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1 kluwer
the language of science
Geometry of CRSubmanifolds
Mathematics and Its Applications (East European Series)
Managing Editor:
M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam. The Netherlands
Editorial Board: A. BIALYNICKIBIRULA, Institute of Mathematics, Warsaw University, Poland H. KURKE, Humboldt University, Berlin, D.D.R. J. KURZWEIL, Mathematics Institute, Academy of Sciences, Prague, Czechoslovakia L. LEINDLER, Bolyai Institute, Szeged, Hungary L. LOVASZ, EetvOs Lordnd University, Budapest, Hungary D. S. MITRINOVI(, University of Belgrade, Yugoslavia S. ROLEWICZ, Polish Academy of Sciences, Warsaw, Poland BL. H. SENDOV, Bulgarian Academy of Sciences, Sofia, Bulgaria I. T. TODOROV, Bulgarian Academy of Sciences, Sofa, Bulgaria H. TRIEBEL, University of Jena, D.D.R.
Aurel Bejancu Department of Mathematics, Polytechnic Institute of lasi, Romania
Geometry of CRSubmanifolds
D. Reidel Publishing Company A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP 000
Dordrecht / Boston / Lancaster / Tokyo
Library of Congress CataloglsgiaPttbIl ado. Data Bejancu, Aurel, 1946Geometry of CRsubmanifolds.
CIP
(Mathematics and its applications. East European series) Bibliography: p. Includes indexes. 1. Submanifolds, CR. 2. Geometry, Differential. 1. Title. 11. Series: Mathematics and its applications (D. Reidel Publishing Company). East European series. QA649.B44 1986 516.3'6 8615614 ISBN 9027721947
Published by D. Reidel Publishing Company P.O. Box 17,3300 AA Dordrecht, Holland
Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, MA 02061, U.S.A. In all other countries sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, Holland
All Rights Reserved m 1986 by D. Reidel Publishing Company, Dordrecht, Holland No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner
Printed in The Netherlands
To Ligia, Rodica  Daniela and Aurelian
TABLE OF CONTENTS
EDITOR'S PREFACE
ix
PREFACE
xi
CHAPTER I : DifferentialGeometrical Structures on Manifolds
1. Linear connections on a manifold 2. The LeviCivita connection 3. Submanifolds of a Riemannian manifold 4. Distributions on a manifold 5. Kaehlerian manifolds 6. Sasakian manifolds 7. Quaternion Kaehlerian manifolds CHAPTER II: CRSubmanifolds of Almost Hermitian Manifolds 1. CRsubmanifolds and CRstructures 2. Integrability of distributions on a CRsubmanifold 3. (pconnections on a CRsubmanifold and CRproducts of almost Hermitian manifolds 4. The nonexistence of CRproducts in S6
1
2 5
7 10 15
18
20 24 30 34
CHAPTER III : CRSubmanifolds of Kaehlerian Manifolds
1. Integrability of distributions and geometry of leaves 2. Umbilical CRsubmanifolds of Kaehlerian manifolds 3. Normal CRsubmanifolds of Kaehlerian manifolds 4. Normal antiholomorphic submanifolds of Kaehlerian manifolds
5. CRproducts in Kaehlerian manifolds
39 43 50 56
63
6. Sasakian antiholomorphic submanifolds of Kaehlerian manifolds 7. Cohomology of CRsubmanifolds
67 73
TABLE OF CONTENTS
CHAPTER IV: CRSubmanifolds of Complex Space Forms
1. Characterization of CRsubmanifolds in complex space forms 2. Riemannian fibre bundles and antiholomorphic submani
77
folds of CP° 3. CRproducts of complex space forms 4. Mixed foliate CRsubmanifolds of complex space forms 5. CRsubmanifolds with semiflat normal connection 6. Pinching theorems for sectional curvatures of CRsubmani
79 84 88 94
folds
CHAPTER V: Extensions of CRStructures to Other Geometrical Structures 1. Semiinvariant submanifolds of Sasakian manifolds 2. Semiinvariant products of Sasakian manifolds 3. Semiinvariant submanifolds with flat normal connection 4. Generic submanifolds of Kaehlerian manifolds 5. QRsubmanifolds of quaternion Kaehlerian manifolds 6. Totally umbilical and toally geodesic QRsubmanifolds of quaternion Kaehlerian manifolds
95
100 106 112 114 115 122
CHAPTER V I: CRStructures and Pseudo Conformal Mappings 1. CRmanifolds and fstructures with complemented frames 2. Generic submanifolds of complex manifolds 3. Antiholomorphic submanifolds of complex manifolds 4. Pseudoconformal mappings
128 134 136 138
CHAPTER VII: CRStructures and Relativity 1. Geometrical Structures of spacetime 2. The twistor space and Penrose correspondence 3. Physical interpretations of CRstructures
143 145 147
REFERENCES
149
AUTHOR INDEX
164
SUBJECT INDEX
167
EDITOR'S PREFACE
Approach your problems from the right a W and begin with the answers. Then one day, perhaps you will find the final question.
It isn't that they can't see the solution. It is that they can't see the probkm
G.K. Chesterton. The Scotch! of Father 'The Hermit Clad in Crane Feathers' in R.
Brown "Ihe point of a Pin'.
van Gulik's The Chinese Maze Mwders.
Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various
sciences has changed drastically in recent years: measure theory is used (nontrivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one
another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in
addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and largescale order", which are almost impossible to fit into the existing classification schemes. gramme,
They draw upon widely different sections of mathematics. This proMathematics and Its Applications, is devoted to new emerging
(sub)disciplines and to such (new) interrelations as exempla grade:
 a central concept which plays an important role in several different mathematical and/or scientific specialized areas;  new applications of the results and ideas from one area of scientific endeavour into another;  influences which the results, problems and concepts of one field of enquiry have and have had on the development of another. The Mathematics and Its Applications programme tries to make available a careful selection of books which fit the philosophy outlined above. With such books, which are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better communication among the practitioners in diversified fields.
Because of the wealth of scholarly research being undertaken in the Soviet Union, Eastern Europe, and Japan, it was decided to devote special attention to the work emanating from these particular regions. Thus it was decided to start three regional series under the umbrella of the main MIA programme.
EDITOR'S PREFACE
x
The present volume in the MIA (Eastern Europe) series deals with a topic in differential geometry: CauchyRiemann submanifolds of Kithlerian manifolds and their (many) applications. This is a new field, (the concept was introduced by the author in 1978); it has the vigorousness characteristic of youth, and in spite of its youth it has already manyfold interactions with other parts of mathematics and substantial applications to (pseudo) conformal mappings and relativity. Also there are interrelations with harmonic maps, deformations of complex structures and more generally the whole field of (real) analysis on complex manifolds. The concept of a CR manifold generalizes both totally real submanifolds and holomorphic submanifolds, both concepts which have proved their worth. However there are not enough of these for many purposes whence the need for the more general notion of CR manifold. This is an uptodate and selfcontained book on the topic.
The unreasonable effectiveness of mathematics in science ... Eugene Wigner
Well, if you know of a better '01C. go to it.
As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company they drew from each other fresh vitality and
thenceforward marched on at a rapid pace towards perfection.
Bruce Bairnsfather Joseph Louis Lagraw
What is now proved was once only ima
William Blake
Bussum, July 1985
Michiel Hazewinkel
PREFACE
The theory of submanifolds of a Kaehlerian manifold is one of the most interesting topics in differential geometry. According to the behaviour of the tangent bundle of a submanifold, with respect to the action of the almost complex structure of the ambient manifold, we have three typical classes of submanifolds: holomorphic submanifolds (see Ogiue [1]), totally real submanifolds (see YanoKon [1]) and CR (CauchyRiemann) submanifolds. The notion of a CRsubmanifold has been introduced by the author in [1] as follows: Let N be an almost Hermitian manifold and let J be the almost complex structure of N. A real submanifold M of N is called a CRsubmanifold if there exists a differentiable distribution D on M satisfying (i)
and
J(Dx)
DX
(ii) J(DX) C TXM
for each x E M, where Dl is the complementary orthogonal distribution to D and TXM1 is the normal space to M at x. Thus holomorphic submanifolds and totally real submanifolds are particular cases of CRsubmanifolds. Moreover, each real hypersurface of N is a CRsubmanifold which is neither a holomorphic submanifold nor a totally real submanifold. The purpose of the book is to introduce the reader to the main problems of geometry of CRsubmanifolds. In order to make it selfcontained as much as possible we arrange in Chapter I most of the required background material. Chapter II is devoted to the differential geometry of CRsubmanifolds of almost Hermitian manifolds. The integrability of both of the distributions D and Dl on a CRsubmanifold are studied and there is obtained a class of linear connections with respect to which the CRstructure is parallel. Also, it is proven that CRproducts do not exist in a sphere S6. In Chapter III we give results on some special classes
xii
PREFACE
of CRsubmanifolds of Kaehlerian manifolds: umbilical CRsubmanifolds, normal CRsubmanifolds, CRproducts, Sasakian antiholomorphic submanifolds. Also, the cohomology of CRsubmanifolds is studied. In Chapter IV we are concerned with CRsubmanifolds of complex space forms. We first discuss the method of Riemannian fibre bundles in the geometry of CRsubmanifolds. We include here various results on mixed foliate CRsubmanifolds, CRproducts, generic submanifolds and CRsubmanifolds with semiflat normal connection. in Chapter V we show that the theory of CRsubmanifolds of Kaehlerian manifolds initiated the study of new structures on submanifolds of several classes of manifolds. We only sketch a study of such structures in Sasakian manifolds and quaternion Kaehlerian manifolds. The interrelation of the geometry of CRsubmanifolds with the general theory of CRmanifolds is studied in Chapter VI. By means of some local fstructures with complemented frames we obtain results on pseudoconformal mappings between CRmanifolds. Finally, we show in Chapter VII an application of CRstructures to relativity. By Penrose correspondence we have an interesting passing from the geometry of a Minkowski space to the geometry of a CRsubmanifold. In concluding the preface I would like to express my sincere gratitude to Professor Michiel Hazewinkel for his valuable suggestions on both the content and the presentation of this book. Special thanks are due to Professor G. D. Ludden (Michigan State University) who edited the manuscript for language and terminology. I am indebted to Professor L. Ver$traelen (Katholieke Universiteit Leuven) for his kind support during the printing of the book. My thanks go to all authors of books and articles, whose ideas we benefited in preparation the manuscript. I express my hearty thanks to my teachers at both Universities of Timi$oara and Ia;i from whom I have learnt the differential geometry. Also, I should like to thank Dr. D. J. Larner for his patience and kind cooperation. It is a pleasant duty for me to acknowledge that D. Reidel Publishing Company took all possible care in the production of the book. March 5, 1985
AUREL BEJANCU
Chapter I
DIFFERENTIALGEOMETRICAL STRUCTURES ON MANIFOLDS
91. Linear Connections on a Manifold
Let N be a real ndimensional connected differentiable manifold. Throughout the book all manifolds and morphisms are supposed to be differentiable of class C°D. Denote by x } a system of coordinate neighborhoods on N, where U {U; is a neighborhood and xh are local coordinates in U, with the indices h, i, j, k,... taking on values in the range {1, ..., n}. TN and F(N) are respectively the tangent bundle to N and the algebra of differentiable functions on N. Also we denote by F(H) the module of differentiable sections of a vector bundle H. Then, for each X E r(TN) and f E F(N) we define Xf E F(N) by
Xf = Xh
ah ax
where Xh are the local components of X with respect to the natural frame {ah = a/axh}. In (1.1) and in the sequel, we make use of the Einstein convention, that is, repeated indices with one upper index and one lower index denote summation over their range. A linear connection on N is a mapping
v
:
r (TN) x I' (TN)
r (TN) ;
(X, Y)
VXY,
satisfying the following conditions: (i)
Y(Z) = fVXZ + VYZ Vfx +
and (ii)
VX(fY + Z)  fVXY + (Xf)Y + VXZ,
for any f E F(N) and X,Y,Z E r(TN). The operator VX is called the covariant differentiation with respect to X. We define the covariant differentiation of a function
2
CHAPTER1
f with respect to X by
Vxf = Xf. Thus for any tensor field S of type (0, s) or (1, s) we define the covariant derivative Vxs of S with respect to X by (VxS)(Xl,...,Xs) = Vx(S(X1,...,xs)) S

