV*yZ - AyV*XZ,

where we have used the fact that A[X,Y] = 0, i.e., n*[X,Y] = 0. Taking the inner product in the equation above with H, we have the first equation. Taking the inner product with V, we obtain

G(R(X,Y)Z,V) = -2G(TYZ,V) + G(VXAYZ,V) - G(V,yAXZ,V) + G(AXVYZ,V) - G(AYVXZ,V).

446

On the other hand, in the proof of Lema 1.7, we have

G(TA_YZ,V) = G(TVZ,AXY).

Moreover, we find

G((VXA)yZ,V) - G((V A)XZ,V) = G(VXAyZ,V) - G(AyVXZ,V) - G(VyAXZ,V) + G(AXVy,Z,V),

because of [X,Y] is vertical. Thus we have

G(R(X,Y)Z,V) = -2g(TVZ,AXY) + G((VXA)YZ,V) - G((VYA)XZ,V).

From this and Lemmas 1.6, 1.7 we have the second equation.

QED.

By the similar argument as that of the preceding theorem we have the following

THEOREM 1.3. If X and Y are horizontal vector fields, and V and W are vertical, then

G(R(X,V)Y,W) = G((VVA)XY,W) - G((VXT)VW,Y) + G(AXV,AYW) - G(TVX,TWY).

We denote by K, K* and k the sectional curvatures of M, B and the fibre, respectively. Then we have THEOREM 1.4. Let X and Y be orthononnal horizontal vectors and V and W be orthonormal vertical vectors. Then

K(V,W) = K(V,W) + G(TVW,TVW) - G(TVV,TWW), K(X,V) = G((VXT)VV,X) + G(AXV,AXV) - G(TVX,TVX), K(X,Y) = K*(n*X,n*Y) - 3G(AXY,AXY).

447

The first equation of Theorem 1.4 is one formulation of the Gauss equation of the fibre. Example 1.1. Let Cn+1 be a ccuplex (n+l)-dimensional number space

with the natural almost complex structure J. We consider a (2n+1)Let N be the outward unit normal of dimensional unit sphere Stn+l Then the integral curves of the tangent vector field in Stn+1.

Cn+l.

JN are great circles in S2n+1 that are the fibres of a bundle mapping : Stn+1 > cpn, & being the complex projective space (see n Example 2.6 of Chapter III). The usual Riemannian structure on CPn is characterized by the fact that n is a submersion. Since the fibres are totally geodesic in S2n+1, the tensor T vanishes. If X and Y are horizontal vector fields, then

AXY = G(X,JY)JN,

AXJN = JX.

Thus we have

G(AXY,JN) = G(X,JY).

Example 1.2. Let G be a Lie group furnished with two-sided invariant Riemannian structure. If K is a closed subgroup, then the usual Riemannian structure on G/K is characterized by the fact that the natural mapping n : G

> G/K is a submersion. The fibres,

left cosets of G mod K, are totally geodesic and hence T = 0. Let X and Y be left invariant horizontal vector fields on G, that is, X and Y be in the orthogonal complement of the Lie algebra of K in

the Lie algebra of G. By Lemna 1.1, AXY = JV[X,Y] is in the Lie algebra of K, and it is known that K(p) = *I[X,Y]I2 for plane section

p spanned by orthonormal vectors X and Y. Then for plane section p* = n*(p) tangent to G/K, Theorem 1.4 implies that

I(p*) = JI[X,Y]I2 +

4IV[X,Y]I2 = JIH[X,Y]I2 + IV[X,Y]I2:

448

Example 1.3. Let F(B) be the frame bundle over B with structure group 0(n). We identify the elements of the Lie algebra o(n) with skew-symmetric matrices, and use the inner product

= -Trace(ab) =

Then there exists a natural Riemannian structure on F(B) such that the projection a : F(B)

> B is a submersion. To define it, let

w be the Riemannian connection form on F(B) taking values in o(n), and let H = Kernelw be the Riemannian connection on F(B). If v is a vertical vector and w(h) = 0, define

IvI = Iw(v)I,

= 0,

IhI = In*(h)I.

Clearly, n is a submersion with H as its horizontal distribution. By a straightforward computation we see that the fibres are totally ,aodesic. Next, we compute A. If X and Y are horizontal vector fields on F(B), then w([X,Y]) = -S2(X,Y), where S2 is the curvature form of B

on F(B). Hence, by Lemma. 1.2, if x and y are horizontal vectors, Axy is the vertical vector such that w(Axy) = -jc2(x,y). If x, y and a vertical vector v are all tangent to F(B) at f = (fit ...,fn), then

_ - = -<w(v),w(Axy)> = i<w(v),SZ(x,y)>

_ i

i IJwij(v)

This implies that Axv is the horizontal vector at f.

2. ALMOST HERMITIAN SUBMERSIONS

Let M be a complex m-dimensional almost Hermitian manifold with Hermitian metric G and almost complex structure J, and B be a ccrrplex

n-dimensional almost Hermitian manifold with Hermitian metric g and almost complex structure J'. We suppose that there exists a submersion

449

IT

: M

> B such that n is an almost cornplex mapping, i.e.,

n*J = J'n*. Then we say that IT is an almost Hermitian submersion. We denote the vertical and horizontal distributions in the tangent bundle of M by V(M) and H(M), respectively. Then T(M) = V(M) 0 H(M). The orthogonal projection mappings are denoted by v : T(M) > H(M) respectively.

