Submersions of CR Submanifolds
O’Neill introduced the notion of Riemannian submersions (cf. O’Neill, Mich. Math. J. 13, 459–469, 1966, [28 ]). For the submersion \(\pi :M\longrightarrow N\) of a CR submanifold M of a Kaehler manifold \(\bar{M}\) onto an almost Hermitian manifold N, Kob
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metry of Cauchy–Riemann Submanifolds
Geometry of Cauchy–Riemann Submanifolds
Sorin Dragomir Mohammad Hasan Shahid Falleh R. Al-Solamy •
Editors
Geometry of Cauchy–Riemann Submanifolds
123
Editors Sorin Dragomir Dipartimento di Matematica e Informatica Università degli Studi della Basilicata Potenza Italy
Falleh R. Al-Solamy Department of Mathematics King Abdul Aziz University Jeddah Saudi Arabia
Mohammad Hasan Shahid Department of Mathematics Faculty of Natural Sciences Jamia Millia Islamia New Delhi, Delhi India
ISBN 978-981-10-0915-0 DOI 10.1007/978-981-10-0916-7
ISBN 978-981-10-0916-7
(eBook)
Library of Congress Control Number: 2016935695 © Springer Science+Business Media Singapore 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Science+Business Media Singapore Pte Ltd.
Preface
The present volume gathers contributions by several experts in the theory of isometric immersions between Riemannian manifolds, and focuses on the geometry of CR structures on submanifolds in Hermitian manifolds. CR structures are a bundle theoretic recast of the tangential Cauchy–Riemann equations in complex analysis in several complex variables. Let X Cn (n ≥ 2) be an open set and let @f
@f dz j ¼ 0 @z j
ð1Þ
be the ordinary Cauchy–Riemann equations in Cn . A function f 2 C1 ðX; CÞ is holomorphic in X if f satisfies (1) everywhere in X. Let M be an embedded real hypersurface in Cn such that U ¼ M \ X 6¼ ; and let us set T1;0 ðMÞx ¼ ½Tx ðMÞ R C \ T 1;0 ðCn Þx ;
x 2 M;
ð2Þ
where T 1;0 ðCn Þ is the holomorphic tangent bundle over Cn (the span of f@=@z j : 1 j ng). Then T1;0 ðMÞ is a rank n − 1 complex vector bundle over M, referred to as the CR structure of M (induced on M by the complex structure of the ambient space Cn ) and one may consider the first order differential operator @ b : C1 ðM; CÞ ! C T0;1 ðMÞ ; ð3Þ ð@ b uÞZ ¼ ZðuÞ;
u 2 C 1 ðM; CÞ;
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