Differential Geometry of Lightlike Submanifolds
This is the first systematic account of the main results in the theory of lightlike submanifolds of semi-Riemannian manifolds which have a geometric structure, such as almost Hermitian, almost contact metric or quaternion Kähler. Using these structures, t
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Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta) Benoît Perthame (Université Pierre et Marie Curie, Paris) Laurent Saloff-Coste (Cornell University, Ithaca) Igor Shparlinski (Macquarie University, New South Wales) Wolfgang Sprössig (TU Bergakademie Freiberg) Cédric Vilani (Ecole Normale Supérieure, Lyon)
Krishan L. Duggal Bayram Sahin
Differential
Geometry of Lightlike
Submanifolds
Birkhäuser Basel . Boston . Berlin
Authors: Krishan L. Duggal Department of Mathematics and Statistics University of Windsor 401 Sunset Avenue Windsor, Ontario, N9B 3P4 Canada e-mail: [email protected]
Bayram Sahin Department of Mathematics Inonu University 44280 Malatya Turkey e-mail: [email protected]
2000 Mathematics Subject Classification: 53B25, 53C50, 53B50, 53C42, 53C15
Library of Congress Control Number: 2009942369
Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .
ISBN 978-3-0346-0250-1 Birkhäuser Verlag AG, Basel · Boston · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2010 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Cover design: Birgit Blohmann, Zürich, Switzerland Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-0346-0250-1
e-ISBN 978-3-0346-0251-8
987654321
www.birkhauser.ch
Contents Preface
vii
Notation
xi
1 Preliminaries 1.1 Semi-Euclidean spaces . . . 1.2 Semi-Riemannian manifolds 1.3 Warped product manifolds . 1.4 Lightlike manifolds . . . . .
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2 Lightlike hypersurfaces 2.1 Basic general results . . . . . . . . . . . 2.2 Screen conformal hypersurfaces . . . . . 2.3 Unique existence of screen distributions 2.4 Induced scalar curvature . . . . . . . . . 2.5 Lightlike Einstein hypersurfaces . . . . . 2.6 Semi-symmetric hypersurfaces . . . . . .
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3 Applications of lightlike hypersurfaces 3.1 Global null splitting theorem . . . . . 3.2 Killing horizons . . . . . . . . . . . . . 3.3 Dynamical horizons . . . . . . . . . . . 3.4 Conformal Killing horizons . . . . . . 3.5 Differential operators on hypersurfaces 3.6 Osserman lightlike hypersurfaces . . .
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