> V(M) and H : T(M)

PROPOSITION 2.1. Let n : M

> B be an almost Hermitian

submersion. Then the horizontal and vertical distributions are Jinvariant, i.e., JH(M) = H(M) and JV(M) = V(M). Proof. Since it is almost carplex, JW is vertical for W E V(M).

Let X be in H(M). Then G(JX,W) = -G(X,JW) = 0 and therefore JX is in H(M).

QED.

From Proposition 2.1 we see that any fibre of the submersion it

: M

> B is a complex subrnanifold of M. We notice here that

if X is a basic vector field, then JX is the basic vector field associated to J'X*.

We prove the following (Watson [1)) THEOREM 2.1. Let it : M

> B be an almost Hermitian submersion.

If M is quasi-Kaehlerian, nearly Kaehlerian, Kaehlerian, or Hermitian, then B has the same property. Proof. Let 4) and V be the fundamental 2-forms of M and B respec-

tively. We first claim

= n*(p' on basic vector fields. If X and Y are

basic vector fields on M, and X* and Y* are their associated vector fields on B, then

O(X,Y) = G(X,JY) =

n*4'(X,Y).

Since n* commutes with d on differential forms, we also see that ft = n*(d'@').

If M is almost Kaehlerian, then n*d'4' = 0. Since n* is a linear isometry, we obtain d'@' = 0 and therefore B is almost Kaehlerian. Suppose that M is nearly Kaehlerian. It is easy to see that the

450

basic vector field associated to p'X*J'X. for any vector field X* on B is JVXJX which vanishes on M. Thus B is nearly Kaehlerian. Similarly, we see that if M is quasi-Kaehlerian, then B is quasi-Kaehlerian. bbreover, the basic vector field on M associated to the Nijenhuis tensor N'(X*,Y*) on B is iN(X,Y). Therefore, if M is Hermitian, B is also Hermitian. Furthermore, when M is Kaehlerian, B is Kaehlerian. QED.

Now we can begin to examin haw the almost Hermitian structure on M places restriction on T and A.

LENIA 2.1. Let Tr : M

> B be a quasi-Kaehlerian submersion,

V and W vertical vectors, and X and Y horizontal vectors. Then

(a) TVJW = TJVW,

(b) TJVX = -TVX,

(c) AXJX = 0,

(d) AXJY = -A1JX.

Proof. (a) follows from the similar result on the second fundamental tensor of complex sutmanifolds. To see (b), note that

G(TJVX,W) = -G(TJVW,X) = -G(TVJW,X) = -G(JTVX,W).

Thus we have TJVX = -JTVX. Since M is quasi-Kaehlerian, we ot;.ain

vxJX - VJXX = JVXX + JVJXJX.

Taking the vertical part of this equation, we find

AXJX - AJXX = JAXX + JAJXTX = 0.

Therefore, we obtain AXJX = 0. Assertion (d). follows from (c) by the standard polarization trick. QED.

451

LQ11MA 2.2. Let n : M

> B be a nearly Kaehlerian submersion.

Then, for all vertical vectors V and W,

(a) TVJW = JTVW,

(c) TVJX = JTVX

(b) TJVW = JTVW,

for all horizontal X.

Proof. For (a) and (b) it will be sufficient to show

that

TVJV = JTVV. By a straightforward computation we have

TVJV = VV_' - OV" = (VVJ)V + JVVV - (VVJ)V - JVVV = J(VVV - VVV) = JTVV.

Assertion (c) is an easy calculation.

QED.

Clearly, we have the following

Lem 2.3. Let Tr : M

> B be a Kaehlerian submersion. If V

is a vertical vector field and E is any vector field, then

TVJE = JTVE.

We prove the following theorem (Watson [1]).

> B be an almost Hermitian submersion

THEOREM 2,2. Let n : M

with M, an almost semi-Kaehlerian manifold. Then B is almost semiKaehlerian if and only if the fibres of Ti are minimal subman;folds

of M. Proof. Let {E1,...'Fln-n`TE1,...,JEM-nF,,...,Fn,JF1,...,JFn} be a local J-basis on M whose horizontal vector fields are basic.

Then

n

60(X)

0 =

=

O(X) i=1

[VF O(Fi,X) + VJF 4V(JFi1X)1 , i i

462

where X is a basic vector field and 30(X) is given by

m-n 6O(X) _ -JI [VEJO(Ej-,X) + VJE s(JE.,X)].

On the other hand, we obtain

VF (D(Fi,X) = -G(VF JFi,X) + G(JVF Fi,X)

= -G(HVF JF.,X) + G(HJVF FL,X)

=

VI Fi* (D'(Fi*,X*).

Similarly, we have

VJ. (D(JFi,X) = V'J'F. (D '(J'Fi*,X*).

Therefore

0 = aVX) + a'V(X*).

Since we have d.D

(X) _

m-n I [G((VE J)EJ,X) + G((VJE J)JEj,X)] J j=1 j rn-n

E G(TE JE.-JTE E.-TJE E.-JTE E.,X) j=1 j J

JJ

JJ

JJ

m-n

_ -2 1 G(JT E

j=l

j

d@(X) = 0 if and only if a fibre of n is minimal. Therefore we have our assertion. THEOREM 2.3. Let M be semi-Kaehlerian, and it : M

QED. > B an

almost Hermitian submersion. Then B is sari-Kaehlerian if and only if the fibres of it are minimal.

