Geometry of CR-Submanifolds Mathematics and Its Applications (East European Series)

Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Editorial Board: A. BIAL YNICKI-BIRULA, 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. LOV Asz, Eiitviis Lorond University, Budapest, Hungary D. S. MITRINOVIC, 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, Sofia, Bulgaria H. TRIEBEL, University oflena, D.D.R. Aurel Bejancu

Department of Mathematics, Polytechnic Institute of Ia~i, Romania

Geometry of CR-Submanifolds

D. Reidel Publishing Company

A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP

Dordrecht / Boston / Lancaster / Tokyo Library of Congress Cataloging-in-Publication Data

Bejancu, Aurel, 1946- Geometry of CR-submanifolds.

(Mathematics and its applications. East European series) Bibliography: p. Includes indexes. 1. Submanifolds, CR. 2. Geometry, Differential. I. Title. II. Series: Mathematics and its applications (D. Reidel Publishing Company). East European series. QA649.B44 1986 516.3'6 86-15614 ISBN -13: 978-94-0 I 0-8545-8 e-ISBN -13: 978-94-009-4604-0 DOl: 10.1 007/978-94-009-4604-0

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All Rights Reserved © 1986 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover I st edition 1986 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 To Ligia, Rodica - Daniela and Aurelian TABLE OF CONTENTS

EDITOR'S PREFACE lX

PREFACE ~

CHAPTER I: Differential-Geometrical Structures on 1. Linear connections on a 1 2. The Levi-Civita connection 2 3. Submanifolds of a Riemannian manifold 5 4. Distributions on a manifold 7 5. Kaehlerian manifolds 10 6. Sasakian manifolds 15 7. Quaternion Kaehlerian manifolds 18

CHAPTER II: CR-Submanifolds of Almost Hermitian Manifolds 1. CR-submanifolds and CR-structures 20 2. Integrability of distributions on a CR-submanifold 24 3. qJ-connections on a CR-submanifold and CR-products of al- most Hermitian manifolds 30 4. The non-existence of CR-products in S6 34

CHAPTER III: CR-Submanifolds of Kaehlerian Manifolds 1. Integrability of distributions and geometry of leaves 39 2. Umbilical CR-submanifolds of Kaehlerian manifolds 43 3. Normal CR-submanifolds of Kaehlerian manifolds 50 4. Normal anti-holomorphic submanifolds of Kaehlerian mani- folds 56 5. CR-products in Kaehlerian manifolds 63 6. Sasakian anti-holomorphic submanifolds of Kaehlerian manifolds 67 7. of CR-submanifolds 73 viii TABLE OF CONTENTS

CHAPTER IV: CR-Submanifolds of Complex Space Forms 1. Characterization of CR-submanifolds in complex space forms 77 2. Riemannian fibre bundles and anti-holomorphic submani- folds of epn 79 3. CR-products of complex space forms 84 4. Mixed foliate CR-submanifolds of complex space forms 88 5. CR-submanifolds with semi-flat normal connection 94 6. Pinching theorems for sectional curvatures of CR-submani- folds 95

CHAPTER V: Extensions of CR-Structures to Other Geometrical Structures 1. Semi-invariant submanifolds of Sasakian manifolds 100 2. Semi-invariant products of Sasakian manifolds 106 3. Semi-invariant submanifolds with flat normal connection 112 4. Generic submanifolds of Kaehlerian manifolds 114 5. QR-submanifolds of quaternion Kaehlerian manifolds 115 6. Totally umbilical and toally geodesic QR-submanifolds of quaternion Kaehlerian manifolds 122

CHAPTER VI: CR-Structures and Pseudo - Conformal Mappings 1. CR-manifolds and f-structures with complemented frames 128 2. Generic submanifolds of complex manifolds 134 3. Anti-holomorphic submanifolds of complex manifolds 136 4. Pseudo-conformal mappings 138

CHAPTER VII: CR-Structures and Relativity 1. Geometrical Structures of space-time 143 2. The twistor space and Penrose correspondence 145 3. Physical interpretations of CR-structures 147

REFERENCES 149

AUTHOR INDEX 164

SUBJECT INDEX 167 EDITOR'S PREFACE

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G.K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders.

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 (non• trivially) 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 us;; Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. This pro• gramme, Mathematics and Its Applications, is devoted to new emerging (sub)disciplines and to such (new) interrelations as exempla gratia: - 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. x EDITOR'S PREFACE

The present volume in the MIA (Eastern Europe) series deals with a topic in : Cauchy-Riemann submanifolds of Kiihlerian 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 sub• stantial 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 submani• folds, 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 up-to-date and self-contained book on the topic.

The unreasonable effectiveness of mathemat• As long as algebra and geometry proceeded ics in science ... along separate paths, their advance was slow and their applications limited. Eugene Wigner But when these sciences joined company they drew from each other fresh vitality and Well, if you know of a better'ole, go to it. thenceforward marched on at a rapid pace towards perfection. Bruce Bairnsfather Joseph Louis Lagrange. What is now proved was once only ima• gined.

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 Yano-Kon [1]) and CR- (Cauchy-Riemann) submanifolds. The notion of a CR-submanifold has been introduced by the author in [1] as follows: Let N be an almost and let J be the almost complex structure of N. A real submanifold M of N is called a CR-submanifold if there exists a differentiable distribution D on M satisfying (i) J(D) D x = x and 1. 1. (ii) J(Dx ) C TxM , 1. . for each x E M, wereh D ~s 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 CR-submanifolds. Moreover, each real hypersurface of N is a CR-submanifold 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 CR-submanifolds. In order to make it self-contained as much as pus sible we arrange in Chapter I most of the required background material. Chapter II is devoted to the differential geometry of CR-submanifolds of almost Hermitian manifolds. The integrability of both of the distributions D and D1. on a CR-submanifold are studied and there is obtained a class of linear connections with respect to which the CR-structure is parallel. Also, it is proven that CR-products do not exist in a sphere 86 . In Chapter III we give results on some special classes PREFACE of CR-submanifolds of Kaehlerian manifolds: umbilical CR• submanifolds, normal CR-submanifolds, CR-products, Sasakian anti-holomorphic submanifolds. Also, the cohomology of CR-submanifolds is studied. In Chapter IV we are concerned with CR-submanifolds of complex space forms. We first discuss the method of Riemannian fibre bundles in the geometry ofCR-submanifolds. We include here various results on mixed foliate CR• submanifolds, CR-products, generic submanifolds and CR-submanifolds with semi-flat normal connection. In Chapter V we show that the theory of CR-submanifolds 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 CR-submanifolds with the general theory of CR-manifolds is studied in Chapter VI. By means of some local f-structures with complemented frames we obtain results on pseudo-conformal mappings between CR-manifolds. Finally, we show in Chapter VII an application of CR-structures to relativity. By Penrose correspondence we have an interesting passing from the geometry of a Minkowski space to the geometry of a CR-submanifold. 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. Verstraelen (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