Contents

Preface xv

Useful Supplements, xvii

Acknowledgements, xvii

How to Use This Book xix

Chapter 1: Surfaces and Straightness 1

Problem 1.1. When Do You Call a Line Straight?, 1

How Do You Construct a Straight Line?, 3

Local (and Infinitesimal) Straightness, 6

Problem 1.2. Intrinsic Straight Lines on Cylinders, 8

Problem 1.3. Geodesics on Cones, 12

Is "Shortest" Always "Straight"?, 16

Locally Isometric Surfaces, 18

Local Coordinates for Cylinders and Cones, 19

Problem 1.4. Geodesics in Local Coordinates, 22

Problem 1.5. What Is Straight on a Sphere?, 24

Intrinsic Curvature on a Sphere, 25

Local Coordinates on a Sphere, 27

Problem 1.6. Strakes, Augers, and Helicoids, 28

Problem 1.7. Surfaces of Revolution, 30

Problem 1.8. Hyperbolic Plane, 31

Problem 1.9. Surface as Graph of a Function z = f (x,y), 34

Chapter 2: Extrinsic Curves 37

Introduction, 37

Problem 2.1. Give Examples of F.O.V.'s, 38

Archimedian Property, 40

Vectors and Affine Linear Space, 40

Problem 2.2. Smoothness and Tangent Directions, 41

Problem 2.3. Curvature of a Curve in Space, 48

Curvature of the Graph of a Function, 50

Problem 2.4. Osculating Circle, 52

Problem 2.5. Strakes, 55

Problem 2.6. When a Curve Does Not Lie in a Plane, 56

Chapter 3: Extrinsic Descriptions of Intrinsic Curvature 61

Problem 3.1. Smooth Surfaces and Tangent Planes, 62

Problem 3.2. Extrinsic Curvature_Geodesic on Sphere, 64

Problem 3.3. Intrinsic Curvature_Curves on Sphere, 65

Intrinsic (Geodesic) Curvature, 67

Problem 3.4. Geodesics on Surfaces_the Ribbon Test, 69

Ruled Surfaces and the Converse of the Ribbon Test, 70

Chapter 4: Tangent Space, Metric, Directional Derivative 73

Problem 4.1. The Tangent Space, 73

Problem 4.2. Mean Value Theorem_Curves_Surfaces, 76

Natural Parametrizations of Curves, 77

Problem 4.3. Riemannian Metric, 79

Riemannian Metric in Local Coordinates on a Sphere, 82

Riemannian Metric in Local Coordinates on a Strake, 83

Problem 4.4. Vectors in Extrinsic Local Coordinates, 85

Problem 4.5. Measuring Using the Riemannian Metric, 87

Directional Derivatives, 89

Directional Derivative in Local Coordinates, 92

Problem 4.6. Differentiating a Metric, 93

Problem 4.7. Expressing Normal Curvature, 94

Geodesic Local Coordinates, 97

Problem 4.8. Differential Operator, 98

Problem 4.9. Metric in Geodesic Coordinates, 100

Chapter 5: Area, Parallel Transport, Intrinsic Curvature 103

Problem 5.1. The Area of a Triangle on a Sphere, 103

Introducing Parallel Transport, 104

The Holonomy of a Small Geodesic Triangle, 106

Problem 5.2. Dissection of Polygons into Triangles, 108

Problem 5.3. Gauss-Bonnet for Polygons on a Sphere, 109

Problem 5.4. Parallel Fields and Intrinsic Curvature, 111

Problem 5.5. Holonomy on Surfaces, 115

Problem 5.6. Holonomy Explains Foucault's Pendulum, 116

Problem 5.7. Intrinsic Curvature of a Surface, 117

Chapter 6: Gaussian Curvature Extrinsically Defined 119

Pep Talk to the Reader, 119

Problem 6.1. Gaussian Curvature, Extrinsic Definition, 120

Problem 6.2. Second Fundamental Form, 123

Problem 6.3. The Gauss Map, 126

Problem 6.4. Gauss-Bonnet and Intrinsic Curvature, 128

Problem 6.5. 2nd Fundamental Form in Coordinates, 129

