Victor Andreevich Toponogov
        with the editorial assistance of
        Vladimir Y. Rovenski
      Differential Geometry
      of Curves and Surfaces
      AConciseGuide
        ¨
      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.
                   AMSSubjectClassification: 53-01, 53Axx, 53A04, 53A05, 53A55, 53B20, 53B21, 53C20, 53C21
                   Library of Congress Control Number: 2005048111
                   ISBN-100-8176-4384-2                 eISBN0-8176-4402-4
                   ISBN-13978-0-8176-4384-3
                   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
               AbouttheAuthor .............................................. ix
               1 TheoryofCurvesinThree-dimensionalEuclideanSpaceandinthe
                 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   TheFrenet 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   TheTangentPlane.......................................... 70
                 2.3   First Fundamental Form of a Surface .......................... 74
                 vi      Contents
                    2.4    Second Fundamental Form of a Surface ........................ 79
                    2.5    TheThirdFundamental Form of a Surface...................... 91
                    2.6    Classes of Surfaces . . . ...................................... 95
                    2.7    SomeClassesofCurvesonaSurface ..........................114
                    2.8    TheMainEquationsofSurfaceTheory ........................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