Foliations on Surfaces
Foliations is one of the major concepts of modern geometry and topology meaning a partition of topological space into a disjoint sum of leaves. This book is devoted to geometry and topology of surface foliations and their links to ergodic theory, dynamica
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A Series of Modern Surveys in Mathematics
Editorial Board S. Feferman, Stanford M. Gromov, Bures-sur-Yvette J. Jost, Leipzig J. Kollar, Princeton H.W. Lenstra, Jr., Berkeley P.-L. Lions, Paris M. Rapoport, Koln }.Tits, Paris D. B. Zagier, Bonn Managing Editor R. Remmert, Munster
Volume 41
Springer-Verlag Berlin Heidelberg GmbH
Igor Nikolaev
Foliations on Surfaces With 23 Figures
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Springer
Igor Nikolaev The Fields Institute for Research in Mathematical Sciences 222 College Street Toronto Ontario MST 3Jl Canada e-mail: [email protected]
Library of Congress Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Nikolaev, Igor: Foliations on surfaces Ilgor Nikolaev. (Ergebnisse der Mathematik und ihrer Grenzgebiete ; Folge 3, VoI. 41)
ISBN 978-3-642-08698-4
ISBN 978-3-662-04524-4 (eBook)
DOI 10.1007/978-3-662-04524-4
Mathematics Subject Classification (2000): Primary: 57Rxx,58Fxx Secondary: 05Cxx, 28Fxx, 30Exx, 34Cxx, 46Lxx
ISSN 0071-1136 ISBN 978-3-642-08698-4 This work is subject to copyright. AII rights are reserved, 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 ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are Iiable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 2001 Originally published by Springer-Verlag Berlin Heidelberg New York in 2001 Softcover reprint of the hardcover lst edition 2001
Typeset by the author. Edited and reformatted by Ludwig Feuchte, Flein, using a Springer TEX macro package. Printed on acid-free paper SPIN 10723139 4413142LK - 5 43210
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If you can dream - and not make dreams your master, If you can think - and not make thoughts your aim ..
Rudyard Kipling: 'If-' ("Poems. Short Stories") We are what we think. All that we are arises with our thoughts, With our thoughts we make the World ... Dhammapada : Choices ("The Sayings of the Buddha")
Foreword
H. Poincare, the founder of the qualitative theory of differential equations, was the first to realize the significance of the simple fact that the trajectories of a smooth vector field, v, determine a geometric picture called the phase portmit of v. From a general viewpoint, such a picture can be considered as a locally oriented one-dimensional foliation, F, with singularities. Actually, these two geometric concepts are equivalent because for a given smooth foliation F one can determine the corresponding vector field v up to a change of time. There is a vast literature devoted to the case of locally oriented foliations on two-dimensional manifolds. Let me only mention two recent texts: S. Aranson, G. Belitsky, E. Zhuzhoma, Introduction