E {S(X1,...,VXXi,...,XS)),
(1.2)
i=1
for any Xi E r(TN), i = 1,...,s. In a similar way we can define the covariant derivative of a tensor field of type (r, s), but for our purpose (1.2) is sufficient. The tensor field S is said to be parallel with respect to the linear connection V if we have V S = 0 x
for any
x E r(TN).
The torsion tensor T of a linear connection V is a tensor field T of type (1, 2) defined by T(X, Y) = V Y  VYx  [x, Y], x
(1.3)
for any X, Y E r(TN), where [X, Y] is the Lie bracket of vector fields X and Y defined by
[x, Y] (f)'  X(Yf)  Y(Xf) , for any f E F(N). A torsionfree connection is a linear connection with vanishing torsion tensor field. The curvature tensor R of a linear connection V is a tensor field of type (1, 3) defined by R(X, Y)Z  V VYZ  VYVxZ  V(X,
Y1Z.
for any X, Y, Z E r(TN).
§2. The LeviCivita Connection A tensor filed g of type (0, 2) is said to be a Riemannian metric on N if the following conditions are fulfilled: (i) g is symmetric, i.e., g(X, Y) = g(Y, X) for any
X, Y E r(TN), (ii) g is positive definite, i.e., g(X, X) > 0 for any X E r(TN) and g(X, X) = 0 if and only if X = 0. The manifold N endowed with a Riemannian metric g is called a Riemannian
DIFFERENTIALGEOMETRICAL STRUCTURES ON MANIFOLDS
3
manifold. The length of a vector X is denoted by 1Xp and it is
defined by
IIX I12
= g(X, X).
A linear connection V on N is said to be a Riemannian connection if the Riemannian metric g is parallel with respect to V, i.e., by (1.2) we have X(g(Y, Z)) = g(VXY, Z) + g(Y, VXZ), for all X, Y, Z E r(TN). The following Theorem is well known.
THEOREM 2.1. On a Riemannian manifold there exists one and only one torsionfree Riemannian connection. The Riemannian connection whose existence and uniqueness are stated in this theorem is called the LeviCivita connection and it is given by 2g(VXY, Z) = X(g(Y, Z)) + Y(g(Z, X))  Z(g(X, Y)) + + g([X, Y), Z) + g([Z, X], Y)  g([Y, z], x), (2.1) for any X, Y, Z E r(TN). From now on, in this section we let V denote the LeviCivita connection on N. Then by using the curvature tensor of V defined by (1.4) we shall introduce some other tensor fields of interest for the geometry of a Riemannian manifold. First, we define the Riemannian curvature tensor of
type (0, 4) by R(X, Y, U, V) = g(R(X, Y)U, V),
(2.2)
for any X, Y, u, v E r(TN). By using (1.4) and (2.2) it is easy to check the following formulas R(X, R(X, R(X, R(X,
Y, Y, Y, Y,
U, U, U, U,
V) V) V) V)
+ + = +
R(Y, R(X, R(U, R(Y,
X, Y, V, U,
U, V, X, X,
V) = 0, U) = 0, (2.3) Y), V) + R(U, X, Y, V) = 0.
Next, we consider a local field of orthonormal frames {E1,...,En} on N. Then n
{g(R(Ei, X)Y, Ei)),
S(X, Y)
(2.4)
i1
defines a global tensor field S of type (0, 2) called the Ricci tensor field. Using S we define a global scalar field p by
4
CHAPTERI
P =
n E {S(Ei, E1 },
(2.5)
i=1
called the scalar curvature of N. For each plane Y spanned by orthonormal vectors X and Y in the tangent space TxN, x E N we define the sectional curvature K(y) by
K(y) = KN(X A Y) = g(R(X, Y)Y, X).
(2.6)
It is not difficult to see that K(Y) is independent of the choice of the orthonormal basis {X, Y} of y. If K(y) is a constant for all planes y in T N and for all points x of N, then N is called a space of constant curvature or a real space form. The following theorem due to Schur is wellknown.
THEOREM 2.2. Let N be a connected Riemannian manifold of dimension n > 2. If the sectional curvature K(Y) depends only on the point x, then N is a real space form. We denote by N(c) a real space form of constant sectional curvature c. Then the curvature tensor of N(c) is given by R(X, Y)Z = c(g(Y, Z)X  g(X, Z)Y),
(2.7)
for any X, Y, Z E r(TN). The Ricci curvature with respect to a nonzero vector X is denoted by k(X) and it is defined by
k(X) = S(X, X)/g(X, X).
(2.8)
If the Ricci curvature at x is independent of the vector X, then the Ricci tensor is given by S = kg,
(2.9)
where k = k(X) for any X E TxN. If this is the case at every point of N and n > 2 then k is a constant on N. If n = 2, the Ricci tensor is always given by (2.9) and k is not necessarily a constant. A Riemannian manifold whose Ricci tensor is given by (2.9) is called an Einstein space. If the curvature tensor R vanishes, that is, N is a space of zero curvature, we say that N is a locally Euclidean space. Now, as it is well known, many applications of differential geometry are concerned with pseudoRiemannian metrics (see Chapter VII for applications to the geometry of a Minkowski space). Let g be a tensor field of type (0, 2) on N. We say that g is of constant index if the dimension of
DIFFERENTIALGEOMETRICAL STRUCTURES ON MANIFOLDS
5
the subspace W C T N on which g is negative definite is the same for all x E N. Then a nondegenerate and symmetric tensor field g of type (0, 2) on N is called a seudoRiemannian metric if it is of constant index. In this case we say that g has a signature of type (p, q) if the canonical form of g has p positive coefficients and q negative coefficients. The length of a vector X is defined as in the Riemannian case, that is, 1XI2 = g(X, X). Taking into account their importance in physics, pseudoRiemannian manifolds have been intensively studied from different points of view. Several results in this field can be found in O'Neill [1].
§3. Submanifolds of a Riemannian Manifold Let N be an ndimensional Riemannian manifold and let M be an mdimensional manifold immersed in N. Since we are dealing with a local study, we may assume that M is imbedded in N. Then M becomes a Riemannian submanifold of N with Riemannian metric induced by the Riemannian metric on N. Denote by TM1 the normal bundle tQ M and by g both metrics on M and N. Also, we denote by V and V the LeviCivita connections on N and M respectively. Then for any X, Y E r(TM) we have V Y = V Y + h(X, Y), X X
(3.1)
where h r(TM) x r(TM)  r(TM1) is a normal bundle valued symmetric bilinear form on r(TM). The equation (3.1) is called the Gauss formula and h is called the second fundamental form of M. Now, for any X E r(TM) and v E r(TMl) we denote by  AvX and VXv the tangential part and normal part of IXV :
respectively. Then we have V V =  AvX + VXV. X
(3.2)
Thus for any v E r(TMI) we have a linear operator AV : r(TM)  r(TM), satisfying
g(h(X, Y), V)  g(AVX, Y).
(3.3)
The linear operator Av is called the fundamental tensor of Weingarten with respect to the normal section V. The equation (3.2) is called the Weingarten formula.
CHAPTERI
6
The differential operator V1 defines a linear connection on the normal bundle TM called the normal connection on M. By using this linear connection and the LeviCivita connection on M we define the covariant derivative of h by
(VXh) (Y, Z) = VX (h (Y, Z) )  h (VXY, Z) 
(3.4)
 h (Y, VXZ) ,
for all X, Y, Z E t(TM). By a straightforward computation, using the Gauss and Weingarten formulas we obtain
R(X, Y)Z = R(X, Y)Z  Ah(Y, Z)X + Ah(X, Z)Y +
+ (VXh) (Y, Z)  (VYh) (X, Z),
(3.5)
for all X, Y, Z E F(TM), where R and R are the curvature tensors of M and N respectively. Then from (3.5) and (3.3) follows the Gauss equation g(R(X, Y)Z, U) = g(R(X, Y)Z, U) +
+ g(h(X, Z), h(Y, U))  g(h(Y, Z) , h(X, U) ) ,
(3.6)
for all X, Y, Z, U E r(TM). Also, taking the normal components of equation (3.5) we obtain the Codazzi equation {R(X, Y)Z}1 = (VXh)(Y, Z)  (VYh)(X, Z).
The curvature tensor R defined by
(3.7)
of the normal connection V1 is
R1(X, Y)V = VX YV  VYVXV  V[X,
Y] V,
(3.8)
for all X, Y E r(TM) and V E r(TM ). We define [AV, AU] = AV*A
 AUAV,
and by using the Gauss and Weingarten formulas obtain the Ricci equation g(R(X, Y) V, U) = g(R1(X, Y)V, U) + + g([AU, AV]X, Y), for any X, Y E r(TM) and V, U E I'(TMi).
(3.9)
DIFFERENTIALGEOMETRICAL STRUCTURES ON MANIFOLDS
7
If RI = 0 we say that the normal connection on M is flat. A normal vector field V on M is called parallel if we have VXv = 0 for any X E r(TM).
THEOREM 3.1 (Chen [1]). Let M be an mdimensional submanifold of an ndimensional Riemannian manifold N. Then the normal connection 0 of M in N is flat if and only if there exist locally n  m mutually orthogonal unit normal vector fields Vi such that each of the Vi is parallel. The submanifold M is said to be totally geodesic in N if its second fundamental form vanishes identically, i.e., h = 0 or equivalently AV = 0, for any V E r(TM1). If, for a normal section V on M, we have AV = al, where a is a differentiable function and I is the identity morphism on r(TM), then M is called totally umbilical with respect to the normal section V. If M is totally umbilical with respect to any normal section we say that M is totally umbilical. Let {E1,...,Em} be an orthonormal basis in TM. Then
M Tr(h) =
F {h(Ei, Ei)}, i=1
is independent of the basis and it is called the trace of h. By means of Tr(h) we define the mean curvature vector H of M by H =
Tr(h).
(3.10)
If H = 0 on M m we say that M is a minimal submanifold of N. Moreover, we obtain that M is totally umbilical if and only
if we have h(X, Y) = g(X, Y)H,
(3.11)
for any X, Y E r(TM). Finally, M is called a submanifold with parallel second fundamental form if we have Vh = 0.
§4. Distributions on a Manifold.
An mdimensional distribution on a manifold N is a mapping D defined on N, which assignee to each point x of N an mdimensional linear subspace Dx of TxN. A vector field X on N belongs to D if we have Xx E Dx for each x E N. When this happens we write x E r(D). The distribution D is said to be differentiable if for any x E N there exist m differentiable
CHAPTER1
g
linearly independent vector fields Xi E r(D) in a neighborhood of x. From now on, all distributions are supposed to be differentiable of class C. The distribution D is said to be involutive if for all vector fields X. Y E r(D) we have [X, Y E r(D). A submanifold M of N is said to be an integral manifold of D if for every point x E M, Dx coincides with the tangent space to M at x. If there exists no integral manifold of D which contains M, then M is called a maximal integral manifold or a leaf of D. The distribution D is said to be integrable if for every x E N there exists an integral manifold of D containing x. Then the well known theorem of Frobenius is stated as follows. THEOREM 4.1. Each involutive distribution is integrable. Moreover, through every point x E N there passes a unique maximal integral manifold of D and every other integral manifold containing x is an open submanifold of the maximal one.
Let V be a linear connection on N. The distribution D is said to be parallel with respect to V if we have for any
V Y E r(D) X
X E r(TN)
and
Y E r(D). (4.1)
Now, suppose N is endowed with two complementary distributions D and b, i.e., we have TN = D e B. Denote by P and Q the projection morphisms of TN to D and b respectively. THEOREM 4.2. All the linear connections with respect to which are parallel, are given by both distributions D and 0
V Y = PVXPY + QVXQY + PS(X, PY) + QS(X, QY), X
(4.2)
for any X, Y E r(TN), where V and S are, respectively, an arbitrary linear connection on N and an arbitrary tensor field of type (1, 2) on N. 0
Proof. Suppose V is an arbitrary linear connection on N. Then any linear connection V on N is given by V Y = VXY + S(X, Y), X
(4.3)
for any X, Y E r(TN), where S is an arbitrary tensor field of type (1, 2) on N. The distributions D and D are both parallel with respect to V if and only if we have
9
DIFFERENTIALGEOMETRICAL STRUCTURES ON MANIFOLDS
Q(VxY) = 0
and
(4.4)
P(VxZ) = 0,
for any X E r(TN), Y E r(D) and Z E F(D). From (4.3) and (4.4) it follows that D and D are parallel with respect to V if and only if we have 0
0
Q7xY + QS(X, Y) = 0
and
PVXZ + PS(X, Z) = 0.(4.5)
Thus, taking account of (4.5) in (4.3), we obtain (4.2). Now, by means of the projection morphisms P and Q we define a tensor field F of type (1, 1) on N by FX = PX  QX,
(4.6)
I. for any x E f'(TN). By direct computation follows f2 We say that F defines an almost product structure on N. The covariant derivative of F is defined by
(VxF)Y = VxFY  F(VxY),
(4.7)
for all X, Y E r(TN). We say that the almost product structure F is parallel with respect to the linear connection V if we have VxF = 0 for each X E F(TN). Then by using (4.4), (4.6) and (4.7) we obtain
THEOREM 4.3. Both distributions D and D are parallel with respect to V if and only if the almost product structure F is parallel with respect to V. Now suppose N is a Riemannian manifold endowed with two complementary orthogonal distributions D and Dl. Denote by V the LeviCivita connection on N. Then we have THEOREM 4.4. Both distributions D and D1 are parallel with respect to LeviCivita connection V if and only if they are integrable and their leaves are totally geodesic in N. Proof. Suppose both distributions D and Dl are parallel with respect to V. Then since V is a torsion free linear connection, we have [x, Y]  VxY  V1x e r(D)
for any
x, Y E r(D)
(u, v]  VUV  VVU E F(Di)
for any
u, v E F(D1).
and
Thus, by the Frobenius Theorem, both distributions D and Dl are integrable. Now, let N be a leaf of D and denote by h the second fundamental form of the immersion of N in N. Then
CHAPTER I
10
by the Gauss formula we have h(X, Y) = VXY  VXY,
(4.8)
for any X, Y E r(TM), where V' is the LeviCivita connection on M. Since VXY E I'(TM) and h(X, Y) E r(Dl), by (4.8) we obtain h = 0, that is M is totally geodesic in N. in a similar way it follows that each leaf of DI is totally geodesic in N. Conversely, suppose D and D'L be integrable and their leaves are totally geodesic in N. Then, by using (3.1) for the immersions of the leaves of D and DL in N, we obtain VXY E T(D)
for any
X, Y E T(D)
VUV E r(D1)
for any
U, V E r(DD).
and
Since g is parallel with respect to V we have g(VUY, V) = g(Y, VUV) = 0
and g(VXV, Y) = g(V, Vx Y) = 0,
for any X, Y E T(D) and U, V E r(D1). Thus both distributions D and Dl are parallel on N. The proof is complete. From Theorem 4.4 it follows that if N is endoyed with two complementary orthogonal distributions D and D that are parallel with respect to the LeviCivita connection, then N is locally a Riemannian product M X Ml, where M and Ml are leaves of D and respectively DJ. Finally, taking account of the fact that g is parallel with respect to V, we obtain PROPOSITION 4.1. The distribution D is parallel with respect if and only if the to the LeviCivita connection is parallel with complementary orthogonal distribution respect to V.
B. Kaehlerian Manifolds Let N be a real differentiable manifold. An almost complex
DIFFERENTIALGEOMETRICAL STRUCTURES ON MANIFOLDS
11
structure on N is a tensor field J of type (1, 1) on N such that at every point x E N we have J` = I, where I denotes the indentity transformation of TEN. A manifold N endowed with an almost complex structure is called an almost complex manifold. Every almost complex manifold is of even dimension and is orientable. Now suppose (N, J) is an almost complex manifold. Then we define the torsion tensor of J or the Nijenhuis tensor of J by [J, J](X, Y) _ [JX, JY]  [X, Y]  J[JX, Y]  J[X, JY]
(5.1)
for any X, Y E r(TN). If the torsion tensor of J vanishes identically on N then we say that J is a complex structure on N and N becomes a complex manifold. From (5.1) we have [J, J](JX, Y) = J([J, J](X, Y)) = [J, J](X, XY). (5.2)
A Hermitian metric on an almost complex manifold N is a Riemannian metric g satisfying g(JX, JY) = g(X, Y),
for any X, Y E r(TN). An almost complex manifold (resp. a complex manifold) endowed with a Hermitian metric is called an almost Hermitian manifold (resp. a Hermitian manifold). Every almost complex manifold with a Riemannian metric g' admits a Hermitian metric. In fact, if we take g(X, Y) = g'(X, Y) + g'(JX, JY), it is easily seen that g is a Hermitian metric on N. The 2form 0 of a Hermitian manifold N is defined by Z(x, Y) = g(X, JY),
for any
X, Y E r(TN).
(5.3)
By a direct computation it follows 3dS2(X, Y, Z) = g((lXJ)Y, Z) + g(VYJ)Z, X) + + g((VZJ)X, Y),
(5.4)
for all X, Y, Z E r(TN), where V is the LeviCivita connection on N. We say that N is a Kaehlerian manifold if its 2form S3 is closed, i.e., we have = 0. It is not difficult to prove that a Hermitian manifold is a Kaehlerian manifold if and only if the almost complex structure J is parallel with
CHAPTERI
12
respect to V, i.e., we have V J  0 for any X E r(TN). An X almost Hermitian manifold is called a nearly Kaehlerian manifold if we have (VXJ)X = 0,
for any
X E r(TN).
(5.5)
From (5.5) we obtain that N is a nearly Kaelerian manifold if and only if we have (VXJ)Y + (VYJ)X  0,
for any
X, Y E r(TN).
(5.6)
PROPOSITION 5.1. Let N be a nearly Kaelerian manifold. Then the Nijenhuis tensor of J is given by (J, J](X, Y)
= 4J(VYJ)X,
(5.7)
for any X, Y E r(TN).
Proof. Taking account of the fact that V is a torsion free connection on N, from (5.1) we obtain (J, J](X, Y) _ (VJXJ)Y  (VJYJ)X + J((VYJ)X)  J((VXJ)Y).
(5.8)
By using (5.6) we have (VJYJ)X = (VXJ)JY = V Y + J(VXJY)  J((VXJ)Y) X
Thus (5.8) becomes (J, J](X, Y)  2(VYX + J(VYJX)  VXY  J(VXJY)) = 2{J(VYJX  J(VYX))  J(VXJY  J(VXY))} ti
2J((VYJ)X  (VXJ)Y) = 4J((VYJ)X),
which proves our assertion, The curvature tensor R of a Kaehlerian manifold N satisfies R(X, Y)J  JR(X, Y)
and R(JX, JY) = R(X, Y), (5.9)
for any X, Y E r(TN). Now we consider a plane Y invariant by the almost complex structure J. In this case we can choose a basis {X, JX} in y, where X is a unit vector in y. Then the sectional curvature K(Y) is denoted by 13(X) and it is called the holomorphic sectional curvature of N determined by the unit vector X. Thus, by using (2.6), we have 13(X)  g(R(X, JX)JX, X)
(5.10)
DIFFERENTIALGEOMETRICAL STRUCTURES ON MANIFOLDS
13
If H(X) is a constant for all unit vectors in T N and for all points x E N, then N is called a space of constant holomorphic sectional curvature. THEOREM 5.1. Let N be a connected Kaehlerian manifold of complex dimension n % 2. If the holomorphic sectional curvature H(X) depends only on x E N, then N is a space of constant holomorphic sectional curvature. A Kaehlerian manifold of constant holomorphic sectional curvature is called a complex space form. THEOREM 5.2. Any two simply connected complete complex space forms of constant holomorphic sectional curvature c are holomorphically isometric to each other. For the proofs of Theorem 5.1 and 5.2 see Kobayashi and Nomizu II, p. 168 and 170. The curvature tensor of a complex space form N of constant holomorphic sectional curvature c is given by 4{g(Y, Z)X  g(X, Z)Y + g(Z, JY)JX 
R(X, Y)Z =
 g(Z, JX)JY + 2g(X, JY)JZ},
(5.11)
for any X, Y, Z E r(TN). The holomorphic bisectional curvature for the pair of unit vectors {X, Y} is given by
HB(X A Y) = g(R(X, JX)JY, Y).
(5.12)
A Typical Example of a Nearly Kaehlerian Manifold Let C be the Cayley division algebra generated by {e0 = 1, ei, 1 < i < 7), over R, and C+ the subspace of C consisting of all purely imaginary Cayley numbers. We may identify C+ with a 7dimensional Euclidean space R7 with the canonical inner product ( , ). The automorphism group of C+ is, by definition, the compact simple Lie group G2. Moreover, is invariant under the action of G, the inner product and hence G2 may be considered as a subgroup of SOM. A ` vector cross product for vectors in R7 C+ is defined by (
,
)
x x y= (x, y) e0 + xy, x, y E C+. Then the multiplication table is given by
14
CHAPTERI 4
7
1
2
3
1
0
e3
e2
e5
e4
e7
e6
2
e3
e1
e6
e7
e4
e5
3
e2
e1
0
e7
e6
e5
e4
4
e5
e6
e7
0
e1
e2
e3
5
e4
e7
e6
e1
0
e3
e,
6
e7
e4
e5
e2
e3
0
e1
7
e6
e5
e4
e3
e2
e1
0
ejXek=
0
5
6
j/k
Considering the unit sphere S6 as {x E C+; (x, x) = 1}, an almost complex structure J on S6 is defined by
JXU=xXU, where x E S6 and U E TX(S6). The almost complex structure J together with the induced metric g on S6 from the inner
product
(
,
)
on R7 = C
gives rise to a nearly Kaehlerian
structure on S6. The group G2 acts on S6 transitively as the group of automorphisms of the nearly Kaehlerian structure (J, g), (see FukamiIshihara (11). Examples of Kaehlerian Manifolds (1). The complex nspace Cn with the metric
ds2
=
E
dzaaI
a=1
where (z1,...,zn) is the natural coordinate system, is a complete, flat Kaehlerian manifold with fundamental 2form n S2
i F dza n dia. a=1
(2). Let CP be the ndimensional complex projective space. Then for any positive number c, CPn carries a Kaehlerian
DIFFERENTIALGEOMETRICAL STRUCTURES ON MANIFOLDS
15
metric of constant holomorphic sectional curvature c given by ds
4
2
(1
+1 zaza)(£ dzadza)  (E zadza) (E zadza)
c
(1 + E zaza ) 2
where (z11...,zn) is an inhomogeneous coordinate system of CPA.
(3). Let Dn be the unit open ball in Cn, i.e., zn);
Dn =
E zaza < 1}.
Then for any negative number c, Dn carries a Kaelerian metric of constant holomorphic sectional curvature c given by 4
( 1  E zaza) (E dzadzz) (T zadza) (E zadza)
(1  E zaa)2
c
For other examples and details of these examples see Kobayashi and Nomizu II, p.159. By using Theorem 5.2 and these examples we obtain that a simply connected complete Kaehlerian manifold of constant holomorphic sectional curvature c can be identified with the complex projective space CPn, the open unit ball Dn in Cn or with Cn, according as c > 0, c < 0 orc = 0 respectively.
96. Sasakian Manifolds
Let N be a real (2n + 1)dimensional differentiable manifold and 4, &, and n be a tensor field of type (1, 1), a vector field and a 1form respectively on N satisfying '"
_ I + fl
;
= 0; n(Ox) = 0; n(E) = 1,
(6.1)
for any x E I'(TN). Then N is called an almost contact E, n) the almost contact structure on N. manifold and Now, suppose on N is given a Riemannian metric tensor field g which satisfies the equations
g(0, $Y)  g(X, Y)  n(X) n(Y) and
n(X) = g(X,
),
(6.2) (6.3)
for any X, Y E I'(TN). Then N is called an almost contact metric manifold and (4', C, n, g) the almost contact metric structure. If we have
16
CHAPTER I
dn(X, Y) = g(X, 4,Y),
(6.4)
for all X, Y E r(TN) then we say that N is a contact metric manifold. The Nijenhuis tensor field of m is defined by [C, 4,](x, Y) = 14x, 4,Y] + 02[x, Y)  4,[X,$ Y]  4,[4,x, Y],
(6.5)
for all Y E r(TN). If N is an almost contact metric manifold and the Nijenhuis tensor of 4, satisfies
[4,, 4,] + 2dr)m& = 0,
(6.6)
then we say that N is a normal almost contact manifold. A normal contact manifold is called a Sasakian manifold. It is known (see Blair [3]) that a contact metric structure is Sasakian if and only if we have
(Vx4)Y = g(X, Y)t  n(y)x,
(6.7)
for any X, Y E r(TN), where V is the LeviCivita connection on N with respect to the Riemannian metric g. Also, on a Sasakian manifold we have I E  OX, x
for any
x E r(TN).
(6.8)
We denote by {0 1 the complementary orthogonal distribution to the 1dimensional distribution spanned by E on N. A plane section in the tangent space TXN is said to be a 4,section if it is spanned by X and OX where X E {E)X. The sectional curvature K(Y) determined by a 4,section y is called a 4,sectional curvature. If a Sasakian manifold has a 4,sectional curvature c which does not depend on the 4,section at each point, then c is a constant on the manifold. A Sasakian space form N(c) is a Sasakian manifold of constant 4, sectional curvature c.
The curvature tensor of a Sasakian space form N(c) is
giver. by R(X, Y)Z = c4 {g(Y, Z)X  g(X, Z)Y} 
 c41 {n(Y) r)(z)X  n(X) n(Z)Y + g(Y, Z) r)(X)i 
 g(x, + 2g(4,X, Y)4,Z}
g(4,Y, z)4,x + g($X, Z)4,Y + ,
for any
x, Y, Z E P(TN).
(6.9)
DIFFERENTIALGEOMETRICAL STRUCTURES ON MANIFOLDS
17
Examples of Sasakian Manifolds (1) Let S`n+1 be a (2n+1)dimensional unit sphere, i.e., Stn+1 = {z E Cn+1= IzI = 1}. Cn+1
and define
Denote by J the almost complex structure on z
S2n+1.
= Jz for any z E
Next we consider the orthogonal
projection IT
:
Tz(Cn+l)  Tz(S2n+1) ,
and define 4) = n°J. Thus we have a Sasakian structure 2n+1
n, g) on S
, where rI is a 1form dual to & and g
is the standard metric tensor field on Moreover, Stn+1 is of constant 4)sectional curvature 1, that is a Sasakian space form N(1). Stn+1.
(2) Let E2n+1 be an Euclidean space with cartesian coordinates (x1....xn, y1...... yn, z). Then we define a 2n+1 Sasakian structure n, g) on E by the following formulas
_ (0,...,0, 2),
gAB =
2n = (y1...... yn, 0,...,0, 1), 0
 1 yil
4 dij
0
0
and
10 4B 0
2n+1
dj
0
0
0
yj
0
4
1
4
Thus E is a Sasakian space form of constant 4)sectional E2n+1(3). curvature c = 3 and it is denoted by More examples and results on the geometry of Sasakian manifolds can be found in Blair (3].
CHAPTER I
18
7. Quaternion Kaehlerian Manifolds Let N be a real ndimensional differentiable manifold. Suppose there exists a vector bundle V consisting of tensors of type (1, 1) over N satisfying the following condition. In any coordinate neighborhood U of N, there is a local basis {J1, J2, J3} of V such that we have (J1)2 = (12)2 = (J3)2
I, (7.1)
J1°J2 = J2°J1 = J3,
where I is the identity tensor field of type (1, 1) in N. Such a local basis {J1, J2, J3} is called a canonical local basis of the vector fundle V in U. The manifold N is called an almost quaternion manifold. From the definition we obtain that the almost quaternion manifold is of real dimension n = 4m (m > 1) and each fibre of the vector bundle is 3dimensional. Let N be an almost quaternion manifold and U, U be two coordinate neighborhoods such that u f) U x Q. Wecongider the canonical local basis {J1, J2, J3} and (jr, J2, J3} in U and U respectively. Then we have 3
Ja =
E
SaJb
(a = 1, 2, 3),
(7.2)
b=1
where Sa are differentiable functions on u fl U. By using
(7.1) for both basis we obtain that at each point the
coefficients Sb(x) define an element of the proper orthogonal group SO(3) of dimension 3. By means of this result it follows that an almost quaternion manifold is orientable. Now we suppose that N is an almost quaternion manifold endowed with a Riemannian metric g satisfying
g(X, Y) + g(4X, Y) = 0,
(7.3)
for any X, Y E f(TN) and any local section 4 of the vector bundle V. In this case we say that N is an almost quaternion metric manifold. Thus if {J1, J2, J3} is a canonical basis of V, each of J1, J2, J3 is almost Hermitian with respect to g. Define three local 2forms 9 a on U by
DIFFERENTIALGEOMETRICAL STRUCTURES ON MANIFOLDS
S2a(X, Y) = g(X, JaY), a = 1, 2, 3,
19
(7.4)
for all vector fields X, Y on U. Then S2 defined by 3
Q=
Z
S2
a
(7.5)
A S2a
a=1
is a 4form globally defined on N. Next, we denote by V the LeviCivita connection on the almost quaternion metric manifold N. We say that N is a quaternion Kaehlerian manifold if, for each local section 4) of V and vector field X on N, +7X is also a section of V. It is not difficult to see that N is a quaternion Kaehlerian manifold if and only if we have 3
VXJa 
I
b=1
Qab(X)Jb,
a = 1, 2, 3,
(7.6)
for any X E r(TN), where Qab are certain local 1forms on N such that Qab + Qba = 0. By using (7.4), (7.5), and (7.6) we obtain that an almost quaternion metric manifold is a quaternion Kaehlerian manifold if and only if M = 0. Now, let N be a quaternion Kaehlerian manifold and X be a unit vector tangent to N at x. We denote by Y(X) the 4dimensional vector subspace of TxN spanned by (X, J1x, J2X, J3X) and call it the quaternion 4space determined by X. A quaternion plane is a 2dimensional vector subspace of a quaternion 4space. The sectional curvature for a quaternion plane is called a quaternion sectional curvature. If the quaternion sectional curvature is a constant c for all quaternion planes and for all points x of N we say that N is a quaternion space form and denote it by N(c). The curvature tensor of a quaternion space form N(c) is given by
(X, Y)Z = 4 {g (Y' Z)X  g(X, Z)Y + 3 +
E {g(JaY, Z)JaX  g(JaX, Z)JaY + a=1
+ 2g(X, JaY)JaZ)),
for any X, Y, z E r(TN).
(7.7)
Chapter II
CRSUBMANIFOLDS OF ALMOST HERMITIAN MANIFOLDS
§1. CRsubmanifolds and CRstructures
Let N be a ndimensional almost Hermitian manifold with almost complex structure J and with Hermitian metric g. Let M be a real mdimensional Riemannian manifold isometrically immersed in N. The differential geometry of M depends on the behaviour of the tangent bundle of M relative to the action of the almost complex structure J. Thus M is called a complex (holomorphic) submanifold if TxM is invariant by J, i.e., we have J(TxM) = TxM,
for each
x E M.
Also, we say that M is a totally real (antiinvariant) submanifold of N if we have
J(TxM) C TXM,
for each
x E M.
These two classes of submanifolds have been extensively investigated in the last decade from different viewpoints. The fundamental results on the geometry of totally real submanifolds can be found in YanoKon [1). Also, a survey of the principal results on the geometry of complex submanifolds is given by Ogiue in [1). In 1978 we initiated in [1) a study of the differential geometry of a new class of submanifolds situated between the above two classes, called CRsubmanifolds. More precisely, M is said to be a CRsubmanifold of N if there exists a differentiable distribution
D : x  Dx C TxM, on M satisfying the following conditions: (i) D is holomorphic, i.e., J(DX) = Dx for each x E M, (ii) the complementary orthogonal distribution D1.
D
: x +
Dx C TXM,
T!
is antiinvariant, i.e., J(DX) C , for each x E H. We denote by p the complex dimension of the distribution
CRSUBMANIFOLDS OF ALMOSTHERMITIAN MANIFOLDS
21
D and by q the real dimension of the distribution Di. Then for q = 0 (resp. p = 0) a CRsubmanifold becomes a complex submanifold (resp. totally real submanifold). If q = dim TXM1 the CRsubmanifold is called an antiholomorphic submanifold. A proper CRsubmanifold is a CRsubmanifold which is neither a complex submanifold nor a totally real submanifold. Each real hypersurface M of N (n > 2) is a proper CRsubmanifold. In fact we define Dl
D
J
ayd take D as the complementary orthogonal distribution to D in TM. Thus M is endowed with a pair of distributions (D, Dl) satisfying the conditions of the definition of a CRsubmanifold. Moreover, we have dim  1 and dim CDx = n 1. X Hence, M is a proper CRsubmanifold. RD'
Remark 1.1. It is easily seen that on a CRsubmanifold the distribution D (reap. Dl) is the maximal distribution invariant by J (reap. antiinvariant by J), i.e., if D' (rasp. D") is an invariant (reap. antiinvariant) distribution on M, then we have DX C Dx (resp. D" c Dx), for each x E M. Now let M be an arbitrary Riemannian manifold isometrically immersed in an almost Hermitian manifold N. For each vector field X tangent to M we put ix  ,X + wX,
(1.1)
where OX and wX are respectively the tangent part and the normal part of JX. Also, for each vector field V normal to M we put JV = BV + CV,
(1.2)
where BV and CV are respectively the tangent part and the normal part of JV. THEOREM 1.1. The submanifold M of N is a CRsubmanifold if and only if we have rank(O)  constant,
(1.3)
woo = 0.
(1.4)
and
Proof. Suppose M is a CRsubmanifold of an Almost Hermitian manifold N. Denote by P and Q respectively the projection morphisms of TM to D and D1. Then we have
22
CHAPTER n
x = JPX
(1.5)
and wX = JQX,
for any
X E r(TM).
(1.6)
Thus from (1.5) it follows rank(4) = 2p and from (1.6), taking account of (1.5), (1.4) follows. Conversely, suppose (1.3) and (1.4) are satisfied. We define the distribution D'by Dx = Im. 4x,
for each
x E M.
Clearly, D is an invariant distribution, since for each X = 0 E T(D) we have ix  ift = 42Y + (wo4))Y =
2
Y E C(Im. 4)) = r(D).
Denote by Dl the complementary orthogonal distribution to D in TM. Then Dl is an antiinvariant distribution. In fact, for any X E r(DL) and Y= U + W where U E T(D) and W E r(Di) we have g(JX, Y) = g(X, JU + JW) = g(X, JW) _ g(X, 4)W) = 0,
since ¢W E r(D). Thus M is a CRsubmanifold of N and the proof is complete.
THEOREM 1.2. The submanifold M of N is a CRsubmanifold if and only if we have rank(B) = constant,
(1.7)
O°B = 0.
(1.8)
and Proof. Suppose M is a CRsubmanifold. First we see that Ii.Bx c DX for each x E M. In fact, we have g(BV, Y) = g(JV, Y) _ g(V, JY) = 0. for any Y E C(D) and V E r(TMI). On the other hand, we have DI c Im.Bx. Indeed, if we take U E Dx then JU E
TOM and we
obtain U = J2U = BJU + CJU.
1 Hence U = BJU E Im.Bx. Thus we have D  Im.B, which implies rank(B)  constant. Next, for each V E r(TM1) we have
23
CRSUBMANIFOLDS OF ALMOST HERMITIAN MANIFOLDS
(JoB) V = (4oB) V + (WoB) V. 0 since both (JOB)V and (WoB)V are normal to Hence M. Thus (1.8) is proven. Conversely, suppose (1.7) and (1.8) be satisfied. Then we define the distribution Dl by DI = Im.Bx. First we note
that Dl is an antiinvariant distribution. In fact, for each X E F(D1) and Y E r(TM) we have g(JX, Y) = g((JOBV, Y) = g(($oB)V, Y) = 0,
by (1.8) .
I Next, the complementary orthogonal distribution D to D in TM is a holomorphic distribution. Indeed, for each x E r(D),
Y E
I'(DI)
and v E r(TMI) we have g(JX, Y)  g(X, JY) = 0,
since
JY E r(TM1)
and
g(JX, V)  g(X, JV)  g(X, BV)  0, since By E r(DL). Thus M is a CRsubmanifold of N. The proof is complete.
Now, suppose M is a CRsubmanifold of the almost Hermitian manifold N. Then from (1.5) we obtain 2
= P
(1.9)
and 03 + m = 0.
(1.10)
On the other hand, applying J to (1.2) and taking the normal part we get C2V + V + WBV = 0, which implies C3 + C = 0.
(1.11)
Thus from (1.10) and (1.11) it follows PROPOSITION 1.1. On each CRsubmanifold M the vector bundle morphisms and C define fstructures on TM and TM respectively. In order to justify the name CRsubmanifold we recall the definition of a CRmanifold. Some results on the geometry of CRmanifolds are found in Chapter VI. Let M be a differentiable manifold and TCM be the complexified tangent bundle to M. A CRstructure (see
24
CHAPTER II
Greenfield [1]) on M is a complex subbundle H of TCM such that H n R = {0} and H is involutive, i.e., for complex vector fields U and V in H, [U, v] is also in H. A manifold endowed with a CRstructure is called a CRmanifold. THEOREM 1.3 (BlairChen [1]). A CRsubmanifold of a Hermitian manifold is a CRmanifold. Proof. Suppose M is a CRsubmanifold of the Hermitian manifold N. Then the Nijenhuis tensor of J vanishes. Hence for any X, Y E r(D) we have 0 = [J, J](X, Y) = IX, Y] + [JX, JY]  J([JX, Y] + [X, JY]).
Thus we obtain 4([Jx, Y] + [X, JY]) = [ix, JY]  [X, Y]
(1.12)
Q([JX, JY]  [X, Y]) = 0.
(1.13)
and
Applying we obtain
to (1.12) and taking account of (1.9) and (1.13)
([X, Y]  [ix, JY]) _ [ix, Y] + [X, JY].
(1.14)
Now define the complex subbundle H of TCM by
Hx = {x  / OX; X E Dx}. OX, V = Y Take U  X computation we obtain
OY and by a direct
[v, V] = [x, Y]  [ix, JY] 
 {[X, jy] + [ix, Y]}.
(1.15)
Thus, the theorem follows from (1.15).
§2. Integrability of Distributions on a CRSubmanifold Let M be a CRsubmanifold of an almost Hermitian manifold N. The purpose of this paragraph is to study the integrability of both of the distributions D and D1 on M. For each vector field Z tangent to N we denote by ZT and Zl its tangent part to M and its normal part to M respectively. is given by The Nijenhuis tensor field of
25
CRSUBMANIFOLDS OFALMOSTHERMITIAN MANffOLDS
01(X, Y) _ [ox, 4Y] + 42[X, y]  ([X, $Y])  4([ox, Y]).
(2.1)
Then, by using (1.1) and (2.1), we obtain [J, J](X, Y) _ [o, m](X, Y)  Q([X, Y]) 
 W([4x, Y] + [x, Y]),
(2.2)
for any X, Y E r(D). Then from (2.2) we have
THEOREM 2.1 (Bejancu [21). Let M be a CR submanifold of an almost Hermitian manifold N. Then the distribution D is integrable if and only if [J, J](X, Y) T = [4, ](X, Y).
(2.3)
for any X, Y E r(D). Taking the normal part in (2.2) we obtain
[j, J](X, Y)1 =  W([$X, Y] + [X, Y]),
(2.4)
for any X, Y E r(D).
THEOREM 2.2 (Bejancu [2]). Let M be a CRsubmanifold of an almost Hermitian manifold N. Then the distribution D is integrable if and only if [J, J1 (X, Y)1 = 0
(2.5)
and
Q[4, ](X, Y) = 0,
for any
X, Y E r(D).
(2.6)
Proof. Suppose D is integrable. Then (2.5) follows from (2.4). By using (1.9) we obtain [o, o](X, Y) _
[ox, $y]  P([X, Y])  $([Ox, Y] + [x, 4Y]),
for any X, Y E r(D). Thus, taking into account that D = Im. we have
[o, ] (x, Y) E r (D)
,
which is equivalent to (2.6). Conversely, suppose (2.5) and (2.6) are satisfied. Then from (2.4) and (2.5) we have Q([Jx, Y] + [X, JY]) = 0, which implies Q([JX, jy]  [x, Y]) = 0.
CHAPTER!
26
Hence Q([J, J](X, obtain
y)T) = 0. On the other hand, from (2.2) we
Q([J, J](X, Y)T) = Q([m; 4>](X, Y))  Q([X, Y]).
Thus by (2.6) we obtain Q([X, Y]) = 0, that is, D is integrable. From Theorem 2.1 we obtain COROLLARY 2.1. Let M be a CRsubmanifold of a Hermitian manifold N. The distribution D is integrable if and only if the Nijenhuis tensor of 4> vanishes identically on D.
Now, we take X, Y E r (Dl) and obtain
I4>, q] (X, Y) = P([X, Y]).
(2.7)
Thus we have
THEOREM 2.3. Let M be a CRsubmanifold of an almost Hermitian manifold N. The distribution is integrable if and only if the Nijenhuis tensor of 4> vanishes identically on Next, we suppose that M is a CRsubmanifold of a nearly Kaehlerian manifold N. Then, by using the formulas of Gauss and Weingarten for the immersion of M in N and (5.6) of Chapter I, we obtain [ix, Y] + [X, JY] = 2 J([J, i](X, Y)) + J([X, Y]) + + Vix Y  VJYX + h(JX, Y) 
 h(X, JY),
(2.8)
for any X, Y F. T(D), where V is the LeviCivita connection on M and h is the second fundamental form of M. Taking into account that V is a torsionfree connection, from (2.8) we get 2 J([J, J](X, Y)) + h(X, JY)  h(JX, Y)
+ J([X, Y]) + VYJX VXJY, (2.9) for any X, Y E T(D).
THEOREM 2.4. (Sato [2]). Let M be a CRsubmanifold of a nearly Kaehlerian manifold N. Then the distribution D is integrable if and only if the following conditions are satisfied: h(X, JY) = h(JX, Y).
(2.10)
CRSUBMANIFOLDS OF ALMOSTHERMITIAN MANIFOLDS
27
and [J, J](X, Y) E r(D),
(2.11)
for any X, Y E r(D).
Proof. Suppose D is integrable, Then (2.8) implies h(X, JY)  h(JX, Y) =
J([J, JIM Y)).
(2.12)
2 From (2.12), taking account of (2.3) and (2.5) and (2.6), we obtain (2.10). Also, from (2.3) and (2.5) and (2.6) it follows that (2.11) holds. Conversely, suppose (2.10) and (2.11) are satisfied. Then by using (2.9) we obtain
J([X, Y]) = VXJY  V JX  2 J([J, J](X, Y)).
(2.13)
We note that for each Z E r(D) there exists V E r(TML) such that Z = JV. Then using (2.11) and (2.13) we obtain g([X, Y], JV) _ g(J([X, Y]), V) = 0.
Hence [X, Y] E r(D) for each X, Y E r(D), that is D is integrable. From Theorem 2.4 and (5.7) of Chapter I we have THEOREM 2.5 (Urbano [1]). Let M be a CRsubmanifold of a nearly Kaehlerian manifold N. Then the distribution is integrable if and only if 0 XJ)(Y) E r(D)
(2.14)
and (2.10) for any X, Y E r(D).
Also, from Theorem 2.5, by again using (5.7) of Chapter I we obtain COROLLARY 2.2. Let M be a CRsubmanifold of a nearly Kaehlerian manifold N. Then the distribution D is integrable if and only if (2.10) is satisfied and [J, J](X, U)T E r(D1),
(2.15)
for any X E F (D) and Ti E r (Dl) . we denote by v the complementary orthogonal subbundle to J(DI) in TMl. Obviously V is invariant by J, i.e., J(vx) = vX for each x E M. Then from (2.9) it follows that
28
CHAPTER R
g(h(X, JY)  h(Y, JX) = 4'9([J, J](X, Y), A0 = 0,
(2.16)
for any X, Y F_ r(D) and E E r(v). Thus from (2.16) we have
PROPOSITION 2.1. The condition (2.10) is satisfied if and
only
if
g(h(X, JY)  h(Y, JX), JZ) = 0,
(2.17)
for any Z, Y E r(D) and z E r(D1). I Now we are concerned with integrability of D on a CRsubmanifold in a nearly Kaehlerian manifold. First by using (5.4) and (5.6) of Chapter I we obtain
4(U, W, X) = g((lUJ)w, X),
(2.18)
for any U, W E r(Dl) and X E r(D). On the other hand, by direct computation we get 3dl(U, W, X) = g((U, w], ix).
(2.19)
Thus from (2.18) and (2.19) we have THEOREM 2.6 (Urbano (1]). Let M be a CRsubmanifold of a nearly Kaehlerian manifold N. Then the distribution D is integrable if and only if g((DUJ)W, X) = 0,
(2.20)
for any U, w E r(DI) and X E r(D).
Taking account of (5.6) from Chapter I and of the Gauss formula, from Theorem 2.6 we obtain COROLLARY 2.3 (Sato (2]). Let M be a CRsubmanifold of a nearly Kaehlerian manifold N. The distribution is integrable if and only if
g(h(U, x), JW) = g(h(W, X), JU),
(2.21)
for any U, W E r(D1) and x 1(D). Thus we have PROPOSITION 2.2. Let M be a CRsubmanifold of a nearly Kaehlerian manifold N. If DL is integrable then each leaf of is immersed in M as a totally geodesic submanifold if and
only if
g (h (U, X), JW)  0,
(2.22)
CRSUBMANIFOLDS OF ALMOST HERMAN MANIFOLDS
29
for any U, W E r(DI) and x E r(D). By using (5.7) of Chapter I and Theorem 2.6 we obtain
COROLLARY 2.4. Let M be a CRsubmanifold of a nearly Kaehlerian manifold N. Then is integrable if and only if
(J, J] (X, U)T E r (D)
(2.23)
,
for any X E r(D) and U E r(DI). Next, combining Corollary 2.2 with Corollary 2.4 we obtain PROPOSITION 2.3. Let M be a CRsubmanifold of a nearly Kaehlerian manifold N. Then both distributions D and D1 are integrable if and only if (2.10) is satisfied and [J, J](X, U)T = 0,
(2.24)
for any X E F(D) and U E r(D1).
A CRsubmanifold of an almost Hermitian manifold is called a mixed geodesic CRsubmanifold if its second fundamental form h satisfies h(X, U) = 0,
for any
x E r(D)
and
u E r(DI).
(2.25) Then, by using (3.3) of Chanter T we obtain PROPOSITION 2.4. Let M be a CRsubmanifold of an almost Hermitian manifold N. Then M is mixed geodesic if and only if both distributions D and DL are invariant with respect to the action of fundamental tensors of Weingarten, i.e., AvX E r(D)
and
AVU E r(D1),.
for any X E F(D), U E r(D1) and V E r(TM1). Finally, by using (2.25) and Corollary 2.3 we obtain COROLLARY 2.5. Let M be a mixed geodesic antiholomorphic submanifold of a nearly Kaehlerian manifold N. Then the distribution is integrable.
30
CHAPCERH
§3. 4)Connections on a CRSubmanifold and CRProducts of Almost Hermitian Manifolds
Let M be a CRsubmanifold of an almost Hermitian manifold N. In §1 we defined by (1.5) a tensor field $ of type (1, 1) on M, which by Proposition 1.1 is an fstructure. A linear connection V on M is called a 4)connection if 4) is covariantly constant with respect to this connection, that is, we have V Y = 0, X
for each
X E r(TM).
(3.1)
THEOREM 3.1 (Bejancu (101 ). All the 4)connections on the CRsubmanifold M of the almost Hermitian manifold N are given by VXY = PVXPY + QQXQY + 4{(VX4))$Y + PK(X, PY) 
$K(X, $Y)} + QS(X, QY),
(3.2)
0
where V is a linear connection with respect to which both distributions D and are parallel, P and Q are the projection morphisms to D and respectively and K and S are arbitrary tensor fields of type (1, 2) on M. for all X, Y E I'(TM)
Remark 3.1. The existence of a linear connection with respect to which both distributions D and Dl are parallel is proven by Theorem 4.2 of Chapter I. Proof of Theorem 3.1. Let V be a 4)connection on M. Then put 0
V Y = V Y + S(X, Y), X X
(3.3)
0
for any x, Y E r(TM), where V is a linear connection on M with respect to which both distributions D and Dl are parallel and S is a tensor field of type (1, 2) on M. Since V has to satisfy (3.1), by using (3.3) we have 0
(VXOY  $S(X, Y)  S(X, 4)Y).
(3.4)
Substituting $Y for Y in (3.4) and taking account of (1.9) we obtain 0
(VX$)Y = 4)S(X, $Y) + S(X, PY).
(3.5)
For each x E r(TM) we denote by SX the tensor field of type (1, 1) on M defined by
SX (Y) = S (X' PY).
(3.6)
31
CRSUBMANIFOLDS OF ALMOST HERMIITIAN MANIFOLDS
Then (3.5) becomes 0
(X')° = SX +
(3.7)
since Po4 = 0. From (3.6) it follows that
SX(Z) = 0,
(3.8)
for any z E r(DL) and X E r(TM). On the other hand, by using (3.7) and taking into account that D is parallel with respect to 8, we obtain Q(SX(Y)) = 0,
(3.9)
for any Y E r(D) and X E r(TM).
Next, we denote by ri(TM) the real vector space of all tensor fields of type (1,
1) on M. Also, we consider the
vector subspace ri(D) of ri(TM) defined by ri(D)
= {H E rl(TM);
HQY = QHPY = 0,
dY E r(TM)}.
(3.10)
Define two linear operators A and Y' on ri(D) by
AH =
2(H + 4)oHo4)
(3.11)
TH =
2(H  oHo4).
(3.12)
and
By a straightforward computation, using (3.10)(3.12), we obtain that A and Y' are complementary projectors on r(D), that is we have A2
= A;
'V
_
'V ;
Ao'V =
4'oA =
0;
A +' = 1,
(3.13)
where I is the identity morphism on ri(D). Now, by means of (3.8) and (3.9) we see that SX E r1 (D). Hence (3.7) becomes A(SX)
(3.14)
We have further that
IV (' 4
4o(VX0)eP)  0,
0
since VXP  0. Thus, there exists SX such that (3.14) is
32
CHAPTER II
satisfied and all solutions of (3.14) are given by SX = 2 (OX4))o4) + K.
 4)oKXod)},
(3.15
where KX is an arbitrary element of )'(D). Thus we have S(X, PY) _
{(VX4))4)Y + K(X, Y)  4)K(X, 4)Y)}, (3.16)
where K is a tensor field of type (1, 2) on M satisfying K(X, QY) = QK(X, PY) = 0.
(3.17)
By using (3.3) we obtain 0
(VX4))QY = OxQY)  4)s(X, QY).
(3.18)
Since V has to be a 4)connection and Dl is parallel with respect to V, from (3.18) we have PS(X, QY) = 0,
for any
X, Y E T(TM).
(3.19)
By using (3.16), (3.17), and (3.19) in (3.3) we obtain (3.2). Finally, it is a simple verification that all connections connections given by (3.2) are 4)connections. This completes the proof of the theorem. From Theorem 3.1 we obtain
COROLLARY 3.1. Both distributions D and Dl on a CRsubmanifold of an almost Hermitian manifold are parallel with respect to any 4)connection. Now, we say that a CRsubmanifold M of an almost Hermitian manifold N is a CRproduct if both distributions D and DL are integrable and M is locally a Riemannian product M X M2, where M1 is a leaf of D and M2 is a leaf of D}. A CRproduct with D x {0} and Dl x {0} is called a proper CRproduct.
THEOREM 3.2 (Bejancu (101). Let M be a CRsubmanifold of an almost Hermitian manifold N. if the LeviCivita connection on M is a 4)connection, then M is a CRproduct.
Proof. By Corollary parallel with respect to assertion of the theorem Chapter I. Remark 3.2. Theorem
3.1 both distributions D and DL are LeviCivita connection. Then the follows from Theorem 4.4 of 3.2 has been obtained by Bejancu
CRSUBMANIFOLDS OF ALMOST HERMITIAN MANIFOLDS
33
KonYano in [1] for antiholomorphic submanifolds of a Kaehler manifold and by Chen in [5] for CRsubmanifolds of a Kaehler manifold. THEOREM 3.3 (Sato [2] ). Let M be a CRsubmanifold of a nearly Kaehlerian manifold N. Suppose the following conditions are satisfied
g (h (X, Y), JZ) = 0,
(3.20)
for any X E r(TM), Y E r(D), Z E r(DI) and g([J, J](X, Y), W) = 0,
(3.21)
for any X, Y E r(D) and W E F(D'). Then M is a CRproduct of N.
Proof. By using (3.3) and (5.6) of Chapter I, (3.20) and the formulas of Gauss and Weingarten for the immersion of M in N we obtain 0 = g(VXY, JZ) = g(JVXY, Z) = = g(JVYX  VXJY  VYJX, Z) = g(h(X, Y), JZ)  g(VXJY, Z) + g(AJXY, Z)
= g(VXJY, z) + g(h(Y, Z), JX) _ g(VXJY, Z),
for any Y E r(D) and X, Z E r(D1). Thus we have VXJY E r(D),
(3.22)
for any x E r(DI) and Y E r(D). On the other hand, by using (5.7) of Chapter 1, (3.20), (3.21) and the formula of Gauss we get ti
0 = g((VXJ)Y, W)  g(VXJY, W) + g(VXY, JW). Hence we have vXJY E r(D),
(3.23)
for any X, Y E r(D). Thus, by (3.22) and (3.23), it follows that D is parallel with respect to the LeviCivita connection. Our assertion then follows from Proposition 4.1 and Theorem 4.4 of Chapter I. Remark 3.3. The notion of CRproduct in a Kaehler
CHAPTER H
34
manifold has been introduced by Chen in [5]. More results on the geometry of CRproducts are given in the next paragraph of this chapter, in Chapter III (see §5) and in Chapter IV (see §3).
§4. The NonExistence of CRProducts in S6
First we obtain some preliminary results on the geometry of the factors of a CRproduct. Then we prove the nonexistence of proper CRproducts in S6. We recall that S6 is endowed with a natural structure of a nearly Kaehlerian manifold (see §5 of Chapter I). Let M = M1 x M` be a proper CRproduct of S6. Denote by h' the second fundamental form of the immersion of M1 in S6. Then we have
h' (X, Y)  h(X, Y) ,
(4.1)
for any X, Y E r(TM1), since M1 is a totally geodesic
submanifold of M. Throughout this paragraph we denote by h the second fundamental form of the immersion of M in S6. Since M1 is a holomorphic submanifold of S6, by a result
of Vanhecke [1], it follows that M1 is a vsubmanifold of S6, i.e., we have h'(JX, Y) = Jh(X, Y),
for any X, Y E r(TM1). Thus by (4.1) we obtain
h(JX, Y) = Jh(X, Y) .
(4.2)
Also, by a result due to Gray [1] we have dim M1 = 2. Next we get g(VXY, Z) _ g(Y, VXZ) = 0,
(4.3)
for any X E r(TM), Y E r(D) and Z E T(D1) since D and Dl are parallel distributions with respect to the LeviCivita connection on M. The Gauss formula for the immersion of M in S6 and (4.3) implies
I + g(h(X, X), h(Z, Z))  g(h(X, Z), h(X, Z))  0, (4.4)
CRSUBMANIFOLDSOFALMOSTHERMITIAN MANIFOLDS
35
g(h(X, JX), h(Z, Z))  q(h(X, Z), h(JX, Z)) = 0, (4.5)
g(h(X, JX), h(X, Z))  g(h(X, X), h(JX, Z)) = 0, (4.6) for all unit vector fields x E r(D) and Z E_ r(DI). By using (4.2) we obtain
g(h(X, Y), JZ) = g(h(JX, Y), Z)  0,
(4.7)
for any X, Y E r(TM1) and z E r(TM2),
By means of the Gauss formula and (5.7) from Chapter I we get
ZJX 4{J([J, Jim X))} + J(VZX + h(Z, X)), VZJX = VZJX + h(Z, JX).
(4.8) (4.9)
Since VZx E r(D), from (4.8) and (4.9) it follows that h(Z, JX) = 4 J([J, J](Z, X))) + Jh(Z, X).
(4.10)
Next, we fix a unit vector Z E DX. Let
g(h(E, E), h(Z, Z) 
Max {g(h(X, NXN=1, XEDx
X), h(Z, Z))). (4.11)
Then we have g(h(E, JE), h(Z, Z))
0.
(4.12)
Next, from (4.5) and (4.12) we get
g(h(E, Z), h(JE, Z)) = 0.
(4.13)
LEMMA 4.1. ME, Z) x 0 and dim M = 3. Proof. Suppose h(E, Z) = 0. Then from (4.4) it follows that
g(h(E, E) , h(Z, Z) ) + 1  0.
(4.14)
From (4.4) and (4.11) we obtain g(h(E, Z), h(E, Z)) 
Max NxN1,
XDD
{g(h(X, Z), h(X, Z))}.
36
CHAFIERII
Then we find g(h(JR, Z), h(JE, Z)) = 0, and hence from (4.2) and (4.4) it follows that 0  g(h(JE, JE), h(Z, Z)) + 1 = _ g(h(E, E), h(Z, Z)) + 1.
(4.15)
But (4.15) contradicts (4.14). Thus we have h(E, Z) x 0. Moreover, we have dim M = 3 since by Proposition 2.2, h(E, Z) is perpendicular to JW for all W E D. Now, from (4.13), taking account of Proposition 2.2 and Lemma 4.1 we obtain
h (JE, Z) = aJh (E, Z) ,
(4.16)
for some a E R. From (4.2), (4.6) and (4.16) we have
(a + 1)g(h(E, E), Jh(E, Z)) = 0, (a + 1)g(h(E, E), h(E, Z)) = 0.
(4.17)
THEOREM 4.1 (Sekigawa [1] ). There does not exist any proper CRproduct in S6. Proof. Throughout this proof we take X, Y E Dx and Z E DI. First, we suppose a e 1 in (4.17). Then from (4.7)
and (4.17) we obtain h(X, Y)  0.
(4.18)
From (4.7), taking account of (3.1), (3.2), and (5.7) of Chapter I and of (4.2), (4.3), and (4.18) we obtain g((VZh)(X, Y), JZ)  0,
(4.19)
where Vh is given by (3.4) of Chapter I. Thus from Proposition 2.2 and taking account of (3.1), (3.7), and (5.7) of Chapter i, (4.3) and (4.19), we get g(4 J([J, J](X, Z)) + Jh(X, Z), h(Y, Z))  0.
Hence by using (5.2) of Chapter I we have
4 g ([J, JIM, Z), h(Y, Z)) = g (Jh (X, Z), MY, Z)). (4.20) Now we take Y = JX in (4.20) and obtain
CRSUBMANTFOLDS OF ALMOSTHERMITIAN MANIFOLDS
37
4 g([J, J](JX, Z), h(JX, Z))
= g(hh(X, Z), h(JX, Z)).
(4.21)
On the other hand, from (4.10) we have
g(hh(X, Z), h(JX, Z)) = g(h(X, Z), h(X, Z))  4 g(h(X, z) , (J, J] (X, Z))
.
(4.22)
From (4.22) it follows that
g(Jh (X, Z), h(JX, Z)) = g(h(JX, Z), h(JX, Z))  4 g(h(JX, Z), [J, J](JX, Z)).
(4.23)
Then (4.21) and (4.23) imply
2g(h(JX, Z), Jh(X, Z)) = g(h(JX, Z), h(JX, Z)). (4.24) From (4.24) it follows that
2g(h(JX, Z), Jh(X, Z)) = g(h(X, Z), h(X, Z)). (4.25)
Thus by using (4.16), (4.24) and (4.25) we obtain (a2
 1)g(h(E, Z), h(E, Z)) = 0
and
(2a  1) g (h (E, Z) , h (E, Z) = 0. But this is impossible by virtue of Lemma 4.1. Next, we suppose a = 1 in (4.16). Then from (4.4), taking account of (4.2) and (4.16), we obtain
1 + g(h(E, E), h(Z, Z))  g(h(E, Z), h(E, Z)) = 0, and 1
 g(h(E, E), h(Z, Z))  g(h(E, Z), h(E, Z)) = 0.
Thus we have
g(h(E, E), h(Z, Z)) = 0,1
g(h(E, Z), h(E, Z))  1. By using (4.12), (4.13), (4.16) and (4.26) we get
(4.26)
g(h(X, Y), h(Z, Z)) = 0,1 g(h(X, Z), h(X, Z)) = 1,
(4.27)
for any unit vectors X, Y E D . From (4.7), taking account of (3.1), (3.2) and (5.6) of 'Chapter I and of (4.2), (4.3) and (4.27) we obtain again (4.19). Thus in a similar way we get a contradiction. The proof is complete. Remark 4.1. By Theorem 4.1 there exist no proper CRsubmanifolds of S6 with both distributions D and Dl parallel with respect to LeviCivita connection. However, Sekigawa constructed in (1] an example of a 3dimensional proper CRsubmanifold such that both distributions D and D are integrable.
Chapter III
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
§1. Integrability of Distributions and the Geometry of Leaves Let N be a Kaehlerian manifold. Then we have (VXJ)Y = 0
and consequently [J, J](X, Y) = 0,
(1.2)
for any x, Y E r(TN). Suppose M is a CRsubmanifold of N. Then by using (1.1) and (1.2) in Theorems 2.4 and 2.6 of Chapter II we obtain THEOREM 1.1. Let M be a CRsubmanifold of a Kaehlerian manifold N. Then we have (i) the distribution D is integrable, (ii) the distribution D is integrable if and only if the second fundamental form of M satisfies h(X, JY)  h(Y, JX),
for any X, Y E r(D).
(1.3)
Remark 1.1. The assertion (i) is due to BlairChen [1] and the assertion (ii) has been obtained by the author in [1]. Also we note that Blair and Chen have constructed in [1] an example of a CRsubmanifold of a Hermitian manifold for which Dl is not integrable. Remark 1.2. By Proposition 2.1 of Chapter II and the assertion (ii) of Theorem 1.1 we obtain that D is integrable if and only if we have g(h(X, JY)  h(Y, JX), JZ) = 0,
(1.4)
for any X, Y E r(D) and Z E r(DI). We say that M is a Dgeodesic CRsubmanifold if its second fundamental form satisfies h(x, Y)  0,
for any
x, Y E r(D).
(1.5)
40
CHAPTER 111
THEOREM 1.2 (Chen [5]). Let M be a CRsubmanifold of a Xaehlerian manifold N. Then (1) the distribution D is integrable and its leaves are totally geodesic in M if and only if g(h(X, Y), JZ) = 0,
(1.6)
for all X, Y E r(D) and Z E r(D1), (ii) the distribution D is integrable and its leaves are totally geodesic in N if and only if M is Dgeodesic. Proof. Suppose D is integrable and each leaf of D is totally geodesic in M. Then we have V Y E r(D) for any x, Y E r(D). By using (1.1) and the formula of Weingarten we obtain g(h(X, Y), JZ) = g(J(VxY), Z) _ g(V{JY, Z) = g(VxJY, Z)= 0,
(1.7)
for any X, Y E r(D) and Z E r(Dl). Thus (1.6) is satisfied. Conversely, from (1.6) and by Remark 1.2. the integrability of D follows. By a similar computation as in (1.7) we obtain VxY E r(D) for all x, Y E r(D), that is, each leaf of D is totally geodesic in M. Thus we get the assertion
(i) . Now, suppose D is integrable and its leaves are totally geodesic in N. Then we have VxY E r(D) for any X, Y E r(D). Thus by using the Gauss formula we obtain
g (h (X, Y) , V) = g (VxY, V) = 0, for any X, Y E r(D) and v E r(TML). Hence M is a Dgeodesic CRsubmanifold. Finally, suppose (1.5) is satisfied. Then by (1.3) it follows that D is integrable and V Y E r(TM) for any x X, Y E r(D). By using (1.1) we have g(VxY, Z)  g(VxJY, JZ) = g(h(X, JY), JZ) = 0, for any X, Y E r(D) and z E r(D1). Thus V Y E r(D), i.e., each leaf of D is totally geodesic in N. xThe proof is complete. The invariant distribution D and the antiinvariant distribution Di are defined respectively by the projectors P and Q. Taking account of (1.1) we obtain
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
41
P(VXOY)  P(AWYX)  (VXY), Q(VXOY)  Q(AAYX) = Bh(X, Y), h(X, 4,Y) + VXwY = W(VxY) + ch(X, Y),
for any X, Y E r(TM), where (1.6) respectively of Chapter Differentiating (1.2) of of the decomposition TN = D
and w are defined by (1.5) and II. Chapter II and taking account Dl ® TMi we obtain
P(VXBV) + (AVX) = P(ACVX),
(1.11)
Q(OXBV)  Q(ACVX) + B(VXV),
(1.12)
h(x, BV) +
(1.13)
XCV + w(AVX)  C(VXV),
for all x E r(TM) and V E r(TM1).
Now, let M2 be a leaf of Dl. Then we have THEOREM 1.3 (BejancuKonYano (1]). A necessary and sufficient condition for the submanifold M2 to be totally geodesic in M is that h(X, Z) E r(v),
(1.14)
for all X E r(D'), and z E r(D), where V is the orthogonal
complementary subbundle to JDl in TMl.
Proof. We take X, Y E r(DI) in (1.8) and for each Z E r(D) we obtain
Z)  q(AwYX, Z)  g(h(X, Z) , WY) . This proves our assertion since M2 is totally geodesic in M if and only if VXY E r(DI) for any X, Y E r(DL). From Theorem 1.3 we have
COROLLARY 1.1. Let M be a mixed geodesic CRsubmanifold of a Kaehlerian manifold N. Then each leaf of DL is totally geodesic in M. COROLLARY 1.2. Let M be an antiholomorphic submanifold of a Kaehlerian manifold N. Then M is mixed geodesic if and only if each leaf of D is totally geodesic in M.
CHAPTER M
42
Concerning the immersion of M2 in N we have THEOREM 1.4 (Bejancu [5]). Let M be a CRsubmanifold of a Kaehlerian manifold N. Then M. is totally geodesic in N if and only if VLJY E r(JD1),
for
any
X, Y E r(D1)
(1.15)
and
h(X, Z) E r(v) ,
for any
x E r(D1)
and Z E r(TM). (1.16)
Proof. By using (3.1) and (3.2) of Chapter I we obtain g(VxY, Z) = g(VxJY, JZ) _ g(h(X, JZ), JY),
(1.17)
g(VxY, JU) = g(h(X, Y), JU)
(1.18)
g(VXY, V) = 9(VXJY, JV) = g(VXJY, JV),
(1.19)
and
for any X, Y, U E r(D1), Z E r(D) and V E r(v). From (1.17)(1.19) our assertion follows since M2 is totally geodesic in N if and only if
VxY E r(D),
for all
x, Y E r(D1).
We say that M is D1geodesic if h(X, Y) = 0,
for all
X, Y E r(D1).
(1.20)
THEOREM 1.5 (Chen [5]). Let M be a CRsubmanifold of a Kaehlerian manifold N. Then M2 is totally geodesic in N if and only if M is D1geodesic and
g(h(X, Z), JY) = 0,
(1.21)
for any X, Y E r(D1) and z E r(D).
Proof. From (1.19) using (3.1) of Chapter I we obtain
q (VXJY, JV) = g (h (X, Y) , V) ,
(1.22)
for all X, Y E r(Di) and v E r(v). Then our assertion follows from Theorem 1.4 using (1.22). From Theorem 1.5 we have COROLLARY 1.3. Let M be an antiholomorphic submanifold of a
43
CRSUBMANIFOLDS OF KAEFILERIAN MANIFOLDS
Kaehlerian manifold N. Then M2 is totally geodesic in N if and only if M is both mixed geodesic and D1geodesic.
§2. Umbilical CRSubmanifolds of Kaehlerian Manifolds First we prove
LEMMA 2.1. Let M be a CRsubmanifold of a Kaehlerian manifold N. Then AJXY = AJYX,
(2.1)
for all X, Y E F(D1).
Proof. By using (3.2) of Chapter I and (1.1) of Chapter II we obtain g(AJXY  AJYX, Z) = g([X, Y], $Z),
(2.2)
for any X, Y E F(DI) and Z E F(TM). Thus our assertion follows from (2.2) taking into account that Di is integrable.
We recall that M is a totally umbilical submanifold if the first and the second fundamental forms are proportional, that is,
h(X, Y) = g(X, Y)H,
(2.3)
for any X, Y E F(TM), where H is the mean curvature vector of M. Then we have the following classification theorem for totally umbilical CRsubmanifolds of a Kaehlerian manifold. THEOREM 2.1. (Chen [4] and Bejancu [81). Let M be a CRsubmanifold of a Kaehlerian manifold N. If M is totally umbilical then we have M is totally geodesic, or (i) M is totally real, or (ii) (iii) the antiinvariant distribution Dl is onedimensional. Proof. Suppose dim D1 > 1. From (2.1) we have AJXBH
AJBHX,
(2.4)
for all X E F(D1). Taking account of (2.3) and (2.4) we obtain g(X, X)q(JBH, H)  g(BH, X)g(JX, H).
(2.5)
44
CHAPTER III
By means of (1.2) of Chapter II we see that (2.5) becomes g(X, X)g(BH, BH) _ g(BH, X)g(JX, H).
(2.6)
We take X orthogonal to BB and from (2.6) obtain BH = 0. Then from (1.11) we obtain P(ACHY) = 4(AHY),
for any
Y E r(TM).
(2.7)
Now suppose M is not totally real, i.e., dim D ) 2. Then take Z E r(D) and obtain
g(PAJHY, Z) = g(AJHY, Z) = g(Y, Z)g(JH, H) = 0 (2.8)
and g(f(AHY), Z) = g(AHY, JZ) = g(Y, JZ)g(H, H).
(2.9)
for any Y E r(TM). Replacing Y by JZ in (2.9) and taking account of (2.7) and (2.8) we obtain H = 0. Thus M is totally geodesic and the proof is complete.
THEOREM 2.2. Let M be a totally geodesic CRsubmanifold of a Kaehlerian manifold N. Then M is a CRproduct. Proof. By using the assertion (i) of Theorem 1.1, Theorem 1.3 and the assertion (i) of Theorem 1.2 we obtain that both distributions D and Dl are integrable and their leaves are totally geodesic in M. Thus M is a CRproduct.
THEOREM 2.3 (BlairChen [11). Let M be a totally umbilical CRsubmanifold of a Kaehlerian manifold N. Then we have K(XAY) = 0,
(2.10)
for any x EI'(D) and Y E r(Dl), where K is the sectional curvature of N.
Proof. By using (3.4) of Chapter I and (2.3) we obtain (Vxh)(Y, Z) = g(Y, Z)VXH,
(2.11)
for any X, Y, Z E r(TM). Equation (3.7) of Chapter I and (2.11) imply R(X, Y, Z, V)  g(Y, Z)g(VXH, V) 
 g (x' Z) g (VlYH, V), for any v E
t(TM ).
Take x E r(D) and Y E r(DI). Hence
(2.12)
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
45
JX E t(D), JY E t(TM1) and from (2.12) obtain R(X, Y, X, Y) = R(X, Y, JX, JY) = 0, which proves our assertion. COROLLARY 2.1. (i) There exist no proper totally umbilical CRsubmanifolds in any positively (or negatively) curved Kaehlerian manifold. (ii) There exist no totally umbilical real hypersurface of any positively (or negatively) curved Kaehlerian manifold. Now we prove a pinching theorem for totally umbilical CRsubmanifolds of Kaehlerian manifolds. THEOREM 2.4 (Chen [3]). Let N be a Kaehlerian manifold 0, satisfying the following pinchings for
(a) (b)
S(X, X) > {1 + (1 
Z)
s  1) }cg(X, X),
where H is the holomorphic sectional curvature of N and S is the Ricci tensor of N. Then every totally umbilical CRsubmanifold M of N with codimension < s is a totally geodesic holomorphic submanifold. Proof. First we suppose that M is of real codimension 2n  m = s and dim D1 = s. In this case we choose {E1,...,E as an orthonormal basis for Dl and obtain an orthonormal basis {JE1,...,JE9) of TM'L. Then, by using (2.4) of Chapter I, we obtain
s S(E1, E1) = H(E1) + E {K(E1tEi) +IG(E1AJEi) }. i=2
By a result of BishopGoldberq [1] we have s
S(E1, E1) = H(E1) + 4 E {H(E1+Ei) + H(E1+JEi) + L2
+ H(E1Ei) + 8(E1JEi)  H(E1)  H(Ei)}. Then, by using (a), we find
S(E1, 81) < {1 + (e1) (1 2) }c,
46
CHAPTER III
which contradicts (b). In a similar way, the theorem follows
if 2n  m = s and dim. Dl < s or 2n  m < s and dim.D1 = 2n  m, or 2n  m < s and dim.DL < 2n  m. Therefore we have dim.D1  0 which means M is a holomorphic submanifold. Moreover, it is known (see Ogiue (1]) that any holomorphic submanifold is minimal. Hence M is totally geodesic in N.
Remark 2.1. The assertion of Theorem 2.4 is also valid if we replace (a) and (b) by
(a') c < H 4 do < 0, and (b') S(X, X) < {1 + (1  2)He  1) }cg(X, X). Now, let M be a CRsubmanifold of a Kaehlerian manifold N. Then the normal bundle to M has the orthogonal decomposition TMl = JDl e V.
(2.13)
Denote by r the complex dimension of v for each x E M. Since v is a holomorphic vector bundle we can take a local field of orthonormal frames {JE1,...,JEQ, V1,...,Vr, Vr+1 = JV1,...,V2r = JVr}
on
TM ,
where {E1,...,Eq} is a local field of orthonormal
frames on Dl. Then we let
Ai = AJE, Aa = AV
i
a
and AC,* = AV
.
a*
Unless otherwise stated, in this paragraph we use the conventions that the ranges of indices are respectively: it j, k, ...  1,...,q; a, Be Y, ... = 1,...,rJ
The CRsubmanifold M is said to be pseudoumbilical if the fundamental tensors of Weingarten are given by
A X = a X + big(X, Ei)Ei, i
i
(2.14)
2
AaX  aaX +
E bag(X, Ei)Ei, is1
and
(2.15)
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
47
q Aa*X = as*X +
F
ba*g(X, Ei)Ei,
(2.16)
i=1
where air bit aa, bi, as*, and ba* are differentiable functions on M and X E r(TM).
PROPOSITION 2.1. Any pseudoumbilical CRsubmanifold of a Kaehlerian manifold is mixed geodesic. Proof. From (2.14)(2.16) it follows that both of the distributions D and Dl are invariant underthe action of AV for any V E r(TMI). Thus by using Proposition 2.4 of Chapter II we have our assertion. PROPOSITION 2.2. Let M be a mixed geodesic CRsubmanifold of a Kaehlerian manifold N. Then we have AJVX = JAVX,
(2.17)
for any X E r(D) and V E r(v).
Proof. Since M is mixed geodesic we have g(AJVX  JAVX, Y) = g(h(X, Y), JV) + g(AVX, JY) = 0, for any X E r(D), Y E r(D1) and V E r(v). On the other hand we have g(AJVX  JAVX, Z) = g(VXJV, Z) + g(AVX, JZ) =
g(IXV + %X, JZ) = g(VXV, JZ) = 0, z E r(D).
Thus by using the decomposition (2.13) we obtain (2.17). PROPOSITION 2.3. Let M be a pseudoumbilical proper CRsubmanifold of a Kaehlerian manifold N. If q 1 then the functions air as and aa* vanish identically on M. Proof. First, by using (2.1) and (2.14) we have ai = g(AiEj, Ej) = g(AjEi, Ej) = 0,
where
j x i.
Next, we take a unit vector field X from the invariant distribution D and by using (2.15), (2.16) and Proposition 2.1 we obtain
CHAPTER III
as = g(AaX, X) = g(JAaX, JX) = g(Aa*X, JX) _ = as*g(X, JX)
0.
In a similar way we get aa* = 0. THEOREM 2.5 (Bejancu [8]). Let M be a pseudoumbilical CRsubmanifold of a Kaehlerian manifold N. If q 1 then M is a CRproduct.
Proof. By using (2.14)(2.16) we obtain q h(X, Y) = E {big(X, Ei)g(Y, Ei)JEi} + i=1
r q E E {bag(X, Ei)g(Y, Ei)Va} + a=1 i=1 2r q + E E {ba*g(x, Ei)g(Y, Ei)Va*}. a*=r+1 i=1 +
(2.18)
for all X, Y E r(TM). Thus we have h(X, Y)  0
for any
x E r(D)
and
Y E r(TM).
Then by using the assertion (i) of the Theorem 1.2 we obtain that D is integrable and its leaves are totally geodesic in M. On the other hand, by Theorem 1.3 we get that each leaf of Dl is totally geodesic in M. This completes the proof. Now, Let M* be a leaf of D1. The restrictions of the vector bundles D, D1 and V to M* will be denoted by the same symbols. Thus the normal bundle to M* is just
)1=D0JD1®V. For each V E r(V) we put JV = B*V + C*V,
(2.19)
where B*V E r(DD) and C*V E r(Dev). Since M* is a totally real submanifold of N, the morphism C* is an fstructure on the normal bundle u (see YanoKon (11). Denote by h* the second fundamental form of M* in N and by AV the fundamental tensor of Weingarten corresponding to the normal section V. For amy v E r(P) and x E r(TM*) we define the covariant derivative of C* by
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS (OXC*)V  V * C * V
49
 C*(V V),
where 0* is the linear connection induced by V on the vector bundle U. Yano and Icon obtained in [1] (VXC*)V  h*(X, B*V)  J(AVX),
(2.20)
for any X E I'(TM*) and v E F(U). The fstructure C* is said to be parallel if we have VXC* = 0 for any X E F(TM*). Also,
the fstructure C on TM1 defined in §1 of Chapter II is saip to be parallel if we have
XCV  C(X V) 0,
(VC)V=
for any x E r(TM) and v E F(TMI).
THEOREM 2.6. Let M be a pseudoumbilical proper CRsubmanifold of a Kaehlerian manifold N with q 1. Then the following assertions are equivalent to each other the fstructure C is parallel on TMl the fstructure C is parallel on U, (iii) all the functions ba and b(%* vanish identically on M. (i)
(ii)
Proof. From (1.X3) we obtain (VXC)V = h(X, BV)  W(AVX),
(2.21)
for any X E I'(TM) and v E r(TM1). Replacing V by JEi and then by Va and Va* in (2.21) and taking account of (2.18) we
obtain the equivalence of assertions (i) and (iii).
Note that Proposition 2.1 and Corollary 1.1 imply that M* is totally geodesic in M. Hence, for any X, Y E F(TM*), we have
h*(X, Y) = h(X, Y).
(2.22)
From (2.22) it follows that AV  0 for any v E F(D). Thus, by (2.20) we obtain that C* is parallel if and only if h*(X, Ei)  J(A**
and
a
va*
for all x E F(TM*).
X0,
X) i
(2.23)
(2.24)
CHAPTER III
so
On the other hand, from (2.18) and (2.21) it follows that V,C = 0 whenever X E r(D). Hence C is parallel on TM1 if and only if
h (X, Ei) = J (AiX)
(2.25)
AAX = Aa*X = 0,
(2.26)
and for any x E r(D1). By using (2.22)(2.26) we conclude that C* is parallel if and only if C is parallel. This completes the proof.
Taking account of (3.4) from Chapter I and (2.22) we obtain PROPOSITION 2.4. Let M be a submanifold of a Kaehlerian parallel second fundamental parallel second fundamental
pseudoumbilical proper CRmanifold N with q 1 and form. Then each leaf of D1 has form.
Now we state a result on the existence of pseudoumbilical proper CRsubmanifolds in Kaehlerian manifolds. THEOREM 2.7. There exist no pseudoumbilical proper CRsubananifolds with q 1 immersed in a positively (or negatively) curved Kaehlerian manifold. Proof. By using (3.7) of Chapter I and (2.18) we obtain g(R(X, Y)JX, JY)  g(h(Y, VxJX), JY),
(2.27)
for any X E r(D) and Y E T(DI). Since h(X, JX)  h(X, X)  0, from (3.1) of Chapter I it follows that VxJX = J(VxX).
(2.28)
Hence VxJX E r(D) and by using (2.27) and Proposition 2.1 we have
q(R(X, Y)X, Y)  g(R(X, Y)JX, JY)  0. Thus, taking account of (2.6) from Chapter I, we obtain the assertion of the theorem.
D. Normal CRSubmanifolds of Kaehlerian Manifolds Let M be a CRsubmanifold of a Kaehlerian manifold N. Then
CRSUBMANIFOLDS OFKAEHLERIAN MANIFOLDS
51
by (1.5) and (1.6) of Chapter II we have an fstructure M and a normal bundlevalued 1form w on M. Taking into account that g is a Hermitian metric on N we get
on
g(oX, Y) = g(X, Y)  g(WX, WY),
(3.1)
for any X, Y E r(TM). Also the covariant derivative of 0 is defined by
(VX0)Y  VX(4Y)  WXY),
(3.2)
for any X, Y E r(TM). Then by using (1.8) and (1.9) we obtain
(VXf)Y = Bh(X, Y) + AwYC. On the other hand the covariant derivative of w is defined by (VXU)Y = VX(WY)  w(VXY),
(3.3)
for any X, Y E r(TM). By (1.10) and (3.3) we have
(VXw)Y = Ch(X, Y)  h(X, Y).
(3.4)
Also by using (1.11)(1.13) we obtain (VXB)V = ACV(  4)(AVX) + B(VX)
(3.5)
(VXC)V = h(X, BV)  w(AVX),
(3.6)
and
for any X E r(TM) and V E t(TMI). The fundamental 2form n of a CRsubmanifold M in a Kaehlerian manifold N is defined by
S2(X, Y)  g(X, 4Y),
(3.7)
for any X, Y E r(TM). By using (1.9) of Chapter II and (3.7) we obtain
S2(0, 4Y)  11 (X, Y)
.
(3.8)
In fact S2 is nothing but the restriction of the fundamental 2form of the Kaehlerian manifold N to TM X TM. Hence (2 is closed, i.e., we have d1 = 0. The exterior derivative of w is given by
du(X, Y) = 2{
X(WY)  VY(wX)  w([X, Y])).
(3.9)
Now we state
PROPOSITION 3.1. Let M be a CRsubmanifold of a Kaehlerian
52
CHAPTER III
manifold N. Then the covariant derivative of $ is given by
2g((VX$)Y, Z)  g(($, $](Y, Z), $X) + 2g(dw($Y, X), WZ) + 2g(dw($Y, z)
 dw($Z, Y), WX)  2g(dw($Z, X), WY),
(3.10)
for all vector fields X, Y, Z tangent to M.
Proof. Since the Riemannian connection of M is given by 2g(VXY, Z)  Xg(Y, Z) + Yg(X, Z)  Zg(X, Y) + + g((X, Y3, Z) + g((Z, X], Y)  g((Y, Z], X), we have 2g((VX$)Y, Z) = 2g(VX$Y, Z) + 2g(VXY, $Z) = = Xg($Y, Z) + $Yg(X, Z)  Zg(X, $Y) + + g((X, $Y], Z) + g((Z, X], $Y)  g([$Y, Z], X) + + Xg(Y, $Z) + Yg(X, $Z)  $Zg(X, Y) + + g((X, Y], $Z) + g(($Z, X3, Y)  g((Y, $Z], X).
Thus, taking into account that g is covariantly constant with respect to V and by using (2.1) of chapter II we obtain 2g((VX$)Y, Z)  g(($, $](Y, Z), $X) + + g(VIYWZ  w(($Y, Z3), wX) + + g(VI WX  W(($Y, X]), WZ) + g(WX, w(($Z, Y])) +
+ g(WY, w(($Z, X]))  $Zg(wX, WY).
(3.11)
Next, from (3.9) it follows that
d)($Y, Z)  2
w(($Y, Z3)).
(3.12)
Since Vl is a Riemannian connection on the normal bundle we have
$Zg(WX, WY)  g(VZWX, WY) + g(WX, V'ZWY).
(3.13)
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
53
Finally, by using (3.12) and (3.13) in (3.11) we obtain (3.10).
We define the tensor field S by S(X, Y)  [0, yJ(X, Y)  2Bdw(X, Y),
(3.14)
for any x, Y E r(TM). Substituting dw from (3.9) and taking into account that V is a torsionfree linear connection, (3.14) becomes
S(X, Y)  (VU4)Y  (V0Y0)X + ${(VY$)X  (Vx$)Y}  B{(Vxw)Y  (VYw)X}.
(3.15)
The CRsubmanifold M is said to be normal if the tensor field S vanishes identically on M. If, in particular, M is a real hypersurface of a Kaehlerian manifold we get the well known notion of a normal real hypersurface of a Kaehlerian manifold (see Okumura [1], [31). THEOREM 3.1. (Bejancu [7]). The CRsubmanifold M is normal if and only if we have
AWYM)  $(AWYX),
(3.16)
for all X E r(D) and Y E r(Dl). Proof. By using (3.2) and (3.4) in (3.15) we obtain
S(X, Y)  (AWYo4  4oAWY)X  (Ae$  $oAwx)Y, (3.17)
for all X, Y E r (TM) . Suppose M is a normal CRsubmanifold of N. Then (3.16) follows from (3.17) since AWx  0 for any X E r(D). Now, if (3.16) is satisfied we shall prove S  0 by means of the Dl. First, from (3.17) we have that decomposition TM  D S(x, Y) = 0 for all X, Y E r(D). Next, for X E r(D) and Y E r(Dl) we obtain S(X, Y) _ (AWY00  moAWY)X
0,
by (3.16). Finally, for X, Y E r(D ), (3.17) becomes S(X, Y)  $(AWXY  AWYX),
since OX = OY  0. By using (2.1) we obtain the vanishing of S on D1 x Dl. The proof is complete.
54
CHAPTER m
Now, suppose {E1,...,Eq} is a local field of orthonormal 1 frames for the antiinvariant distribution D . Denote by Ai the fundamental tensor of Weingarten with respect to Vi = JEi (i = 1,...,q). Then from Theorem 3.1 we have COROLLARY 3.1. The CRsubmanifold M is normal if and only if the fundamental tensors of Weingarten Ai commute with on the invariant distribution, that is we have Ai04 _ 4oAi
(i = 1,...,q).
(3.18)
Remark 3.1. Corollary 3.1 is a generalization of a result due to Okumura (3] for normal real hypersurfaces of a Kaehlerian manifold. Using (3.1) and (3.2) of Chapter I for the immersion of M in N we obtain
VXEi = (AiX)  B(VXVi)
(3.19)
and
XVi
= W(VxEi) + Ch(X, E1),
(i = 1,...,q),
(3.20)
for all X E r(TM). It is well known that X is a Killing vector field if and only if we have
g(VZX, Y) + g(Z, VYX) = 0, for any Y, Z E r(TM). We introduce here a weaker condition for X. Thus we say that X is a DKilling vector field if we have g(VZX, Y) + g(Z, VYX) = 0,
(3.21)
for any Y, Z E T(D). Now we state
THEOREM 3.2. A necessary and sufficient condition for the CRsubmanifold to be normal is that Ei (1 = 1,...,q) be DKilling vector fields. Proof. Using (3.19) we obtain g(VZEi, Y) + g(Z, VYEi) = g((4,°Ai 
Y), (3.22)
for all Y, Z E T(D). Thus our assertion follows from (3.22) and Corollary 3.1.
CRSUBMANIFOLDS OFKAEHLERIAN MANIFOLDS
55
The Lie derivative of $ with respect to Y E r(TM) is given by (9Y$)X = [Y, $X]  $([Y, x)),
(3.23)
for any X E r(TM). Then normal CRsubmanifolds can be characterized by another tensor field S* defined by S*(Y, X) =
(C Y$)X,
(3.24)
for any X, Y E r(TM).
THEOREM 3.3. Suppose M is a CRsubmanifold of a Kaehlerian manifold N and we have Q(VXY) = 0,
(3.25)
for all x E r(D) and Y E r(DL). Then M is a normal CRsubmanifold of N if and only if S*(Y, X) = 0,
(3.26)
for all X E r(D) and Y E r(DL).
Proof. From the proof of Theorem 3.1 follows that M is a normal CRsubmanifold if and only if S(X, Y) = 0 for any X E r(D) and Y E r(Dl). By using (3.4) and (3.9) in (3.14) we obtain S(X, Y) = $($([X, Y])  [$X, Y])  Bh($X, Y). (3.27)
Next, from (1.10) we have
h($X, Y)  w(VYX) + Ch (x, Y) ,
which implies Bh($X, Y) _ Q(Vyx).
Hence (3.27) becomes S(X, Y) _ $($([X, Y])  [$X, Y]) t Q(VYX).
(3.28)
Using (3.23), (3.24), and (3.28) we obtain S(X, Y) = 4)(S*(Y, X)) + Q(VYX).
(3.29)
Now suppose M is a normal CRsubmanifold. Then from (3.29) we obtain
PS*(Y, X) = 0 and
(3.30)
Q(VYX)  0.
(3.31)
Also, by using (3.23), (3.24), (3.25), and (3.31), we have
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CHAPTER M
QS*(Y, X)  0. Therefore (3.26) is satisfied. Conversely, suppose (3.26) is satisfied. Then from (3.23) and (3.24) it follows that Q([X, Y]) = 0,
(3.32)
for any x c r(D) and Y E r(D1). Thus, by using (3.25), (3.26) and (3.32) in (3.29), we obtain S(X, Y) = 0. Thus M is a normal CRsubmanifold. By using Corollary 3.1 it is easy to verify that totally geodesic CRsubmanifolds, totally umbilical CRsubmanifolds and pseudoumbilical CRsubmanifolds are examples of normal CRstibmanifolds. Moreover, each normal real hypersurface of a Kaehlerian manifold is also an example of normal CRsubmanifold.
§4. Normal AntiHolomorphic Submanifolds of Kaehlerian Manifolds In this paragraph we consider an antiholomorphic submanifold M of a Kaehlerian manifold N. Thus the dimension of the antiinvariant distribution on M is the same as the codimension of the immersion of M in N. We introduce some tensor fields which are generalizations of tensor fields introduced by Sasaki and Hatakeyama in [1] for an almost contact structure on a manifold. We shall obtain characterizations of normal antiholomorphic submanifolds by means of these tensor fields and Ricci curvatures. Suppose dim Dl = q and choose a local field of orthonormal frames {E1,...,Eq} on the antiinvariant
distribution Dl. Then define the following tensor fields 4](X, Y)  2Jdw(X, Y),
(4.1)
S(2)(X, Y) = 2J{dw(4X, Y)  dw(gY, x)},
(4.2)
S(1) (X, Y) _
S(3)
(x)  (CE )W, (i = 1,...,q),
i
(4.3)
for any X, Y tangent to M.
Remark 4.1. The tensor field S(1) is just the tensor field S defined on each CRsubmanifold by (3.14). Also, we remark that the tensor fields
S(3)
depend on the local field
57
CRSUBMANTFOLDS OF KAEHLERIAN MANIFOLDS
of orthonormal frames {E1,...,Eg1. However, in this section, since we are dealing only with antiholomorphic submanifolds, we prove their independence of this field of frames. Since M is an antiholomorphic submanifold, from (3.2) and (3.4) we obtain
(VX))Y = PubX, Y) + AWYX
(4.4)
(VXw)Y = h(X, qY),
(4.5)
and for any X, Y E r(TM).
PROPOSITION 4.1. Let M be a mixed geodesic antiholomorphic vanishes submanifolds of a Kaehlerian manifold N. Then g identically on M. Proof. By using (3.9) and (4.5) in (4.2) we obtain S(2)(X, Y) = J{h(QX, PY)  h(QY, PX)).
Thus our assertion follows from (2.25) of Chapter II.
COROLLARY 4.1. Let M be a mixed geodesic antiholomorphic submanifold of a Kaehlerian manifold N. Then dw is invariant by 4, that is, we have
dw(X, for any X, Y E r(TM).
(4.6)
Proof. Since S(2)  0, by (4.2) we have
dw(4X, Y)  dw(X, PY),
(4.7)
for any X, Y E r(TM). On the other hand, by using (3.9) and (4.5), we obtain
dw(X, QY)  2 h(QY, X)  h(X, 4QY))  0.
(4.8)
Thus (4.6) follows from (4.7) and (4.8).
PROPOSITION 4.2. Each normal antiholomorphic submanifold of a Kaehlerian manifold is mixed geodesic. Proof. By using (3.3) of Chapter I we have g(h(Ej, X), JEi)  g(AiX, Ej), ij j  1,....4,
for any X E T(D). By Theorem 1.1 of Chapter II we have = D. Hence there exist Y E r(m) such that X  $Y. Im
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CHAPTER 111
Thus, by using (3.18), we obtain Ej) = g(Ai4Y, Ej) = 0.
g(h(X, Ej), JEi) = This proves our assertion.
In the remaining part of this section we are dealing with antiholomorphic submanifolds with flat normal connection, i.e., the curvature tensor Rl of Vl vanishes identically on M.
PROPOSITION 4.3. Let M be a normal antiholomorphic submanifold of a Kaehlerian manifold N with flat normal connection. Then we have (i)
(ii)
S(2) vanishes identically on M, 3) vanishes identically on M for any local field of
orthonormal frames on Dl and i = 1,...,q. Proof. By Propositions 4.1 and 4.2 we have the assertion (i). In order to prove the assertion (ii) we choose a distinguished field of frames on the antiinvariant distribution. By Theorem 3.1 of Chapter I there exists a local field of orthonormal frames {V10...,Vq} such that Vi is parallel with respect to the normal connection. Then the distinguished field of frames on D is defined by = JVq}.
{E1 = JV1,....Eq
(4.9)
By using (4.3) the tensor fields S(3) corresponding to the field of frames (4.9) are given byi S(3)(X) i
_ (VE 4))X  (VX4))Ei 
(4.10)
i
Now take X E r(D) and by (4.4) and Proposition 4.2 obtain (VE 4))X = Jh(X, E
)
i
= 0
and
(VX0)Ei = AiX.
Thus, by using (3.18), (3.19), and (4.10), we have
Si3) (X) AiX  (4)°Ai°4))X  0. The tensor fields S(3) vanish on the antiinvariant distribution if and only if S(3)(Ei) = 0 for any Bj from the distinguished field of frames (4.9). By (3.19) and (4.3)
CRSUBMANIFOLDS OFKAEHLERIAN MANIFOLDS
59
we obtain S(3)(Ej)
Ej  VE Ei) _
_ 4((Ei, En]) = 4(V i
_ 4(4AjEi  AiEj) = 0. Hence the tensor fields
Si3)
vanish identically on M.
Next, suppose {Ei} (i= 1,...,q) is an arbitrary field of orthonormal frames on
D.
Denote by Si3) the corresponding
tensor fields given by (4.3). First we claim that
813)(x) E r(D) for each x E r(TM). In fact, this follows from the equalities
g(VEiox, Ej) _ g(4X, VEiEj) _ g($X, 4A4Ei) = 0
and g(V oxEi, Ej) _ g(Ei, VoXEj)
AjoX) = 0,
by means of (4.3). Finally we put q E fiEj i = E j=i
Then we have S(3)(X)
q E {fisj3)(X)  X(fi)Ej}
0,
j=1
since S(3)(X) = 0 and Si3)(X) E r(D). The proof is complete.
THEOREM 4.1. Let M be an antiholomorphic submanifold of a Kaehlerian manifold N with flat normal connection. Then the following assertions are equivalent to each other: (i) M is a normal antiholomorphic submanifold of N; the vector fields Ei from the distinguished field of (ii) frames (4.9) are Killing vector fields; (iii) the tensor fields Si3) vanish identically on M.
Proof. By (3.19), taking account of flatness of the normal connection we obtain g(OZEi, Y) + g(VYEi, Z) = g(($°Ai  Ai°4)Y, Z), (4.11)
for all vector fields Y, Z tangent to M. Thus (4.11) proves the equivalence of assertions (i) and (ii) by means of Corollary 3.1.
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CHAPTER III
Next, by Proposition 4.3, we see that (i) implies (iii). Suppose S(3) = 0. Using (3.19) and (4.4) in (4.3) we obtain 0 = S(3)(X)
= Jh(Ei, X) + 4)(4AiX  Ai0),
for any X E r(D). Thus by Corollary 3.1 M is a normal antiholomorphic submanifold. This completes the proof. Now combining Theorems 3.2 and 4.1 we obtain COROLLARY 4.2. Let M be an antiholomorphic submanifold of a Kaehlerian manifold N with flat normal connection. Then the vector fields E from the distinguished field of frames are Killing vector fields if and only if they are DKilling vector fields. Let M be a mixed geodesic antiholomorphic submanifold of a Kaehlerian manifold N. Then by Proposition 2.4 of Chapter II we have the invariance of the holomorphic distribution by the action of fundamental tensors of Weingarten Ai = AJE . Denote by AiID the restriction of Ai to D.
i
THEOREM 4.2 (Bejancu [14]). Let M be a mixed geodesic antiholomorphic submanifold of a Kaehlerian manifold N with flat normal connection. Then M is a normal antiholomorphic submanifold if and only if the Ricci curvatures with respect to the distinguished field of frames (4.9) are given by k(Ei) = trace (AiID)2'
(4.12)
Proof. First, by using (3.19), we obtain the curvature tensor R of M. It is given by
R(Ei, X)Ei  VE AiX  Ai([Ei, X]),
(4.13)
i
for all X tangent to M. For any x E T(D), by (3.19) and Proposition 2.4 of Chapter II we have g(VE X, Ej) _ g(X, VE Ej) = 0,
i
i
i, j = 1,...,q.
Hence VE X E1'(D). Then replacing X by Ei and Y by PZ in (4.4) we obtain V
Ei
4z = j(V
Ei
PZ),
(4.14)
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
61
for any Z E T(TM). We take X E r(D) and by (4.13) and (4.14) obtain g(R(Ei, X)Ei, X) = g(J(VE AiX), X) i
 g(JAi(VE X), X)  g(JAiX, AiJX).
i
(4.15)
Thus we have g(R(Ei, X)Ei, X) + g(R(Ei, JX)Ei, JX) =
 g(J(VE AiX) , X)  g(JAi(VE X)' X) +
i
i
+ g(VE AiJX, X)  g(Ai(VE X), X) 
i
i
 2g(JAiX, AiJX) = g(VE (JAi + AiJ)X, X) + + g((JAi + AiJ)X, VE X)  2g(JAiX, AiJX).
(4.16)
By direct computation it follows that g((JAi + AiJ)X, X)  0.
Hence, (4.16) becomes g(R(Ei, X)Ei, X) + g(R(Ei, JX)Ei, JX)
 2g(JAiX, AJX)
(4.17)
for any X E r(D). Now, for any X E r(Dl) from (4.13) it follows that
R(Ei, X)Ei  0.
(4.18)
since Dl is integrable and invariant by Ai. Further, we choose the following local field of orthonormal frames on M
(F1,...,Fp, F1 = JF1,...,Fp = JFp, s1,...,Eq (4.19)
where Fa c T(D), a  1,...,p, dim x = p and (E1,...,Eq} is the distinguished field of frames (4.9) on Dl. Then by using (4.17)(4.19), the Ricci curvature k(Ei)
becomes k(Eil  2
E a1
(g(JAiFa, AiJFa)).
(4.20)
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By a straightforward computation we obtain g((AiJ  JAi) X, (AiJ  JAi)X) = g(AiX, AiX) +
+ g(AJX, AiJX)  2g(AiJX, JA X),
(4.21)
for all X E r(D). Next, taking into account that Ai are symmetric linear operators and g is a Hermitian metric we obtain that A J  JAi are symmetric linear operators too. Denote by i
(AiJ  JAi)
ID
the restrictions of A J  JAi to D. Then by i
using (4.21) and (4.20) we obtain k(Ei) = trace (AjID)2 
trace (AiJ  JAi)21D. 2
(4.22)
Now suppose M is a normal antiholomorphic submanifold. Then by using (3.18) and (4.22) we obtain (4.12). Conversely, if (4.12) is satisfied, by (4.22) we have trace (AiJ  JAi)2ID = 0.
(4.23)
All the eigenvalues of AiJ  JAi are real since they are symmetric linear operators. Hence the eigenvalues of (AiJ  JAi)2ID are nonnegative. Thus (4.23) implies
(AiJ  JAi) I D = 0. Therefore, by Corollary 3.1 M is a normal antiholomorphic submanifold. The proof is complete. THEOREM 4.3. Let M be a mixed geodesic antiholomorphic submanifold of a Kaehlerian manifold N with flat normal connection. Then each leaf of the antiinvariant distribution is a locally Euclidean space. Proof. Suppose M* is a leaf of Dl. Denote by h* the second fundamental form of the immersion of M* in M and by R* the curvature tensor of M*. Then the Gauss equation with respect to the immersion of M* in M becomes
g(R(X, Y)Z, W) = g(R*(X, Y)Z, W) +
(4.24)
+ g(h*(X, Z) , h*(Y, W))  g(h*(Y, Z) , h*(X, W)) ,
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
63
for all X, Y, z, w E r(TM*). By using (3.19) we obtain R(X, Y)Z = 0, for any X, Y, Z E r(DL). On the other hand, by Corollary 1.2 we obtain the vanishing of h* on M*. Then from (4.24) it follows that R* = 0, which means M* is a locally Euclidean space. Combining Theorem 4.3 with Proposition 4.2 we have COROLLARY 4.3. Let M be a normal antiholomorphic submanifold of a Kaehlerian manifold with flat normal connection. Then each leaf of the antiinvariant distribution is a locally Euclidean space.
§5. CRProducts in Kaehlerian Manifolds By Theorem 3.2 of Chapter II we found a class of CRsubmanifolds which are CRproducts in almost Hermitian manifolds. We show here that if the ambient space is a Kaehlerian manifold this is the only class of CRproducts. More precisely we have PROPOSITION 5.1. Let M be a CRproduct in a Kaehlerian manifold N. Then V0 = 0,
(5.1)
that is, the LeviCivita connection is a 4)connection.
Proof. First, for any U E I'(TM) and Y E T(D) we have V Y E F(D). By using (3.1) of Chapter I and (1.2) we obtain U S
I
g(h(U, JY), V) = g(VYJY, V) = g(J(VUY), V)
= g (Jh (U, Y) , V) , for any V E F(TMy ). Hence Bh(U, Y) = 0 and by (3.2) we have (VU4))Y = 0. Finally, for x E r(D) and U E r(TM) we get VUx E r(DL). Hence
0 and we obtain our assertion.
THEOREM 5.1 (Chen [5]). Let M be a CRsubmanifold of a Kaehlerian manifold N. Then M is a CRproduct if and only if (5.1) is satisfied. The proof follows by combining Theorem 3.2 of Chapter II and Proposition 5.1.
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CHAPTER M
THEOREM 5.2. Let M be a CRsubmanifold of a Kaehlerian manifold N. Then M is a CRproduct if and only if
Bh(X, Y) = 0,
(5.2)
for any X E r(TM) and Y E r(D). Proof. Suppose M is a CRproduct. Then by using (3.2) and Theorem 5.1 we obtain (5.2). Conversely, suppose (5.2) be satisfied. Then from (3.2) it follows that (VxO)Y = 0
for any
X E r(TM)
and
Y E r(D).
Now take Z E F(Di) and by using (1.8) obtain
(Vx4) Z = 0 (DXZ) = P(AJZX) , for any x E r(TM). Finally, (3.3) of Chapter I and (5.2) imply g(AJZX, Y) = g(h(X, Y), JZ) = g(Bh(X, Y), Z) = 0, for any Y E r(D). Thus w e have (Vx )Z = 0. Hence by Theorem 3.2 of Chapter II M is a CRproduct. From Theorem 5.2 it follows that
COROLLARY 5.1. Let M be an antiholomorphic submanifold of a Kaehlerian manifold N. Then M is a CRproduct if and only if
h(X, Y) = 0,
(5.3)
for any X E r (TM) and Y E r (D) . Now we state THEOREM 5.3. Each CRproduct in a Kaehlerian manifold is a normal CRsubmanifold. Proof. Let M be a CRproduct in a Kaehlerian manifold N. Then by Theorem 5.2 we have
g(h(X, Y), WZ) = 0, for any X E r(TM), Y E r(D) and z E r(D1). Thus by using (5.4) we obtain g(AAZOY  O(A,ZY), X) = g(h(X,O Y), WZ) + + g(h(Y, $X), WZ) = 0.
Hence by Theorem 3.1 M is a normal CRsubmanifold.
(5.4)
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
65
LEMMA 5.1. Let M be a CRproduct in a Kaehlerian manifold N. Then for any unit vector fields X E r(D) and Y E F(DI) we have
HB(X AY) = 2Nh(X, Y) q2
(5.5)
where H (X AY) is the holomorphic bisectional curvature of N determined by 1X, Y . Proof. By using (3.7) of Chapter I, (5.2) and taking into account that D is a parallel distribution we obtain R(X, JX, Y, JY) = g(VX (JX, Y)  Vxh(X, Y), JY). (5.6)
Next, by using (5.4) we see that (5.6) becomes
R(X, JX, Y, JY) = g(h(X, Y), V,JY) 
 g (h (JX, Y), VXJY) = g (h (X, Y) , A (JX, Y) )  g(h(JX, Y), Jh(X, Y)) = 2g(h(X, Y), Jh(JX, Y)). (5.7)
for any X E r(D) and Y E r(D1). By using (3.3) of Chapter I and (5.2) we obtain
h(JX, Y) = Jh(X, Y).
(5.8)
Then (5.5) follows from (5.7) and (5.8) and (5.12) in Chapter I. From Lemma 5.1 we have THEOREM 5.4 (Chen [51). Let N be a Kaehlerian manifold with negative holomorpnic bisectional curvature. Then every CRproduct in N is either a holomorphic submanifold or a totally real submanifold. We say that a CRproduct M is an antiholomorphic CRproduct if the dimension of a fibre of the antiinvariant distribution is just the codimension of M in N. Then we have THEOREM 5.5. Let M be an antiholomorphic CRproduct of a Kaehlerian manifold N with positive or negative holomorphic bisectional curvature. Then M is a totally real submanifold. Proof. Since M is an antiholomorphic CRproduct each leaf of the antiinvariant distribution is totally geodesic
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CHAPTER III
in M. By Corollary 1.2 M is a mixed geodesic CRsubmanifold. Hence by (5.5) we have
HB(XAY) =0,
forany XET(D) and YET(DD),
from which our assertion is proved. We say that the antiholomorphic submanifold M is a cosymplectic antiholomorphic submanifold if it is normal and the differential form W is closed, that is, we have dw = 0.
(5.9)
Also we say that w and 4) are parallel on M if we have Vw = 0 and V4) = 0 respectively.
PROPOSITION 5.2. Let M be an antiholomorphic submanifold of a Kaehlerian manifold N. Then the fstructure 4) is parallel if and only if the differential form w is parallel. Proof. By Theorem 5.1 Q is parallel if and only if M is a CRproduct. On the other hand, from (4.5) we see that w is parallel if and only if (5.3) is satisfied. Thus the assertion follows by using Corollary 5.1. THEOREM 5.6 (Bejancu [7]). Let H be an antiholomorphic submanifold of a Kaehlerian manifold N. Then M is a cosymplectic submanifold if and only if M is a CRproduct. Proof. Suppose M is a cosymplectic antiholomorphic submanifold of N. Since M is normal and dw = 0, from (3.14) we obtain [4), 4)) = 0. Thus by (3.10) we have V4)  0, that is, by Theorem 5.1 M is a CRproduct. Conversely, suppose M is a CRproduct. Then by Propositions 5.1 and 5.2 we have V$  0 and Vw = 0. Hence dw = 0. On the other hand, from (3.14) we have S = 0. Thus M is a normal antiholomorphic submanifold. THEOREM 5.7. An antiholomorphic submanifold M of a Kaehlerian manifold N is cosymplectic if and only if we have (VX4))X = 0,
(5.10)
for any X tangent to M.
Proof. Suppose M is a cosymplectic antiholomorphic submanifold of N. Then (5.10) is satisfied since, by Theorems 5.6 and 5.1, 4) is parallel on M.
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
67
Conversely, suppose (5.10) is satisfied on M. Then replacing X and Y from (4.4) by X + Y and taking account of (5.10) we obtain 2Jh(X, Y) + AWYX + AwXY = 0,
(5.11)
for any X, Y E r(TM). If we take X, Y E r(D) then (5.11) implies h(X, Y) = 0.
(5.12)
Next, for X E r(DI) and Y F r(D), (5.11) becomes. 2Jh(X, Y) + AwXY = 0.
(5.13)
On the other hand, from (1.9) we obtain Jh(X, Y) + Q(AAXY) = 0.
(5.14)
Thus from (5.13) and (5.14), taking account that Jh(X, Y) E r(D1) we obtain h(X, Y) = 0,
(5.15)
for any X E P(D1) and Y E r(D). Finally, we see that (5.12) and (5.15) imply (5.3). Thus M is a CRproduct and by Theorem 5.6 a cosymplectic manifold. The proof is complete.
§6. Sasakian AntiHolomorphic Submanifolds of Kaehlerian Manifolds
Let M be a proper antiholomorphic submanifold of a Kaehlerian manifold N. Take the local field of orthonormal frames on D given by {F1,.... Fp, JF1,..., p}.
(6.1)
Define a normal vector field HD by
P
HD = 2 E
{h(Fi, Fi) + h(JFi, JFi)}. (6.2) P i=1 Thus H is a welldefined normal vector field to M and it is called Dthe Dmean curvature vector of M. We say that M is a contact antiholomorphic submanifold of N if HD ;d 0 and we have dw(X, Y) = n (X, Y)HD,
(6.3)
for all vector fields X, Y tangent to N. By using (3.9) and (4.5) we see that (6.3) is equivalent to
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CHAPTER M
h(4X, Y)  h(4Y, X) = 252(X, Y)HD.
(6.4)
PROPOSITION 6.1. Let M be a contact antiholomorphic submanifold of a Kaehlerian manifold N. Then M is mixed geodesic and D is not involutive. Proof. We take X E r(D ) and Y E r(DI) in (6.4) and obtain that M is mixed geodesic. Take a nonnull vector field X E F(D) and Y = JX in (6.4). Then we obtain
h(JX, JX) + h(X, X)  2g(X, X)HD x 0, that is (1.3) is not satisfied. Thus D is not involutive.
A Sasakian antiholomorphic submanifold is a normal contact antiholomorphic submanifold of N. Any Sasakian real hypersurface of a Kaehlerian manifold is an example of a Sasakian antiholomorphic submanifold. Suppose M is an antiholomorphic submanifold with flat normal connection. Then denote by {E1,...,Eq} the distinguished field of frames (4.9) on Dl induced by parallel unit normal sections {V1,...IVq}. Thus (3.19) becomes VXEj = 4(A
ix),
(i = 1,...,q),
(6.5)
for any X tangent to M.
THEOREM 6.1. Let M be an antiholomorphic submanifold of a Kaehlerian manifold N with flat normal connection. Then the following assertions are equivalent to each other: M is a Sasakian antiholomorphic submanifoldi (i) (ii) the covariant derivative of each vector field from the distinguished field of frames (4.9) is given by V E = g(HD, Vi)4)X, X i
(6.6)
for any X E r(TM) and i = 1,...,q. (iii) the fundamental tensors of Weingarten satisfy
Ai(") = g(HD, Vi)$X,
(6.7)
for any X E r(TM) and i = 1,...,q.
Proof. Suppose M is a Sasakian antiholomorphic submanifold of N. Then by using (3.3) of Chapter I, (3.18), (4.5) and (6.5) we obtain
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
69
q dw(X, Y)
{g(VXEit Y)vi}.
(6.8)
i=1
for any X, Y E r(TM). On the other hand, since M is a contact antiholomorphic submanifold we have dw(X, Y) =
Y)
E {g(HD, Vi)V1}.
(6.9)
i=1
Thus(6.6) follows from (6.8) and (6.9). Now suppose (6.6) holds. Then for all Y, Z E r(TM) we have
g (VYEi, Z) + g (Y,VZE1) = = g(HD, Vi){g(cY, Z) + g(4Z, Y)} = 0. Hence by Theorem 4.1 M is a normal antiholomorphic submanifold. We note that in order to obtain (6.8) we made use only of the fact that M is a normal antiholomorphic submanifold. Thus, by using (6.5) in (6.8), we obtain (6.3), that is, M is a contact antiholomorphic submanifold. Hence the equivalence of assertions (i) and (ii) is proven. Further, we suppose (ii) is satisfied. Then from (6.5) and (6.6) it follows that
(AiX) = g(HD. Vi)4X.
(6.10)
On the other hand, we have seen that (6.6) implies the normality of M. Hence from (6.10) and (3.18) it follows that (6.7) holds.
Conversely, suppose (6.7) is satisfied. Then by Proposition 2.4 of Chapter II M is a mixed geodesic antiholomorphic submanifold. If we take x E,r(D) then from (6.5) it follows that VXEj = 0. Hence (6.7) implies'(6.6) in this case. For any X E r(D) there exists Y E r(D) such that OY = X. Then by (6.7) we have
(AiX) = (Ai0Y) = g(HD, V1)02Y = g(HD, Vi)oX. (6.11)
From (6.11) and (6.5) it follows that (6.6) holds for each X E r(D). The proof is complete. THEOREM 6.2. Let M be an antiholomorphic submanifold of a Kaehlerian manifold N with flat normal connection. Then M
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CHAPTER III
is a Sasakian antiholomorphic submanifold if and only if (VX4))Y = g(4X, OY)JHD + g(WY, HD)PX,
(6.12)
for any X, Y E t(TM).
Proof. Suppose (6.12) is satisfied and replacing Y by Ei we obtain f(OXEi) = g(HD, Vi)PX.
Thus by (6.5) we have (6.6). Hence, by Theorem 6.1, M is a Sasakian antiholomorphic submanifold of N. Conversely, suppose M is a Sasakian antiholomorphic submanifold of N. Then from (4.1) we have
g((4, WY, Z), 4X) = 0, for any X, Y, Z E r(TM). Thus, by (3.10) and (6.3), we have g((VX4)Y, Z) = g(4Z, X)g(HD, WY) 
 g(4Y,4X)g(HD WZ) = g(Z, g(HD, WY)PX) + + g(4X, 4Y)g(JHD, Z) = g(g(HD, WY)PX +
+ g(oX, 4Y)JIID, Z). Hence we have (6.12) and the proof is complete. Now let S2 be the fundamental 2form of M defined by (3.7). Then, by using (1.2) of Chapter I, we obtain
(VXS)) (Y, Z) = g((VX4) Y, Z),
(6.13)
for any X, Y, Z E r(TM). Thus from (6.12) and (6.13) we obtain THEOREM 6.3. Let M be an antiholomorphic submanifold of a Kaehlerian manifold N with flat normal connection. Then M is a Sasakian antiholomorphic submanifold if and only if
(Vxsl) (Y, Z) = g(4X, Z) g (HD, WY)  g(OX, +Y)g(HD, WZ),
(6.14)
for any X, Y, Z E r(TM).
Now we state THEOREM 6.4. Let M be an antiholomorphic submanifold of a
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
71
Kaehlerian manifold N with flat normal connection. Then M is a Sasakian antiholomorphic submanifold if and only if there exist q(q+1)/2 differentiable sections Lij(Lij = Lji) of the normal bundle such that
q E
h(X, Y) = g(0, $Y)HD +
{g(Y, Ej)g(WX, Lij)Vi)
i,j=1
(6.15)
for any X, Y E r(TM).
Proof. By direct computation we obtain
(VXQ) (Y, Z) = (VX(2) (Y, Z)  H (h (X, Y) ,
Z) 
 Q (Y, h (X, Z)). Since N is Kaehlerian we have
(VX0) (Y, Z) = g ((lXJ) Y, Z) = 0. Hence we get (VXQ) (Y, Z) = g(h(X, Z) , JY)  g(h(X, Y) , JZ), (6.16)
for any X, Y, Z E r(TM). Now, suppose M is a Sasakian antiholomorphic submanifold. Then from (6.14) and (6.16) it follows that
g(h(X, Z), JY)  g(h(X, Y), JZ) _ = g(4X, Z)g(HD, WY)  g(4X, 4Y)Q(HD. WZ).
(6.17)
On the other hand, by (2.1) we have
g (h (X, Z), JY)  g (h (X, Y), JZ) = 0,
(6.18)
for any X E F(TM) and Y, Z E r(D1). Next, we take Z = E in (6.17) and taking account of (6.18) and of Proposition 16.1 we obtain
g(h(X, Y), JEi)  g($X, Y)g(HD, Vi) + + q(h(QY, Ei), WX).
Thus we have
q h(X, Y) = g(O X, 4Y)HD +
E {g(h(QY, EWX)Vi). i=1 (6.19)
Substituting QY
q E {YJE ) j=1
i
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CHAPTER M
in (6.19) we obtain h(X, Y) = g(lX, 4Y)HD +
q E {Yjg(h(Ej, Ei), wx)Vi
i,j=1 which implies (6.15) with Lij = h(Ei, Ej). Conversely, suppose (6.15) is satisfied. Then (6.14) follows from (6.16) since we have q E {(g(Y, E4)g(Vi, JZ) i,j=1
 g(Z, Ej)g(Vi, JY))g(Lij1 wX)} = 0.
Hence by Theorem 6.3 M is a Sasakian antiholomorphic submanifold. The proof is complete. Now, from (6.7) it follows that on each Sasakian antiholomorphic submanifold with flat normal connection we have
A X = g(HD, Vi)X, i
(6.20)
for any X E r(D) and i = 1,...,q. Thus, by using (4.12) and (6.20) we obtain PROPOSITION 6.2. Let M be a Sasakian antiholomorphic submanifold of a Kaehlerian manifold N with flat normal connection. Then the Ricci curvatures of M with respect to the distinguished field of frames (4.9) are given by k(Ei) = 2pg(HD, Vi)2.
(6.21)
With respect to the sectional curvatures of M we have PROPOSITION 6.3. Let M be a Sasakian antiholomorphic submanifold of a Kaehlerian manifold N with flat normal connection. Then we have
KM(Ei A X) = g (HD, vi) 2
(6.22)
KM(Ei AE j) = 0,
(6.23)
and
for any unit vector field x E r(D) and i, j = 1,...,g, i x J. Proof. By using (6.6) we obtain
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
R(Ei, X)Ei = V
73
(g(HD, Vi)U) i
 g(HD, Vi)4(IEi, X]) = Ei(g(HD, Vi))fX +
+ g(HD, Vi)(VE )X  g(HD, V1)2X,
(6.24)
for any unit vector field X E r(D). But from (4.4) it follows that 0, since M is mixed geodesic. By using (2.6) of Chapter I and (6.24) we have KM(Ei AX) = g(R(Ei, X)X, Ei) = g(HD,
g(R(Ei, X)Ei, X)
Vi)2
Finally, by (6.6) we have VEiEj = 0 which implies R(Ei, E.)Ei = 0. This completes the proof. From Proposition 6.3 we have
COROLLARY 6.1. There exist no positively or negatively curved Sasakian antiholomorphic submanifolds with flat normal connection and q 1 in any Kaehlerian manifold.
V. Cohomology of CRSubmanifolds First we give some remarks on distributions on Riemannian manifolds (see §4 of Chapter I). Let H be a differentiable distribution on a Riemannian manifold M with LeviCivita connection V. We put
h(X, Y) _ (VXY)(7.1) for any X, Y E r(H), where (VXY)1 denotes the component of V Y in the orthonormal complementary distribution H1 on M. X Let {x1,...,Xr} be an orthonormal basis of H, r = dim.RH.
We put 0
H =
1
r
r
,
Z h(Xi, Xi).
(7.2)
i=1
Then H is a welldefined H1valued vector field on M, called the mean curvature vector of H. The distribution H is called minimal if the mean curvature vector H of H vanishes identically.
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CHAPTER III
Let M be a CRsubmanifold of a Kaehlerian manifold N. Then we have LEMMA 7.1. The holomorphic distribution D on M is minimal.
Proof. By using the formulas of Gauss and Weingarten for the immersion of M in N we obtain g(Z, VxX) = q(JZ, VxJX) = g(VxJZ, JX) _ = g(AJZX, JX),
(7.3)
for any x E r(D) and z E r(Dl). Then we have g(Z, VJxJX) = g(AJZJX, X) = g(AJZX, JX).
(7.4)
Thus by (7.3) and (7.4) we obtain g(Z, VxX + VJxJX) = 0,
which proves our assertion.
{F
Now, we choose an orthonormal local field of frames 1,...,wp, .,F , JF ,...,JF } of D. We denote by {w
+I,,
w
1forms ,...,wp2p} the 1 P on M. satisfying wj(z) = 0, wi(Fj)
= aj,
i, j = 1,...,2p,
(7.5)
for any Z E r(D'1) where Fp+j = JFj. Then W
=w1
n ...
(7.6)
nw2p
is a globally welldefined 2pform on M because D is orientable. THEOREM 7.1 (Chen [7]). Let M be a closed CRsubmanifold of a Kaehlerian manifold N. Then w is closed and thus defines a canonical de Rham cohomology class given by c(M) _ [w] E H2p(M: R),
p = dim. CD.
(7.7)
Moreover, this cohomology class is nontrivial if D is integrable and DL is minimal. Proof. From (7.6) we obtain 2p
dw =
E
i=1
(1)iwl
A... ndwiA... Aw2p
(7.8)
By using (7.5) and (7.8) we obtain that dw  0 if and only if
CRSUBMANIFOLDS OF KAEHLERIAN MANIFOLDS
75
dw(Z1, Z2, X1,...,X2p1) = 0
(7.9)
dw(Z1, X1,...,X2p) = 0,
(7.10)
and
for any Z1, Z2 E r(D) and X1,...,X2p E r(D). By a straightforward computation it follows that (7.9) holds if and only if Dl is integrable and (7.10) holds if and only if D is minimal. But by Theorem 1.1 and Lemma 7.1 Dl is integrable and D is minimal. Hence the 2pform W is closed. Now, let {F2p+1'...'F2p+q} be an orthonormal local field of frames of Dl and let 1forms on M satisfying wa(x)
and
= 0
{w2p+l,...,w2p+1}
be the
wa(FB) = ds,
for any X E r(D) and a, S = 2p+1,...,2p+q. Then iq a similar way we may conclude that if D is integrable and D is 2p+l
minimal, then the qform wl = w A... A W2p+q is closed. Thus the 2pform w is coclosed, i.e. 6w = 0. Since H is a closed submanifold, w is harmonic. Because w is nontrivial, the cohomology class [w] represented by w is nontrivial in H2p(M; R). Thus the proof is complete. Next we choose an orthonormal local field of frames (F1,...,Fp, ,
P+1'...,Fp+q,
P+q+l'...,F'no
JF1,...,JFn)
on N in such a way that restricted to No (F ,...,F , JF,...,JF ) are in D and {F } are in ,...,F p+q 1 p * 1 p p+l Dl. We denote by {wl,...,wn, w1 ,...,wn* } the dual frame to {Flo...,Fn JF1,...,JFn}. We put
wA*, eA = wA 
A = wA +
wA*,
A = 1,...,n. Then restricting WA's and SA's to H we have 9a
go = wa for
a = p+1.... 'p+q,
9r
Sr = 0
r = p+q+l,...,n.
(7.11)
for
The Kaehler form n of N is a closed 2form on N given by n E AA A gA. (7.12) 2
A= 1
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CHAPTER III
Now we can state
THEOREM 7.2 (Chen [7]). Let M be a closed CRsubmanifold of a Kaehlerian manifold N. If H2k(M; R) = 0 for some k 4 dim CD1 then either D is not integrable or D1 is not minimal.
Proof. Let S2 = i*S2' be the 2form on M induced from via the immersion i : M . N. Then from (7.11) and (7.12) we have
P
it = 2 E
01 A91.
(7.13)
1=1 It is clear that 9 is a closed 2form on M and it defines a cohomology class [SI] in H2(M; R). Then (7.6) and (7.13) imply that the canonical class c(M) and [S2] satisfy
[slip  (1)p(pI)c(M)
(7.14)
If D is integrable and Dl is minimal, then by Theorem 7.1 and (7.14) we obtain H2k(M; R) x 0 for k = 1,...,p. The proof is complete.
Let M be a CRproduct of N. Then the distribution D is integrable. Moreover Dl is minimal because we have g(X, VYZ) = 0
for all
X E r(D) and Y, Z E r(D1).
Hence the assumption on the cohomology group in Theorem 7.2 is necessary.
Chapter IV
CRSUBMANIFOLDS'OF COMPLEX SPACE FORMS
§1. Characterization of CRSubmanifolds in Complex Space Forms Let N(c) be a ndimensional complex space form of constant holomorphic sectional curvature c. Then the curvature tensor field R of N(C) is given by (see §5 of Chapter I) R(X, Y)Z =
4{g(Y,
Z)X  g(X, Z)Y + g(JY, Z)JX 
 g(JX, Z)JY + 2q(X, JY)JZ},
(1.1)
for any vector fields X, Y, Z tangent to N(c). We give here a characterization of CRsubmanifolds in a complex space form in terms of the curvature tensor field of the ambient space. THEOREM 1.1.(BlairChen{1]). Let M be a submanifold of a complex space form N(c) with c X 0. Then M is a CRsubmanifold if and only if the maximal holomorphic subspaces
Dx = TxM fl J(TXM), x E M define a nontrivial differentiable distribution D on M such that R(D, D, Dl , D l) a We (1.2) where Dl denotes the orthogonal complementary distribution of D in TM. Proof. Suppose M is a CRsubmanifold of N(c). Then by using (1.1) we obtain R(X, Y)Z = 2 g(X, JY)JZ,
for any x, Y E r(D) and Z E r(D1). Thus we have (1.2) since JZ is normal to M for any Z E r(Dl). Conversely, if the maximal holomorphic subspaces D define a nontrivial distribution D such that (1.2) holds, then (1.1) implies
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CHAPTERIV
0 = R(JX, X, Z, W) = 2 g(X, X)g(JZ, W), for any X E r(D) and Z, W E r(Di). Thus JDX is perpendicular to DX. Since D is invariant by J, JDX is also perpendicular 1 1 to Dx. Therefore JDX C TXM and M is a CRsubmanifold. The
proof is complete.
Taking into account that the curvature tensor field of N(c) is given by (1.1) we have special forms for the structure equations of Gauss, Codazzi and Ricci for the immersion of M in N(c). Thus, the equation of Gauss becomes
g(R(X, Y)Z, U) = 4{g(Y, Z)g(X, U)  g(X, Z)g(Y, U) + + g(o, Z)g(u, U)  g(M Z)g(4Y, U) + + 2g(X, cY)g(CZ, W)) + g(h(Y, Z), h(X, U))  g(h(X, Z), h(Y, U)),
(1.3)
for all X, Y, Z tangent to M, where R is the curvature tensor of M. The equation of Codazzi is given by (VXh)(Y, Z)  (VYh)(X, Z)  4{g(4Y, Z)WX 
 g(X,
(1.4)
for any X, Y, Z tangent to M. Finally, the Ricci equation becomes
g(RI(X, Y)V, W) + g([p, AV]X, Y) _ = 4{g(wY, V)g(WX, W)
 g(WX, V)g(wY, W) +
+ 2g(X, Y)g(CV, W)),
(1.5)
for any X, Y tangent to M and V, W normal to M, where R1 is the curvature tensor of the normal connection on M. The special form of the curvature tensor field of N(c) requires a study of the existence of special classes of CRsubmanifolds in N(c). Thus we have THEOREM 1.2 (Bejancu [4]). There exist no totally umbilical proper CRsubmanifolds of N(c) with c * 0. Proof. Suppose M is a totally umbilical proper CR
79
CRSUBMANIFOLDS OF COMPLEX SPACE FORMS
submanifold of N(c) with c x 0. Let X and Z be two nonnull vector fields from D and D1 respectively. Replacing Y by OX in (1.4) and taking into account that M is totally umbilical, the lefthand side of (1.4) becomes
g(JX, Z)VX  g(X, Z) VI H = 0. On the other hand, the right hand side of (1.4) becomes 2 g(X, X)wZ x 0.
Thus we have a contradiction. From this theorem we have COROLLARY 1.1 (Okumura [1]). There exist no totally umbilical real hypersurfaces of N(c) with c x 0. COROLLARY 1.2. There exist no totally geodesic proper CRsubmanifolds in N(c) with c x 0.
§2. Riemannian Fibre Bundles and AntiHolomorphic Submanifolds of CP Let N be a (2n+1)dimensional regular Sasakian manifold with n, q). Then there exists a fibering structure tensors n : N , N/ = N, where N denotes the set of orbits of t and is a real 2ndimensional Kaehlerian manifold. We denote by (J, q) the Kaehlerian structure on N. Also we denote by * the horizontal lift with respect to the connection n. Then we have (2.1) (JX)* _ W; g(X*, Y*) = (g(X, Y))*,
for any X, Y tangent to N. We denote byV (reap. V) the LeviCivita connection with respect to g (reap. g). Then (VXY)* =
VAX*y*
g(,pX*,
+
y*)E _
(2.2)
We denote by K and R the Riemannian curvature tensors of N and respectively N. Then by using (2.2) we obtain (R(X, Y)Z)* = K(X*, Y*)Z* + g(Z*, 4Y*)$X*
g(Z*, X*)0Y*
 2q(Y*, X*)PZ*.