463 We next consider the integrability of the horizontal distribution of a Kaehlerian submersion (see Johnson [1], Watson [1]). THEOREM 2.4. The horizontal distribution of a Kaehlerian submersion is carpletely integrable. Proof. Let Tr : M

> B be a Kaehlerian submersion, X a basic

horizontal vector field on M, Y horizontal and V vertical. Then

G(AJXY,V) = G(AXJY,V) _ -G(JY,HVVX)

= G(Y,HVVJX) = G(Y,AJXV) = -G(AjXY,V).

Therefore AJXY = 0, and hence AxY = 0. From this we see that the distribution is completely integrable. Moreover, the integral manifold is totally geodesic.

QED.

In the sequel, we consider the Riemannian curvature tensors of an almost Hermitian submersion.

The holomorphic bisectionaZ curvature of an almost Hermitian manifold M is defined for any pair of unit vectors E and F on M by (Goldberg-Kobayashi [1])

B(E,F) = G(R(E,JE)JF,F).

Then the holcmorphic sectional curvature of M is given by H(E) _ B(E,E). We have the following theorems.

THEOREM 2.5. Let v : M

> B be an almost Hermitian submersion.

Let X and Y be horizontal unit vectors, and V and W be unit vertical vectors. Then

B(V,W) = B(V,W) + G(TvJW,TJVW) - G(TVW,TJVJW), B(X,V) = G((VVA)XJX,JV) - G(AXJV,AJXV) + G(AXV,AJXJV) - G((V, A)XJX,V) + G(TjVX,TVJX) - G(TVX,TJVJX),

464

B(X,Y) = B' (X*,Y*) - 2G(I JX,A.JY) + G(AJ Y,AXJY) + G(AX,AJJY).

> B is a quasi-Kaehlerian submersion,

THEOREM 2.6. If n : M then

B(V,W) = B(V,W) + ITVJW12 + JTVW12,

B(X,V) = G((VVA)XJX,JV) - G(AJV,AJXV) + G(AXV,AjXJV)

- G((V

1)XJX,V) - 2G(TVX,TjJX),

B(X,Y) = B'(X*,Y*) + IA__N1 2 + JAXYJ2.

From this we have THEOREM 2.7. Let n : M

B be a quasi-Kaehlerian submersion.

Then we have the following: (a) B(V,W) > B(V,W), and equality holds if and only if the fibres of it are totally geodesic;

(b) B(X,Y) > B'(X*,Y*), and equality holds if and only if the horizontal distribution is completely integrable.

We also have the following THEOREM 2,8. If it : M

> B is a nearly Kaehlerian submersion.

then

B(V,W) = B(V,W) + 2ITVWI2, B(X,V) = G((VVA)XJX,JV) - G(AXJV,AJXV) + G(AXV,AJ JV)

- G((VNA)X.JX,V) + 2 JTVXJ 2.

THEORBI 2.9. If n : M

> B is a Kaehlerian submersion, then

B(X,Y) = B'(X*,Y*).

B(X,V) = 2ITVXI2,

an this we obtain THEOREM 2.10. Let n : M

> B is an almost Hermitian submer-

sion. If M is of constant holomorphic bisectional curvature b, then B is of constant holorrorphic bisectional curvature b.

THEOREM 2.11. If n : M

> B is an almost Hermitian submersion,

then

H(V) = H(V) + ITVJVI2 - G(TVV,TJVJV), H(X) = H'.(X*) - 3IAXJXI2.

THEOREM 2.12. If n : M

> B is a quasi-Kaehlerian submersion,

then

H(V) = H(V) + ITVVI2 + ITVJVI2, H(X) = H'(X*).

THEOREM 2.13. If it : M

> B is a nearly Kaehlerian submersion,

then

H(V) = H(V) + 2ITVVI2.

3. SUBMERSIONS AND SUBMANIFOLDS

Let M be a (2m+1)-dimensional Sasakian manifold with structure tensors

such that there exists a fibering n :

Id

> M/!;

= N, where N denote the set of orbits of E and is a real 2m-dimensional Kaehlerian manifold (see §6 of Chapter V). This is a special case of a submersion -with totally geodesic fibres. We denote by (J,g) the

456

Kaehlerian structure of N. We denote by * the horizontal lift with respect to the connection n. Then we have

-G(X*,Y*) = g(X,Y)

(JX) * = q,X*,

g(X,Y)

for any vector fields X and Y on !, where we write

by

the operator of

covariant differentiation with respect to G (resp. g). Then we obtain (see Example 1.1)

(0'XY)* _ -¢2VX*Y* = V Y* - G(OX*Y*,E)E = OX*Y* - G(Y*,$X*)E.

Let R and R' be the Riemannian curvature tensors of R and N respectively. Then we have PROPOSITION 3.1. The Riemannian curvature tensors R and R' satisfy

(R'(X,Y)Z)* = R(X*,Y*)Z* + G(Z*,¢Y*)OX* -

2G(Y*,4,X*)OZ*

for any vector fields X, Y and Z on M. Proof. First of all we obtain

(0'XV'yZ)* = OX*(0'YZ)* - G((0'YZ)*,$X*)C = OX*DY*Z* -

G(Z*,OoX*Y*)C

+ G(Z*,4)Y*)4)X* - G(0Y*Z*,4)X*)E.

From this we have t4 similar expression of (D'YO'XZ)*. Moreover, we have

0'MY]Z)* = V[X*,Y*]Z* + 2G(Y*,$X*)4Z* - G(Z*,[X*,Y*))l;.

Fran these equations we have the equation of proposition.

QED.

Let S and S' be the Ricci tensors of M and N respectively. Then Proposition 3.1 implies

S'(X,Y) = S(X*,Y*) + 2G(X*,Y*).

