Differential Geometry of Curves and Surfaces

Transcription

1 Victor Andreevich Toponogov with the editorial assistance of Vladimir Y. Rovenski Differential Geometry of Curves and Surfaces A Concise Guide Birkhäuser Boston Basel Berlin

2 Victor A. Toponogov (deceased) Department of Analysis and Geometry Sobolev Institute of Mathematics Siberian Branch of the Russian Academy of Sciences Novosibirsk-90, Russia With the editorial assistance of: Vladimir Y. Rovenski Department of Mathematics University of Haifa Haifa, Israel Cover design by Alex Gerasev. AMS Subject Classification: 53-01, 53Axx, 53A04, 53A05, 53A55, 53B20, 53B21, 53C20, 53C21 Library of Congress Control Number: ISBN eisbn ISBN Printed on acid-free paper. c 2006 Birkhäuser Boston All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser 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 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 in the United States of America. (TXQ/EB)

3 Contents Preface... About the Author... vii ix 1 Theory of Curves in Three-dimensional Euclidean Space and in the Plane Preliminaries Definition and Methods of Presentation of Curves Tangent Line and Osculating Plane Length of a Curve Problems: Convex Plane Curves Curvature of a Curve Problems: Curvature of Plane Curves Torsion of a Curve The Frenet Formulas and the Natural Equation of a Curve Problems: Space Curves Phase Length of a Curve and the Fenchel Reshetnyak Inequality Exercises to Chapter Extrinsic Geometry of Surfaces in Three-dimensional Euclidean Space Definition and Methods of Generating Surfaces The Tangent Plane First Fundamental Form of a Surface... 74

4 vi Contents 2.4 Second Fundamental Form of a Surface The Third Fundamental Form of a Surface Classes of Surfaces Some Classes of Curves on a Surface The Main Equations of Surface Theory Appendix: Indicatrix of a Surface of Revolution Exercises to Chapter Intrinsic Geometry of Surfaces Introducing Notation Covariant Derivative of a Vector Field Parallel Translation of a Vector along a Curve on a Surface Geodesics Shortest Paths and Geodesics Special Coordinate Systems Gauss Bonnet Theorem and Comparison Theorem for the Angles of a Triangle Local Comparison Theorems for Triangles Aleksandrov Comparison Theorem for the Angles of a Triangle Problems to Chapter References Index...203

5 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 geometry 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 problems (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, Theorems , the problems (Sections 2.7.4, 2.8.3) and the appendix (Section 2.9). The second stream contains more difficult and additional material and formulations 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 surfaces, 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 arbitrary dimension is known as Toponogov s theorem.

6 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 exceptional 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 author 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 comparison 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.

7 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 Toponogov s childhood, his father was subjected to Soviet repression. After finishing school in 1948, Toponogov entered the Department of Mechanics and Mathematics 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 became a research scientist at the Institute of Radio-Physics and Electronics, then directed by the well-known physicist Y.B. Rumer. In December 1958, Toponogov defended his Ph.D. thesis at Moscow State University. In his dissertation, the Aleksandrov convexity condition was extended to multidimensional Riemannian manifolds. Later, this theorem came to be called the Toponogov (comparison) theorem. 2 In April 1961, Toponogov moved to the Institute of Mathematics and 1 Aleksandr Danilovich Aleksandrov ( ). 2 Meyer, W.T. Toponogov s Theorem and Applications. Lecture Notes, College on Differential Geometry, Trieste

8 x About the Author Computer Center of the Siberian Branch of the Russian Academy of Sciences at its inception. All his subsequent scientific activity is related to the Institute of Mathematics. In 1968, at this institute he defended his doctoral thesis on the theme Extremal problems for Riemannian spaces with curvature bounded from above. From 1980 to 1982, Toponogov was deputy director of the Institute of Mathematics, and from 1982 to 2000 he was head of one of the laboratories of the institute. In 2001 he became Chief Scientist of the Department of Analysis and Geometry. The first thirty years of Toponogov s scientific life were devoted to one of the most important divisions of modern geometry: Riemannian geometry in the large. From secondary-school mathematics, everybody has learned something about synthetic methods in geometry, concerned with triangles, conditions of their equality and similarity, etc. From the Archimedean era, analytical methods have come to penetrate geometry: this is expressed most completely in the theory of surfaces, created by Gauss. Since that time, these methods have played a leading part in differential geometry. In the fundamental works of A.D. Aleksandrov, synthetic methods are again used, because the objects under study are not smooth enough for applications of the methods of classical analysis. In the creative work of V.A. Toponogov, both of these methods, synthetic and analytic, are in harmonic correlation. The classic result in this area is the Toponogov theorem about the angles of a triangle composed of geodesics. This in-depth theorem is the basis of modern investigations of the relations between curvature properties, geodesic behavior, and the topological structure of Riemannian spaces. In the proof of this theorem, some ideas of A.D. Aleksandrov are combined with the in-depth analytical technique related to the Jacobi differential equation. The methods developed by V.A. Toponogov allowed him to obtain a sequence of fundamental results such as characteristics of the multidimensional sphere by estimates of the Riemannian curvature and diameter, the solution to the Rauch problem for the even-dimensional case, and the theorem about the structure of Riemannian space with nonnegative curvature containing a straight line (i.e., the shortest path that may be limitlessly extended in both directions). This and other theorems of V.A. Toponogov are included in monographs and textbooks written by a number of authors. His methods have had a great influence on modern Riemannian geometry. During the last fifteen years of his life, V.A. Toponogov devoted himself to differential geometry of two-dimensional surfaces in three-dimensional Euclidean space. He made essential progress in a direction related to the Efimov theorem about the nonexistence of isometric embedding of a complete Riemannian metric with a separated-from-zero negative curvature into three-dimensional Euclidean space, and with the Milnor conjecture declaring that an embedding with a sum of absolute values of principal curvatures uniformly separated from zero does not exist.

9 About the Author xi Toponogov devoted much effort to the training of young mathematicians. He was a lecturer at Novosibirsk State University and Novosibirsk State Pedagogical University for more than forty-five years. More than ten of his pupils defended their Ph.D. theses, and seven their doctoral degrees. V.A. Toponogov passed away on November 21, 2004 and is survived by his wife, Ljudmila Pavlovna Goncharova, and three sons.

10 Differential Geometry of Curves and Surfaces