(2.3)
Now, let M be an (m+l)dimensional submanifold immersed
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CHAPTER IV
in N such that the structure vector field & of N' is tangent to tM and let M be an mdimensional submanifold immersed in N. N + M such that We assume that there exists a fibration it the diagram
nI M
i
n
(C)
N
is commutative and the immersion i' is a diffeomorphism on the fibres. We denote by the same symbols q and g the induced Riemannian metric on rA and M respectively. Let V' (reap. V) be the LeviCivita connection on M (resp. M). We denote by h and h the second fundamental forms of the immersions i' and i and by A and A respectively the associated Weingarten maps. Then for any X, Y tangent to M, the Gauss' formulas are given by V Y = QXY + h(X, Y) X From (2.2) we have
and oX*Y* = OX*Y* + h(X*, Y*).
(VXY)* = ,P2(OX*Y*)
(2.4)
(h(X, Y))* = h(X*, Y*).
(2.5)
and
Now let V and V1 be the Riemannian connections on the normal bundles to M and M respectively. Then we have Weingarten's formulas VXV = AVX + VXV; VXV = AV*X* + X*V*,
(2.6)
and from (2.2) and (2.6) we obtain
42
(A,X)
and
(VV)*
=
(2.7)
(AV*X*)
X*V*.
(2.8)
From (6.8) of Chapter I we have
($X) T = VXE;
(OX)1

