Curves and Surfaces

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S Title


Curves and Surfaces, SECOND EDITION


Copyright

     © 2009 by the American Mathematical Society.

     ISBN 978-0-8218-4763-3

     QA643. M6613 2009 516.3'62-dc22

     LCCN 2009008149


Dedication


Contents


Preface to the Second Edition

Preface to the English Edition
Preface


Chapter 1  Plane and Space Curves

     1.1. Historical notes

     1.2. Curves. Arc length

     1.3. Regular curves and curves parametrized by arc length

     1.4. Local theory of plane curves

     1.5. Local theory of space curves

     Exercises

     Hints for solving the exercises


Chapter 2  Surfaces in Euclidean Space

     2.1. Historical notes

     2.2. Definition of surface

     2.3. Change of parameters

     2.4. Differentiable functions

     2.5. The tangent plane

     2.6. Differential of a differentiable map

     Exercises

     Hints for solving the exercises


Chapter 3  The Second Fundamental Form

     3.1. Introduction and historical notes

     3.2. Normal fields. Orientation

     3.3. The Gauss map and the second fundamental form

     3.4. Normal sections

     3.5. The height function and the second fundamental form

     3.6. Continuity of the curvatures

     Exercises

     Hints for solving the exercises


Chapter 4  Separation and Orientability

     4.1. Introduction

     4.2. Local separation

     4.3. Surfaces, straight lines, and planes

     4.4. The Jordan-Brouwer separation theorem

     4.5. Tubular neighbourhoods

     Exercises

     Hints for solving the exercises

     4.6. Appendix: Proof of Sard's theorem


Chapter 5  Integration on Surfaces

     5.1. Introduction

     5.2. Integrable functions and integration on S x R

     5.3. Integrable functions and integration on surfaces

     5.4. Formula for the change of variables

     5.5. Fubini's theorem and other properties

     5.6. Area formula

     5.7. The divergence theorem

     5.8. The Brouwer fixed point theorem

     Exercises

     Hints for solving the exercises


Chapter 6  Global Extrinsic Geometry

     6.1. Introduction and historical notes

     6.2. Positively curved surfaces

     6.3. Minkowski formulas and ovaloids

     6.4. The Alexandrov theorem

     6.5. The isoperimetric inequality

     Exercises

     Hints for solving the exercises


Chapter 7  Intrinsic Geometry of Surfaces

     7.1. Introduction

     7.2. Rigid motions and isometries

     7.3. Gauss's Theorema Egregium

     7.4. Rigidity of ovaloids

     7.5. Geodesics

     7.6. The exponential map

     Exercises

     Hints for solving the exercises

     7.7. Appendix: Some additional results of an intrinsic type

          7.7.1. Positively curved surfaces.

          7.7.2. Tangent fields and integral curves.

          7.7.3. Special parametrizations.

          7.7.4. Flat surfaces.


Chapter 8  The Gauss-Bonnet Theorem

     8.1. Introduction

     8.2. Degree of maps between compact surfaces

     8.3. Degree and surfaces bounding the same domain

     8.4. The index of a field at an isolated zero

     8.5. The Gauss-Bonnet formula

     8.6. Exercise: The Euler characteristic is even

     Exercises: Steps of the proof


Chapter 9  Global Geometry of Curves

     9.1. Introduction and historical notes

     9.2. Parametrized curves and simple curves

     9.3. Results already shown on surfaces

          9.3.1. Jordan curve theorem.

          9.3.2. Sard's theorem, the length formula, and some consequences

          9.3.3. The divergence theorem

          9.3.4. The isoperimetric inequality.

          9.3.5. Positively curved simple curves.

     9.4. Rotation index of plane curves

     9.5. Periodic space curves

     9.6. The four-vertices theorem

     Exercises

     Hints for solving the exercises

     9.7. Appendix: One-dimensional degree theory


Bibliography


Index


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