Topics in Extrinsic Geometry of Codimension-One Foliations
Extrinsic geometry describes properties of foliations on Riemannian manifolds which can be expressed in terms of the second fundamental form of the leaves. The authors of Topics in Extrinsic Geometry of Codimension-One Foliations achieve a technical tour
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Vladimir Rovenski • Paweł Walczak
Topics in Extrinsic Geometry of Codimension-One Foliations
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Vladimir Rovenski Department of Mathematics University of Haifa 31905 Haifa, Israel [email protected]
Paweł Walczak Department of Mathematics University of Lodz 90238 Lodz, Poland [email protected]
ISSN 2191-8198 e-ISSN 2191-8201 ISBN 978-1-4419-9907-8 e-ISBN 978-1-4419-9908-5 DOI 10.1007/978-1-4419-9908-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011931089 Mathematics Subject Classification (2010): 53C12, 35L45, 26B20 © Vladimir Rovenski and Paweł Walczak 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
V. Rovenski dedicates this book to his parents: Ira Rushanova and Yuzef Rovenski P. Walczak dedicates this book to his family.
Foreword
The authors of this work asked me to read it and write a foreword. I did so with pleasure because differential geometry of foliations was one of my research subjects decades ago. Foliations, i.e., partitions into submanifolds of a constant, lower dimension, are beautiful structures on manifolds that encode a lot of geometric information. The topological study of foliations was initiated by Ch. Ehresmann and G. Reeb in the 1940s and soon became a research subject of many mathematicians. In particular, the study of the smooth case and of the differential geometric aspects became an important part of foliation theory, developed in the early stages by B. Reinhart, R. Bott, F. Kamber, Ph. Tondeur, P. Molino, and many others. The present work is a research monograph and is addressed to readers who have enough knowledge of differential and Riemannian geometry. Its first two chapters are devoted to the development of a computational machinery that provides integral and variational formulas for the most general, extrinsic invariants of the leaves of a foliation of a Riemannian manifold. The third chapter defines a very general notion of extrinsic geometric flow and studies the evolution of the leaf-wise Riemannian metric along the trajectories of this flow. The authors give existence theorems and estimations of the maximal evolution time and make a study of soliton solutions. The authors of the present
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