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Victor Andreevich Toponogov with the editorial assistance of Vladimir Y. Rovenski Differential Geometry of Curves and Surfaces A Concise Guide Birkhauser¨ Boston • Basel • Berlin Victor A. Toponogov (deceased) With the editorial assistance of: Department of Analysis and Geometry Vladimir Y. Rovenski Sobolev Institute of Mathematics Department of Mathematics Siberian Branch of the Russian Academy University of Haifa of Sciences Haifa, Israel Novosibirsk-90, 630090 Russia Cover design by Alex Gerasev. AMS Subject Classification: 53-01, 53Axx, 53A04, 53A05, 53A55, 53B20, 53B21, 53C20, 53C21 Library of Congress Control Number: 2005048111 ISBN-10 0-8176-4384-2 eISBN 0-8176-4402-4 ISBN-13 978-0-8176-4384-3 Printed on acid-free paper. c 2006 Birkhauser¨ Boston All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkhauser¨ Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA) and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and re- trieval, 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 in the United States of America. (TXQ/EB) 987654321 www.birkhauser.com Contents Preface ....................................................... vii About the Author .............................................. ix 1 Theory of Curves in Three-dimensional Euclidean Space and in the Plane ....................................................... 1 1.1 Preliminaries . ............................................ 1 1.2 Definition and Methods of Presentation of Curves . .............. 2 1.3 Tangent Line and Osculating Plane . .......................... 6 1.4 Length of a Curve .......................................... 11 1.5 Problems: Convex Plane Curves .............................. 15 1.6 Curvature of a Curve . ...................................... 19 1.7 Problems: Curvature of Plane Curves .......................... 24 1.8 Torsion of a Curve .......................................... 45 1.9 The Frenet Formulas and the Natural Equation of a Curve . ........ 47 1.10 Problems: Space Curves ..................................... 51 1.11 Phase Length of a Curve and the Fenchel–Reshetnyak Inequality . 56 1.12 Exercises to Chapter 1 ...................................... 61 2 Extrinsic Geometry of Surfaces in Three-dimensional Euclidean Space ...................................................... 65 2.1 Definition and Methods of Generating Surfaces . .............. 65 2.2 The Tangent Plane .......................................... 70 2.3 First Fundamental Form of a Surface .......................... 74 vi Contents 2.4 Second Fundamental Form of a Surface ........................ 79 2.5 The Third Fundamental Form of a Surface . .................... 91 2.6 Classes of Surfaces . ...................................... 95 2.7 Some Classes of Curves on a Surface ..........................114 2.8 The Main Equations of Surface Theory ........................127 2.9 Appendix: Indicatrix of a Surface of Revolution .................139 2.10 Exercises to Chapter 2 ......................................147 3 Intrinsic Geometry of Surfaces .................................151 3.1 Introducing Notation . ......................................151 3.2 Covariant Derivative of a Vector Field .........................152 3.3 Parallel Translation of a Vector along a Curve on a Surface . ......................................153 3.4 Geodesics .................................................156 3.5 Shortest Paths and Geodesics . ................................161 3.6 Special Coordinate Systems . ................................172 3.7 Gauss–Bonnet Theorem and Comparison Theorem for the Angles of a Triangle . ............................................179 3.8 Local Comparison Theorems for Triangles . ....................184 3.9 Aleksandrov Comparison Theorem for the Angles of a Triangle ....189 3.10 Problems to Chapter 3 . ......................................195 References ..................................................199 Index .......................................................203 Preface This concise guide to the differential geometry of curves and surfaces can be recommended to first-year graduate students, strong senior students, and students specializing in geometry. The material is given in two parallel streams. The first stream contains the standard theoretical material on differential geom- etry of curves and surfaces. It contains a small number of exercises and simple problems of a local nature. It includes the whole of Chapter 1 except for the prob- lems (Sections 1.5, 1.7, 