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Birkhäuser

Differential Geometry of Curves and Surfaces

A Concise Guide

  • Textbook
  • © 2006

Overview

Authors:
  1. Victor A. Toponogov

    1. Department of Analysis and Geometry, Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Russia

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Editors:
  1. Vladimir Y. Rovenski

    1. Department of Mathematics, University of Haifa, Haifa, Israel

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  • Central topics covered include curves, surfaces, geodesics, intrinsic geometry, and the Alexandrov global angle comparision theorem
  • Many nontrivial and original problems (some with hints and solutions)
  • Standard theoretical material is combined with more difficult theorems and complex problems, while maintaining a clear distinction between the two levels

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Table of contents (3 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Theory of Curves in Three-dimensional Euclidean Space and in the Plane

    Pages 1-63

  3. Extrinsic Geometry of Surfaces in Three-dimensional Euclidean Space

    Pages 65-150

  4. Intrinsic Geometry of Surfaces

    Pages 151-197

  5. Back Matter

    Pages 199-206
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Keywords

About this book

This concise guide to the differential geometry of curves and surfaces can be recommended to ?rst-year graduate students, strong senior students, and students specializing in geometry. The material is given in two parallel streams. The ?rst stream contains the standard theoretical material on differential ge- 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 pr- 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, T- orems 2.8.1–2.8.4, the problems (Sections 2.7.4, 2.8.3) and the appendix (S- tion 2.9). The second stream contains more dif?cult and additional material and for- 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 1 surface, formulations of A.V. Pogorelov’s theorem about rigidity of convex s- 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) ofnonstandard andoriginalproblems that introduce the student into the real world of geometry.

Reviews

From the reviews:

"This book by the late author covers … the subjects which are normally taught in a course on the differential geometry of curves and surfaces. … It can be recommended for first-year graduate students and also for use in the classroom. … the book is rich in geometry and concrete examples. … the book is very welcome since it is an original contribution in various aspects and gives a number of geometric insights … . Numerous illustrations make the reading enjoyable." (Wolfgang Kühnel, Mathematical Reviews, Issue 2006 m)

"Toponogov’s ‘concise guide’ to elementary differential geometry has the potential to be a useful reference and/or review book … ."(Fernando Q. Gouvèa, MathDL, March, 2006)

“This book, written by one of the Russian masters, offers a comprehensive introduction to classical differential geometry of curves and surfaces. Only a basic background in analysis and linear algebra is needed to follow the presentation. … A distinctive feature of the book is that, apart from a large number of exercises, it also contains many original problems with outlines of solutions.” (A. Cap, Monatshefte für Mathematik, Vol. 155 (1), September, 2008)

Authors, Editors and Affiliations

  • Department of Mathematics, University of Haifa, Haifa, Israel

    Vladimir Y. Rovenski

  • Department of Analysis and Geometry, Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Russia

    Victor A. Toponogov

Bibliographic Information

  • Book Title: Differential Geometry of Curves and Surfaces

  • Book Subtitle: A Concise Guide

  • Authors: Victor A. Toponogov

  • Editors: Vladimir Y. Rovenski

  • DOI: https://doi.org/10.1007/b137116

  • Publisher: Birkhäuser Boston, MA

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Birkhäuser Boston 2006

  • Softcover ISBN: 978-0-8176-4384-3Published: 05 December 2005

  • eBook ISBN: 978-0-8176-4402-4Published: 10 September 2006

  • Edition Number: 1

  • Number of Pages: XIV, 206

  • Number of Illustrations: 70 b/w illustrations

  • Topics: Differential Geometry

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