(3.1)

Therefore, the scalar curvature r of M and the scalar curvature r' of

N satisfy (3.2)

r'

+ 2m.

Moreover, the sectional curvatures of M and N determined by orthonormal vectors X and Y on N satisfy

K'(X,Y) = K(X*,Y*) + 3G(X*,g*)2.

(3.3)

Especially, we have

R'(X,JX) = K(X*,cX*) + 3.

Thus we have THEOREM 3.1, M is of constant 4-sectional curvature c if and only if N is of constant holomorphic sectional curvature (c+3) Let M be an (n+l)-dimensional suYmanifold tangent to the structure vector field ( of M and N be an n-dimensional sutrmanifold of N.

Throughout in this section we assume that the diagram

M

1

>f

N

1

>N

canmutes. let V (resp. V') be the operator of covariant differentiation

458

in M (resp. N). We denote by the same G and g the induced metric tensor fields on M and N respectively. We denote by B (resp. B') the second fundamental form of the immersion i (resp. i') and the associated second fundamental tensors of B and B' will be denoted by A and A'

respectively. Let X and Y be vector fields tangent to N. Then we have the following Gauss formulas:

V'XY = V'XY + B'(X,Y)

and

pX*Y* = OX*Y* + B(X*,Y*).

Therefore we obtain

(V'XY)* + (B'X,Y))* = -02VX*Y* + B(X*,Y*).

Comparing the tangential and normal parts of this equation, we have respectively

(V'XY)* = -c2VX*Y* = VX*Y* -

(3.4)

(3.5)

(B'(X,Y))* = B(X*,Y*).

Let D and D' be the operators of covariant differentiation with respect to the linear connection induced in the normal bundles of M and N respectively. For any vector field X tangent to N and vector field V normal to N, we obtain

V'XV = -A'VX + D'XV

and

VX*V* = -AV*X* + DX*V*.

Thus we have

-(A'VX)* + (D'XV)* = 2AV,.X* + DX*V*,

from which

(3.6)

(A'VX)* = -$2AV*X* =

n(AV*X*)E,

4W (D'XV)* = DX*V*.

(3.7)

For any vector field X tangent to M we put

4)X = PX + FX,

where PX is the tangential part of OX and FX the normal part of qX. Similarly, for any vector field V normal to M, we put

OV=tV+fV, where tV is the tangential part of 4V and fV the normal part of V (see 93 of Chapter IV). We can define the operators P', F', t' and f' on N corresponding respectively to P, F, t and f (see §4 of Chapter VI). Then we have

(P'X)* = PX*,

(F'X)* = FX*

for any vector field X tangent to N. Moreover, we obtain

(t'V)* = tV*,

(f'V)* = fV*

for any vector field V normal to N. Thus we have PROPOSITION 3.2. (1) M is a contact CR submanifold of M if and only if N is a CR sulmanifold of N;

(2) M is a generic sutmanifold of M if and only if N is a generic sutmanifold of N;

(3) M is an anti-invariant submanifold of rd tangent to C if and only if N is an anti-invariant sub manifold of N; (4) M is an invariant submanifold of M if and only if N is an invariant submanifold (a complex sulmanifold) of N. We now study the relation between covariant differentiations of the second fundamental forms B and B'. From (3.4), (3.5) and (3.7)

460

we have

((VXB')(Y,Z))* - G(Y*,4X*)BQ,Z*) -G(Z*,4X*)B(Y*,E) = (VX*B) (Y*, Z*).

On the other hand, we see that

VXE = -PX,

B(X,O = -FX,

0.

Thus we obtain

(3.8)

(VX*B)(Y*,Z*) = [(VXB')(Y,Z)+g(Y,P'X)F'Z+g(Z,P'X)F'Y]*.

Moreover, from (3.7) of Chapter IV, we obtain

(VX*B)(Y*,E) = (VX*F)Y* - B(Y*,PX*) = fB(X*,Y*) - B(X*,PY*) - B(Y*,PX*),

from which

(3.9)

(VX*B)(Y*,E) = [f'B'(X,Y)-B'(X,P'Y)-B'(Y,P'X)]*.

From (3.8) and (3.9) we see that B is parallel if and only if B' satisfies

(VXB')(Y,Z) = g(X,P'Y)F'Z + g(X,P'Z)F'Y

and

f'B'(X,Y) = B'(X,P'Y) + B'(Y,P'X).

This is the proof of Lemma 3.1 of Chapter IV.

'E6I In the next place, we consider the normal connections of M and N. We denote by K` and R` the curvature tensors of the normal bundles of

M and N respectively. We give the relation of Kl and R. Let X and Y be vector fields tangent to N and U and V be vector fields normal to N. Then (3.7) implies

(D'XD'YV)*

DX*DY*V*,

(D'YD'XV)* =

DY*DX*V*.

Since [X,Y]* = [X*,Y*] - 2G(Y*,PX*)E, we find

(D'[X,Y]V)* = D[X* Y*]V* - 2G(Y*,PX*)DV*.

From these equations we have

(3.10)

G(KL(X*,Y*)V*,U*) = [g(R'(X,Y)V,U)+2g(Y,P'X)g(f'V,U)]*.

On the other hand, the Ricci equation of M implies

G(KL(X*,E)V*,U*) = G([AU*,AV*]X*,E) = G(AV*X*,tU*) - G(A. X*,tV*) _ -[g(F'A'VX,U) + g(B'(X,t'V),U)]*.

From this and (3.9) of Chapter IV we obtain

(3.11)

G(KL(X*, )V*,U*) = g((VXf')V,U)*.