1.10) and Section 1.11, about the phase length of a curve, and the whole of Chapter 2 except for Section 2.6, about classes of surfaces, The- orems 2.8.1–2.8.4, the problems (Sections 2.7.4, 2.8.3) and the appendix (Sec- tion 2.9). The second stream contains more difficult and additional material and formu- lations of some complicated but important theorems, for example, a proof of A.D. Aleksandrov’s comparison theorem about the angles of a triangle on a convex surface,1 formulations of A.V. Pogorelov’s theorem about rigidity of convex sur- faces, and S.N. Bernstein’s theorem about saddle surfaces. In the last case, the formulations are discussed in detail. A distinctive feature of the book is a large collection (80 to 90) of nonstandard and original problems that introduce the student into the real world of geometry. Most of these problems are new and are not to be found in other textbooks or books of problems. The solutions to them require inventiveness and geometrical intuition. In this respect, this book is not far from W. Blaschke’s well-known 1 A generalization of Aleksandrov’s global angle comparison theorem to Riemannian spaces of ar- bitrary dimension is known as Toponogov’s theorem. viii Preface manuscript [Bl], but it contains a number of problems more contemporary in theme. The key to these problems is the notion of curvature: the curvature of a curve, principal curvatures, and the Gaussian curvature of a surface. Almost all the problems are given with their solutions, although the hope of the author is that an honest student will solve them without assistance, and only in excep- tional cases will look at the text for a solution. Since the problems are given in increasing order of difficulty, even the most difficult of them should be solvable by a motivated reader. In some cases, only short instructions are given. In the au- thor’s opinion, it is the large number of original problems that makes this textbook interesting and useful. Chapter 3, Intrinsic Geometry of a Surface, starts from the main notion of a covariant derivative of a vector field along a curve. The definition is based on extrinsic geometrical properties of a surface. Then it is proven that the covariant derivative of a vector field is an object of the intrinsic geometry of a surface, and the later training material is not related to an extrinsic geometry. So Chapter 3 can be considered an introduction to n-dimensional Riemannian geometry that keeps the simplicity and clarity of the 2-dimensional case. The main theorems about geodesics and shortest paths are proven by methods that can be easily extended to n-dimensional situations almost without alteration. The Aleksandrov comparison theorem, Theorem 3.9.1, for the angles of a triangle is the high point in Chapter 3. The author is one of the founders of CAT(k)-spaces theory,2 where the com- parison theorem for the angles of a triangle, or more exactly its generalization by the author to multidimensional Riemannian manifolds, takes the place of the basic property of CAT(k)-spaces. Acknowledgments. The author gratefully thanks his student and colleagues who have contributed to this volume. Essential help was given by E.D. Rodionov, V.V.Slavski, V.Yu.Rovenski, V.V.Ivanov, V.A.Sharafutdinov, and V.K.Ionin. 2 The initials are in honor of E. Cartan, A.D. Aleksandrov, and V.A. Toponogov. About the Author Professor Victor Andreevich Toponogov, a well-known Russian geometer, was born on March 6, 1930, and grew up in the city of Tomsk, in Russia. During To- ponogov’s childhood, his father was subjected to Soviet repression. After finish- ing school in 1948, Toponogov entered the Department of Mechanics and Math- ematics at Tomsk University, and graduated in 1953 with honors. In spite of an active social position and receiving high marks in his studies, the stamp of “son of an enemy of the people” left Toponogov with little hope of continuing his education at the postgraduate level. However, after Joseph Stalin’s death in March 1953, the situation in the USSR changed, and Toponogov became a postgraduate student at Tomsk University. Toponogov’s scientific interests were influenced by his scientific advisor, Professor A.I. Fet (a recognized topologist and specialist in variational calculus in the large, a pupil of L.A. Lusternik) and by the works of Academician A.D. Aleksandrov.1 In 1956, V.A. Toponogov moved to Novosibirsk, where in April 1957 he be- came a research scientist at the Institute of Radio-Physics and Electronics, then directed by the
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