Curved Spaces: From Classical Geometries to Elementary Differential Geometry Book

Preface ix

1 Euclidean geometry 1

1.1 Euclidean space 1

1.2 Isometries 4

1.3 The group O (3, R) 9

1.4 Curves and their lengths 11

1.5 Completeness and compactness 15

1.6 Polygons in the Euclidean plane 17

Exercises 22

2 Spherical geometry 25

2.1 Introduction 25

2.2 Spherical triangles 26

2.3 Curves on the sphere 29

2.4 Finite groups of isometries 31

2.5 Gauss-Bonnet and spherical polygons 34

2.6 Mobius geometry 39

2.7 The double cover of SO(3) 42

2.8 Circles on S[superscript 2] 45

Exercises 47

3 Triangulations and Euler numbers 51

3.1 Geometry of the torus 51

3.2 Triangulations 55

3.3 Polygonal decompositions 59

3.4 Topology of the g-holed torus 62

Exercises 67

Appendix on polygonal approximations 68

4 Riemannian metrics 75

4.1 Revision on derivatives and the Chain Rule 75

4.2 Riemannian metrics on open subsets of R[superscript 2] 79

4.3 Lengths of curves 82

4.4 Isometries and areas 85

Exercises 87

5 Hyperbolic geometry 89

5.1 Poincare models for the hyperbolic plane 89

5.2 Geometry of the upper half-plane model H 92

5.3 Geometry of the disc model D 96

5.4 Reflections in hyperbolic lines 98

5.5 Hyperbolic triangles 102

5.6 Parallel and ultraparallel lines 105

5.7 Hyperboloid model of the hyperbolic plane 107

Exercises 112

6 Smooth embedded surfaces 115

6.1 Smooth parametrizations 115

6.2 Lengths and areas 118

6.3 Surfaces of revolution 121

6.4 Gaussian curvature of embedded surfaces 123

Exercises 130

7 Geodesics 133

7.1 Variations of smooth curves 133

7.2 Geodesics on embedded surfaces 138

7.3 Length and energy 140

7.4 Existence of geodesics 141

7.5 Geodesic polars and Gauss's lemma144

Exercises 150

8 Abstract surfaces and Gauss-Bonnet 153

8.1 Gauss's Theorema Egregium 153

8.2 Abstract smooth surfaces and isometries 155

8.3 Gauss-Bonnet for geodesic triangles 159

8.4 Gauss-Bonnet for general closed surfaces 165

8.5 Plumbing joints and building blocks 170

Exercises 175

Postscript 177

References 179

Index 181