Lie Sphere Geometry With Applications to Submanifolds
This book provides a clear and comprehensive modern treatment of Lie sphere geometry and its applications to the study of Euclidean submanifolds. It begins with the construction of the space of spheres, including the fundamental notions of oriented contac
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S. Axler K.A. Ribet
Thomas E. Cecil
Lie Sphere Geometry With Applications to Submanifolds
Second Edition
Thomas E. Cecil Department of Mathematics and Computer Science College of the Holy Cross 1 College Street Worcester, MA 01610 [email protected] Editorial Board (North America): K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA [email protected]
S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
ISBN: 978-0-387-74655-5
e-ISBN: 978-0-387-74656-2
Library of Congress Control Number: 2007936690 Mathematics Subject Classification (2000): 53-02, 53A07, 53A40, 53B25, 53C40, 53C42 ©2008 Springer Science+Business Media, LLC 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, U.S.A.), 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. 9 8 7 6 5 4 3 2 1 springer.com
(JLS/SB)
To my sons, Tom, Mark, and Michael
Preface to the First Edition
The purpose of this monograph is to provide an introduction to Lie’s geometry of oriented spheres and its recent applications to the study of submanifolds of Euclidean space. Lie [104] introduced his sphere geometry in his dissertation, published as a paper in 1872, and used it in his study of contact transformations. The subject was actively pursued through the early part of the twentienth century, culminating with the publication in 1929 of the third volume of Blaschke’s [10] Vorlesungen über Differentialgeometrie, which is devoted entirely to Lie sphere geometry and its subgeometries. After this, the subject fell out of favor until 1981, when Pinkall [146] used it as the principal tool in his classification of Dupin hypersurfaces in R 4 . Since that time, it has been employed by several geometers in the study of Dupin, isoparametric and taut submanifolds. This book is not intended to replace Blaschke’s work, which contains a wealth of material, particularly in dimensions two and three. Rather, it is meant to be a relatively brief introduction to the subject, which leads the reader to the frontiers of current research in this part of submanifold theory. Chapters 2 and 3 (chapter numbers from the second edition) are accessible to a beginning graduate student who has taken courses in linear and abstract algebra and projective geometry. Chapters 4 and 5 contain the applications to submanifold theory. These chapters require a
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