Curves and Surfaces

✍ Scribed by Sebastian Montiel and Antonio Ros

• Publisher: American Mathematical Society
• Year: 2009
• Tongue: English
• Leaves: 395
• Series: Graduate Studies in Mathematics, Vol. 69
• Edition: 2
• Category: Library

✦ Synopsis

This introductory textbook puts forth a clear and focused point of view on the differential geometry of curves and surfaces. Following the modern point of view on differential geometry, the book emphasizes the global aspects of the subject. The excellent collection of examples and exercises (with hints) will help students in learning the material. Advanced undergraduates and graduate students will find this a nice entry point to differential geometry.

In order to study the global properties of curves and surfaces, it is necessary to have more sophisticated tools than are usually found in textbooks on the topic. In particular, students must have a firm grasp on certain topological theories. Indeed, this monograph treats the Gauss-Bonnet theorem and discusses the Euler characteristic. The authors also cover Alexandrov's theorem on embedded compact surfaces in R3 with constant mean curvature. The last chapter addresses the global geometry of curves, including periodic space curves and the four-vertices theorem for plane curves that are not necessarily convex.

Besides being an introduction to the lively subject of curves and surfaces, this book can also be used as an entry to a wider study of differential geometry. It is suitable as the text for a first-year graduate course or an advanced undergraduate course.

Readership: Undergraduate students, graduate students, and research mathematicians interested in the geometry of curves and surfaces.

✦ Table of Contents

Cover

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

Back Cover