h(X,
(2.9)
is the for any X tangent to M, where (q,)T (resp. tangent (resp. normal) part of 4X. From the second equation in (2.9) we obtain h(&, E) = 0.
LEMMA 2.1. Let M be an antiholomorphic submanifold of a be a submanifold of a Kaehlerian manifold N and let
CRSUBMANIFOLDS OF COMPLEX SPACE FORMS
81
Sasakian manifold N satisfying the condition (C). Then we have Z*) = ((VXh)(Y, Z) + g(Y, X)WZ + (VIX*h)(Y*,
+ g(Z, +X)wY)*
(2.10)
and (vX*h)(Y*, c) _ (h(Y, 4X) + h(X, $Y))`.
(2.11)
for all X, Y, Z tangent to M.
Proof. By using (6.1) of Chapter I, (2.4), (2.5) and (2.7) we obtain ((VXh)(Y, Z))*  (VX*h)(Y*, Z*) + ;(Y*, ox Y*).
h(E, Z*) + g(Z*,
Therefore, by (2.1) and (2.9) we have (2.10). By using (2.9) we obtain (VX*h)(Y*, E)  VX*(4DY*)l + [(VX*Y")]1 + + h(y*,
(,DX*)T)
Taking into account that VX*($Y*)1 =
(4VX*Y*)1  h(X*,
0 we obtain (mY*)F ).
Thus we have
(VX*h)(Y*, ) = h(X*, (oY*)T) + h(Y*, and (2.11) follows by means of (2.1) and (2.5).
THEOREM 2.1 (YanoKon 151). The second fundamental form h of M is parallel if and only if the second fundamental form h of M satisfies the conditions
(VXh)(Y, Z)  g(X, Y)wZ + q(X, Z)wY and
AV o
=
o AV,
(2.12)
(2.13)
for any vector fields X, Y, Z tangent to M and any vector field V normal to M. Proof. By Lemma 2.1 h is parallel on M if and only if (2.12) holds and we have
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CHAPTERIV
h(X, 4Y) + h(Y, 4X) = 0,
(2.14)
for all X, Y E r(TM). The equivalence of (2.13) and (2.14) is a simple verification using (3.3) of Chapter I.
LEMMA 2.2. Kaehlerian manifold N connection
Let M be an antiholomorphic submanifold of a manifold N and M be a submanifold of a Sasakian which satisfies the condition (C). Then the normal of M is flat if and only if the normal connection
of Al is flat.
Proof. By using (2.8) we obtain [g(R1(X, Y)V, U))* = g(Kl(X*, Y*)VU*)  2g(4X, Y E r(TM) and U, V E r(TMI), where RI and K are the curvature tensor fields of the normal connections on M and M respectively. Since JV is tangent to M, (2.15) becomes
[g(R'(X, Y)V, U)]* = g(Kl(X*, Y*)V*, U*).
(2.16)
From (2.9) we obtain AV*& _ (4V*)T. Thus (2.7) implies
;([AU*, AV*]x, E) _ (g(h(X, JV), U)  g(h(X, JU), V))* _ (g(AUJV  AVJU, X))*.
Taking into account Lemma 2.1 from Chapter III we obtain g([AU*,
Av*]x*, E) = 0.
Hence, by using the Riaci equation for the immersion of M in N, we have g(KI(X*, &)V*, U*) = 0.
(2.17)
Thus our assertion follows from (2.16) and (2.17). In the submanifold holomorphic dimensional the diagram
following we assume that M is an mdimensional of complex projective space CPnAwith constant sectional curvature 4 and that M is an (m+1)submanifold of the unit sphere S2n+1 such that it
M 7r
1 :N
1
M
i
y Pn
CRSUBMANIFOLDS OFCOMPLEX SPACE FORMS
83
commutes. Suppose dim CDx = p and dim RDX = q. Then we have LEMMA 2.3. Let M be an mdimensional antiholomorphic > 2. If submanifold of CPn with flat normal connection and p (2.12) holds then the second fundamental form h of M is parallel. LEMMA 2.4. Let M be an mdimensional antiholomorphic submanifold of CPn. Then we have g(Vh, Vh) > 4pq,
(2.18)
and the equality holds if and only if (2.12) holds. For the proofs of these lemmas see YanoKon [3]. In the same paper the authors proved that the diagram Sm1(r1)x...xS
(rk)
i 'Sm+k
1
m! IT (S
+
(r1)x...xS
c (rk))CP( i m+k1 ) 2
k
k
i=1
i=1
m + 1= E mi,
E
(ri) 2 = 1,
commutes. Moreover Tr(sm1(r1)x...XS
(rk))
is an example of antiholomorphic submanifold with flat normal normal connection in a complex projective space and
Sm1(r1)x...xS(rk) is a semiinvariant submanifold (or contact CRsubmanifold) Sm+k,
We shall give some results on semiinvariant submanifolds in Chapter V. Now we recall in
THEOREM 2.2 (YanoKon [31). Let M be a complete mdimensional submanifold of Sn 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 number of spheres. Moreover, if M is of essential codimension n  m then M is a Pythagorean product of the form
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CHAPTER IV
Sm1(r1)X...XSmk (rk),
k E
(r i)2
1, k  nm+1,
i=1 or a pythagorean product of the form
m S 1(rI)X...XSmk (rk,) c
k' E
Sn1
= Sn,
(ri) 2 = r2 < 1, k' = n  m.
i=1 Then we can state THEOREM 2.3 (YanoKon [3]). Let M be a complete mdimensional antiholomorphic submanifold of CPn with flat normal connection. If p 2 and g(Vh, Vh) = 4pq, then M is
k n(Sm1(r1)x...xS"(rk)), m + 1 = k F
£ mi, i=1
(ri)
= 1,
i=1 2
k < m + 1, where ml,...,mk are odd numbers and
n = (m+k  1)12.
Proof. We consider the diagram M. Since the normal connection of M is flat, by Lemma 2.2 the normal connection of M is flat. Therefore Lemmas 2.3 and 2.4 imply that the second fundamental form of M is parallel. On the other hand, if M admits a geodesic section, that is, if there exists a normal vector V such that g(h(X, Y), V) = 0 for all X, Y tangent to M, then (1.5) implies that the holomorphic sectional curvature of the ambient manifold is zero. This is a contradiction. Consequently, the immersion i is full. Thus by (2.5) the immersion i' is full too. Therefore Theorem 2.2 and the example above prove our assertion. More results on the geometry of antiholomorphic submanifolds in CPn via Riemannian fibre bundles are obtained by Yano and Kon in [3]. Moreover, they obtained in [5] and [6] results on CRsubmanifolds of CPn by using the same method of Riemannian fibre bundles.
§3. CRProducts of Complex Space Forms
In §5 of Chapter III we obtained the main results on CRproducts of Kaehlerian manifolds. The aim of this paragraph
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CRSUBMANIFOLDS OF COMPLEX SPACE FORMS
is to study the existence of CRproducts in complex space forms.
First, from Theorems 5.4 and 5.5 of Chapter III we have PROPOSITION 3.1. (i) There exist no proper CRproducts in any complex hyperbolic space N(c), c < 0. (ii) There exist no antiholomorpnic CRproducts in any eliptic complex space form N(c), c > 0. The first statement is due to Chen [5] and the second to BejancuKonYano [1]. The Proposition 3.1 leads us to a study of CRproducts in Cn and to a study of proper CRproducts (but not antiholomorphic) in CPA. For CRproducts in Cn we have
THEOREM 3.1 (Chen [5]). Every CRproduct in Cn is the Riemannian product of a holomorphic submanifold in a linear complex subspace Cs and a totally real submanifold of Cns locally, i.e.,
M=M1 XM2CCs XCns=Cn. Proof. From Lemma 5.1 of Chapter III we obtain h(X, Y)  0,
(3.1)
for any X E T(D) and Y E r(Dl). Thus by applying a lemma of Moore (1] we see that m = M1 X M2 is a product submanifold in
Rr
X
R2nr.
Since M1 is a holomorphic submanifold of Cn
we may choose Rr to be a complex linear subspace of Cn. THEOREM 3.2. (BejancuKonYano 11]). Every mdimensional antiholomorphic product M of Cn is the Riemannian product Cp X M2 of Cn, where M2 is an (2n  m)dimensibnal totally real submanifold of Cn Proof. Since M is an antiholomorphic product, by Corollary 5.1 of Chapter III we have h(U, V)  0 for any v, v E ('(D). Then by Theorem 1.2 of Chapter III each leaf of D is a totally geodesic submanifold of Cn. Hence each leaf of D is itself a Cp in Cn and our assertion follows. Now we give some examples of CRproducts in CFO. In order to do this we define a mapping
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CHAPTER IV
:
S
CPp X CPq ., CPpPq
pq
by
(ZO,...,zpj UOr...,Uq) . (Z0U01...,Ziuj,...,Zpuq),
where (z0,...,zp) (resp. (u0,...,uq)) are the homogeneous coordinates of CPp (resp. CPq). The mapping S is a Kaehler Cp+q+pq. imbedding of the Riemannian product CPp X CPq into
Let M2 be a real qdimensional totally real submanifold of CPq. Then CPp X M2 is a CRproduct M1 X M2 in CPp+q+pq via Spq, in which M1 = CPp is a totally geodesic holomorphic CPp+q+pq.
submanifold and M2 is a totally real submanifold in With these examples in mind we give the following definition.
A CRproduct M1 X M2 in CP is called a standard CRproduct if we have: (i)
n  p + q + pq, where p = dim.CDX, q = dim.RDX; and
(ii) M is a totally geodesic holomorphic submanifold of CPn. We shall prove that in fact n = p + q + pq is the smallest dimension of CPn for admitting a (2p+q)dimensional CRproduct. First we have
LEMMA 3.1. Let M be a CRproduct in CPn. Then
{h(Xir Za)}, i  1,...,2ps of = 1,...,q, are orthonormal vectors in V
(TM
=
JD ® v), where
{X11...Ox2p} and {Z1,.... Zq} are orthonormal basis in Dx and DX respectively.
Proof. By Lemma 5.1 of Chapter III we obtain
I h(X, Z) A  1,
(3.2)
for any unit vector fields x E T(D) and Z E r(D1). Hence we obtain by linearity g(h(Xi, Z), h(Xj, Z)) a 0, i x J.
(3.3)
By Theorem 5.2 of Chapter III we have h(X, Z) E r(v). Hence if q  I the lemma is proved.
CRSUBMANIFOLDS OF COMPLEX SPACE FORMS
87
Now, suppose q> 1. Then from (3.3) it follows that
g(h(Xi, Za), h(Xj. ZS)) + g(h(Xi, ZS),
h(Xj, Za))  0
(3.4)
for i ^ j and a x (3. Since M = M1 X M2 is a CRproduct we have R(Xi, Xj, Za, ZS) = 0.
(3.5)
On the other hand, by (1.1) we have R(Xi, Xj, Za, ZQ)
(3.6)
0.
Thus by (1.3), (3.5) and (3.6) we obtain g(h(Xi, Za), h(xj, Z8)) = g(h(xj, Za),
h(Xi, ZS)).
(3.7)
Finally our assertion follows from (3.2), (3.4), and (3.7). This lemma implies THEOREM 3.3 (Chen 151). Let M be a CRproduct in CPA. Then
nap+q+pq.
(3.8)
Remark 3.1. Since the standard CRproducts satisfy the equality sign in (3.8) the estimate of n in (3.8) is the best possible. With respect to this topic we give here other two important results due to Chen. THEOREM 3.4 (Chen [5]). Every (2p+q)dimensional CRproduct in CPn with n = p + q + pq is a standard CRproduct. THEOREM 3.5 (Chen [5]). Let M = M
x M be a (2p+q)
dimensional CRproduct in CPn. Then II h
112
> 4pq.
(3.9)
Now, we can ask whether there exist proper CRsubmanifolds in CPn which are neither real hypersurfaces nor CRproducts. we shall give here a procedure due to Shimizu in order to get such CRsubmanifolds. Let G/K be an irreducible Hermitian symmetric space of S2n+1  CPn the Hopf fibration. compact type. Denote by W :
88
CHAPTER IV Stn+1
For a point p E denote by N the Korbit of p and put M = 7;(N). We say that p is regular if N has the maximal dimension. THEOREM 3.6 (Shimizu [1]). Let r be the rank of G/K. Suppose Stn+1
p is a regular element of
and r is greater than one.
Then: (i)
r  1;
M is a proper CRsubmanifold in CPn of codimension
M is not a CRproduct; (iii) M has parallel mean curvature; (iv) M has flat normal connection. (ii)
The proof requires some knowledge of symmetric spaces, so we omit it here.
94. Mixed Foliate CRSubmanifolds of Complex Space Forms
Let M be a CRsubmanifold of a Kaehlerian manifold N. Then M is called mixed foliate if D is integrable and M is mixed geodesic, i.e., the second fundamental form h of M satisfies
h (X, Y)  0,
(4.1)
for any X E P(D) and Y E r(D1). The geometry of mixed foliate CRsubmanifolds has been studied by Chen in [5] and (6), BejancuKonYano in (1] and Bejancu in [4]. In this section we gather some results from the papers quoted above. First we have
LEMMA 4.1. Let M be a mixed foliate CRsubmanifold of a Kaehlerian manifold N. Then
AV o
+ 4)
o AV = 0,
(4.2)
for any vector field V normal to M. Proof. It is easy to check that (4.2) is equivalent to
g (h (X, +Y) , V) = g (h (Y, 4X) , V) ,
(4.3)
for any X, Y E r(TM) and V E r(TMl). Since D is integrable we obtain (4.3) for any X, Y E r(D) (see Theorem 1.1 of Chapter III). On the other hand, 4X and Y belong to D for any X, Y E r(TM). Hence if we take x E r(Di), (4.3) follows from (4.1). In a similar way we have (4.3) for any Y E r(Dl).
CRSUBMANIFOLDS OF COMPLEX SPACE FORMS
89
Concerning the existence of mixed foliate CRsubmanifolds in complex space forms we have THEOREM 4.1 (BejancuKonYano (1]). If M is a mixed foliate proper CRsubmanifold of a complex s ace form N(c) then we 0. have c
Proof. Let X, Y E r(D) and Z E r(DI). Then we have (Vxh) (Y, Z)  (VYh) (X, Z) = h(X, VYZ)  h(Y, VxZ) . (4.4)
Take a vector field V E r(JDI) such that Z  JV = BV and obtain VYZ =  4(AVY) + B(VYV).
(4.5)
Thus by (4.3)(4.5) we have (VXh)(Y, Z)  (VYh)(X, Z) = h(OY, AVX) + h(X, AVOY)
(4.6) Now let X = 4Y and using (4.6) and (1.4) we obtain
2h(41, AV(4Y)) _  2 cg(OY, Y)V. Therefore 0 G 2g(AV(QY), AV($Y))
2 cg(4Y, 4Y)g(V, V) (4.7)
which proves our assertion.
COROLLARY 4.1. Let M be a mixed foliate CRsubmanifold of a complex space form N(c). If c 0 then M is either a holomorphic submanifold or a totally real submanifold of N(c). By Theorem 4.1 we need only to study the geometry of mixed foliate CRsubmanifolds in N(c) with c < 0. THEOREM 4.2 (Chen (51). Let M be a CRsubmanifold of Cn. Then M is mixed foliate if and only if M is a M product. Proof. Suppose M is a mixed foliate CRsubmanifold of Cn. Then we have c 0 in (4.7) and obtain
AVX  0, Thus we have
for any v e r(JD)
and
x e r(D).
Bh(X, Y)  0, for any X E r(D) and Y E r(TM), Hence, by Theorem 5.2 of
(4.8)
90
CHAPTER N
Chapter III, M is a CRproduct. Conversely, suppose M is a CRproduct of Cn. Then D is an integrable distribution and (4.1) follows from (4.8), that is, M. is mixed geodesic. The proof is complete.
Remark 4.1. Theorem 4.2 has also been proved by BejancuKonYano in 1 for antiholomorphic submanifolds. in the remaining part of this paragraph we shall study mixed foliate CRsubmanifolds in complex hyperbolic spaces. For simplicity, we consider an ndimensional complex hyperbolic space H with constant holomorphic sectional curvature 4. First we prove some lemmas due to Chen [6]. LEMMA 4.2. Let M be a mixed foliate CRsubmanifold in H. Then for any unit vectors x E r(D) and Z E r( DL) we have tl
uAJZX12 = 1
(4.9)
N h 12 .4 2pq.
(4.10)
and
The equality sign in (4.10) holds if and only if we have: (a) the leaves of DL are totally geodesic in H; and
(b) Im h = Proof. We put c = 4 in (4.7) and obtain (4.9). Then taking account of (4.9) we obtain (4.10). Again using (4.9) we see that the equality sign in (4.10) holds if and only if we have
Im h = JD and
AD1
1
= {0}.
(4.11) (4.12)
Thus our assertion follows from Theorem 1.5 of Chapter III.
Let M be a mixed foliate CRsubmanifold in H and M be a leaf of the holomorphic distribution D. Then M is a etc., Kaehlerian submanifold of H. We denote by h, the second fundamental ,form, the normal connection,.... etc., ,..., etc., the corresponding for M in H and by h', quantities for M in M. Then h(X, Y) = h'(X, Y) + h(X, Y),
(4.13)
for any X, Y E F(TM).
LEMMA 4.3. Let M be a mixed foliate CRsubmanifold in H. Then
CRSUBMANIFOLDS OFCOMPLEX SPACE FORMS
AZS
=
AJZX
AJZJX,
=
91
(4.14)
AJZX,
VIJZ = VXJZ = w(VXZ),
(4.15)
CXZ = VX'Z,
(4.16)
AZX = AZX,
for any unit vector fields x E r(TM) and Z E r(D
1 ). InI
Proof. Since M is a Kaehlerian submanifold of H, by using (4.13) we obtain g(AZX, Y) = g(Jh(X, Y), JZ) = g(h(JX, Y), JZ)
= g(AJZJX, Y)
and
g(AJZX, Y) = g(h(X, Y), JZ) = g(AJZX, Y), , VY E r(TM).
On the other hand, by Proposition 2.4 of Chapter II, the distribution D is invariant with respect to the action of
A. Hence we have (4.14). JZ
Next, for any X E r(TM) and z E r(Dl) we have J(VXZ) = VXJZ = AJZX + VXJZ.
(4.17)
Thus we obtain
VIiz = w(VXZ). Also OXJZ =  AJZX + VXJZ,
which together with (4.17) implies dxJZ =
XJZ.
Thus we have (4.15). Finally, by using equations of Gauss and Weingarten for the immersions of R in H, R in M and M in H we obtain  AzX +
1X1Z
 VXZ  VXZ =  AZX + VXJZ.
Hence we have (4.16). The proof is complete. LEMMA 4.4. Let M be a mixed foliate CRsubmanifold of H. Then Im h = Dl
JDl
(4.18)
and A7, AJZ E 0(2p), r(
for any z E
DI), where 0(2p) is the orthogonal group.
(4.19)
CHAPIERIV
92
proof. Let U be a normal vector field in v. Then by using (1.5) and (4.15) we obtain (4.20)
[AJZ, AU] = 0.
Taking into account that M is a Kaehlerian submanifold of H we have (see Ogiue (1)).
AJU = JAU = AUJ
Thus, by using (4.20), we obtain
0 = AJZ
U
u JZ =
JO(AU
JZ + AJZ U)
Since J is nonsingular this gives
AUaJZ+AJz=0.
(4.21)
Thus from (4.20) and (4.21) we get AJZAU = 0.
(4.22)
By using (4.9) and (4.14) we obtain
1,
NAZX1=1AJZX1
for any unit vectors X E r(TM) and Z E r(D1). Thus we have (4.19). In particular AJZ is nonsingular. Hence from (4.22) we have AU = 0 which implies (4.18). Now we can state THEOREM 4.3. Let M be a mixed foliate CRsubmanifold of H. Then M is an antiholomorphic submanifold of a complex (p+q)dimensional complex submanifold H' which is totally geodesic in H.
Proof. Taking into account that AU = 0 for each U E r(v) we obtain
1h
12
=
E
a,i
{IAZ Eil2 + dAJZ EiI
a
a
where {E1,...RE2p} (reap. {Z1,...,Zq}) is a basis of TxM (reap. DX). Then by using (4.9) and (4.14) we have
8 h N` =
E {I AJZ JE112 +q AJEi q2} = 4pq. Za a a,i
Next, from (4.9), (4.14), and (4.16) it follows that
(4.23)
CRSUBMANIFOLDS OF COMPLEX SPACE FORMS
h'112 =
f {I AZ Ei 12)
a' j
=
a
93
E {I AJZ JE
aj
a
N2}
= 2pq. (4.24)
Thus, from (4.13) we obtain 11h11
= 2pq.
(4.25)
Then by using Lemma 4.2 and the equation of Gauss and Weingarten for the immersion of M in H we get J(V Z) = DYJZ = AJZY +
for any Y E F(TM) and Z E r(D1). Hence DYJZ = w(V Z).
(4.26)
By (4.26) we see that JD1 is a parallel subbundle of the normal bundle of M in H. Thus, by assertion (b) of Lemma 4.2, we see that M lies in a complex (p+q)dimensional complex submanifold H' which is totally geodesic in H. Obviously, M is an antiholomorphic submanifold of H'. LEMMA 4.5. Let M be a mixed foliate proper CRsubmanifold of 0 and q 1. 0. Then c a complex space form M(c), c Proof. By Theorem 4.1 we have c < 0. If q = 1 then for any unit vector fields z E I'(D1) and X E r(TM) by (4.16) we _ obtain = 0. pXZ =
XZ
Hence Z is a parallel normal vector field of the complex submanifold M in N(c), c < 0. This contradicts a theorem of ChenOgiue [2]. PROPOSITION 4.1. Let M be a mixed foliate proper CRsubmanifold of H. Then each leaf M of D is an EinsteinKaehler submanifold of H with Ricci tensor 9 given by S(X, Y)
2(p+q+1)g(X, Y).
(4.27)
Moreover, we have p > 1 and q > 1. Proof. Since M is a Kaehlerian submanifold of H, by a direct computation we obtain
S(X, Y) _ 2(q+1)g(X, Y)  F{g(A X, A Y)}, a a a for any X, Y E T(TM), where {4a} is an orthonormal basis of
CHAPTERIV
94
TM''. Thus, taking into account that AU = 0 for any U E r(v) and using (4.9) and (4.14) we obtain (4.27). By Lemma 4.5 we have q > 1. It remains to prove p > 1. Suppose p = 1. Then by (4.27) M is of constant curvature 2(q+2). Hence a theorem of Calabi [1] gives q = 0 which is a contradiction.
By using Theorem 4.3 and taking into account that for a proper mixed foliate CRsubmanifold of H we have p > 1 and q > 1 we obtain THEOREM 4.4. Let M be a mixed foliate CRsubmanifold of H. If dim RM < 5 then M is a totally real submanifold. Thus we have
COROLLARY 4.2. There exist no mixed foliate proper CRsubmanifolds of dimension 4 5 in a hyperbolic complex space form.
Remark 4.2. It is an open problem if Corollary 4.2 is valid for any dimension of M. If M is a compact manifold, Chen obtained in [10] theorems on nonintegrability of the holomorphic distribution.
§5. CRSubmanifolds with SemiFlat Normal Connection Let M be a CRsubmanifold of a complex space form N(c). Then we say that M has semiflat normal connection if its normal curvature R'1 satisfies
R1 (X,
Y, V, W) = 2 g(X, Y)g(JV, W) ,
(5.1)
for any X, Y E r(TM) and V, W E r(TMI). By using (1.5) and (5.1) we obtain that a CRsubmanifold M of a complex space form N(c) has semiflat normal connection if and only if g([AW, AV]X, Y) = 4{g(wY, V)g(wX, W) 
 g(WX, V)g(wY, w) ).
(5.2)
CRsubmanifolds with semiflat normal connection have been defined by Yano and Kon in [5]. A complete classification of CRsubmanifolds with semiflat normal connection immersed in N(c) with c * 0 has been obtained by Chen in [6]. More precisely we have
CRSUBMANIFOLDS OFCOMPLEX SPACE FORMS
95
THEOREM 5.1. Let M be a CRsubmanifold in a complex space form N(c), c x 0. Then M has semiflat normal connection in N(c) if and only if M is one of the following: a totally geodesic complex submanifold; (i) (ii) a flat totally real submanifold of a totally geodesic complex submanifold 9(c) in N(c); (iii) a proper antiholomorphic submanifold with flat normal connection in a totally geodesic complex submanifold N'(c) of N(c); (iv) a space of positive constant sectional curvature immersed in a totally geodesic complex submanifold N"(c) of N(c) with flat normal connection as a totally real submanifold. CRsubmanifolds with semiflat normal connection in CPn have been studied by Yano and Kon in [5] and [6]. The method of Riemannian fibre bundles was also the main tool in obtaining interesting results. We only mention here a theorem which is a generalization of Theorem 2.3. THEOREM 5.2 (YanoKon [5]). Let M be a complete mdimensional CRsubmanifold of CPn with semiflat normal connection and p 1. If the fstructure C in the normal bundle to M is parallel and if g(9h, Ch) = 4pq, then M is a totally geodesic holomorphic submanifold CPm/2 of CPn or M is an antiCP(m+q)/2
holomorphic submanifold of 1T(S1(r1)x...XS k (rk)), E
i1
in CPn and it is
m + 1 = iE1 mi,
(r i)2 = 1,
where m1,...,mk are odd numbers. More results on the geometry of CRsubmanifolds with semiflat normal connection can be found in YanoKon [6] and Chen [6].
§6. Pinching Theorems for Sectional Curvatures of CRSubmanifolds Let M be a real mdimensional CRsubmanifold in an ndimensional complex space form N(c). A plane section A of the tangent bundle TM is called an antiholomorphic plane
CHAPTERIV
96
section if J(A) is orthogonal to A. Then the sectional curvature K(A) of M is called an antiholomorphic sectional curvature of M. if the plane section belongs to the holomorphic distribution we say that K(A) is a Dantiholomorphic sectional curvature of M. The distribution Di is called Dparallel if we have V
x
Y E r (Dl) ,
(6.1)
for all x E r(D) and Y E r(D').
LEMMA 6.1. Let M be a CRsubmanifold of N(c). Then the distribution is Dparallel if and only if
g(h(X, Y), JZ) = 0, for any X, Y E T(D) and z E rah.
(6.2)
Proof. Since g is Hermitian, by (3.2) of chapter III we obtain
g(h(X, Y) , JZ) _ g(Bh(X, Y) , Z) = g((VX0 Y, Z) . Since V is a Riemannian connection we get g(h(X, Y), JZ) = g(4Y, VxZ), and the lemma follows.
LEMMA 6.2. Let M be a CRsubmanifold of N(c). If D is Dparallel then the holomorphic distribution is integrable and its leaves are totally geodesic in M. The proof of the lemma follows from Theorem 1.2 of Chapter III and Lemma 6.1. Now let (VIP ....V2nm} be a local field of orthonormal frames in the normal bundle TMt and let Act = AV
(a = 1,...,
2n  m) be the fundamental tensors of Weingart n. The mean curvature vector H is given by
H  m E Tr (a) Va. a Then by a direct computation the scalar curvature p of M is given by
p= where we have
4
(m2
 m + 6p) c
+ m2M H f
lh
f,
NHN2 =1 a(Tr(Aa))2 and 1h 12 = a Tr(A2)
(6.3)
CRSUBMANIFOLDS OF COMPLEX SPACE FORMS
97
Also, the sectional curvature of M determined by orthonormal vectors {X, Y} is given by
4{1 + 3g(PX, Y)2} +g(h(X, X), MY, Y)) 
KM(X AY) =
 g(h(X, Y), h(X, Y)).
(6.4)
Let {E1,...,Em} be a local field of orthonormal frames tangent to M. Then we put h(Ei, Ej) = F hijVa,
i, j = 1,...,m.
(6.!)
CL
By using (6.4) and (6.5) we obtain II h
and
aij (hi)2
(6.6)
F
KM(Ei AE j) =
4{1 + 3q(PEi,
+ E {hijhjj for all i x J. Now we can state
2} +
(hi)2(6.7)
a
THEOREM 6.1 (Bejancu (18]). Let M be a CRsubmanifold of a complex space form N(c). Suppose the following conditions are fulfilled: the distribution Dl is Dparallel; (i) (ii) there exists k > 0 such that the second fundamental form h of M satisfies
(6.8)
II h 112 s k.
Then all the Dantiholomorphic sectional curvatures of
M are > (c  k)/4. Proof. Let A be an antiholomorphic plane section which belongs to the distribution D. Suppose {E , E } is an orthonormal field of frames of A. Then choose 2the orthonormal field of frames {E1,...,Ep, E1*  JE1*,...,Ep*  JEp, F1,...,Fq F1*  JF1,...OFq* = JFq, V1,...,Vs, V1* = JV1,...,Vs*  JVs} on N(c) such that EL E r(D)
(i = 1,...,p), Fa E r(DL)
(a  1,...,q) and Vp, E Nv) (a  1,...,s). By using this
(6.9)
field of frames, from (6.7) we obtain s
KM(El AE 2)
T
1h
*
h
q
*
(h 12)2 
*
h h 4 + a=1ll22 + il22
K12
(h12)2} +
E {h11h22  (h12)2}
(6.10)
a=1
Taking account of Lemmas 6.1 and 6.2 we have ha* = ha* hi*j*t
hij =
(6.11)
i*j*
ij
and
(6.12)
Hence by
(6.8), (6.11)
and (6.12) it follows that s
k$Mh12>2
E {(h11)2 + (h22)2 + 2(ha12)2 + a=1
+ (ha*)2 + (ha*)2 + 2(h a*)2} 11 22 12
> 4 E { (ha ) 2 + (ha*) a ha 12 2 h 11 22 12 a=i