*Problem 6.6. Mean Curvature and Minimal Surfaces, 131

Celebration of Our Hard Work, 133

Chapter 7: Applications of Gaussian Curvature 135

Problem 7.1. Gaussian Curvature in Local Coordinates, 135

*Problem 7.2. Curvature on Sphere, Strake, Catenoid, 140

Problem 7.3. Circles, Polar Coordinates, and Curvature, 141

Problem 7.4. Exponential Map & Shortest Is Straight, 142

Problem 7.5. Ruled Surfaces and Ribbons, 147

Problem 7.6. Surfaces with Constant Curvature, 148

Problem 7.7. Curvature of the Hyperbolic Plane, 149

Chapter 8: Intrinsic Local Descriptions and Manifolds 151

Problem 8.1. Covariant Derivative and Connection, 151

*Problem 8.2. Manifolds_Intrinsic and Extrinsic, 154

Problem 8.3. Christoffel Symbols, Intrinsic Descriptions, 161

Problem 8.4. Intrinsic Curvature and Geodesics, 164

Problem 8.5. Lie Brackets, Coordinate Vector Fields, 165

Problem 8.6. Riemann Curvature Tensors, 167

Calculation of Curvature Tensors in Local Coordinates, 171

Problem 8.7. Intrinsic Calculations in Examples, 173

Appendix A: Linear Algebra_a Geometric Point of View 175

A.0. Where Do We Start?, 175

A.1. Geometric Affine Spaces, 176

A.2. Vector Spaces, 180

A.3. Inner Product_Lengths and Angles, 182

A.4. Linear Transformations and Operators, 185

A.5. Areas, Cross Products, and Triple Products, 192

A.6. Volumes, Orientation, and Determinants, 194

A.7. Eigenvalues and Eigenvectors, 197

A.8. Introduction to Tensors, 198

Appendix B: Analysis from a Geometric Point of View 201

B.1. Smooth Functions, 201

B.2. Invariance of Domain, 202

B.3. Inverse Function Theorem, 202

B.4. Implicit Function Theorem, 203

Appendix C: Computer Scripts 205

Standard Functions, 205

Computer Exercise 1.6: Strake, 208

Computer Exercise 1.7: Surfaces of Revolution, 209

Computer Exercise 1.9: Surfaces as Graph of a Function, 210

Computer Exercise 2.2: Tangent Vectors to Curves , 210

Computer Exercise 2.3: Curvature and Tangent Vectors, 211

Computer Exercise 2.4a: Osculating Planes, 212

Computer Exercise 2.4b: Osculating Circles, 213

Computer Exercise 2.6: Frent Frame, 214

Computer Exercise 3.1: Tangent Planes to Surfaces, 215

Computer Exercise 3.2a: Curves on a Surface, 216

Computer Exercise 3.2b: Extrinsic Curvature Vectors, 217

Computer Exercise 3.3: The Three Curvature Vectors, 219

Computer Exercise 3.4: Ruled Surfaces, 221

Computer Exercise 5.2: Non-dissectable Polyhedron, 222

Computer Exercise 5.5: Sign of (Gaussian) Curvature, 223

Computer Exercise 6.1: Multiple Principle Directions, 223

Computer Exercise 6.3: Gauss Map, 224

Computer Exercise 6.6: Helicoid to Catenoid, 226

Bibliography 227

A. Ancient Texts, 227

AD. Art and Design, 227

An. Analysis, 228

DG. Differential Geometry, 228

DS. Dimensions and Scale, 230

GC. Geometry in Different Cultures, 231

Hi. History, 232

LA. Linear Algebra and Geometry, 233

Mi. Minimal Surfaces, 234

MP. Models, Polyhedra, 234

Na. Nature, 235

NE. Non-Euclidean Geometries (Mostly Hyperbolic), 235

Ph. Philosophy, 236

RN. Real Numbers, 236

SP. Spherical and Projective Geometry, 237

SG. Symmetry and Groups, 238

SE. Surveys and General Expositions, 238

Tp. Topology, 240

Tx. Geometry Texts, 240

Z. Miscellaneous, 241

Notation Index 243

Subject Index 245