CR Submanifolds of Kaehlerian and Sasakian Manifolds
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Edited by J. Coates and S. Helgason
Birkhauser Boston· Basel· Stuttgart
Kentaro Yano Masahiro Kon
CR Submanifolds of Kaehlerian and Sasakian Manifolds
1983
Birkhauser Boston • Basel • Stuttgart
Authors: Kentaro Yano Department of Mathematies Tokyo Institute of Teehno109Y Ohokayama, Meguro-ku Tokyo, 152 Japan Masahiro Kon Hirosaki University Hirosaki Japan
Library of Congress Cata10ging in Pub1ieation Data Yano, Kentaro, 1912CR submanifo1ds of Kaeh1erian and Sasakian manifo1ds. (Progress in mathematies v. 30) Bib1iography: p. 198 Inde~udes indexes. 1. Kah1erian manifo1ds. 2. Sasakian manifo1ds. 3. Submanifo1ds, CR. I. Kon, Masahiro. 11. Tit1e. 111. Tit1e: C.R. submanifo1ds of Kaeh1erian and Sasakian manifo1ds. IV. Series: Progress in mathematies (Cambridge, Mass.) ; v. 30. QA649.Y29 1983 516.3'604 82-22752 ISBN 978-1-4684-9426-6
CIP-Kurztitelaufnahme der Deutschen Bibliothek Yano, Kentaro: CR submanifolds of Kaeh1erian and Sasakian manifolds / Kentaro Yano ; MasHhiro Kon. - 30ston ; Basel ; Stuttgart : Birkhauser, 1983 (Progress in mathematics ; Vo1. 30)
ISBN 978-1-4684-9426-6 ISBN 978-1-4684-9424-2 (eBook) DOI 10.1007/978-1-4684-9424-2
NE: Kon, Masahiro:; GT
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or ötherwise, without prior permission of the copyright owner. ©Birkhiluser Boston, Ine., 1983 Softcover reprint ofthe hardcover 1st edition 1983 ISBN 978-1-4684-9426-6
IN1RODUCTION
Let M be an almost Hermitian manifold with almost complex structure tensor J. We consider a submanifold M of M and denote by T (M) and x
Tx (MO~ the tangent space and the normal space of M at x E M respectively. If Tx(MO is invariant under the action of J for each x e M, that is, if JTx(M) = Tx(M) for each x holomorphic) submanifold of
~
M.
M, then M is called an invariant (or
On the other hand, if the transform of
Tx (M) by J is contained in the normal space Tx (M)~ for each xE M, that is, if JTx (M) C Tx (M)",- for each x E M, then M is called an anti - invariant (or totally real) submanifold of M. Similar definitions apply to submanifolds of an almost contact metric manifold.
An invariant submanifold inherits almost all properties of the ambient almost Hermitian manifold and so the study of invariant submanifolds is not so interesting from the point of view of the geometry of submanifolds. On the other hand, the theory of anti-invariant submanifolds proved to be a very nice topic in modern differential geometry and has been studied by many authors in 1970's (see our Lecture Notes [63]). In 1978, generalizing these ideas, A. Bejancu [2], [4] defined CR (Cauchy-Riemann) submanifolds of an almost Hermitian manifold as follows: Let M be an almost Hermitian manifold with almost complex structure tensor J. A submanifold M of M is called a CR submanifold of M if there exists a differentiable distribution
D :
x
~
satisfying the two conditions: (i) D is invariant, that is, J
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