Consequently, the normal connection of M is flat if and only if

R1(X,Y)V = 2g(X,P'Y)f'V

and

(y, )V = 0.

This gives the proof of Leama 3.2 of Chapter IV.

462

PROPOSITION 3.3. Let M and N be invariant suhmanifolds of M and N respectively. Then the second fundamental form B of M is n-parallel if and only if the second fundamental form B' of N is parallel.

Proof. From the assumption we see that F' = 0. Thus (3.8) shows that (VX*B)(Y*,Z*) = ((VXB')(Y,Z))*. From this we have our assertion. QED.

When M is anti-invariant in M, the normal connection of M is

pseudo-flat if and only if K(X*.Y*)V* = 0 (see Exercise E of Chapter VI). Thus, fran (3.10), we have PROPOSITION 3.4, If M and N are anti-invariant submanifolds of R and N respectively, then the normal connection of M is pseudo-flat if and-only if the normal connection of N is flat.

When M is a generic submanifold of M, f vanishes identically, and hence f' on N vanishes identically. Therefore, we have PROPOSITION 3.5. If M and N are generic submanifolds of M and IN respectively, then the normal connection of M is flat if and only if the normal connection of N is flat.

3.1. Let u and u' be the mean curvature vectors of M and N respectively. Then

n+l

Proof. If we take an orthonormal basis {ei} for Tx(N). Then {et,E} forms an orthonormal basis for Ty(M) (a(y) = x). Thus (3.5)

implies (u')* = n(TrB')* = 1[EB(et,et) + B(E,E)] = 1TYB = nnl u.

In view of Learna 3.1 we obtain

PROPOSITION 3.6. M is a minimal sutmanifold of M if and only if N is a minimal submanifold of N.

MK From (3.7) and Lemma 3.1 we have DX*u

n+1

*(u')*

nnl(D'XU')*.

Therefore we have PROPOSITION 3.7. If the mean curvature vector u of M is parallel, then the mean curvature vector u' of N is also parallel. We consider the converse of this proposition. We prove the follow-

ing LE4 A 3,2, Let M be a sutmanifold tangent to E of a Sasakian

manifold M. Then DEu = -fu.

Proof. We first notice that the second fundamental form B satis-

fies (VXB)(E,Y) = (VB)(X,Y). Then we have

(n+1)DEu = J(VEB)(ei,ei) = J(ve B)(E.ei) i

= J[ D B(E,ei) - B(Ve E.ei)) = -j(Ve F)ei i

= -7.[-B(ei,Pei) + fB(ei,ei)]

In the above we have denoted by the same ei local, orthonormal vector fields on M which extend ei of the orthonormal basis {ei} and which

are covariant constant with respect to V at x of M.

QED.

From Lemma 3.2 we have PROPOSITION 3.8. Let M and N be generic submanifolds of &'S and N

respectively. Then the mean curvature vector U.of M is parallel if and only if the mean curvature vector p' of N is parallel.

464

When M is invariant in M, the mean curvature vector p of M is pseudo-parallel if and only if DX*v = 0 (see Exercise E of Chapter VI). Thus we have PROPOSITION 3.9. Let M and N be anti-invariant suYmanifolds of M and N respectively. Then the mean curvature vector p of M is pseudoparallel if and only if the mean curvature vector u' of N is parallel. Let R and R' be the Riemannian curvature tensors of M and N respectively. From Proposition 3.1 we obtain

(R'(X,Y)Z)* - (A'B,(Y Z)X)* + (A'B'(X Z)Y)* ((OXB')(Y,Z))* - ((VyB)(X,Z))*

= R(X*,Y*)Z* - AB(Y*,Z*)X* + %(X*,Z*)Y* + (oX*B)(Y*,Z*) - (VY*B)(X*,Z*) + G(Z*,PY*)¢X* - G(Z*,PX*)OY* - 2G(Y*,PX*)¢Z*.

Taking the tangential part of this equation and using (3.6), we obtain PROPOSITION 3.10. The Riemannian curvature tensors R and R' satisfy

(R'(X,Y)Z)* = R(X*,Y*)Z* + G(Y*,tB(X*,Z*))E - G(X*,tB(Y*,Z*))E + G(Z*,PY*)PX* - G(Z*,PX*)PY* - 2G(Y*,PX*)PZ*.

We denote by S and S' the Ricci tensors of M and N respectively. Then we have

(3.12)

S'(X,Y) = S(X*,Y*) + 2G(PX*,PY*).

Let {ei} be annorthonormal basis of N. Then, by (3.12), the scalar curvature r of M and the scalar curvature r' of N satisfy

40 (3.13)

r' = r + YG(Pet,Pet),

r' = r + n - IG(Fei,Fei).

Therefore we obtain the following PROPOSITION 3.11, (1) M and N are anti-invariant sub manifolds;

if and only if r = r';

(2) M and N are invariant sukmanifolds if and only if r = r' - n. We now ccapute the square of the length of the second fundamental forms. We obtain

G(A,A) _

2JG(B(et,E),B(et,E)) i

g(B'(ei,e),B'(ei,ej)) + 21G(Fet,Fet)

= g(A',A') + 21g(F'ei,F'ei), i

that is,

(3.14)

1A12 = JA1I2 + 21g(F'ei,F'ei) IA'12 + 2n - 21g(F'ei,F'ei).

i

From (3.14) we have PROPOSITION 3,12. (1) M and N are invariant sutmanifolds if and only if 1A12 = JA'12;

(2) M and N are anti-invariant sutmanifolds if and only if JA!2 = IA'12 + 2n. As an application of submersions with totally geodesic fibre, we consider sulmanifolds of a complex projective space (see Lawson [1]). THEOREM 3.2. Let N be an n-dimensional compact minimal sutmanifold

of &n. If the scalar curvature r' of N satisfies the inequality

p

r' > n(n+2) - 2-i

(p = 2m-n),

466

then M is a totally geodesic eamplex projective space

CPn/2.

