Preface

Contents

1

Curves in the Plane and in Space

1.1

What is a Curve?

1

1.2

Arc-Length

9

1.3

Reparametrization

13

1.4

Closed Curves

19

1.5

Level Curves versus Parametrized Curves

23

2

How Much Does a Curve Curve?

2.1

Curvature

29

2.2

Plane Curves

34

2.3

Space Curves

46

3

Global Properties of Curves

3.1

Simple Closed Curves

55

3.2

The Isoperimetric Inequality

58

3.3

The Four Vertex Theorem

62

4

Surfaces in Three Dimensions

4.1

What is a Surface?

67

4.2

Smooth Surfaces

76

4.3

Smooth Maps

82

4.4

Tangents and Derivatives

85

4.5

Normals and Orientability

89

5

Examples of Surfaces

5.1

Level Surfaces

95

5.2

Quadric Surfaces

97

5.3

Ruled Surfaces and Surfaces of Revolution

104

5.4

Compact Surfaces

109

5.5

Triply Orthogonal Systems

111

5.6

Applications of the Inverse Function Theorem

116

6

The First Fundamental Form

6.1

Lengths of Curves on Surfaces

121

6.2

Isometries of Surfaces

126

6.3

Conforrnal Mappings of Surfaces

133

6.4

Equiareal Maps and a Theorem of Archimedes

139

6.5

Spherical Geometry

148

7

Curvature of Surfaces

7.1

The Second Fundamental Form

159

7.2

The Gauss and Weingarten Maps

162

7.3

Normal and Geodesic Curvatures

165

7.4

Parallel Transport and Covariant Derivative

170

8

Gaussian, Mean and Principal Curvatures

8.1

Gaussian and Mean Curvatures

179

8.2

Principal Curvatures of a Surface

187

8.3

Surfaces of Constant Gaussian Curvature

196

8.4

Flat Surfaces

201

8.5

Surfaces of Constant Mean Curvature

206

8.6

Gaussian Curvature of Compact Surfaces

212

9

Geodesies

9.1

Definition and Basic Properties

215

9.2

Geodesic Equations

220

9.3

Geodesies on Surfaces of Revolution

227

9.4

Geodesies as Shortest Paths

235

9.5

Geodesic Coordinates

242

10

Gauss’ Theorema Egregium

10.1

The Gauss and Codazzi-Mainardi Equations

247

10.2

Gauss’ Remarkable Theorem

252

10.3

Surfaces of Constant Gaussian Curvature

257

10.4

Geodesic Mappings

263

11

Hyperbolic Geometry

11.1

Upper Half-Plane Model

270

11.2

Isometries of H

277

11.3

Poincare Disc Model

283

11.4

Hyperbolic Parallels

290

11.5

Beltrami-Klein Model

295

12

Minimal Surfaces

12.1

Plateau’s Problem

305

12.2

Examples of Minimal Surfaces

312

12.3

Gauss Map of a Minimal Surface

320

12.4

Conformal Parametrization of Minimal Surfaces

322

12.5

Minimal Surfaces and Holomorphic Functions

325

13

The Gauss—Bonnet Theorem

13.1

Gauss-Bonnet for Simple Closed Curves

335

13.2

Gauss-Bonnet for Curvilinear Polygons

342

13.3

Integration on Compact Surfaces

346

13.4

Gauss-Bonnet for Compact Surfaces

349

13.5

Map Colouring

357

13.6

Holonomy and Gaussian Curvature

362

13.7

Singularities of Vector Fields

365

13.8

Critical Points

372

A0

Inner Product Spaces and Self-Adjoint Linear Maps

Al

Isometries of Euclidean Spaces

A2

Mobius Transformations

Hints to Selected Exercises

Solutions

Index