ha*ha*}.
11 22
(6.13)
Finally, from (6.10) and (6.13) and using (6.12) we obtain K12 ) (c  k)/4, that is, the assertion of the theorem. Next we need an algebraic lemma LEMMA 6.3 (Okumura [2]). Let al,...,am and t be real numbers satisfying the inequality
(E a i )
i
2
> (m  1) {E (ai) 2 + t}.
i
Then for any pair of distinct i and j we have 2aiaj > t. Now we state THEOREM 6.2 (Bejancu [18]). Let M be a CRsubmanifold of a complex space form N(c), c a 0. If the scalar curvature p of H satisfies
p)
4(m2
 3m + 6p + 2)c + (m  2) 1h12
+ 2(m  1)k, then all sectional curvatures of M are a k. Proof. We note first that from (6.4) we have
+
(6.14)
CRSUBMANIFOLDS OFCOMPLEX SPACE FORMS
99
4 + g(h(X, X) , h(Y, Y) ) 
KM(X AY)
 g (h (X, Y), h (X, Y))
(6.15)
for all orthonormal vectors X, Y tangent to M. Now, by (6.3) and (6.14) we obtain
m2.IIH12 > (m
 1){Ih12 + 2k  i}.
Since we have to prove a pointwise theorem, we arbitrary point x E M. Then we have either IHI minimal at x. Suppose JJHII x 0 at x. Then we can choose orthonormal frames {v1,...,v2nm} in TM such
Vi =I
.
FHI
(6.16) shall take an x 0 or M is the field of that
H.
Hence (6.16) becomes
m
Ih
1
2 > (m  1){IhM2 + 2k  2}.
(6.17)
i=1
Next, by using Lemma 6.3 and a computation similar to the one made by Chen and Okumura for the proof of Theorem I of (1] we obtain 2nm (6.18) (hij)2} > k, 4 +
ai {hiihjj
for arbitrary i x j (i, j  1,...,m). Thus, in this case the theorem follows from (6.15) and (6.18). If H = 0 at x, then from (6.16) we have 2nm
E {(hi)2+ (hj2+
a=1
2nm + 2(hij)2}
(hij)2}
4
+ a1 {hiihjj
(6.19)
for all i x J. Hence by (6.19) and (6.15) the proof of the theorem is complete.
Chapter V
EXTENSIONS OF CRSTRUCTURES TO OTHER GEOMETRICAL STRUCTURES
§1. SemiInvariant Submanifolds of Sasakian Manifolds Let N be a real (2n + 1)dimensional almost contact metric E, n, g), where is a manifold with structure tensors tensor field of type (1, 1), E is a vector field, n is a 1form and g is a Riemannian metric on N. These tensor fields are related by (see §6 of Chapter I) 2x = x +
4& = o; n() = 1; n(ix)  0 (1.1
gw, 4Y) = g(x, Y) 
n(X) = gm E), (1.2)
for any vector fields X, Y tangent to N. Let M be a real mdimensional submanifold of N. We assume that the vector field & is tangent to M. Denote by the 1dimensional distribution spanned by t on M. Then M is called a semiinvariant submanifold of N if there exist two differentiable distributions D and Dl on M satisfying
(i)
TM = D ®
Dl a W,
where D, D1 and {E} are mutually orthogonal to each other; the distribution D is invariant by 0, that is, (ii)
Wx) = Dx for each x E M; (iii) the distribution D'i is antiinvariant by , that is, (DX) c TTMl for each X E M.
Remark 1.1. The study of semiinvariant submanifolds in Sasakian manifolds has been initiated by BejancuPapaghiuc in [1]. The same concept was studied under the name "contact CRsubmanifold" by YanoKon [7], [8], Kobayashi M. [11141, Hsu [1], Matsumoto [2], ArcaRogca [1]. Remark 1.2. A semiinvariant submanifold is nothing but the extension of the concept of a CRsubmanifold of a Kaehler manifold to submanifolds of almost contact metric manifolds.
EXTENSIONS OF CRSTRUCTURES TO OTHER GEOMETRICAL STRUCTURES
101
We denote by 2p and q the real dimensions of Dx and Dx
x respectively, x E M. Then we see that for p  0 we obtain an antiinvariant submanifold tangent to F (see YanoKon E1]) and for q = 0 we obtain an invariant submanifold (see Kon [21). On the other hand, it is easy to check that each hypersurface of N which is tangent to F inherits the structure of a semiinvariant submanifold of N. In this paragraph we are concerned with semiinvariant submanifolds of Sasakian manifolds (see §6 of Chapter I for Sasakian manifolds). More precisely, we shall outline a study of the integrability of the distributions on M and of the immersion of their leaves in N or N. Let M be a semiinvariant submanifold of a Sasakian manifold N. Then g(X, Y)F  n(Y)X
(1.3)
and VXE  Ox,
(1.4)
for any X, Y tangent to N, where V is the LeviCivita connection on N. We denote by P and Q the projection morphisms of TM on D and Dl respectively. Then
X =PX+QX+n(X)F,
1.5)
for any x E r(TM). Also we put
OV  BV + CV,
(1.6)
for any V E F(TMi), where BV is the tangent part of V and CV is the normal part of ¢V. We define two tensor fields
TM . TM and W : TM
TMl by
ox  On
(1.7)
and respectively wx  4QX,
for any
x E r(TM).
(1.8)
Now, by using (1.1)(1.8) and the equations of Gauss and Weingarten for the immersion of M in N, we obtain the following lemmas (see BejancuPapaghiuc (1]). LEMMA 1.1. Let M be a semiinvariant submanifold of a Sasakian manifold N. Then AWYX  *(VXY) + Bh(X, Y)  n(Y)X, n(VX1Y) 
$Y) + n(AWYX).
(1.9) (1.10)
CHAPTER V
102
h(X,
(1.11)
V1WY  Ch(X, Y) + W(VXY),
for any X, Y E r(TM). LEMMA 1.2. Let M be a semiinvariant submanifold of a Sasakian manifold N. Then h(X, E) = 0,
VX _ 'X,
for any
x F T(D)
V C = 0, Y
for any
Y E
(1.12)
and
h(Y, ) _ 4Y,
r(D ).(1.13)
LEMMA 1.3. Let M be a semiinvariant submanifold of a Sasakian manifold N. Then AgXY  A0YX
(1.14)
and [X, Y] E t(D ® Dl),
for any
x, Y E r(D ).
(1.15)
Remark 1.3. In these lemmas we have made use of notations from §3 of Chapter I with respect to the immersion of M in N. Remark 1.4. From (1.13) it follows that a totally umbilical semiinvariant submanifold of a Sasakian manifold is an invariant submanifold. By using these lemmas we obtain the covariant derivatives of *, W, B and C. PROPOSITION 1.1. Let M be a semiinvariant submanifold of a Sasakian manifold N. Then (VXJ)Y = Bh(X, Y) + AWYX + g(X, Y)E 
(1.16)
 T1(Y)X,
(V)W)Y = Ch(X, Y)  h(X, tjY),
(1.17)
(VXB)V = ACVX  4)(AVX),
(1.18)
(VXC)V = W(AVX)  h(X, BV),
(1.19)
for any X, Y E r(TM) and y E Now we can state
T(TM
THEOREM 1.1 (BejancuPapaghiuc 111). Let M be a semiinvariant submanifold of a Sasakian manifold N. Then the distribution I>L is integrable. Proof. Take X, Y E r(DI) and using (1.3) and the
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103
equations of Gauss and Weingarten we obtain
f(DYX) + h(X, Y) + g(X, Y)E _ _
VYmX.
(1.20)
By changing X and Y in (1.20) and then subtracting the obtained relation from (1.20) we haves
((Y, X]) = AO C  A,XY + VYfX  VX4Y.
(1.21)
By (1.1) and Lemma 1.3, (1.21) becomes (X, Y] = B(VYoX  VX4Y).
(1.22)
Since BV E F(D for any V E r(TM ), the assertion follows from (1.22). Using (1.3) and Lemma 1.2 we obtain g([X, E], Y) = 0,
(1.23)
for any X E r(D1) and Y E r(D). Hence, by Theorem 1.1, we have COROLLARY 1.1. Let M be a semiinvariant submanifold of a Sasakian manifold N. Then the distribution ® t&T is integrable. Now we state LEMMA 1.4. Let M be a semiinvariant submanifold of a Sasakian manifold N. Then
g(h(X, Y), Z) = g(VXZ, 4Y)
(1.24)
and (Y,
] E r(D
{}),
(1.25)
for any X E r(TM), Y E r(D) and z E t(D1). Proof. Using the equations of Gauss and Weingarten we obtain (1.24). Next, by (1.4) and Lemma 1.2 we have g((Y,
], Z)  g(VCY, Z) = g(VCY, Z) _
= g (y, VCZ) ,
(1.26)
for each Y E r(D) and z E r(D'). Now we take x E )'(D) such that Y = 0 and using (1.24) and (1.12) obtain
g(Y, V&Z)  g(4X, V&Z)  g(h(&, X), Z) 0. Thus, by (1.26) and (1.27), we have (1.25).
(1.27)
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THEOREM 1.2 (BejancuPapaghiuc (13). Let M be a semiinvariant submanifold of a Sasakian manifold N. Then is integrable if and only if the distribution D ®
g(h(X, Y)  h(Y, OX), Z) = 0,
(1.28)
for all X, Y E r(D) and Z £ r(D?). Proof. By (1.11) we have
h(X, qY)  Ch(X, Y) + P(VxY),
(1.29)
for all X, Y E r(D). Since h is symmetric, if follows that h(X, OY)  h(Y, OX) = w([X, Y]).
(1.30)
(o) if and only if (1.28) is satisfied. Thus [X, Y] E r(D Therefore, by (1.25) the proof is complete. By a direct computation we obtain
g((X, Y], &) = 2g(Y, X), for any x, Y E I'(D). Hence we have
PROPOSITION 1.2. Let M be a semiinvariant submanifold of a Sasakian manifold N with D x W). Then the distributions D DL are not integrable. and D From these results we conclude that it might be interesting to study the geometry of the leaves of the {l;} and Dl. distributions D First we state
THEOREM 1.3. Let M be a semiinvariant submanifold of a Sasakian manifold N. Then { } is integrable and its The distribution D (i) leaves are totally geodesic in M if and only if
g(h(X, Y), Z)  0,
(1.31)
for any X, Y E r(D O {t}); (ii) The distribution D 49 { } is integrable and its leaves are totally geodesic in N if and only if
h(X, Y)  0, for any x, Y E r(D
(1.32)
{t}),
Proof. The distribution D S {l;} is integrable and its leaves are totally geodesic in M if and only if Thus the V Y E r(D 0 {E}) for any x, Y E r(D x
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EXTENSIONS OF CRSTRUCTURES TO OTHER GEOMETRICAL STRUCTURES
assertion (i) follows by using (1.29) and (1.12). Now suppose D e {C} is integrable and its leaves are totally geodesic in N. Then we have VXY E r(D i {U}) for any X, Y E r(D o {0). By using the equation of Gauss we obtain
g(h(X, Y) , V) = g(IXY, V) = 0, for any v E r(m ), that is, (1.32) is satisfied. Conversely, suppose (1.32) is satisfied. Then by Theorem 1.2, D 0 { } is integrable. Let M* be a leaf of {&} and denote by h* (reap. h') the second fundamental D form of the immersion of M* in N (reap. M). Then by using (1.32) we have h*  h'. On the other hand, by the assertion (i) M* is totally geodesic in M. Hence h*  h' a 0, that is, M* is totally geodesic in N.
THEOREM 1.4 (BejancuPapaghiuc (1]). Let M be a semiinvariant submanifold of a Sasakian manifold N. Then any leaf of Er' is totally geodesic in M if and only if
g(h(X, Y), Z) = 0,
(1.33)
for any Y E r(D) and X, Z E r(D1).
Proof. By using (1.9) we obtain
Y)  g(h(X, Y), +Z)
(1.34)
be a leaf of Dl. Denote by I the LeviCivita connection Let on M and by h the second fundamental form of the immersion of M in N. Hence the Gauss formula is given by VXZ = IXZ + h(X, Z),
(1.35)
for any X, z E r(Ti). By using (1.34) and (1.35) we obtain
g(h (X, Z), Y)  g (h (X, Y), $Z).
(1.36)
On the other hand, by (1.13) we have n(VXZ)  g(VXZ. 0 _ g(Z, V, E)  0.
Thus by (1.35) we obtain n(h(X, Z))  0.
(1.37)
Therefore, the assertion follows from (1.36) and (1.37). Now we say that M is a (D, DL)geodesic (rasp. Dlgeodesic) semiinvariant submanifold if
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CHAPTER V
h(D, D1) _ {0}
(reap. h(D ,
D1)
(1.38)
Then by Theorem 1.4 we obtain COROLLARY 1.2. submanifold of each (i) (ii) each only if M is We denote to D1 in TM'.
1 Let M be a (D, D )geodesic semiinvariant a Sasakian manifold N. Then we have leaf of is totally geodesic in M; leaf of D is totally geodesic in N if and geodesic. by v the complementary subbundle orthogonal Then we state
THEOREM 1.5 (BejancuPapaghiuc (1]). Let M be a semiinvariant submanifold of a Sasakian manifold N. Then any leaf of DL is totally geodesic in M if and only if (i)
E
for all X, Y E r(Dl); and
(ii) h (x, Z) E r (v) , for all x E r w1) and z E r (D 0 Dl). Proof. Adding (1.9) and (1.11) we obtain h(X, Y) = W(A.yX)  (OXWY) = w(AWYX)  C(OXwY),
(1.39)
for any X, Y E N D1). Now, let i be a leaf of Dl. Denote by hh (resp. R) the second fundamental form of the immersion of M in N (resp. M). Then we have h(X, Y) = h(X, Y) + h(X, Y),
(1.40)
for any X, Y E r(TM). Suppose M is totally geodesic in N. Then M is totally geodesic in M and by Theorem 1.4 and (1.40) we obtain (ii). Also, from (1.39) it follows that
C(Vl Y) = 0 which is just M. Conversely, suppose (i) and (ii) are satisfied. Then from (1.39) we obtain h(X, Y) = 0 for all X, Y E r(D1). On the other hand, by (ii) and Theorem 1.4, M is totally geodesic in M. Hence by (1.40) we have h = 0, that is, M is totally geodesic in N.
§2. SemiInvariant Products of Sasakian Manifolds Let M be a semiinvariant submanifold of a Sasakian manifold N. We say that M is a semiinvariant product if the
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EXTENSIONS OFCRSTRUCTURES TO OTHER GEOMETPJCAL STRUCTURES
{&) is integrable and locally M is a distribution D Riemannian product M1 X M2, where M1 (resp. M2) is a leaf of D 0 {&} (resp. D1). If we have pq x 0 we say that M is a proper semiinvariant product. As is well known, Riemannian decomposable spaces are characterized by linear connections (see §4 of Chapter I and Yano [2] p. 219). We give first such a characterization for semiinvariant products.
THEOREM 2.1. Let M be a semiinvariant submanifold of a Sasakian manifold N. Then M is a semiinvariant product if and only if
vYx E r(D
for any
Y E r(TM)
and x E r(D). (2.1)
Proof. Suppose M is a semiinvariant product locally represented by M1 x M2. Then M1 and M2 are totally geodesic in M and the Gauss formula implies (1)
V YX E r(D
V X = Y
and
{E})
for any
{F}) (2.2)
(2)
V U =
V
(1)
X, Y E r(D
1 V VU E r(D ),
for any
1 u, v E r(D ),
(2.3)
(2)
where V amd V are the Riemannian connections on M1 and M2 respectively. On the other hand, by using (2.3) and (1.12) we obtain g(VVX, U)  g(X, VVU)  0, (2.4)
g(VEX, U)  g(4X + [E, x], U)  0 for any X E r(D W) and U, v E r(D1). Thus from (2.2) and (2.4) it follows that (2.1) holds. Conversely, suppose (2.1) is satisfied. Then the distribution D i { } is integrable since we have [y, x]  v x  vxY E r(D 1 (F)), Ex, E]  VxE  VEX  fix  V x E r(D E
w),
for any X, Y E r(D). Moreover, if M1 is a leaf of D e {} then from (2.L) and the Gauss formula for the immersion of M in M it follows that M1 is totally geodesic in M. Finally, from (2.1) it follows that VYU E r(D') for any Y E r(TM) and
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CHAPTER V
U E r(D1). By using again the equation of Gauss for a leaf M2 of Dl we obtain that M2 is totally geodesic in M. The proof is complete.
THEOREM 2.2 (Bejancu (13]). Let M be a semiinvariant submanifold of a Sasakian manifold N. Then M is a semiinvariant product if and only if its second fundamental form satisfies Bh(X, Y) = 0
(2.5)
h (X, ft) = Ch (X, Y) ,
(2.6)
or
for any X E r(TM) and Y E r(D).
Proof. From (1.9) and (1.11) it follows that 1P(VxY) + Bh(X, Y)
(2.7)
and
h(X, Y) = Ch(X, Y) + w(VxY),
(2.8)
for any x E r(TM) and Y E r(D). Thus our assertion follows from (2.7) and (2.8) by means of Theorem 2.1. Now, using the formulas of Gauss and Weingarten, we obtain g(A0zX, Y) = g(Bh(X, Y), Z),
(2.9)
for any x E r(D), Y E r(TM) and z E r(D ). Then, by using (2.9) and Theorem 2.2, we get COROLLARY 2.1. Let M be a semiinvariant submanifold of a Sasakian manifold N. Then the following assertions are equivalent to each other: M is a semiinvariant product, (i) (ii) the fundamental tensors of Weingarten satisfy A,zX = 0,
(2.10)
for any z E r(D') and x E r(D), (iii) the second fundamental form of M satisfies
h(Y, X)  mh(Y, X),
(2.11)
for anyx E r(D) andY E r(TM). Let N(c) be a Sasakian space form of constant sectional curvature c. Then the curvature tensor R of N(c) is given by (see §6 of Chapter 1)
EXTENSIONSOFCR TRUCTURESTOOTHERGEOMETRICALS7RUCfURES
109
R(X, Y)Z = 4 (c + 3){g(Y, Z)X  g(X, Z)Y} +
+ 4 (c  1){n{X) fl(Z)Y  n(Y) n(Z)X + + g(X,
(Y)
 g(Y,
g(Z, Y)4X 
 g(Z, $X)4Y + 2g(X, $Y)#},
(2.12)
for any X, Y, Z tangent to N(c). Suppose M is a semiinvariant submanifold of N(c). Then, by using (2.10) and (2.12), the equations of Codazzi and Ricci respectively become
(VXh) (Y, Z)  (VYh) (X, Z) = 4 (c  1) x x {g(Z, *Y)WX  g(Z, >IPX)wY + 2g(X, 4Y)WZ}
(2.13)
and
4 (c  1){g(U, Y)g(V, X)  g(U, $X)g(V, Y) +
+ 2g(X, Y)g(V, U)}  g(R1(X, Y)U, V) + + g([AV, AU]X, Y),
for any X, Y, Z E r(TM) curvature of the normal Now we shall prove semiinvariant products
(2.14)
and U, v E r(TM1'), where Rl is the connection on M. the nonexistence of proper in some Sasakian space forms.
THEOREM 2.3 (Bejancu [13]). There exist no proper semiinvariant products in Sasakian space forms N(c) with c < 3. Proof. Suppose M is a semiinvariant product in N(c). Then by (2.11) and (2.13) we obtain
2 (1  c)g(X, X)g(Z, Z) = g((VXh)(+X, Z), Z) 
 g((VoXh) (X, Z) , 4Z)
(2.15)
for any x E r(n)'and z E r(D1). On the other hand, by using (2.1) we have g((OXh)(OX, Z)  (VoXh) (X, Z), +Z) 
= g(VX (OX, Z)  Vx (X, Z) , Z) + + g(Bh(VXox  VoxX, Z), Z).
(2.16)
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Next, by using (2.11), (1.5), and (1.12) we get g(VngXh(4)X,
Z), 4)Z) _ g(h(X, Z), h(X, Z))
(2.17)
Replacing X by $X in (2.17) and taking account of (2.11) we have
= g(h(X, Z), h(X, Z)). Then by using (1.2), (1.5) and (1.12) we obtain Z), 4)Z)
g(Bh(VXox  VoxX, Z), Z) = 2g(X, X)g(Z, Z).
(2.18) (2.19)
Thus, from (2.16)(2.19), it follows that
g ((VXh) (OX, Z)  (V4Xh) (X, Z), 4)Z) _ = 2{g(X, X)g(Z, Z)  g(h(X, Z), h(X, Z))}. (2.20) Finally, (2.15) and (2.20) imply
4 (c + 3)g(X, X)g(Z, Z) = g(h(X, Z), h(X, Z)).(2.21) Since M is a proper semiinvariant product, from (2.21) it follows that c ) 3. The proof is complete. Let M be a semiinvariant product in N(c). If q = dim TXM± we say that M is a generic semiinvariant product. Then by using Theorem 2.3 we obtain
COROLLARY 2.2. There exist no proper generic semiinvariant products in a Sasakian space form N(c) with c x 3. Proof. Suppose M is a proper generic semiinvariant product in N(c). Since we have B  4), (2.5) implies that h(X, Z) = 0 for any X E r(D) and z e r(r)j.). Thus from (2.21) we have c = 3. Let M be a semiinvariant submanifold of a Sasakian manifold N. Then we say that M is totally contactumbilical if there exists a normal vector field H such that the second fundamental form of M is given by h(X, Y) = g(4)X, 4)Y) H + n(X)h(Y, &) + n(Y)h(X, (2.22)
for any vector fields X, Y tangent to M. If we have H = 0 in (2.22), that is the second fundamental form of M is given by
h(X, Y)  n(X)h(Y, E) + fl(Y)h(X, ), then we say that M is totally contactgeodesic.
(2.23)
EXTENSIONS OF CRSTRUCTURES TO OTHER GEOMETRICAL STRUCTURES
III
THEOREM 2.4. (Bejancu [131). Any proper totally contactumbilical semiinvariant submanifold of a Sasakian manifold 1 is a totally contactgeodesic submanifold. with q Proof. From (1.14) it follows that
A,X(BH)  AOH(X),
(2.24)
for any x E r(D1). Since M is totally contactumbilical, from (2.24) we obtain g(X, X)g(BH, BH) = g(BH, X)2. From (2.25) we get BH Next, by using BH Weingarten we obtain
(2.25)
0 since we supposed q > 1. 0 and the formulas of Gauss and
PA,HY = tj(ANY),
(2.26)
for any Y E r(TM). We have further
Z)  g(Y, Z)g(H, H) = 0
Z) =
(2.27)
and
gMAHY), Z) _ g(AHY, Z) _ g(Y, OZ)g(H, H), (2.28)
for any Y E r(TM) and z E r(D). Since M is a proper semiinvariant submanifold, (2.26)(2.28) imply that H  0. Hence M is totally contactgeodesic. The proof is complete. THEOREM 2.5 (Bejancu [131). Any totally contactgeodesic semiinvariant submanifold M of a Sasakian manifold N is locally a Riemannian product M1 x M2 where M1 is a totally geodesic invariant submanifold of N and M2 is a totally geodesic antiinvariant submanifold of N and
is normal to
M2.
Proof. First, from (2.23) we obtain h(X, Y) = 0,
for any
x e r(D)
and Y E r(TM).
Thus, by Theorem 2.2, M is a semiinvariant product locally represented by M1 x M2. On the other hand, by direct computation, using (2.23) and Corollary 1.1 we obtain that M1 and M2 are both totally geodesic in N. From Theorems 2.4 and 2.5 the following holds
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COROLLARY 2.3. Any proper totally contactumbilical semiinvariant submanifold M of the Sasakian manifold N with q 1 is locally the Riemannian product M1 X M, where M and M have the properties of Theorem 2.5.
§3. SemiInvariant Submanifolds with Flat Normal Connection S2n+1
be a real (2n+1)dimensional sphere endowed with Let the standard Sasakian structure (see §6 of Chapter I). The main purpose of this paragraph is to give some results on the geometry of semiinvariant submanifolds with flat normal connection in S2n+1. First we have S2n+1
LEMMA 3.1. Let M be a semiinvariant submanifold of with flat normal connection. Then ACV  0 for any vector field V normal to M.
S2n+1
LEMMA 3.2. Let M be a semiinvariant submanifold of with flat normal connection. If the mean curvature vector is parallel and if OAV =AV'4J for any vector field V normal to M, then the second fundamental form of M is parallel. Now we can state THEOREM 3.1 (YanoKon [71). Let M be an mdimensional with flat normal complete semiinvariant submanifold of S n connection. If the mean curvature vector of M is parallel and if kAV _AVo for any vector field normal to M, then M is
an S}° or Mis ink
Sm1(r1)X...XS k
k (rk),
in =
F m
2 < k < in,
i=1
F (r i)2 = 1,
i=1
in some S m+q , where m1,...,mk are odd numbers. Proof. First we assume w  0, that is, M is an invariant
Then the second fundamental form of M submanifold of satisfies **AV + AVoo = 0 (see Kon [2]). Thus by assumption S2n+1.
we have *'AV  0 which implies AVX = 0 for each x E r(D) since D is invariant by AV and iP is an automorphism of D. On
EXTENSIONS OFCRSTRUCTURES TO OTHER GEOMETRICAL STRUCTURES
113
the other hand, by using the Gauss formula, we obtain g(AVE, X) = g(h(X, c), V) = g(4X, V) = 0, for any X E r(D ® {E}). Hence AVM = 0. Therefore M is a totally geodesic submanifold of Stn+1 that is, M is an S=° and m is odd. Now we assume w x 0. Then it follows that A x 0 for each Y E F(TM). Hence by Lemma 3.1 the first normal space is of dimension q. It is easy to check that the product of spheres from the assertion is a semiinvariant submanifold with flat normal connection and parallel mean curvature vector. Then the assertion follows by using Theorem 2.2 of Chapter IV and Lemma 3.2. From this theorem we have
COROLLARY 3.1. Let M be a (2p+q+1)dimensional complete S2n+1
submanifold of with flat normal connection and q + p = n. If the mean curvature vector of M is parallel and if oAV = A4 for any vector field V normal to M, then M is S m1 (r1)x...xS
Mk
k 2p + q + 1 = E mil i=1 k
(rk),
E (ri) 2= 1,
2 6 k s 2p + q + 1,
i=1
where m1,...,mk are odd numbers.
Semiinvariant submanifolds with flat normal connection have also been studied in other Sasakian space forms. We note such a result here. THEOREM 3.2 (M. Kobayashi [2]). Let M be a (2p+q+1)dimensional semiinvariant submanifold of a (2n+1)dimensional Sasakian space form N(c) with c > 1. Suppose M has flat normal connection. Then we have: (i) if 2p + q < n then M is an antiinvariant submanifold;
(ii) if 2p + q > n then we have n=p+q.
More results on the geometry of semiinvariant submanifolds (contact CRsubmanifolds) of Sasakian manifolds can be found in YanoKon [7], Kobayashi [1][4], Hsu [1], Roqca [1], Papaghiuc [2], BejancuPapaghiuc [1], [2] and sejancu [12], [13]. Taking into account the large variety of problems in the geometry of submanifolds, some other results
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CHAPTER V
are expected to be obtained.
§4. Generic submanifolds of Kaehlerian manifolds
The purpose of this section is to show that CRsubmanifolds have an interesting generalization even in the case of almost Hermitian manifolds. Let (N, J, g) be an almost Hermitian manifold and let M be a real submanifold of N. Then we say that M is a generic submanifold of N if the maximal complex subspaces Dx = TXM n J(TxM) determine on M a distribution D : x + Dx c TXM. It is easy to prove that D is a
differentiable distribution. The concept of generic submanifolds was first introduced by Chen in [9] for submanifolds in almost complex manifolds. we denote by DL the orthogonal complementary distribution to D in TM and note that JDl fl Dl = {0}. If in particular, JDL C TML we have the concept of a CRsubmanifold. When D = {0} we say that M is a purely real submanifold of N. We call D and D1 the holomorphic and purely real distributions on M respectively. With respect to the geometry of generic submanifolds interesting results were obtained by Chen in (9] and ChenLuddenMontiel in [i]. On the other hand, the concept has also been considered for submanifolds in Sasakian manifolds by Verheyen [1] and BejancuPapaghiuc (3). We shall sketch here some problems on the integrability of the distributions on a generic submanifold. First, for each vector field X tangent to M we put ix  4X + WX,
where $X and WX are the tangential and normal components of JX, respectively. As in the case of CRsubmanifolds, W is a normalbundlevalued 1form of TM and is an endomorphism of TM.
Now we state THEOREM 4.1 (Chen [9]). Let M be a generic submanifold of a Kaehler manifold N. Then we haves the holomorphic distribution D is integrable if (i) and only if the second fundamental form of M satisfies
g (h (X, JY) , JZ)  g (h (JX, Y) , JZ) ,
EXTENSIONS OF CRSTRUCTURES TO OTHER GEOMETRICAL STRUCTURES
115
for any X, Y E r(D) and Z E I'(DL);
(ii) the purely real distribution Dl is integrable if and only if 17
Z
(w  VU
AWZU  AWUZ E I' (D
for all Z, U E I'(D1). Proof. The proof of (i) is analogous with that of Theorem 1.1 of Chapter III. For the second assertion, by using the equations of Gauss and Weingarten we get J(VZU) + Jh(Z, U) = Vz(4U) + h(Z, 4>U) 
 AWUZ + VZWU.
Thus we obtain [Z, U] = P(VU(0Z)  VZ($U) + AWUZ  AWZU) + + B(VUWZ  VZWU)
+ h(U, Z)  h(Z, 4U)
where BV is the tangent part of JV for a certain V E r(TMI). Thus the assertion (ii) follows since in fact BV E r(D1) for each V E T(TM1). Next, we say that a real sbumanifold M of a Kaehler manifold N is a generic product if it is locally the Riemannian product of a complex submanifold MT and a purely real submanifold M of N. of course, if we take a complex submanifold M of Cp and a purely real submanifold MT of Cq, then MT x Mi is a generic product in CP+q. Chen obtained in [9] examples of generic products in CPn. On the other hand, there exist no proper generic products in N(c) with c < 0, that is, in a such space a generic product is either a complex manifold or a purely real submanifold. Finally, we note that Theorems 4.1 and 4.2 of Chapter IV and Theorem 5.2 of Chapter III were generalized by ChenLuddenMontiel [1] to the case of generic products.
55. QRSubmanifolds of Quaternion Kaehlerian Manifolds
Let N be a 4ndimensional manifold and g be a Riemannian metric on N. Then N is said to be a quaternion Kaehlerian manifold (see §7 of Chapter I), if there exists a 3dimensional vector bundle V of tensors of type (1,1) with
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CHAPTER V
local basis of almost Hermitian structures J1, J2, J3 satisfying
J1 u J2  J2 c J1  J3 and
(5.1)
3
E Q_(X)J, (a = 1, 2, 3), (5.2) b=1 for all vector fields X tangent to N, where V is the LeviCivita connection determined by q on N and Qab are certain 0 J = X a
1forms locally defined on N such that Qab + Qba = 0. suppose {J1, J2, J3} is a local basis for the vector bundle V in a coordinate neighbourhood U on N. We take another coordinate neighbourhood 0 with the corresponding local basis {J1, J2, J3} and u fl U
0. Then we have
3
E SabJb, (a = 1, 2, 3), (5.3) b=1 where [sab] is an element of the special orthogonal group Ja =
SO(3). Now, let M be an mdimensional Riemannian manifold isometrically immersed in N. The geometry of M depends on the behaviour of the tangent space to M under the action of the local basis {J1, J2, J3}. Then we say that M is a quaternionreal submanifold (QRsubmanifold) if there exists a vector subbundle v of the normal bundle TMl such that we have Ja(vx)  vx
(5.4)
Ja(Vi) c TxM,
(5.5)
and
for each x E M and a orthogonal subbundle Of course, each is a QRsubmanifold have
= 1, 2, 3 where V1 is the complementary to V in TM1. quaternion submanifold (see Chen [2]) with V = TML and vl = M. Moreover we
PROPOSITION 5.1. Let M be a real hypersurface of a quaternion Kaehlerian manifold N. Then M is a QRsubmanifold. Proof. Since T M1 is 1dimensional and J are Hermitian a structures we obtain Ja(TxMl) c TxM,
for each x E M.
EXTENSIONS OFCRSTRUCTURES TO OTHER OEOMETRICALSTRUCRIRES
117
Hence M is a QRsubmanifold with V = {0) and V1 = TM1. Also, each antiquaternion submanifold of N (see Pak [2] and Bejancu [221) is an example of a QRsubmanifold. Let M be a QRsubmanifold of N. Then we denote Dax = Ja(VX) and remark that D1X, D2x' D3x are mutually orthogonal vector subspaces of TX M. We consider J
D
X =
D
lx
D
® D
2x
3x
1 1 : x + Dx and by (5.3) obtain a 3sdimensional distribution D globally defined on M, where s = dim \X. Also we have
Ja(D
)
Ja(Dbx) = Dcx,
= VX;
(5.6)
for each x E M, where (a, b, c) is a cyclic permutation of (1, 2, 3). Next, we denote by D the complementary orthogonal distribution to D1 in TM and obtain that D is an invariant distribution on M, i.e., we have Ja(Dx) = Dx,
for any
x E M,
a = 1, 2, 3.
(5.7)
We call D the quaternion distribution on M. Now denote by P the projection morphism of TM to the quaternion distribution D and choose a local field of orthonormal frames {V1,...,Vs} on the vector subbundle V1 in
TM . Then on the distribution D we have the field of orthonormal frames {E11,...,E1s, E21,...,E2s, E31,...,E3s},
(5.8)
where Eai = JaVi, a = 1, 2, 3 and i = 1,...,s. Thus any vector field Y tangent to M can be written locally as follows y = PY +
3
s
E
E wbi (Y)E bi
,
(5.9)
b=l i=1
where wbi are 1forms locally defined on M by wbi(Y) = g(Y. Ebi).
Applying Ja to (5.9) and taking account of (5.1) we obtain s
Jay  JaPY +
iE1{wbi(Y)Eci  wci(Y)Ebi 
 wai(Y)Vi} On the other hand, for any normal section V we put
(5.10)
its
CHAPTER V
J V = B V + Ca V, a a
a = 1, 2, 3,
(5.11)
where BaV and CaV are respectively the tangent part and the normal part of JaV. By using (5.10), (5.11) and equations of Gauss and Weingarten in (5.2) and then taking the normal parts we obtain h(X, JaPY)  Cah(X, Y)
+
s i1: 1{wbi(Y)h(X, Eci)