S2m+1 Proof. We consider the fibering n : > CPm. By S2m}1 Proposition 3.6 the submanifold N of is minimal. Thus the scalr curvature r of N is given by

r = n(n+1) - G(A,A).

From this and (3.13) we obtain

G(A,A) = n(n+2) - r' - JG(Fet,Fei) < n(n+2) - r'. i

Therefore, by the assumption, we have

G(A,A) < n+l - 2-1/p

Hence Theorem 5.4 of Chapter II implies that A = 0 or JA{2= S2m+l (n+l)/(2-1/p). If A = 0, then M is totally geodesic in and hence N is a totally geodesic complex projective space

Cpn/2.

If

JA12 = (n+1)/(2-1/p), then F = 0, which shows that M is an invariant S2m+1. Moreover, from Theorem 5.8 of Chapter II, M is sub manifold of Sn+2.

a Veronese surface in S4 or a Clifford minimal hypersurface in S2m+1 But the ambient manifold is odd dimensional and any hypersurface S2m+1 of is not invariant. From these considerations we have our

assertion.

QED.

467 EXERCISES

A. POSITIVE RICCI CURVATURE: Nash [1] construct complete metrices of positive Ricci curvature on a large class of fibre bundles. THEOREM. Let it

: M -> B be a vector bundle over B, a compact

manifold admitting a metric of positive Ricci curvature. If the fibre dimension is greater than two, M admits a carrplete metric of positive Ricci curvature.

B. FIBRE BUNDLES AND SUBMERSIONS: A mapping f : M

> B of

Riemannian manifolds is said to be totally geodesic if for each geodesic xt in M the image f(xt) is a geodesic in B. Vilms [1] proved THEOREM 1. A Riemannian submersion is totally geodesic if and only if the fibres are totally geodesic sulznanifolds and the horizontal

subbundle is integrable. THEOREM 2. Let IT : M

> B be a fibre bundle with standard

fibre F and Lie structure group G. Assume the bundle admits a connection in the sense of Ehresman [1]. Endow B and F with Riemannian metrices, and assume F is G invariant. Then there exists a natural metric on M such that it is a Rienannian submersion with totally geodesic fibres.

We also have (Hermann [1], Muto [1], Nagano [1])

THEOREM 3. Let it : M

> B be a Riemannian submersion, and

assume M to be connected. If M is complete, so is B, and n is locally trivial fibre space. If, in addition, the fibres are totally geodesic, then it is a fibre bundle with structure group the Lie group of iso-

metries of the fibre. For a geodesic we have (Hermann [1], O'Neill [3]) THEOREM 4. Let it : M

> B be a Riemannian submersion. If x

is a geodesic of M which is borizontal at one point, then it is always

horizontal, and hence

is a geodesic of B.

40 C. EQUIVALENCE PROBLEM: Let Trl and

n2

be Riemannian submersions

from a complete M onto B. Assume the fibres of Trl and Tr2 are totally geodesic. Trl and Tr2 are said to be equivalent provided there exists

an isometry f of M which induces an isometry f' of B, so that the

following diagram commutes: M Trl

> M

f

I

1112 tl

.

B

>B

Escobales [1] proved THEOREM 1. Let Trl and r2 be Riemannian submersions from M onto B

satisfying the above hypotheses. Suppose f is an isometry of M satisfying the following two conditions: (1) f*p : ylp

> y2f(p) is an isometry from the horizontal

distribution yip of Trl at p onto the horizontal distribution 92f(p) of ?2 at f(p);

(2) For E, F E Tp(M), f*(AIEF) = A2f*Ef*F, where Ai are the integrability tensors of Tri.

Then f induces an isometry f' of B so that Trl and Tr2 are equiva-

lent. O'Neill [2] gives the following

THEOREM 2, Let Trl and 72 be submersions of a Riemannian manifold M and B. If Tr1 and Tr2 have the same tensors T and A, and if their derivation maps agree at one point of M, then Tr1 = 7r2. D. SUBMERSIONS WITH TOTALLY GEODESIC FIBRE: Escobales [1] gives the classification those B for which there is a Riemannian submersion TT

: SP

desic.

> B, where Sn is a square and the fibres are totally geo-

0 > B be a Riemannian submersion with

THEOREM 1. Let n : Sn

connected totally geodesic fibres, and assume 1 < dim fibre < n-i. Then, as a fibre bundle, n is one of the following types:

(a)

S1

S2m+1

(b)

>

S3

(c)

(e)

S1

S7

HPm

for n > 2

> S3

>

S4m+3

In

lIT

Cpm

>

(d)

S3

for n > 2

> S7

In

In

S2(i)

S4(1)

S15

In case (a) and (b), B is isometric to complex and quaternion projective space with sectional curvature 1 < K, < 4. In case (c), (d) and (e), B is isometric to a sphere of curvature 4.

Moreover, if

1 and

_

2 are say two submersions both in class

(a), (b) or Cc), then nl is equivalent to n2.

