Differential Geometry of Curves and Surfaces

This volume covers local as well as global differential geometry of curves and surfaces.

Preface

v

Some Remarks on Using this Book

vii

Curves

1

(50)

Introduction

1

(1)

Parametrized Curves

2

(3)

Regular Curves; Arc Length

5

(6)

The Vector Product in R3

11

(5)

The Local Theory of Curves Parametrized by Arc Length

16

(11)

The Local Canonical Form

27

(3)

Global Properties of Plane Curves

30

(21)

Regular Surfaces

51

(83)

Introduction

51

(1)

Regular Surfaces; Inverse Images of Regular Values

52

(17)

Change of Parameters; Differential Functions on Surfaces

69

(14)

The Tangent Plane; the Differential of a Map

83

(9)

The First Fundamental Form; Area

92

(10)

Orientation of Surfaces

102

(7)

A Characterization of Compact Orientable Surfaces

109

(5)

A Geometric Definition of Area

114

(20)

Appendix: A Brief Review on Continuity and Differentiability

118

(16)

The Geometry of the Gauss Map

134

(83)

Introduction

134

(1)

The Definition of the Gauss Map and Its Fundamental Properties

135

(18)

The Gauss Map in Local Coordinates

153

(22)

Vector Fields

175

(13)

Ruled Surfaces and Minimal Surfaces

188

(29)

Appendix: Self-Adjoint Linear Maps and Quadratic Forms

214

(3)

The Intrinsic Geometry of Surfaces

217

(98)

Introduction

217

(1)

Isometries; Conformal Maps

218

(13)

The Gauss Theorem and the Equations of Compatibility

231

(7)

Parallel Transport; Geodesics

238

(26)

The Gauss-Bonnet Theorem and its Applications

264

(19)

The Exponential Map. Geodesic Polar Coordinates

283

(15)

Further Properties of Geodesics. Convex Neighborhoods

298

(17)

Appendix: Proofs of the Fundamental Theorems of The Local Theory of Curves and Surfaces

309

(6)

Global Differential Geometry

315

(156)

Introduction

315

(2)

The Rigidity of the Sphere

317

(8)

Complete Surfaces. Theorem of Hopf-Rinow

325

(14)

First and Second Variations of the Arc Length; Bonnet's Theorem

339

(18)

Jacobi Fields and Conjugate Points

357

(14)

Covering Spaces; the Theorems of Hadamard

371

(9)

Global Theorems for Curves; the Fary-Milnor Theorem

380

(28)

Surfaces of Zero Gaussian Curvature

408

(7)

Jacobi's Theorems

415

(10)

Abstract Surfaces; Further Generalizations

425

(21)

Hilbert's Theorem

446

(25)

Appendix: Point-Set Topology of Euclidean Spaces

456

(15)

Bibliography and Comments

471

(4)

Hints and Answers to Some Exercises

475

(22)

Index

497

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Differential Geometry of Curves and Surfaces

Summary

Table of Contents

Supplemental Materials