 wci(Y)h(X, Ebi)  Wai(Y)OXVi  X(wai(Y))Vi + + wai(VXY)Vi + Qab(X)Wbi(Y)Vi +
+ Qac(X)wci(Y)Vi} =
0,
(5.12)
for any X, Y tangent to M, with usual notations for geometrical objects on a submanifold (see §3 of Chapter V. Next, from (5.12) we obtain h(X, JaY)  Cah(X, Y) +
E {wai(VXY)V1}
0, (5.13)
i=1
for any X, Y E P(D). We say that M is a Dgeodesic QRsubmanifold if we have
h(X, Y) = 0, for any x, Y E r(D).
(5.14)
Then we can state
THEOREM 5.1. Let M be a QRsubmanifold of the quaternion Kaehlerian manifold N. Then the following assertions are equivalentthe second fundamental form of M satisfies U) h(X, Jay) = h(Y, JaX),
for any
X, Y E T(D); (5.15)
M is Dgeodesic; (iii) the distribution D is integrable. (ii)
Proof. (i) s+ (ii). By (5.15) and (5.1) we obtain
h (J3X, Y) = h (X, J3Y) = h (X, (JIDJ2) Y) = = h(J1X, J2Y) = h((J2eJ1)X, Y) = h(J3X, Y). (5.16)
Thus, from (5.16) it follows that (5.14) holds since J3 is
EXTENSIONS OFCRSTRUCTURES TO OTHER GEOMETRICAL STRUCIURES
119
an automorphism of D. (ii) * (iii). Taking account of (5.14) in (5.13) we obtain
a = 1, 2, 3
x, Y E r(D),
for any
g(VXY, Eai) = 0
and
i  1,...,s.
Thus D is integrable and each leaf of D is totally geodesic in M.
(iii)  (i). Suppose D is integrable. Then we have Wai((X, Y]) = 0, a = 1, 2, 3;
for all
x, Y E T(D),
i = 1,...,s.
Thus from (5.13) taking into account that V is a torsionfree connection we obtain (5.15). The proof is complete. We denote by Ai the fundamental tensor of Weingarten with respect to Vi. Then we.have LEMMA 5.1. Let M be a QRsubmanifold of a quaternion Kaehlerian manifold N. Then AiEaj = AjEai,
a = 1, 2, 3;
i, j = 1,...,s.(5.17)
Proof. First, from (5.2) we obtain OXJaZ = Ja(VXZ) + Qab(X)JbZ + Qac(X)JcZ,
(5.18)
for any X, Z tangent to N, where (a, b, c) is a cyclic permutation of (1, 2, 3). Next, we take X tangent to M and replace Z by Eai. By using the formulas of Gauss and Weingarten, (5.18) becomes h(X, Eai)
Ja(VXVi)  Ja(AiX)  VXEai +
+ Qab(X)Ebi
+ Qab(X)Ebi.
(5.19)
Finally, by (5.19) and (3.3) of Chapter I we obtain g(AjEai, X) = g(h(X, Eai). Vj)  g(J3(VXVi)  Ja(AiX), Vj)
g( XVi, Eai) +
+ g(AiX, Eaj)  g(AiEaj, X), for any X tangent to M. Thus we have the assertion of the lemma.
120
CHAPTER V
COROLLARY 5.1. Let M be a QRsubmanifold of a quaternion Kaehlerian manifold N. Then g(V E
ai
Eaj, X)
g(VE
Eai, X),
(5.20)
aj
for any x E r(D) and a = 1, 2, 3. Proof. By using (5.2) and the equation of Gauss we obtain g(V E Eaj , X) = g(VE Ja'1j' X) = g(Ja(VE Vi) + ai ai ai + Qab(X)Ebj + Qac(X)Ecj, X) = g(VE
Vj, JaX)
ai
= g(AjEai, JaX).
Thus by Lemma 5.1 we obtain the assertion.
Now we define differential 1forms Baij by Baij (X) = g(VE i,
Eaj , X),
a = 1, 2, 3;
ai j = 1,...,s,
(5.21)
for any X E r(TM) and state
THEOREM 5.2. Let M be a QRsubmanifold of a quaternion Kaehlerian manifold N. Then the following assertions are equivalent: I the distribution D is integrable; (i) Baij (X) = 0 for all a = 1, 2, 3; i, j = 1,...,s; (ii)
x F r(D); (iii) h(D, Dl) c V.
Proof. First, by the definition of a QRsubmanifold we
have Eai
Jb$ci
JCEbi'
(5.22)
for i  1,...,s and (a, b, c) a cyclic permutation of (1, 2, 3). By using (5.2), (5.20) and (5.22) we obtain g((Eai, Ebj], JcX) = Baij(X) + Bbij(X),
(5.23)
forany x F, I'(D). On the other hand, by using (5.17), (5.2) and the Weingarten equation we infer
g((Eai' Eaj1, X) = 0.
(5.24)
EXTENSIONS OFCRSTRUCTURES TOOTHERGEOME71UCALSTRUCTUPES
121
Thus by (5.23) and (5.24) we get the equivalence of (i) and (ii). Next, by using (5.2) we get
Bai j (X) = g(h(JaX, Eai) . Vi) .
(5.25)
Hence the equivalence of (ii) and (iii) follows from (5.25). The proof is complete.
From this theorem we obtain COROLLARY 5.2. Let M be a real hypersurface of a quaternion Kaehlerian manifold N. Then the distribution D is integrable 0 . = if and only if we have h(D, )
1 Now, we say that a QRsubmanifold of N is D geodesic if we have h(D1, D1) _ (0). Then with respect to the geometry of the leaves of D and D1 we have
THEOREM 5.3. Let M be a QRsubmanifold of a quaternion Kaehlerian manifold N. Then we have the following assertions: if the quaternion distribution D is integrable, (i) then each leaf of D is totally geodesic in N; is integrable then we have: (ii) if the distribution is totally geodesic in M; (a) each leaf of (b) a leaf of D is totally geodesic in N if and only if M is geodesic. Proof. Suppose D is integrable. Then since D is a quaternion distribution, each leaf of D is a quaternion submanifold of N. Thus the assertion (i) follows from Lemma 4 in Chen [2]. Now, we assume D1 be integrable. Then by using (3.2) and Theorem 5.2 we obtain g(VXEai' Z) = g(VXEai' Z) = g(Ja(VXVi). Z) g(AiX, JaZ) = g(h(X, JaZ), Vi) = 0, a  1, 2, 3;
i 
for any X E F(D1) and Z E T(D). Thus we have q(VXY, Z)  0,
for any
x, Y E r(D1
and z E T(D),
(5.26)
that is, each leaf of D' is totally geodesic in M. Next, from (5.26) it follows that
122
g(IXY, Z) = 0,
for any
X, Y E r(Di)
and Z E r(D).
On the other hand, we have 9(OXY, V) = g(h(X, Y), V), for any X, Y E r(DI) and v E r(TM'). Let M* be a leaf of Di and h* the second fundamental form of the immersion of M* in N. Then, from (5.27) and (5.28) we have
g (h* (X, Y), Z) = 0
(5.29)
g (h* (X, Y), V) = g (h (X, Y), V).
(5.30)
and Thus the assertion (ii) b follows from (5.29) and (5.30).
By means of Theorems 5.15.3 we obtain THEOREM 5.4. Let M be a QRsubmanifold of a quaternion Kaehlerian manifold N. Then M is locally a Riemannian product
M X M , where M
M
and are leaves of D and Dl respectively, if and only if the second fundamental form of M satisfies h(D, D) = {0}
(5.31)
h(D, Dl) c v.
(5.32)
and From this theorem we infer COROLLARY 5.3. Let M be a real hypersurface of a quaternion Kaehlerian manifold N. Then M is locally a Riemannian product
M
MT X Ml, where MT and are leaves of D and D respectively, if and only if the second fundamental form of M satisfies h(D, TM) = 10 .
§6. Totally Umbilical and Totally Geodesic QRSubmanifolds of a Quaternion Kaehlerian Manifold. Let M be a totally umbilical QRsubmanifold of a quaternion Kaehlerian manifold, that is, we have h(X, Y) = g(X, Y)H,
(6.1)
for any X, Y tangent to M, where H is the mean curvature vector defined by B  (1/m) Trace (h) (see 93 of Chapter I).
GEOMETRICALSTRUC URES
EXTENSIONS
123
THEOREM 6.1. Let M be a totally umbilical QRsubmanifold of a quaternion Kaehlerian manifold N. Then we have the following assertions: the distribution Dl is involutive; (i) (ii) the distribution D is involutive if and only if M is totally geodesic. Proof. These follow by using (6.1) and Theorems 5.1 and 5.2.
From Theorem 5.4 we have a complete characterization of totally geodesic QRsubmanifolds. THEOREM 6.2. Each totally geodesic QRsubmanifold of a quaternion Kaehlerian manifold is locally a Riemannian product M T x Ml where M T and Ml are leaves of D and Dl
respectively. Now we state
THEOREM 6.3. Let M be a totally umbilical QRsubmanifold of 1 a quaternion Kaehlerian manifold N. If we have dim yx then M is totally geodesic. The proof is similar to that of Theorem 2.1 from Chapter III so we omit it here. The remaining part of this section is devoted to the study of the existence of totally umbilical QRsubmanifolds in curved quaternion Kaehlerian manifolds and of curved QRsubmanifolds. First we state LEMMA 6.1. Let M be a totally umbilical QRsubmanifold of a quaternion manifold N. Then g(R(X, Eai)X, Eai) + g(R(X, Eai)JaX, Vi) = 0, (6.2)
for any x c r(D), a = 1, 2, 3 and i = 1,...,s, where R is the curvature tensor of N. Proof. It suffices to prove (6.2) for a = 1. By using (5.2) and (5.22) we have
124
CHAPTER V ti N
J1X, Vi) _  g(VXVE
g(QXVE li
fi
 g(VXVE
J1 X, J1E1i) =
X, Vi) + Q12(E1i)g(VXX, E li
21
)
+
X, E 21) + Q13(X)g(VEli X, E31) + Q12(X)g(VEli
+ Q13(Eli)g(VXX, E3i),
for any x E T(D).
(6.3)
By Theorem 6.1 the distribution Dl is involutive and thus by Theorem 5.3 any leaf of D1 is totally geodesic in M. Thus we have g(X, VE E 14) = 0 g(VE X' E21) 1i
1i
and
X, E
g(VE li
31
)
E
g(X, VE li
31
)
= 0.
On the other hand, taking account of (5.2) and (6.1) we obtain g(VXX, E 21) = g(X, VXE21) _  g(X, J2(VXVi)) = =  g(J2X, AIX)   g(h(X, J2X), Vi) = 0
and
ti
g(VXX, E 31) = 0.
Hence (6.3) becomes g(VXVE
ii
J1X, Vi) + g(VXVE
X, Eli) = 0.
(6.4)
li
In a similar way it follows that g(VE 11
and g (V
VXJiX' Vi) + g(VE
VXX' Eli) = 0
(6.5)
li
IX, E1i]JIX' Vi) + g(VN, E111X, E11)  0. (6.6)
Thus the assertion of the lemma follows from (6.4)(6.6) by using (1.4) of Chapter I. Now we say that M is a proper QRsubmanifold if D x {0} and V1 * {0}. Then we have THEOREM 6.4. There exist no proper totally umbilical QRsubmanifolds in positively or negatively curved
EXTENSIONS OF
125
TO OTHER GEOMETRICAL STRUCTURES
quaternion Kaehlerian manifolds. Proof. Suppose M is a proper totally umbilical QRsubmanifold of N with KN x 0. Taking account of (6.1) and of the Codazzi equation we obtain g(R(X, Y)Z, V) = g(Y, Z)g(VXH, V) 
(6.7)
 g (X, Z) g (VYH, V) ,
for any X, Y, Z tangent to M and V normal to M. Now we take X E r(D), Z = J1X, Y = E1i and V = V i in (6.7) and obtain g(R(X, E1i)J1X, Vi) = 0.
(6.8)
By using (6.2) and (6.8) we obtain KN(X AE li) = 0 which is a contradiction. The proof is complete.
PROPOSITION 6.1. Let M be a totally umbilical QRsubmanifold of a quaternion Kaehlerian manifold N. Then KM(X A Eai) = 1H12,
a = 1, 2, 31
i = 1,...,s,(6.9)
for any unit vector X from the quaternion distribution D. Proof. By the structure equation of Gauss (see (3.6) of Chapter I) we have
KM(X AY) = KN(X AY) +
11112,
(6.10)
for any orthonormal vectors X and Y tangent to M. Then we put X E r(D), Y = Eai in (6.10) and taking into account that
KN (X AE ai) = 0 obtain (6.9). From Proposition 6.1 we have COROLLARY 6.1. There exist no proper totally umbilical negatively curved QRsubmanifolds in quaternion Kaehlerian manifolds. COROLLARY 6.2. Any proper totally geodesic QRsubmanifold of constant sectional curvature is flat. Now, suppose M is an antiquaternion manifold, i.e., M is a QRsubmanifold with v = W. Then we say that M is a proper antiquaternion product if both distributions D and are integrable, D x 0 and M is locally a Riemannian product
126
CHAPTER V
MT x M1 where MT is a leaf of D and ML is a leaf of D1.
THEOREM 6.5. Let M be a proper antiquaternion product of a quaternion space form N(c). Then c = 0. Proof. By using Theorem 5.4 we obtain h(D, TM) = {0}. Hence we get (VXh)(Y, Z) = VXh(Y, Z)  h(VXY, Z) 
h(Y, VXZ) = 0 for any X, Y E V(TM) and Z E r(D) since the quaternion distribution D is parallel with respect to the LeviCivita connection on M. Thus the Codazzi equation for the immersion of M in N(c) becomes [R(X, Y)Z]1 = 0.
(6.11)
Taking account of the special form of the curvature tensor field of N(c) (see §7 of Chapter I) and of (6.11) we get 3
4
E {g(JaY, Z)JaX  g(JaX, Z)JaY) = 0.
(6.12)
a=1
Now we choose Y E r(D) and X F_ r(D1). Then from (6.12) we have either c = 0 or 3
F {g(JaY, Z)JaX} = 0.
(6.13)
a=1
We show that (6.13) leads to a contradiction. Since J1X, 12X and J3X are mutually orthogonal, from (6.13) we get g(JaY, Z) = 0, for a = 1, 2, 3. Finally, we take Z = J1Y and obtain D = {0} which is a contradiction.
COROLLARY 6.3. There exist no antiquaternion products in a quaternion space form N(c) with c x 0.
M. Barros, B.Y. Chen and F. Urban [1] have introduced the notion of quaternion CRsubmanifold of a quaternion Kaehlerian manifold as a natural generalization of the concept of CRsubmanifold of a Kaehlerian manifold. More precisely, we say that M is a quaternion CRsubmanifold of N if M is endowed with two differentiable orthogonal distributions D and DL such that
EXTENSIONS OFCRSTRUCTURES TO OTHER GEOMETRICAL STRUCTURES
TM = D
Dli Ja(Dx) = Dx
and
127
Ja(DX) c TxMl,
for each x E M and a = 1, 2, 3. If Dl = (0) (resp. D = {0}) a quaternion CRsubmanifold becomes a quaternion submanifold (resp. totally real submanifold). On the other hand, the concept of generic submanifold has been also considered in the quaternion case by Martinez and Santos [1].
Remark 6.1. Between the two classes of QRsubmanifolds and quaternion CRsubmanifolds there exists no inclusion relation because a real hypersurface is a QRsubmanifold and it is not a quaternion CRsubmanifold and a totally real submanifold is a quaternion CRsubmanifold and it is not a QRsubmanifold. On the other hand, quaternion submanifolds lie in the intersection of the above classes. We conclude this chapter by saying that the concept of CRsubmanifold has been also considered in: (i) locally product spaces (see Matsumoto [1] and Bejancu [211), (ii) paraSasakian manifolds (see IanusMihai [1]), (iii) locally conformal Kaehler manifolds (see Matsumoto [3]), (iv) Kenmotsu manifolds (see Papaghiuc [1]), (v) fstructures with complemented frames (see Mihai [1] and Ornea [1]), (vi) coKaehler manifolds (see BejancuSmaranda [1]). Also we note that Tashiro and Kim [1] starting from the structures induced on a submanifold of a Kaehlerian manifold introduced and studied metric compound structures on Riemannian manifolds.
Chapter VI
CRSTRUCTURES AND PSEUDOCONFORMAL MAPPINGS
§1. CRManifolds and fStructures with Complemented Frames
CRmanifolds were intensively studied from the analytic point of view (see Wells [1], [21). As is well known, complex manifolds and normal almost contact manifolds (see Blair [3], p. 62) are examples of CRmanifolds. Nontrivial CRmanifolds appeared as boundaries of domains in complex spaces, which in fact are real hypersurfaces (i.e., particular CRsubmanifolds). On the other hand, in 1963, Yano introduced in [1] the concept of manifold endowed with an fstructure which is a generalization of both a complex manifold and an almost contact manifold. Important results on the geometry of fstructures with complemented frames have been obtained by GoldbergYano [1], [2), Blair [1], [2], BlairLuddenYano
W. It is the purpose of this section to show the relationship between CRmanifolds and fstructures with complemented frames. Then we shall apply this relationship to the study of submanifolds of complex manifolds and pseudoconformal mappings. Let M be a (2n + s)dimensional real differentiable manifold and D be a differentiable distribution on M of real dimension 2n. Suppose D is endowed with a morphism J : D  D of vector bundles satisfying j2 = I, where I is the identity morphism on D. Then we say that M is endowed with an almost complex distribution (D, J). If the following conditions are fulfilled: [ix, Jy]  [x, Y] E F(D),
(1.1)
[J, J](X, Y) _ fix, JY]  [X, Y)  J([X, JY] + + [ix, Y))  0,
(1.2)
for all X, Y E F(D), then we say that (D, J) defines a real CRstructure on M.
CRSTRUCTURES AND PSEUDOCONFORMAL MAPPINGS
129
Next, we denote by T M the complexified tangent bundle to M. A complex CRstructure on M is a complex subbundle H of TCM satisfying_conditions (see §1 of Chapter II)
(i)
H n H = {0},
(ii) H is involutive, i.e., for any complex vector fields U and V in H, [U, V] is also in H.
THEOREM 1.1. A differentiable manifold has a real CRstructure if and only if it has a complex CRstructure. Proof. Suppose M has a real CRstructure (D, J). Then we
define H = {x  /I ix; X E r(D)}.
(1.3)
Of course we have H n H  {0}. Moreover, if we take
U  X  /JX and V = Y 
JY from H we obtain
(U, V]  [x, Y]  [ix, JY] 
.{[x, JY] + fix, Y]}.
(1.4)
Taking account of (1.2), (1.4) becomes [u, vl = [X, Y]  fix, JY] 
 TJ{[x, Y]  [ix, JY]}.
(1.5)
Thus, by using (1.1) and (1.3), we obtain that [U, V] belongs to H. Consequently M has a complex CRstructure. Conversely, suppose M has a complex CRstructure. Then we define the distribution D by D = {X = Re(U); U E H}
and J : D . D
(1.6)
and
u E H.(1.7)
given by JX  Re (V71  U),
where
X  Re(U)
Then it is easy to check that we have j2  I. On the other hand, by using (1.6) and (1.7) we get (JX, JY]  [X, Y]   Re([U, V]) E r(D),
(1.8)
where X = Re(U) and Y = Re(V), that is, condition (1.1) is satisfied. Substituting Y by JY in (1.8) we obtain [JX, Y] + [X, JY]  Re(/I[U, V]) E r(D).
(1.9)
The condition (1.2) follows from (1.8) and (1.9). The proof is complete.
130
CHAPTER VI
By this theorem we can say that we have a CRstructure on M either when M has a real CRstructure or a complex CRstructure. A manifold endowed with a CRstructure is called a CRmanifold. Now, suppose M is a real (2n+s)dimensional manifold endowed with an fstructure 4), that is, there exists a tensor field 4> of type (1, 1) and of rank 2n on M satisfying
3
+ 4) = 0.
(1.10)
Besides, we suppose there exist s vector fields {1'" ''s} and s 1forms {ni.... ,fl i satisfying s
2=i+ E aCas
Ea=0,
a=1
n(
b
)
= aa bI n a° , = 0 ,
where I is the identity morphism on TM. Then we say that M is endowed with an fstructure with c lemented frames (see GoldbergYano L11) given by tensor fields a , n Denote by D the distribution defined by (
D  {x E P(TM)t na(X) = 0, a = i,...,s}.
and by using the first equality in (1.11) we obtain that the restriction J of 4> to D is an almost complex structure on D. Thus we have an almost complex distribution (D, J) on M with dim D = 2n for each x E M. x The torsion tensor field S of an fstructure with na, ta) is defined by complemented frames s
S = [4), $] + 2
E dna ®$
(1.12)
a=1 where [4), 4>] is the Nijenhuis tensor field of 4). We say that (4), na,
vanishes
is Dnormal if the torsion tensor field S D, i.e., we have
on S(X, Y)  0,
for any
x, Y E T(D).
(1.13)
PROPOSITION 1.1. Let M be a manifold endowed with a Dnormal fstructure with complemented frames. Then M is a CRmanifold. Proof. As we have seen above we hate an almost complex distribution (D, J) on M. By applying n to (1.13) and taking account of (1.11) and (1.12) we obtain
CRSTRUCTURES AND PSEUDOCONFORMAL MAPPINGS
0 = nb([C ](X, Y) 
131
F na([X. Y]1&a1 = 1
= nb([JX, JY]