Escobales [2) proved the following

> B with connected complex

THEOREM 2, Any submersion it : CPn

totally geodesic fibres and with 2 < dim fibre < 2n-2 must fall into one of the following two classes:

(a)

S2

>

Cpl lIT BPFv

(b)

CP3

> CP7 lIT

S8()

In fact, 1 < K < 4, where K,4 denotes the curvature of BPFv, and S8( ) denotes the sphere of radius J. Moreover, class (a) is not empty. Finally, if n > 2, any two subversions in class (a) are equivalent.

41U

E. SUBMERSION WITH MINIMAL FIBRE: When a submersion has minimal fibres we obtain

THEOREM 1. Let n : M

> B be a Riemannian submersion with

minimal fibres. Then a closed hypersurface P of B has constant mean

curvature in B if and only if

a-1(P)

has constant mean curvature in

M.

We also have (see Escobales [21) THEOREM 2. Let it : M

> B be a Riemannian submersion with

totally geodesic fibres. Assume P is a totally geodesic submanifold of B. Then n-1(P) is totally geodesic provided AxY = 0 whenever Y is horizontal and tangent to n 1(P) and X is normal to n_1 (p).

For the submersion with totally umbilical fibres see Bishop [1]. F. REAL SUBMANIFOLDS AND SUBMERSIONS: Let 1r

:

> Nn be

Mn+1

a submersion with 1-dimensional fibre. Then the fundamental tensor

P of the submersion is a skew symmetric tensor of type (1,1) on N defined by

Vx*Y* = (0'XY)* + g(PY,X)*V,

where V is a unit vertical vector. Then Okumura [1] proved the following

THEOREM, Let Nn be a hypersurface of CP(n+l)/2 and

:

it

n+1

_

-> Nn the submersion which is compatible with the fibration > Sn+2 > CP (n+l)/2. In order that the second fundamental Si tensor A' of Nn commutes wit:, the fundamental tensor P of the submersion it, it is necessary and sufficient that the second fundamental Mn+1 tenor A of is parallel.

G. QUATERNION KAEHLERIAN MANIFOLDS AND SASAKIAN 3-STRUCTURE: Let it

: M

> B be a Riemannian submersion. Then we have the following

(see Ishihara-Konishi [1])

471

THEOREM. (1) If M admits a K-contact 3-structure, then B admits an almost quaternion structure;

(2) If M admits a Sasakian 3-structure, then B admits a quaternion Kaehlerian structure; (3) If M admits a Sasakian structure with constant curvature c,

then c = 1 and the induced quaternion Kaehlerian structure of B is of constant Q-sectional curvature 4.

473

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455

INDEX

A Adapted frame of f-structure, 385 Affine

connection, 18 transformation, 20

Almost canplex manifold, 107 canplex mapping, 109 canplex structure, 107 contact manifold, 252 contact metric manifold, 254 contact metric structure, 254 contact structure, 252

Hermitian manifold, 124 Hermitian submersion, 449 Kaehlerian manifold, 128

product manifold, 423

product Riemannian manifold, 423 product structure, 423 quaternion manifold, 159 quaternion metric manifold, 160 quaternion metric structure, 160

quaternion structure, 159 semi-Kaehlerian manifold, 129 symplectic manifold, 129

Tachibana manifold, 129 Alternate mapping, 6 Analytic function, 3

manifold, 2

0 Analytic

structure, 2 vector field, 114 Anti-invariant sutmanifold, 199, 329 Associated second fundamental form, 66

vector field to n, 256

B Base space, 48

Basic vector field, 440

Bianchi's 1st identity, 31 2nd identity, 31 Brieskorn manifold, 291

Bundle space, 48 C

Canonical complex structure, 118 flat connection, 57 form, 58

global basis, 159 local basis, 159 Chart, 1 Clifford

minimal hypersurface, 90 torus, 90

Codazzi equation, 68 Codifferential, 42

Complete

manifold, 30 vector field, 14

C Complex analytic mapping, 3 conjugation, 105

Lie algebra, 118 Lie group, 119 projective space, 120 quadric, 142 r-form, 110 space form, 134

structure, 105 tangent space, 109 tangent vector, 109 torus, 120

Complexification, 105 Component of vector, 4

Concircular curvature tensor, 54 Conjugate vector, 105 Conformal

change, 40 Killing vector, 41 transformation, 41

Conformally flat space, 40 Connection, 56 Connection form, 57

Constant curvature, s5 holomorphic sectional curvature, 134

Q-sectional curvature, 169

Contact Bochner curvature tensor, 308 CR sutmanifold, 353 distribution, 269 form, 255

499 Contact

manifold, 255 manifolds in the wider sense, 258

metric manifold, 256 metric structure, 256 structure, 255 structure in the wider sense, 257 totally umbilical submanifold, 374 transformation, 257, 272 Contraction, 9 Contravariant tensor, 6 Coordinate neighborhood, 2 Cosymplectic manifold, 372 Covariant

constant framed metric manifold, 402 differentiation, 18 tensor, 6

Covering space, 50 CR product, 239 CR structure, 217

CR sutrnanifold, 216 Curvature tensor, 25

tensor of the normal bundle, 69 transformation, 31

D Derivation, 16 Derivative, 16

D-hamthetic deformation, 281 Diffeomorphisn, 13 Differentiable curve, 3 function, 3

499

Differentiable

manifold, 2 mapping, 2

structure, 2 transformation, 13 vector field, 4

Differential form, 5

Distance, 28 Distribution, 17

Divergence, 43 E Effictive action, 47 Einstein manifold, 38 Elliptic space, 39 F,quation

of Codazzi, 68 of Gauss, 68 of Ricci, 69

Equivalence submersion, 468 TI-Einstein manifold, 285

r) parallel Ricci tensor, 321

second fundamental form, 314 Euclidean n-space, 39 Exponential mapping, 22 Exterior differentiation, 11

