Projective and Cayley-Klein Geometries
Projective geometry, and the Cayley-Klein geometries embedded into it, were originated in the 19th century. It is one of the foundations of algebraic geometry and has many applications to differential geometry. The book presents a systematic introduction
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Arkady L. Onishchik • Rolf Sulanke
Projective and Cayley-Klein Geometries With 69 Figures
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Arkady L Onishchik Faculty of Mathematics Yaroslavl State University Sovetskaya 14 150000 Yaroslavl, Russia e-mail: [email protected] RolfSulanke Institut fiir Mathematik Humboldt-Universitat zu Berlin Rudower Chaussee 25 10099 Berlin, Germany e-mail: [email protected]
Library of Congress Control Number: 2006930592
Mathematics Subject Classification (2000): 51-01,51N15,51N30, 51M10,51A50,57S25 ISSN 1439-7382 ISBN-10 3-540-35644-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-35644-8 Springer Berlin Heidelberg New York This workis subject to copyright. All 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 microfilm or in any other way, 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. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 The use of general descriptive names, registered names, trademarks, 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. Typesetting by the authors using a Springer TgX macro package Production: LE-TgX Jelonek, Schmidt & Vockler GbR, Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper
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Preface
Projective geometry, and the Cayley-Klein geometries embedded into it, are rather ancient topics of geometry, which originated in the 19th century with the work of V. Poncelet, J. Gergonne, Ch. v. Staudt, A.-F. Mobius, A. Cayley, F. Klein, S. Lie, N. L Lobatschewski, and many others. Ahhough this field is one of the foundations of algebraic geometry and has many applications to differential geometry, it has been widely neglected in the teaching at German universities — and not only there. In the more recent mathematical literature these classical aspects of geometry are also scarcely taken into account. In the present book, the synthetic projective geometry and some of its recent applications, e.g. of the finite geometries, are mentioned only in passing, i.e., they form the content of some remarks. Instead, we intend to present a systematic introduction of projective geometry as based on the notion of vector space, which is the central topic of the first chapter. In the second chapter the most important classical geometries are systematicaUy developed foUowing the principles founded by A. Cayley and F. Klein, which rely on distinguishing an absolute and then studying the resulting invariants of geometric objects