[XaY]),
for any
x, Y E r(D).
Thus (1.1) is satisfied and we have [Jx, Y] + [x, JY] Er(D), for any X, Y E r(D). Finally, by using (1.11) and (1.12) in (1.13) we obtain (1.2). Hence M is a CRmanifold. Let M be a (2n+s)dimensional CRmanifold and (D, J) be the CRstructure on M with 2n = dim. Dx, x E M. We can choose s linearly independent vector fields
defined on a coordinate neighborhood U of M, such that each &a does not belong to the distribution D. Then we define s differential 1forms {n1,...,no) on U by s
F a Eb) = aa,
na(X +
(1.14)
b=1
where as (a = 1,...,s) are differentiable functions on U and X E r(D). It is easy to check that na are s linearly independent 1forms on U defining locally the distribution D, i.e., we have na(X) = 0 for each X E r(D). Moreover, we
have na(b)
= ab'
Next, for each vector field X E r(TM) the vector field
Y=X 
8
E
nb(X)
b=1 belongs to D since na(Y) = 0, a = 1,...,s. Thus a tensor field 0 of type (1, 1) is well defined on U by s
E nb(X)%), for any x E r(TM). (1.15) b=1 Finally, by a direct computation we can verify (1.11) and (1.13). Thus we obtain ox  J(X 
PROPOSITION 1.2. On each coordinate neighborhood U of a CRmanifold M there exists a Dnormal fstructure with complemented frames. na, Ea) is called an associate The fstructure fstructure with complemented frames to the CRstructure is another associate na, (D, J) on U. T1ow, suppose fstructure to the same CRstructure (D, J) on U. Then we have )
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CHAPTER VI
LEMMA 1.1. The associate fstructures with complemented a, ,a ) and (j, ;a,_,
frames (4),
a
s
Aa +
a
b=E1
)
are related by
aafb,
s
abnb,
na
a = 1,...,s,
(1.16)
b=1 S +
E a=1
on U, where
na 41 JA
=
4.
a
a] is an invertible matrix of differentiable
functions, [ $] is the inverse matrix of [ab] and Aa E r(D). For the fstructure (4), na, ) associate to the CRstructure (D, J) on U we define the tensor fields Ga (a  1,...,s) of type (0, 2) by Ga(X, Y)
(1.17)
= drla(4)X, Y),
for any X, Y tangent to M on U. Then we have
PROPOSITION 1.3. The tensor fields Ga are symmetric and Hermitian on the almost complex distribution D, i.e., we have Ga(X, Y)  Ga(Y, X)
(1.18)
and Ga(JX, JY)  Ga(X, Y),
for all
X, Y E r(D). (1.19)
Proof. By (1.17) we obtain Ga(X, Y)
_  2 na([JX, Y])
for any
X, Y E T(D), (1.20)
since the restriction of 4) to D is just J and
na(JX) = that
a(Y) = 0. On the other hand, from (1.1) it follows na([JX, y] + [X, JY]) = 0,
for any
x, Y E r(D). (1.21)
Thus we obtain our assertion from (1.20) and (1.21). nay
Now we consider another fstructure ( associate to the CRstructure (D, J) and denote bya Ga the tensor fields defined by (1.17). Then by using (1.20) and
CRSTRUCTURES AND PSEUDOCONFORMAL MAPPINGS
133
the second equality in (1.16) we obtain Ga(X, y) 
F AaaGb(X, Y), b=1
for any
X, Y E r(D). (1.22)
PROPOSITION 1.4. Let M be a (2n + s)dimensional CRmanifold and let (D, J) be the CRstructure on M with 2n = dim RDx, x E M. Suppose for each X E r(D) there exists a E 1,...,s} such that Ga(X, X) x 0. Then the distribution D is not integrable and the dimension of an integral submanifold of D is less than n + 1. Proof. By the hypothesis and (1.20) we obtain that D is not integrable. Now suppose M is an integral submanifold of D of dimension n + 1. Then there exists a local field of frames {X1,..#x, JX1} on M* since otherwise it follows
dim RDx > 2n. But, there exists a E (1,...,s} such that Ga(X1, X1) x 0, that is fa([JX1, X1]) x 0. Thus we have [JX1, X1] E r(D) which contradicts [JX1, X1] E r(TM*). From this proposition we have
COROLLARY 1.1. Let M be a (2n + 1)dimensional CRmanifold such that G given by (1.17) is positive or negative definite. Then the distribution D is not integrable and the dimension of an integral submanifold of D is less than
n + 1. Now, let M be a (2n + s)dimensional manifold endor d with an fstructure of rank 2n. Then we define P  4 and
Q = I + 2 and we have
P+Q=I,
P2
=P, Q2=Q, PQ=QP0,
that is, P and Q are two complementary projection morphisms on TM. Thus we have two complementary distributions D and D defined by D = (x E r(TM): Qx  o],
D'  (x E r(TM): Px = o]. (1.23)
We say that the fstructure $ is Dnormal if we have
$](x, Y) =Q((X, YJ), for any x, Y E r(D). (1.24)
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134
It is easy to check that each normal almost contact manifold (see Blair [3]) has a Dnormal fstructure. Thus we have a justification for the above definition. On the other hand, we see that each fstructure with complemented frames na, &a) satisfying (1.13) is also a Dnormal fstructure in this sense. Now we state THEOREM 1.2. Any real manifold endowed with a Dnormal fstructure is a CRmanifold. Proof. Suppose M is endowed with a Dnormal fstructure . Then we have the distribution D given by (1.23) and the restriction of 0 to D is just an almost complex structure J on D. Thus M is endowed with an almost complex distribution (D, J). Moreover, (1.24) becomes [JX, JY]  [X, Y]  4){[X, JY] + [JX, Y]} = 0,(1.25)
for any X, Y E r(D). Now, from (1.25) taking into account the definition of the projection morphisms P and Q we obtain (1.1) since we have Q4) = 0 + 0 = 0. Then (1.25) becomes (1.2). Thus by Theorem 1.1 the proof is complete. From this theorem we get
COROLLARY 1.2 (Blair [3]). Any normal almost contact manifold is a CRmanifold.
§2. Generic Submanifolds of Complex Manifolds
Let M be a complex manifold of complex dimension p and let M be a submanifold of M of real dimension m. Then, denoting by 3 the almost complex structure on M, we have [J, J](X, Y) = 0,
for any
X, Y E r(TM).
(2.1)
Now let Dx = TXM f) J(TX M), x E M so that D x is the maximal invariant subspace Dx of TxM under the action of J. Then we say that M is a generic submanifold of M if D : x  Dx C TxM
is a distribution on M. Chen proved in [S] that in fact D is a differentiable distribution. We denote by J the restriction of J to the distribution D. Then (D, J) is an almost complex distribution on M
CRSTRUCTURES AND PSEUDOCONFORMAL MAPPINGS
135
of real dimension 2n. Moreover, we have THEOREM 2.1. Each generic submanifold of a complex manifold is a CRmanifold. Proof, we show that (D, J) defined above is a CRstructure on M. First, we consider a coordinate neighborhood U on M and take a complementary distribution D to D on U. Denote by P and P the projection morphisms of TM to D and respectively D. Then X = PX + PX,
for any
x E r(TM).
(2.2)
Next, we see that JPX = JPX E r(D) and JPX is not tangent to M, otherwise D is not the maximal holomorphic distribution on M. Thus, by using (2.1) and (2.2), we obtain [JX, JY]  [X, Y]  JP([JX, Y] + [X, JY]} = 0 (2.3) and
P{ [Jx, Y] + [X, JY] } = 0,
for any X, Y E IJD) . (2.4)
Replacing X by JX in (2.4) we obtain (1.1) and then (1.2) is a consequence of (2.3) taking account of (2.2) and (2.4). The proof is complete. Let M be a (2n +s)dimensional generic submanifold of the complex manifold M. Suppose (D, J) is the CRstructure stated by Theorem 2.1. By the theory in §1 on each coordinate neighborhood U on M there exists an associate fstructure with complemented frames (0, na, Ca), where Ca does not belong to D and na are 1forms defining locally the distribution D. Thus we obtain on U the vector fields Na = Jca, a = i,...,s, which are not tangent to the generic submanifold M. For this reason we call them the affine na, ta). If normals induced by the fstructure (4, ra, Via) is another fstructure with complemented frames
associated to (D, J) and Na are the corresponding affine normals, by (1.16) we have
s
&a  Aa +
E
J1a&b
b=1 Na
JAa +
a b1
E
71aNb
b1 _
(2.5)
s _b E
b1
136
CHAPTER VI
If s = codim M we say that M is an affine antiholomorphic submanifold. It is interesting to note that in this case (2.5) is sufficient for a complete study of the geometry of M.
93. AntiHolomorphic Submanifolds of Complex Manifolds As we have seen, the Riemannian metric on the ambient space was an important aspect to the definition of a CRsubmanifold. It is the purpose of this section to define CRsubmanifolds in complex manifolds, that is, without using a Riemannian metric, and to sketch the main lines of a study for antiholomorphic submanifolds. Let M be a complex manifold of complex dimension p and M be a real submanifold of M of real dimension in. Then we say that M is a CRsubmanifold of MNif there exist two differentiable distributions D and D on M satisfying Dx; T M Dx (i) (ii) J(Dx) = Dxj
J(Dx) f) TxM = {0},
for any x E M, where J is the almost complex structure on M. A CRsubmanifold M with m = 2n + a, where s = dim RDx and 2n = dim RDx immersed in a complex manifold M with p  n +s is called an antiholomorphic submanifold. From now on, in this section we suppose that M is an antiholomorphic submanifold of a complex manifold M. Then NN by the definition, JD is a vector bundle satisfying N NN TM = D 0 D 0 JD. (3.1) Hence JD should be considered as the normal bundle to M. In order to study the geometry of M we fix a symmetric affine connection I on M such that we have N N VXJ = 0, for any x E r(TM). (3.2) For the existence of a such affine connection see walker [1) or Yano [2]. Then we put I Y  V Y + h(X, Y),
for any x, Y E T(TM), (3.3) X X where VXY E I'(TM) and h(X, Y) E r(JD). Certainly, by using the properties of I we get that V is a symmetric affine connection on M and h is a normal bundlevalued symmetric bilinear form on M. We call h the second fundamental form
of M.
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CRSTRUCTURES AND PSEUDOCONFORMAL MAPPINGS
We denote by P and Q the projection morphisms of TM to D and D respectively. Then we define the morphisms
4: TM  TM and w : TM P JD by ox  JPX
(3.4)
WX = JQX,
(3.5)
and respectively, for any X E r(TM). Thus we have
,ix = ,x + WX.
(3.6)
Next, let u E r(JD) and X E r(TM). Then we put V U   AUX + VXU, X
(3.7)
1 where Aux E r(TM) and V U E r(JD). By the properties of V it X
follows that AU is an endomorphism of TM and Vl is an affine connection on JD.
THEOREM 3.1. Let M be an antiholomorphic submanifold of a complex manifold M. Then we have: the distribution D is integrable if and only if (i) the second fundamental form h of M satisfies h(X, JY) = h(Y, JX),
for any X, Y E r(D)i
(3.8)
(ii) the distribution D is integrable if and only if
AJVW = AV, for any v, w E f(D).
(3.9)
Proof. We take X, Y E r(D) and by using (3.2), (3.3) and (3.6) we obtain
VXJY + h(X, JY) _ (VXY) + W(VXY) + Jh(X, Y).(310) Since h(X, Y) E r(JD) we_infer that Jh(X, Y) E r(D). Thus, taking the component in JD in (3.10), we obtain
h (X, JY) = w(VXY) . Then taking into account that V is a torsionfree affine connection we get N ti (3.11) h(X, JY)  h(Y, JX) = w((X, Y]). Now it is easy to see that assertion (i) follows from (3.11). Next, by using (3.3) and (3.7) in (3.2) we obtain VVJW  w(VvW),
for any
v, w e r(D).
138
Thus by again using (3.7) we get
VVJW  VWJV = AaVW  A w + w([V, W]). On the other hand, we have VVJW  Vw1V = J([V, W]).
Hence by (3.12), (3.13), and (3.6) we infer that
4) ([V, W]) = A3VW  A3V, which proves the assertion (ii) since V E T(D) if and only if 4,V = 0. The proof is complete. By Theorem 2.1, each antiholomorphic submanifOld is a CRmanifold. In this case the torsion tensor S defined by (1.12) is given by S(X, Y) _ [4), 4)](X, Y)  2Jdw(X, Y),
(3.14)
for any X, Y E r(TM), where we have dw(X, Y) = 2 {VXwY  VYwx  w([X, Y])}. Of course, by Proposition 1.2 the fstructure 4, is Dnormal. According to the terminology of YanoIshihara [3] this means D is a torsionless distribution. When S vanishes identically on M we say that M is a normal antiholomorphic submanifold.
THEOREM 3.2. An antiholomorphic submanifold M of the is normal if and only if complex manifold
AN °0 = OAN
a = 1,...,s, a where Na is the local basis in JD. a
This theorem should be considered as a version of Theorem 3.1 of Chapter III for the nonmetrical case.
§4. PseudoConformal Mappings
Let M and M' be two CRmanifolds of the same real dimension 2n + s. we denote by (D, J) and (D', J') the CRstructures on M and respectively M'. A diffeomorphism 0 : M oMI is
CRSTRUCTURES AND PSEUDOCONFORMALMAPPINGS
139
called a pseudoconformal mapping if @*X E r(D') for each J'*O* where '* is the differential
X E r(D) and O*oJ mapping of (5.
M  M' be a pseudoconformal mapping. Now, suppose a, ) be an fstructure with complemented frames Let associated toa the CRstructure We J) on a coordinate neighborhood U of M. Then U' _ (U) is a coordinate neighborhood on M' on which we define
0*'0°0*1;
and
Ea = O*Ea
na = rla°It*1.
(4.1)
rla, is an fstructure with It is easy to check that complemented frames associated to a(D', J') on U'. We take another fstructure na', C) associated to (D', J') on U' and by using (4.1) and (1.16)awe obtain )
s * a a
=A
+
a
E
b=1 s
a nat* 1 =
Aa% a
Xbn be
E
(4.2)
b=1 s
t*000%
= ' 
E
(nao,t*
® JA
a=1
is an invertible matrix of differentiable
where [Aab ]
functions on U', (Aa] is the inverse matrix of (Ab] and a
A E r(D').
conversely, suppose 0 M + M' is a diffeomorphism satisfying (4.2) for two fstructures with complemented frames (¢, na, &a) and (01, nay. Ea) associated to (D, J)
and (D', J') respectively. Take x E r(D) and from the second equality in (4.2) we obtain s
be ((P* X) =
E Aaa(X) = 0. a=1
Hence O*X E r(DI) for each X E r(D). Next, by using the third equality in (4.2), we get 0*(OX) = $'(l*X),
for any x E T(D).
But we know that the restrictions of $ and 0' to D and D' respectively are just J and it. Consequently, we have *eJ = J'°' that is, 0 is a pseudoconformal mapping. Thus
CHAPTER VI
140
we have the following important result (see Ishihara [2] for the case s = 1). THEOREM 4.1. Let 0 : M , M' be a diffeomorphism of two CRmanifolds M and M' with (D, J) and respectively (D', J') as CRstructures. Then 0 is a pseudoconformal mapping if and only if there exist two fstructures with complemented frames
and (0', na', E
na
)
a)
associated locally to (D, J)
and (D', J') respectively such that (4.2) is satisfied. Now, suppose 0 : M . M is a pseudoconformal mapping on the CRmanifold. Then we say that 0 is a pseudoconformal transformation on M. Let X be a vector field on M. Then we say that X is a pseudoconformal vector field if any local (E t transformation 0) of M spanned by X is c, c a pseudoconformal transformation. By using Theorem 4.1 we obtain
THEOREM 4.2. field if and frames ( , n a coordinate
A vector field X is a pseudoconformal vector only if for any fstructure with complemented ) associated to the CRstructures (D, J) on , neighborhood of M we have
I nb a JBb,
.Cx
lXna =
b=1
I abnb b=1
(4.3)
s
rX a a  Ba where ( tX
is
a]
th
I aaF,b, bal
are differentiable functions on U, Bb E r(D) and Lie derivative operator with respect to X.
On the other hand, we have THEOREM 4.3. Let X be a pseudoconformal vector field on a CRmanifold N and (D J) be the CRstructure on M. If X (Y, X) a 0, a a 1,...,s for any vector belongs to D then
field Y on N. Proof. By (1.14) we see that mY E r(D) for any
Y E MO. Then (1.17) implies Ga(Y, X)
{4)Y(n*X)
=
2
na((4Y, X]))

X(na()Y))

2 na((sy, X]).
(4.4)
141
MAPPINGS
CR.STRL'C'iIJRES AND
Next, from the first equality in (4.3) we get
E nb(Y) [X, y]  $([X, Y])   b'1 J%_ Hence [x, :Y] E T(0) and our assertion follows from (4.4). We say that the CRstructure (D, J) is nondegenerate if at least one of the tensor fields Ga given by (1.17) is nondegenerate on D, that is, from Ga(Y, X) on 0 for any Y E F(D) and a certain x E T(D) it follows that X  0. From (1.22) we see that the definition of a nondegenerate CRstructure does not depend on the associate fstructure with complemented frames. A CRmanifold whose CRstructure is nondegenerate is called a nondegenerate CRmanifold. Then by Theorem 4.3 we have THEOREM 4.4. Let X be a pseudoconformal vector field on a nondegenerate CRmanifold M. If X belongs to D then X vanishes identically on M. If in particular, M is a real hypersurface of a complex space and the CRstructure on M is nondegenerate, Theorem 4.4 can be found in Tanaka (1) or Ishihara (2). The remaining part of this section is devoted to the pseudoconformal transformations on generic submanifolds. Let M be a complex manifold with almost complex M be a holomorphic transformation on structure a and f : M M. Then we have 4*'3  J4*, where 4* denotes the Next, we consider two generic differential mapping of subm3nifolds M and No in M such that No to (N). Denote to N and by (D, J) and (0', J') the restrictior. of by the CRstructures on N and No respectively. We see that that is, 4*X E i'(0') for each x E C(0) and #, *j is a pseudoconformal mapping. Therefore we have M . M of the THEOREM 4.5. Any holomorphic transformation : ambient complex manifold M induces a pseudoconformal mapping M', where M and M' are generic submanifolds such 1, M :
:
that M'
=
Next from Theorems 2.1 and 4.1 we obtain
COROLLARY 4.1. Let M and M' be two generic submanifolds of M  M' be a diffeomorphism. and : a complex manifold Then ' is a pseudoconformal transformation if and only if the equalities in (4.2) are satisfied. :
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CHAPTER VI
The following theorem can be obtained in the same fashion as Theorem 4.6 of YanoIshihara [3]. THEOREM 4.6. Let M and M' be two affine antiholomorphic submanifolds analytically immersed in a complex manifold M and assume 4) : M  M' is an analytic homeomorphism. Then int x is pseudoconformal if and only if, for any and ' of x and belonging to M there exist neighborhoods ''(x) in 14 respectively and a holomo hic homeomor hism + 6' such that 4) is the restriction of to u n M. '
As a final remark we note that CRmanifolds of dimension 2n + 1 have been intensively studied by means of associated almost contact structures (see Ishihara [2], YanoIshihara [3], SakamotoTakemura [1], [2], Takemura [1]). The author believes that more results might be obtained on the geometry of CR manifolds of dimension 2n + s, via associate fstructures with complemented frames.
Chapter VII
CRSTRUCTURES AND RELATIVITY
The main purpose of this chapter is to show that CRstructures on real hypersurfaces of a complex manifold have an interesting application to relativity. It is the merit of Roger Penrose to discover a correspondence between points of a Minkowski space and projective lines of a certain real hypersurface in a complex projective space (see (2.4)).
§1. Geometrical Structures of SpaceTime In this section we summarize briefly the geometry of both the Minkowski space and the Einstein space, which in fact are the most important spac$time models. A Minkowski space is R endowed with a flat pseudoRiemannian metric g of Lorentzian signature (1, 3), (see §2 of Chapter I). we denote by M the Minkowski space defined above and by TpM the tangent space to M at p. Then a tangent vector X E T M is said to be null if and only if 1X1  0, where 11.11 ispthe norm defined by g. The set of null vectors in TpM is called the null cone or light cone at p and it is denoted by Cp. In the case of Minkowski geometry it is convenient to work with null vectors as basis vectors. Suppose {X0, xl, X2, X3} is a basis in TM. Then any X E TpM is given by xiX and x = (x0, xl, x , x3) gives a system of coordinates on M. We can choose the basis {Xi) such that
g(X, Y) 
x0y0  x I I y
x2 y2
 x3 y3 x)}1/2.
and denote g(X, Y) by g(x, y) and 1x1  {g(x, Now we construct a set of coordinates in M ® RC by the following equalities
144
CHAPTER VII
u
72
(x0+x1)
T' (x2
 ix3)
2 (x2+ix3) 1
T
v = 1 (x0
 x1),
where i = . Then we obtain fluff = NO  UEfl = fl&fl = 0, where we considered the extension by complex linearity of g to M ! RC. The complex matrix A = has the determinant det(A) = uv  C& = 1 flxfl2 and it is a Hermitian matrix. On the other hand, each (2X2)Hermitian matrix is of this form with some (u, v, E, C). As we shall see later, it is convenient to identify the space H(2) of (2X2)Hermitian matrices endowed with the determinant as norm with the Minkowski space in which we have choosen an origin and a basis for the tangent space at the origin. Of course, both of the pseudometrics on M and M RC induce a distance in such spaces. We denote by d the squared distance function and then we have d(x, x ' )
_ (x0  x01)2  (x1 
x1,)2
 (x2  x2')
rru  u'  (x3  x3')2 = det
{111CC'
' 
vv'
,])
If we have d(x, x') > 0, d(x, x') = 0 or d(x, x') < 0 we say that the separation between x and x' is timelike, null or spacelike respectively. The group of isometries of M is denoted by P and it is the 10parameter Poincax group and the Lorentz group at p E M is the subgroup Lp of P which leaves p fixed. There exist four otthher alternatives of spacetime structures: Aristotelean spacetime, Galilean spacetime, Newtonian spacetime and Einsteinian spacetime. For the geometrical structures of the first three spacetime structures consult Penrose [2]. We give here only the geometrical structure of the Einsteinian spacetime, which in fact can be regarded as the best model for gravity. As is well known, the mathematics of Einstein spaces is of great interest for physics. An Einsteinian spacetime is a 4dimensional real manifold E endowed with a pseudometric g of signature (1, 3), but in this case g is generally non flat. The inertial
145
motions in E are given by geodesics of the unique torsionfree connection V induced by g. But this connection has a curvature R whose physical interpretation is obtained from the Jacobi equations. In order to provide a physical theory of gravity, Einstein had to postulate the so called field equations given by
2
pg + Ag s XT,
where S is the Ricci tensor of q, p is the scalar curvature, X is the Einstein's constant, A is the cosmological constant and T is the energymomentum tensor. In fact we have here a system of ten partial differential equations of second order in which the ten components of the pseudometrie g are the unknown functions. Einstein's model (whose particular case is Mirkowski's model) is still the best of all spacetime models known up to date.
§2. Th
Twistor Space and Penrose Correspondence
Let T be a 4dimensional complex vector space and let 0
(z
3
,
i, z`, z) be the coordinates of an element z of T.
Suppose on T there is given the Hermitian form signature (2, 2) defined by
(?
z0i2 + z1i3 +
z2
z
+ z3i1.
of (2.1)
The space T endowed with the Hermitian form is called the twistor space (see Penrose [1), (2) and wells (1)). Two points z C T and x E M (where M is a Minkowski space) are said to be incident whenever we have
(z2, z3) 0 11 (z0, z1)
We see that (2.2) can only hold if (z) a 0. Now, we denote by PT the complex 3dimensional projective space associated with T. In this complex manifold we define the open complex submanifolds
PT+ a {z E PT, (Z) > 0) and PT_  {z E PT, (Z) < 0} and a real Sdimensional manifold
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CHAPTER VII
PT0 = {z E PT: O(z) = 0). Of course, PTO is the common boundary of PT and PT and thus it inherites a natural CRstructure as+ a real hypersurface of a complex manifold (see §1 of Chapter II). In this way PT0 contains the points of PT for which (2.2) holds for a certain x, but not only them. As we can easily see, the points of projective line I, given by ZO = Z1 = 0 admit no solution for x in (2.2). For each Z E PT  I we can solve (2.2) for a certain 0 point x E M. Moreover, the general solution is given by Z'Z1
fu
`&
J
Z1ZOl (2.3)
f
Zo
+ k`ZO21
where
v is a solution and k is an arbitrary real number. All of these solutions are null vectors in M. Therefore, the points x incident with the given Z constitute a null geodesic (straight line) in M. Thus we can consider PT0  I as the space of null geodesics in M. On the other hand, for each x E M fixed in (2.2) we obtain a complex 2dimensional linear subspace of T, that is, a complex projective line in PT. Moreover, this line lies in PT0  I and each line lying entirely in PT0  I is obtained from points in M. Hence we have the correspondence {points of Minkowski space M} $
(2.4)
{projective lines lying entirely in PT0  I}.
This correspondence will be called the Penrose correspondence. It is an interesting means of passing from the geometry of a Minkowski space to the geometry of a CRmanifold. By the Penrose correspondence the family of light rays through a fixed point x of the Minkowski space M represents the family pf.points of the corresponding complex projective line Lx in PTO  I. Taking account of the topology of L we conclude' that the topology of an observer's field of vision is just S2. More important is the holomorphic structure of L which shows up in the fields of vision of several observers passing through the same point x
CRSTRUCTURES AND RELAT[V1TY
147
and which are conformally related to one another.
§3. Physical Interpretations of CRStructures As we have seen in the previous section, by the Penrose correspondence there appears the physical interpretation of the differential geometry determined by the CRstructure on the real 5dimensional real hypersurface PTO in PT. First, let us see what we can say about the almost complex distribution (D, J) on PTO (for the notations see §1 of Chapter VI). For any point Z of PTO we have a complex 2dimensional holomorphic subspace Dz on which the almost complex structure J is induced by that of PT. The point Z is interpreted as a light ray in the Minkowski space M. Now, suppose xO = 0, that is, time takes a specific value. Thus the geometrical picture becomes a Euclidean space E of real oton" described by Z is represented dimension 3. Then the by a unit vector z at some point q of E. Hence we can imagine the tangent space to PTO as small displacements of q and z. if a displacement is such that q is moved in a direction which is orthogonal to the direction of z we get an element of DZ. The subspace Dz divides the tangent space of PTO at Z hp
into two remaining pieces representing light rays that are slightly ahead or lag slightly behind the original ray Z. All of these properties are independent of time x0, that is, the above interpretation of DZ is independent of the Euclidean hyperplane E. Next, for the physical interpretation of J we imagine a 2plane element n at q, which is orthogonal to the direction of the unit vector z. We are interested in studying displacements of q and z such that q is moved within n, since these correspond to real vectors in D. Such displacements are represented by pairs of vectors (r, v) in n, where r gives the displacement of q and v measures the change in z. Now keep the light ray Z and the neighbouging ray to which it is displaced fixed and vary the time x . Then the functions r(x$) and v(x0) are related by
and
0
(3.1)
0 The action of the operator J means to rotate both r and v through a right angle in the plane n, in a lefthanded sense about the direction of z. Moreover, (3.1) are invariant
148
CHAPTER VII
with respect to this action. In this way, we obtained for the CRstructure of PT0 an interpretation in terms of physical space  time geometry. On the other hand, PT0 is just S2 x S3 (see wells [11) and thus we know completely the geometry of PT0. With this example in mind we can think of the following problem of differential geometry. Let N be a real 5dimensional CRmanifold. Then under what conditions does N become a Penrose hypersurface, that is, when can N be immersed in CP3 as a hypersurface which is pseudoconformal (see §4 of Chapter VI) with PTO? CRstructures can also be discussed in the general context of a Einsteinian spacetime manifold. But in this case we have to remark that the induced CRstructures might be locally distinct from one another. As we have seen in this chapter, the CRmanifolds which appear in relativity by means of the Penrose correspondence are in fact real hypersurfaces of a certain complex manifold. However, counterexamples given by Nirenberg [1] show that not all CRmanifolds can be realized as a real hypersurface in a complex manifold. The CRstructures of such CRmanifolds are called nonrealizable CRstructures and they are now intensively studied by many people.
REFERENCES
Andreotti A. and Hill C.D.: 1. 'Complex characteristic coordinates and tangential CauchyRiemann equations', Ann. Scuola Norm. Sup. Pisa Sci. Fis. Nat. 26 (1972), 299324. Arca G. and Ro$ca R.: 1. 'Contact CRsubmanifolds of a Sasakian manifold admitting a contact concircular vector pairing', Tensor N.S. 40 (1983), 280284.