F

F-anti-invariant submanifold, 426 Fibre, 48, 439 Fibre bundle, 50

F-invariant submanifold, 424 First normal space, 77

600

F Flat

connection, 58 normal connection, 70 space, 39

Framed f-structure, 402

manifold, 402 metric manifold, 402 Free action, 47 f-structure, 379 Fubini-Study metric, 141 Fundamental

2-form, 125, 256, 402 vector field, 48 G

Gauss

equation, 68 formula, 66

Gaussian curvature, 37 Generic sutxnanifold, z!6, 353

Geodesic, 21

Geodesic hypersphere, 249 Globally

framed f-structure, 402

0-synmetric manifold, 308 Great hypersphere, 40 H

Harmonic form, 42

501

H Hermitian manifold, 124 metric, 124 Hodge manifold, 291

Holomorphic bisectional curvature, 453

foam, 116 mapping, 3 sectional curvature, 134 vector field, 117

Homogeneous coordinate system, 120 Homomorphism, 49

Homothetic, 40 Horizontal lift, 57, 440 subspace, 56

vector, 56, 439

Hybrid tensor, 145, 417 Hyperbolic space, 39 Hypersphere, 39

I

Imbedded sutmanifold, 12 Imbedding, 12, 49 Immersed submanifold, 12 Immersion, 12

Induced connection, 65

metric, 62 Infinitesimal

affine transformation, 41

S02 Infinitesimal

automorphism, 114

isametry, 41 Inhcmogeneous coordinate system, 120 Injection, 49 Integrable, 113, 160, 391 Integral

curve, 13

manifold, 18 Invariant

submanifold, 180, 312 affine connection, 20 Involutive distribution, 17 Isometric

imbedding, 30 immersion, 30 Isometry, 30 Iscnorphic, 50

Isoperimetric section, 101 K

Kaehlerian

manifold, 129 metric, 128

submanifold, 180 K-contact

manifold, 267 structure, 267 Killing vector, 41 K-space, 129 L

Laplacian, 44 Left invariant vector field, 47

L Levi-Civita connection, 29

Lie algebra, 47

differentiation, 14

group, 46 transformation group, 47

Lift, 57 Local coordinate, 2

coordinate system, 2 1-parameter group, 13

Locally decomposable Riemannian manifold, 419 flat space, 38

¢-symmetric manifold, 308 product manifold, 415 product structure, 415 product Riemannian manifold, 419 syim tric space, 38

M Maximal integral manifold, 18

Mean curvature vector, 67 Metric connection, 29 Minimal section, 101

sulYnanifold, 67 Mixed tensor-algebra, 9 Multilinear mapping, 6

04 N

Nearly Kaehlerian manifold, 129 Noninvariant hypersurface, 408 Non-trivial

contact CR submanifold, 353 CR submanifold, 216 Normal

almost contact structure, 263 connection, 62

contact metric manifold, 272 contact metric structure, 272 f-structure, 233, 401 neighborhood, 22 variation, 408 0

1-parameter group, 13 Orbit, 13

P

Parallel

f-structure, 333 normal vector, 67

second fundamental form, 37 Partially integrable, 220 Period function, 286 0-section, 277 cu-sectional curvature, 277

Planer geodesic immersion, 102 Principal fibre bundle, 47 Product manifold, 3 Projection, 48

SM P

Projective curvature tensor, 54 Proper CR submanifold, 216

PseudoEinstein hypersurface, 376

Einstein real hypersurface, 249 flat normal connection, 373

parallel mean curvature vector, 373 umbilical hypersurface, 375

Pure tensor, 145, 416, 417

Q Quasi-Kaehlerian manifold, 129 Q-sectional curvature, 169

Quaternion Kaehlerian manifold, 161 Kaehlerian structure, 161 R

Rank of mapping, 12 of complex hypersurface, 191

Reduced bundle, 49

Reduction, 49 Regular contact structure, 286 coordinate neighborhood, 286

Restriction, 48

Ricci equation, 69 tensor, 37

Right invariant vector field, 47

SM R Riemannian

connection, 29 manifold, 27 metric, 27 submersion, 439 S

Sasakian

manifold, 272 space form, 281 structure, 272 3-structure, 307

Scalar curvature, 37 Second fundamental form, 65 Semiflat normal connection, 224 Kaehlerian manifold, 129 Separable coordinate system, 415 Skew-derivarion, 16 Small hypersphere, 40 Space form, 35

of constant curvature, 35 of constant holomorphic sectional curvature, 134

Standard Cat product, 240 Strict contact transformation, 258, 272 Structure group, 48 Subbundle, 49 Submersion, 439 Symmetric space, 38

0 S

Symplectic

manifold, 129, 287 structure, 287

T

Tangent space, 4

vector, 3

Tensor field, 6 Torsion tensor, 25

Total differential, 5 space, 48

Totally geodesic mapping, 467 geodesic submanifold, 67

real submanifold, 199, 250 umbilical subaanifold, 67 Transformation, 13 Transitive function, 49 Transversal form, 392

Trivial normal connection, 70 principal bibre bundle, 48 U

Umbilical section, 67 Universal covering space, 50

500

V

Vector, 3

Vector field, 4 Veronese surface, 90 Vertical subspace, 56 vector, 56, 439

Weingarten formula, 66 Weyl conformal curvature tensor, 40