Barros M. and Urbano F.: 1. 'CRsubmanifolds of generalized complex space forms', An. St. Univ. Al. I. Cuza Iasi 25 (1979), 355365
2. 'Submanifolds of complex Euclidean space which admit a holomorphic distribution', Quart. J. Math. Oxford 34 (1983), 141143.

3. 'Topology of quaternion CRsubmanifolds', Bollettino U.M.I. 6 (1983), 103110. Barros M., Chen B.Y., and Urbano F.: 1. 'Quaternion CRsubmanifolds of quaternion manifolds', Kodai Math. J. 4 (1981), 399418. Bejancu A.: 1. 'CRsubmanifolds of a Kaehler manifold I', Proc. Amer. Math. Soc. 69 (1978), 134142.
2. 'On the integrability conditions on a CRsubmanifold', An. St. Univ. Al. I. Cuza Ia$i 24 (1978), 2124. 3. 'Une classe de sousvari4t4s d'une varidtd kdhlerienne', C. R. Acad. Sc. Paris 286 (1978), (S4rie A), 597599. 4. 'CRsubmanifolds of a Kaehler manifold II', Trans. Amer. Math. Soc. 250 (1979), 333345.
REFERENCES
150
5. 'On the geometry of leaves on a CRsubmanifold', An. St. Univ. Al. I. Cuza Ia4pi 25 (1979), 393398. 6. 'Real hypersurfaces of a complex projective space', Rendiconti di Mat. 12 (1979), Serie VI, 439445. 7. 'Normal CRsubmanifolds of a Kaehler manifold', An. St. Univ. Al. I. Cuza Ia$i 26 (1980), 123132. 8. 'Umbilical CRsubmanifolds of a Kaehler manifold', Rendiconti di Mat. 13 (1980), 431446. 9. 'On a class of mixed totally geodesic CRsubmanifolds', An. Univ. Timi?oara 28 (1980), 1123. 10. 'Fconnections on a CRsubmanifold', Bul. Inst. Politehnic Iasi 27 (1981), 3340. 11. 'Hypersurfaces of quaternion space forms', An. St. Univ. Al. I. Cuza Iayi 27 (1981), 291296. 12. 'On semiinvariant submanifolds of an almost contact metric manifold', An. St. Univ. Al. 1. Cuza Ia$i, Supliment tom 27 (1981), 1721. 13. 'Umbilical semiinvariant submanifolds of a Sasakian manifold', Tensor N.S. 37 (1982), 203213. 14. 'Sasakian antiholomorphic submanifolds of a Kaehler manifold', Glasnik Matematicki 17 (1982), 115130. 15. 'Antiholomorphic submanifolds of almost Hermitian manifolds', Math. Rep. Toyama Univ. 6 (1983), 179196. 16. 'A theorem of classification for semiinvariant submanifolds of a Sasakian space form', An. St. Univ. Al. 1. Cuza Iayi 29 (1983), 5964. 17. 'Hypersurfaces of quaternion manifolds', Revue Roumaine de Math. Pures et Appl. 28 (1983), 567576. 18. 'Pinching theorems for sectional curvature of a CRsubmanifold', Rendiconti di Mat. 3 (1983), 6571. 19. 'On contact umbilical submanifolds of Sasakian space forms', An. St. Univ. Al. I. Cuza Ia$i 30 (1984), 8994. 20. 'Hypersurfaces of manifolds with a Sasakian 3structure', Demonstratio Math. 17 (1984), 197209. 21. 'Semiinvariant submanifolds of locally product Riemannian manifolds', An. Univ. Timipoara 22 (1984), 311.

REFERENCES
151
22. 'Antiquaternion submanifolds of quaternion manifolds', LucrArile Conf. Geom. Top., P. Neamt, (1984), 141144. 23. 'CRstructures and pseudoconformal mappings', Seminarul Geom. Top. nr. 78, Univ. Timi$oara, 1984. 24. 'QRsubmanifolds of quaternion manifolds', to appear. 25. 'Generic submanifolds of manifolds with a Sasakian 3structure', Math. Rep,. Toyama Univ. 8 (1985), 75101. Bejancu A., Kon M., and Yano K.: 1. 'CRsubmanifolds of a complex space form', J. Differential Geometry 16 (1981), 137145.
Bejancu A. and Papaghiuc N.: 1. 'Semiinvariant submanifolds of a Sasakian manifold', An. St. Univ. Al. I. Cuza Ia$i 27 (1981), 163170. 2. 'Semiinvariant submanifolds of a Sasakian space form', Colloquium Math. 48 (1984), f.2 , 229240. 3. 'Almost semiinvariant submanifolds of a Sasakian manifold', Bull. Math. Soc. Sci. R.S. Roumanie 28 (1984), 321338. Bejancu A. and Smaranda D.: 1. 'Semiinvariant submanifolds of a coKAhler manifold', An. St. Univ. Al. I. Cuza Ia$i 29 (1983), 2732. Bishop R.L. and Goldberg S.I.: 1. 'Some implications of the generalized GaussBonnet theorem', Trans. Amer. Math. Soc. 112 (1964), 508535. Blair D.E.: 1. 'Geometry of manifolds with structural group U(n) x 0(s)', J. Differential Geometry 4 (1970), 155167.
2. 'On a generalization of the Hopf fibration', An. St. Univ. Al. I. Cuza Ia$i 17 (1971), 171177. 3. 'Contact manifolds in Riemannian geometry', Lecture Notes in Math. 509, Springer Verlag, Berlin, 1976. 4. 'Three lectures on complex differential geometry', Bull. Soc. Mat. de Belgique 35 (1983), 2538. Blair D.E. and Chen B.Y.: 1. 'On CRsubmanifolds of Hermitian manifolds', Israel J.
152
REFERENCES
Math. 34 (1979), 353363. Blair D.E., Ludden G.D., and Yano K.: 1. 'Differential geometric structures on principal toroidal bundles', Trans. Amer. Math. Soc. 181 (1973), 175184.

Buchner K. and Rogca R.: 1. 'Sasakian manifolds having the contact quasiconcurrent property', Rendiconti del Circolo Mat. di Palermo 32 (1983), 388397.

Calabi E.: 1. 'Metric Riemannian surfaces', Annals of Math. Studies No. 30, Princeton University Press, Princeton.
2. 'Isometric imbedding of complex manifolds', Annals of Math. 58 (1953), 123. Calapso M.T. and Rooca R.: 1. 'Sous varidtds gdndriques pseudoombilicales de contact dune varidtd sasakienne', Rendiconti del Circolo Mat. di Palermo 32 (1983), 6975. Cartan E.: 1. 'Sur la gdomdtrie pseudoconforms des hypersurfaces de deux variables complexes, I, II', Ann. Math. Pura Appl. 11 (1932), 1790; Ann. Scuola Norm. Sup. Pisa 1 (1932), 333354.
Chen B.Y.: 1. Geometry of Submanifolda, M. Dekker Inc., New York, 1973.
2. 'Totally umbilical submanifolds of quaternion space forms', J. Austral. Math. Soc. 26 (1978), 154162. 3. 'Totally umbilical submanifolds', Soochow Journal of Math. 5 (1979), 937. 4. 'Totally umbilical submanifolds of Kaehler manifolds', Archiv der Mathematik 36 (1981), 8391. 5. 'CRsubmanifolds of a Kaehler manifold I', J. Differential Geometry 16 (1981), 305323. 6. 'CRsubmanifolds of a Kaehler manifold II', J. Differential Geometry 16 (1981), 493509.
REFERENCES
153
7. 'Cohomology of CRsubmanifolds', An. Facultd des Sciences Toulouse 3 (1981), 167172. 8. Geometry of Submanifolds and Its Applications, Science University of Tokyo, 1981.
9. 'Differential geometry of real submanifolds in a Kaehler manifold', Monat. f:3r Math. 91 (1981), 257274. 10. 'Some nonintegrability theorems of holomorphic distributions', to appear. Chen B.Y., Ludden G.D., and Montiel S.: 1. 'Real submanifolds of a Kaehler manifold', Algebras, Groups and Geometries 1 (1984), 176212. Chen B.Y. and Lue H.S.: 1. on normal connections of Kaehler submanifolds', J. Math. Soc. Japan 27 (1975), 550556. Chen B.Y. and Ogiue K.: 1. 'On the scalar curvature and sectional curvature of a Kaehler submanifold', Proc. Amer. Math. Soc. 41 (1973), 247250.

2. 'Some extrinsic results for Kaehler submanifolds', Tamkang J. Math. 4 (1973), 207213. 3. 'on totally real submanifolds', Trans Amer. Math. Soc. 193 (1974), 257266. 4. 'Two theorems on Kaehler manifolds', Michigan Math. J. 21 (1974), 225229. Chen B.Y. and Okumura M.: 1. 'Scalar curvature, inequality and submanifolds', Proc. Amer. Math. Soc. 38 (1973), 605608. Chern S.S. and Moser J.K.: 1. 'Real hypersurfaces in complex manifolds', Acta Math. 133 (1974), 219271. Cihodariu Gh. and Smaranda D.: 1. 'Semiinvariant submanifolds of a manifold with Sasakian (f, g, u, v)structure', LucrArile Coloc. Nat. Geom. Top., Busteni (1981), 5464.
154
REFERENCES
Fukami T. and Ishihara S.: 1. 'Almost Hermitian structure on S6', Tdhoku Math. J. 7 (1955), 151156. Gheorghiev Gh. and Oproiu V.: 1. 'Finite and infinite dimensional differentiable manifolds I, II', Ed. Acad. R.S.R., Bucuregti, 1976, 1979, (in Romanian). Goldberg S.I. and Yano K.: 1. 'On normal globally framed fmanifolds', T6hoku Math. J. 22 (1970), 362370.
2. 'Globally framed fmanifolds', Illinois J. of Math., 15 (1971), 456474. Goldberg V.V. and Rogca R.: 1. 'Mixed isotropic submanifolds and isotropic cosymplectic structures', Soochow J. of Math. 9 (1983), 7184.
Gray A.: 1. 'Kaehler submanifolds of homogeneous almost Hermitian manifolds', T8hoku Math. J. 21 (1969), 190194. 2. 'Six dimensional almost complex manifolds defined by means of threefold vector cross products', Tdhoku Math. J. 21 (1969), 614620. 3. 'Almost complex submanifolds of the six sphere', Proc. Amer. Math. Soc. 20 (1969), 277279. Greenfield S.: 1. 'CauchyRiemann equations in several variables', An. della Scuola Norm. Sup. Pisa 22 (1968), 275314. Hsu C.J.: 1. 'On CRsubmanifolds of Sasakian manifolds I', Math. Research Center Reports, Symposium Summer 1983, 117140.
2. 'On some properties of CRsubmanifolds of Kaehler manifolds', Chinese J. of Math. 22 (1984), 727. 3. 'Two theorems on CRsubmanifolds of Kaehler manifolds', Bull. of Inst. Math. Acad. Sinica 12 (1984), 9599. Ianu , S.: 1. 'Sulle variet& di CauchyRiemann', Rend. dell'Accademia
REFERENCES
155
di Science Fisiche Mat. Napoli 33 (1972), 191195. Ianu$ S. and Mihai I.: 1. 'Semiinvariant submanifolds of an almost paracontact manifold', Tensor N.S. 39 (1982), 195199.
Ianu$ S., Mihai I., and Matsumoto K.: 1. 'Almost semiinvariant submanifolds of some almost paracontact Riemannian manifolds', Bull. of the Yamagata Univ. 11 (1985), 121128. Ishihara S.: 1. 'Quaternion Kaehlerian manifolds', J. Differential
Geometry 9 (1974) , 483500. 2. 'Distributions with complex structure', Kodai Math. J. 1 (1978), 264276. Ki U.H. and Jin D.H.: 1. 'Generic submanifolds with parallel Ricci curvature of S2n+1(1)', J. Korean Math. Soc. 19 (1982), 5560.
Ki U.H., Pak J.S., and Kim Y.H.: 1. 'CRstructures of submanifolds immersed in complex space forms', Kyungpook Math. J. 23 (1983), 155168. Ki U.H. and Pak J.S.: 1. 'Generic submanifolds of an evendimensional Euclidean space', J. Differential Geometry 16 (1983), 293303. Kim I.B.: 1. 'Submanifolds of Kaehlerian manifolds and metric compound structures', Hiroshima Math. J. 13 (1983), 401443.

Kobayashi S. and Nomizu K. 1. Foundation of Differential Geometry, I, II, Interscience, New York, 1963, 1969. Kobayashi M.: 1. 'CRsubmanifolds of a Sasakian manifold', Tensor N.S. 35 (1981), 297307.

2. 'CRsubmanifolds of a Sasakian space form with flat normal connection', Tensor N.S. 36 (1982), 207213.
REFERENCES
156
3. 'Integrabilities of CRsubmanifolds of a nearly Sasakian manifold', Tensor N.S. 36 (1982), 215221. 4. 'Contact CRproducts of Sasakian manifolds', Tensor N.S. 36 (1982), 281287. 5. 'Symmetric twofold CRsubmanifolds in a Euclidean space R4n,
to appear.
Kon M. 1. 'On some complex submanifolds in Kaehler manifolds', Canadian J. Math. 26 (1974), 14421449.
2. 'Invariant submanifolds in Sasakian manifolds', Math. Ann. 219 (1976), 277290. Lawson H.H. Jr.: 1. 'Rigidity theorems in rank1 symmetric spaces', J. Differential Geometry 4 (1970), 349357. Ludden G.D.: 1. 'Submanifolds of cosymplectic manifolds', J. Differential Geometry 4 (1970), 237244.
Ludden G.D., Okumura M., and Yano K.: 1. 'Totally real submanifolds of complex manifolds', Atti della Acad. Naz. Lincei 58 (1975), 346353. Maeda Y.: 1. 'On real hypersurfaces of a complex projective space', J. Math. Soc. of Japan 28 (1976), 529540.
Maeda S. and Sato N.: 1. 'On submanifolds all of whose geodesics are circles in a complex space form', Kodai Math. J. 6 (1983), 157166. Martinez A., Pdrez J.D., and Santos F.G.: 1. 'On the normal connection of quaternion CRsubmanifolds', Hull. of Inst. Math. Acad. Sinica 12 (1984), 237247. Martinez A. and Santos F.G.: 1. 'Generic submanifolds of quaternion Kaehlerian manifold', to appear.
REFERENCES
157
Matsumoto K.: 1. 'On submanifolds of locally product Riemannian manifolds', TRU Mathematics 182 (1982), 145157.
2. 'On contact CRsubmanifolds of Sasakian manifolds', Internat. J. Math. sci. 6 (1983), 313326. 3. 'on CRsubmanifolds of locally conformal KBhler manifolds', J. Korean Math. Soc. 21 (1984), 4961. Mihai I.: 1. 'CRsubmanifolds of a framed fmanifold', Stud. Cerc. Mat. 35 (1982), 127136. Miron R.: 1. Differential Geometry, Ed. Did. Ped., Bucure$ti, 1976, (in Romanian). Moore J.D.: 1. 'Isometric immersions of Riemannian products', J. Differential Geometry 5 (1971), 159168. Naitza D.: 1. 'CRsousvaridtds de contact & forma simple verticale fermdes dune vari6td sasakienne', Rend. Sem. Fac. Sci. Univ. Cagliari 52 (1982), 3134.
Nirenberg L.: 1. Lectures on Linear Partial Differential Equations, CBMS Regional Conf. Ser. in Math. No 17, Amer. Math. Soc., Providence R.I., 1973. Ogiue K.: 1. 'Differential geometry of Kaehler submanifolds',
Advances in Math. 13 (1974), 73114: Okumura M.: 1. 'Certain almost contact hypersurfaces in Kaehlerian manifolds of constant holomorphic sectional curvature', TBkohu Math. J. 16 (1964), 270284.
2. 'Submanifolds and a pinching problem on the second fundamental form', Trans. Amer. Math. Soc. 178 (1973), 285291. 3. 'On some real hypersurfaces of a complex projective space', Trans. Amer. Math. Soc. 212 (1975), 355364.
158
REFERENCES
4. 'Submanifolds of real codimension p of complex projective space', Atti della Accad. Naz. Lincei 58 (1975), 544555. 5. 'Compact real hypersurfaces of a complex projective space', J. Differential Geometry 12 (1977), 595598. O'Neill B.: 1. SemiRiemannian Geometry, Academic Press, New York, 1983. Oproiu V.: 1. Variety di CauchyRiemann, Institute di Mat. dell Univ. di Napoli, Relatione No. 20, 1972. Ornea L.: 1. 'Generic CRsubmanifolds of Smanifolds', Stud. Cerc. Mat. 36 (1984), 435443. Pak, J.S.: 1. 'Real hypersurfaces in quaternion Kaehlerian manifolds with constant Qsectional curvature', Kodai Math. Sem. Rep. 29 (1977), 2261.
2. 'Antiquaternion submanifolds of a quaternion projective space', Kyungpook Math. J. 18 (1981), 91115.

3. 'ContactthreeCRsubmanifolds', J. Korean Math. Soc. 19
(1982), 110). 4. 'CR and contact CRsubmanifolds', Research Review of Kyungpook Nat. Univ. 35 (1983), 225230. 5. 'Quaternionic CRsubmanifolds of quaternionic space form', to appear.
Pak J.S. and Kang T.H.: 1. 'Generic submanifolds of a quaternionic projective space with parallel mean curvature vector', J. Korean Math. Soc. 17 (1981), 175192. Papaghiuc N.: 1. 'Semiinvariant submanifolds in Kenmotsu manifolds', Rendiconti di Mat. 3 (1983), 607622.
2. 'Semiinvariant products in Sasakian manifolds', An. St. Univ.Al. I. Cuza Iagi 30 (1984), 6978. Papuc D.: 1. Differential Geometry, Ed. Did. Ped., Bucuregti (1982),
REFERENCES
159
(in Romanian).
Penrose R.: 1. 'The complex geometry of the natural world', Proc. of Internat. Congres of Math., Helsinki (1978), 189194.
2. 'Physical spacetime and nonrealizable CRstructures', Proc. of Symposia in Pure Math. 39 (1983), 401422. Ros
A.:
1. 'Spectral geometry of CRminimal submanifolds in the complex projective space', Kodai Math. J. 6 (1983), 8899. Ros A. and Verstraelen L.: 1. 'Solution of a conjecture by K. Ogiue concerning the sectional curvature of Kaehler submanifolds', to appear. Ro$ca. R.: 1. 'CRhypersurfaces A champs normal covariant decomposable inclues dans une variete pseudoriemannierine neutre', C.R. Acad. Sc. Paris, (Serie A) 292 (1981), 287290.
2. 'Sous varietes generiques dune variete sasakienne A champ vectorial antiinvariant recurrent', Riv. Mat. Univ. Parma 9 (1983), 125131. Sakamoto K. and Takemura Y.: 1. 'On almost contact structures belonging to a CRstructure', Kodai Math. J. 3 (1980), 144161.
2. 'Curvature invariants of CRmanifolds', Kodai Math. J. 4 (1980), 251265. Sasaki S.: 1. Almost Contact Manifolds, Lecture Notes, Tdhoku Univ., 1965.
Sasaki S. and Hatakeyama Y.: 1. 'On differentiable manifolds with certain structures which are closely related to almost contact structures II', T6hoku Math. J. 13 (1961), 281294. Sato N.: 1. 'Certain antiholomorphic submanifolds of almost Hermitian manifolds', Science Reports of Niigata Univ. 18 (1982), 19.
160
REFERENCES
2. 'Certain CRsubmanifolds of almost Hermitian manifolds', to appear. 3. 'On certain CRsubmanifolds of 6dimensional nearly Kaehlerian manifolds', to appear. 4. 'On antiholomorphic and CRsubmanifolds of a complex projective space', to appear. Sekigawa K.: 1. 'Some CRsubmanifolds in a 6dimensional sphere', Tensor N.S. 41 (1984), 1320.
Shimizu Y.: 1. 'On a construction of homogeneous CRsubmanifolds in a complex projective space', Commentarii Math. Univ. Sancti Pauli 32 (1983), 203207. Simionescu S.: 1. 'CRsubmanifolds of codimension two and (f, g, u, v, A)structures', to appear. Smyth B.: 1. 'Differential geometry of complex hypersurfaces', Ann. of Math. 85 (1967), 247266.
2. 'Homogeneous complex hypersurfaces', J. Math. Soc. Japan 20 (1968), 643647. Stong R.E.: 1. 'The rank of an fstructure', Kodai Math. Sem. Rep. 29 (1977), 207209.

Takagi R.: 1. 'On homogeneous real hypersurfaces in a complex projective space', Osaka J. Math. 10 (1973), 495506.
2. 'Real hypersurfaces in a complex projective space with constant principal curvatures, I, II', J. Math. Soc. Japan 27 (1975), 4353, 507516. Takemura Y.: 1. 'On the invariant submanifold of a CRmanifold', Kodai Math. J. 5 (1982), 416425. Tanaka N.: 1. 'On the pseudoconformal geometry of hypersurfaces of
REFERENCES
161
the space of n complex variables', J. Math. Soc. Japan 14 (1962), 397429. 2. 'On nondegenerate real hypersurfaces, graded Lie algebras and Cartan connections', Jap. J. Math., New Ser. 2 (1976), 131190. Tashiro Y.: 1. 'on contact structure of hypersurfaces in complex manifolds I, II', T6hoku Math. J. 15 (1963), 6278; 167175.
Tashiro Y. and Kim I.B.: 1. On almost contact metric compound structure, Kodai Math. J. 5 (1982), 1329. Teleman C.: 1. Local and Global Differential Geometry, Ed. TehnicA, Bucuresti, 1974 (in Romanian). Urbano F.: 1. CRsubmanifolds. Totally real submanifolds of quaternion manifolds, Thesis Univ. Granada, 1979 (in Spanish). Vaisman I.: 1. 'on locally conformal almost Kaehler manifolds', Israel J. Math. 24 (1976), 338351. Vanhecke L.: 1. 'On immersions with trivial normal connection in some almost Hermitian manifolds', Rendiconti di Mat. 10 (1977), 7586.

Verheyen P.: 1. Generic Submanifolds of Sasakian Manifolds, Mededelingen uit het Wiskundig Instituut, Katholieke Univ. Leuven, No. 157, 1982.
Verstraelen L.: 1. Three Types of Results in Differential Geometry of Submanifolds, Mededelingen uit het Wiskundig Instituut, Katholieke Univ. Leuven, No. 154, 1982. Walker A.G.: 1. 'Connections for parallel distributions in the large I, II', Quart. Math. Oxford 6 (1955), 3013081 9 (1958), 221231. 
REFERENCES
162
Wells R.O. Jr.: 1. 'Complex manifolds and mathematical physics', Bull. Amer. Math. Soc. (N.S.) 1 (1979), 296336.
2. Differential Analysis on Complex Manifolds, New York, Berlin, Springer, 1980. 3. 'The CauchyRiemann equations and differential geometry', Proc. of Symposia in Pure Math. 39 (1983), 423435. Willmore T.J.: 1. 'Parallel distributions on manifolds', Proc. London Math. Soc. 6 (1956), 191204.
2. Total Curvature in Riemannian Geometry, Ellis Horwood Limited, 1982. Yano K. 1. 'On a structure defined by a tensor field of type (1, 1)
satisfying f3 + f = 0, Tensor N.S. 14 (1963), 99109. 2. Differential Geometry on Complex and Almost Complex Spaces, Pergamon, New York, 1965. Yano K. and 1. 'Almost complex Reports
Ishihara S.: contact structures induced on hypersurfaces in and almost complex spaces', Kodai Math. Sem. 17 (1965), 222249.
2. 'Submanifolds with parallel mean curvature', J. Differential Geometry 6 (1971), 95118.
3. 'Real hypersurfaces of a complex manifold and distributions with complex structure', Kodai Math. J. 1 (1978), 289303. Yano K. and Kon M.: I.Antiinvariant Submanifolds, M. Dekker Inc., New York, 1976.
2. 'CRsousvaridtds d'un espace projectif complexe', C.R. Acad. Sc. Paris 288 (1979), 515517.
3. 'Generic submanifolds', Ann. di Mat. Pura Appl. 123 (1980), 5992. 4. 'Generic submanifolds of Sasakian manifolds', Kodai Math. J. 3 (1980), 163196. 5. 'Differential geometry of CRsubmanifolds', Geometriae
REFERENCES
163
Dedicata 10 (1981), 369391, 6. 'CRsubmanifolds of a complex projective space', J. Differential Geometry 16 (1981), 431444. 7. 'Contact CRsubmanifolds', Kodai Math. J. 5 (1982), 238252.

8. CRSubmanifolds of Kaehlerian and Sasakian Manifolds, BirkhAuser, Boston, 1983. Yoo I.Y.: 1. 'Einstein CRsubmanifolds of a flat Kaehlerian manifold', Honam Math. J. 6 (1984), 117122.
AUTHOR INDEX
Andreotti, A., 149 Arca, G., 100, 149 Barros, M., 126, 149 Bejancu, A., 25, 30, 32, 41, 42, 43, 48, 53, 60, 66, 78, 85, 88, 89, 90, 97, 98, 100, 101, 102, 104, 105, 106, 108, 109, 111, 113, 114, 117, 127, 149 Bishop, R. L., 45, 151 Blair, D. E., 16, 17, 24, 39, 44, 77, 128, 134, 151 Buchner, K., 152 Calabi, E.,94, 152 Calapso, M.T., 152 Cartan, E., 152
Chen, B. Y., 7, 24, 33, 34, 39, 40, 42, 43, 44, 45, 63, 65, 74, 76, 77, 85, 87, 88, 89, 93, 94, 95, 99, 114, 115, 126, 134, 149, 151, 152, 153
Chern, S. S., 153 Cihodariu, Gh., 153 Fukami, T., 14, 154 Gheorghiev, Gh., 154 Goldberg, S. I., 45, 128, 130, 151, 154 Goldberg, V. V., 154 Gray, A., 34, 154 Greenfield, S., 24, 154
Hatakeyama, Y., 56, 159 Hill, C. D., 149 Hsu, C. J., 100, 113, 154 Ianu$, S., 127, 154, 155 Ishihara, S., 14, 138, 141, 142, 155, 162 Jin, D. H., 155
AUTHOR INDEX
165
Kang, T. H., 158 Ki, U. H., 155 Kim, I. B., 127, 155, 161 Kim, Y. H., 155 Kobayashi, M., 100, 113, 155 Kobayashi, S., 13, 15, 155 Kon, M., 20, 33, 41, 48, 81, 83, 84, 85, 88, 89, 90, 94, 95, 100, 101, 112, 113, 151, 156, 162 Lawson, H. B. Jr., 156 Ludden, G. D., 114, 128, 152, 153, 156 Lue, H. S., 153 Maeda, S., 156 Maeda, Y., 156 Martinez, A., 127, 156 Matsumoto, K., 100, 127, 155, 157 Mihai, I., 127, 155, 157 Miron, R., 157 Montiel, S., 114, 115, 153 Moore, J. D., 85, 157 Moser, J. K., 153 Naitza, D., 157 Nirenberg, L., 148, 157 Ogiue, K., 20, 46, 93, 153, 157 Okumura, M., 53, 54, 79, 98, 99, 153, 156, 157 O'Neill, B., 5, 158 Oproiu, V., 154, 158 Ornea, L., 127, 158 Pak, J.S., 117, 155, 158 Papaghiuc, N., 100, 101, 102, 104, 105, 106, 113, 114, 127, 151, 158 Papuc, D., 158 Penrose, R., 143, 144, 159 P4rez, J. D., 156 Ros, A., 159 Rogca, R., 100, 113, 149, 152, 154, 159 Sakamoto, K., 142, 159 Santos, F. G., 127, 156 Sasaki, S., 56, 159
166
AUTHOR INDEX
Sato, N., 26, 28, 33, 156, 159 Sekigawa, K., 36, 38, 160 Shimizu, Y., 88, 160 Simionescu, S., 160 Smaranda, D., 127, 151, 153, 160 Smyth, B., 160 Stong, R.E., 160 Takagi, R., 160 Takemura, Y., 142, 159, 160 Tanaka, N., 141, 160 Tashiro, Y., 127, 161 Teleman, C., 161 Urbano, F., 27, 28, 126, 149, 161 Vaisman, I., 161 Vanhecke, L., 34, 161 Verheyen, P., 114, 115, 161 Verstraelen, L., 159, 161
Walker, A.G., 136, 162 Wells, R. 0. Jr., 128, 148, 162 Willmore, T.J., 162
Yano, K., 20, 33, 41, 48, 81, 83, 84, 85, 88, 89, 90, 94, 95, 100, 101, 107, 112, 113, 128, 130, 136, 138, 142, 151, 152, 154, 156, 162 Yoo, I. Y., 163
SUBJECT INDEX
affine antiholomorphic submanifold, 136 normal, 135 almost complex distribution, 128 complex manifold, 11 complex structure, 10 contact manifold, 15 contact metric manifold, 15 contact structure, 15 Hermitian manifold, 11 product structure, 9 quaternion manifold, 18 quaternion metric manifold, 18 antiholomorphic plane section, 95  holomorphic sectional curvature, 96 holomorphic submanifold, 21, 136 Codazzi equation, 6 complex CRstructure, 129 manifold, 11 space form, 13 structure, 11 submanifold, 20 contact antiholomorphic submanifold, 27 CRsubmanifold, 100 metric manifold, 16 cosymplectic antiholomorphic submanifold, 66 covariant derivative, 2 CRmanifold, 24 CRproduct, 32 CRsubmanifold, XI, 20 curvature tensor, 2 Dantiholomorphic sectional curvature, 96 Dgeodesic CRsubmanifold, 39 Dgeodesic QRsubmanifold, 118 Dmean curvature vector, 67
168
Dnormal fstructure, 133 distribution, 7 flat normal connection, 7 fundamental tensor of Weingarten, 5 fstructure with complemented frames, 130 Gauss equation, 6 formula, 6 generic semiinvariant product, 110 submanifold, 114, 134 Hermitian manifold, 11 metric, 11 holomorphic sectional curvature, 12 integrable distribution, 8 integral manifold, 8 involutive distribution, 8 Kaehlerian manifold, 11 leaf of distribution, 8 LeviCivita connection, 3 light cone, 143 linear connection, 1 locally Euclidean space, 4 mean curvature vector, 7 minimal distribution, 73 submanifold, 7 mixed foliate CRsubmanifold, 88 geodesic CRsubmanifold, 29 nearly Kaehlerian manifold, 12 Nijenhuis tensor, 11 nondegenerate CRmanifold, 141 normal almost contact manifold, 16 connection, 6 CRsubmanifold, 53 
parallel almost product structure, 9 distribution, 8 fstructure, 49 tensor field, 2 Penrose correspondence, 146 proper antiquaternion product, 125
SUBJECT INDEX
SUBJECT INDEX
CRsubmanifold, 21 CRproduct, 32 semiinvariant product, 110 pseudoconformal mapping, 139 conformal transformation, 140 conformal vector field, 140 Riemannian metric, 5 umbilical CRsubmanifolds, 46 purely real submanifold, 114 

quaternion distribution, 117 Kaehlerian manifold, 19 real submanifold, (QRsubmanifold), 116 sectional curvature, 19 space form, 19 real CRstructure, 128 space form, 4 Ricci curvature, 4 equation, 6 tensor field, 3 Riemannian connection, 3 curvature tensor, 3 manifold, 3 metric, 2 Sasakian antiholomorphic submanifold, 68 manifold, 16 space form, 16 scalar curvature, 4 second fundamental form, 5 sectional curvature, 4 semiflat normal connection, 94  invariant product, 106  invariant submanifold, 100 standard CRproduct, 86 torsionfree connection, 2 tensor, 2 totally contactgeodesic submanifold, 110 contactumbilical submanifold, 110 geodesic submanifold, 7 real submanifold, 20 totally umbilical submanifold, 7 twistor space, 145 Weingarten formula, 5
169
Mathematics and Its Applications
Managing Editor: MICHIEL HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Mathematics and Its Applications is devoted to interrelations such as:  a central concept which plays an important role in several different specialized areas of mathematics and/or science;  new applications of the results and ideas from one area of scientific endeavour into another;  the influence which the results, problems and concepts of one field of enquiry have, and have had, on the development of another. Presentday (applied) mathematics has some noteworthy features compared with even a few decennia ago:  Rapidly increasing mathematization of a large number of areas such as, e.g., physical chemistry, electrical engineering, and geology.  The enormous power of mathematical modeling which, when combined with powerful computers, enables us to carry out experiments which otherwise "in reality" would be too dangerous, too expensive, take too long, and be simply impossible.  The era of increasing specialization seems to be over. The time has come to start using all the powerful tools that have been forged.
All in all it looks like the next decennia will witness an unequalled flowering in the mathematical sciences. Based on these facts and this philosophy, the Mathematics and Its Applications book series is being developed quite vigorously. Given its success we plan to continue along the same lines often by means of specially commissioned volumes covering new areas of interaction likely to generate new kinds of problems, or covering a more established topic together with all its interrelations with others.
MAIN SERIES W. KUYK. Complementarity In Mathematics. A First Introduction to the Foundations of Mathematics and Its History. P. H. SELLERS. Combinatorial Complexes. A Mathematical Theory of Algorithms. J. CHAILLOU. Hyperbolic Differential Polynomials and Their Singular Perturbations. S. FUCIK. Solvability of Nonlinear Equations and Boundary Value Problems. W. L. MIRANKER. Numerical Methods for Stiff Equations and Singular Perturbation Problems.
P. M. COHN. Universal Algebra. V. 1. ISTRATESCU. Fixed Point Theory. An Introduction. N. E. HURT. Geometric Quantization In Action. Applications of Harmonic Analysis in Quantum Statistical Mechanics and Quantum Field Theory. P. M. ALBERTI and A. UHLMANN. Stochastkity and Partial Order. Doubly Stochastic Maps and Unitary Mixing. L. LANGOUCHE, D. ROEKAERTS, and E. TIRAPEGUI. Functional Integration and Semiclassical Expansions.
C. P. BRUTER, A. ARAGNOL, and A. LICHNEROWICZ (eds). Bifurcation Theory, Mechanics and Physics. Mathematical Developments and Applications. J. ACZEL (ed.). Functional Equations: History, Applications and Theory.
A. V. ARKHANGEL'SKII and V. I. PONOMAREV. Fundamentals of General Topology. Problems and Exercises. P. E. T. JORGENSEN and R. T. MOORE. Operator Commutation Relations. R. BELLMAN and G. ADOMIAN. Partial Differential Equations. New Methods for Their Treatment and Solution.
N. K. BOSE (ed.). Multidimensional Systems Theory. Progress, Directions and Open Problems in Multidimensional Systems.
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Assay of Spatially Random Material.
Y. CHERRUAULT. Mathematical Modelling In Biomedicine: Optimal Control of Biomedical Systems.
R. E. BELLMAN and R. S. ROTH. Methods In Approximation. Techniques for Mathematical Modelling. V. KOMKOV. Variational Principles of Continuum Mechanics with Engineering Applications. Volume I: Critical Points Theory. C. CUVELIER, A. SEGAL, and A. A. VAN STEENHOVEN. FiniteElement Methods and Equations. J. D. LOUCK and N. METROPOLIS. Symbolic Dynamics of Trapezoidal Maps. A. G. RAMM. Scattering by Obstacles.
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F.H. VASILESCU. Analytic Functional Calculus and Spectral Decompositions. W. KECS. The Convolution Product and Some Applications. K. REKTORYS. The Method of Discretization in Time and Partial Differential Equations. J. MOCKOR. Groups of Divisibility. L. IXARU. Numerical Methods for Differential Equations and Applications.
H. WALTHER. Ten Applications of Graph Theory. A. WAWRZYNCZYK. Group Representations and Special Functions. D. S. MITRINOVIC and J. KECKIC. The Duchy Method of Residues. Theory and Applications. V. BARBU and T. PRECUPANU. Convexity and Optimization I. Bunch Spaces. 1. BUCUR. Selected Topics in Algebra and its Interrelations with Logic, Number Theory and Algebraic Geometry. 1. M. STANCUMINASIAN. Stochastic Programming with Multiple Objective Functions.
J. SZEP and F. FORGO. Introduction to the Theory of Games. L. RERAN. Orthomodular Lattices. Algebraic Approach. A. MARCINIAK. Numerical Solutions of the NBody Problem. S. ROLEWICZ. Metric Linear Spaces. A. PAZMAN. Foundations of Optimum Experimental Design. K. WAGNER and G. WECHSUNG. Computational Complexity.
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