This self-contained textbook presents an exposition of the well-known classical two-dimensional geometries, such as Euclidean, spherical, hyperbolic, and the locally Euclidean torus, and introduces the basic concepts of Euler numbers for topological triangulations, and Riemannian metrics. The careful discussion of these classical examples provides students with an introduction to the more general theory of curved spaces developed later in the book, as represented by embedded surfaces in Euclidean 3-space, and their generalization to abstract surfaces equipped with Riemannian metrics. Themes running throughout include those of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the link to topology provided by the Gauss-Bonnet theorem. Numerous diagrams help bring the key points to life and helpful examples and exercises are included to aid understanding. Throughout the emphasis is placed on explicit proofs, making this text ideal for any student with a basic background in analysis and algebra.
Author(s): P. M. H. Wilson
Edition: 1
Publisher: Cambridge University Press
Year: 2008
Language: English
Pages: 196
Cover......Page 1
Half-title......Page 0
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 10
1.1 Euclidean space......Page 12
1.2 Isometries......Page 15
1.3 The group O(3, R)......Page 20
1.4 Curves and their lengths......Page 22
1.5 Completeness and compactness......Page 26
1.6 Polygons in the Euclidean plane......Page 28
Exercises......Page 33
2.1 Introduction......Page 36
2.2 Spherical triangles......Page 37
2.3 Curves on the sphere......Page 40
2.4 Finite groups of isometries......Page 42
2.5 Gauss--Bonnet and spherical polygons......Page 45
2.6 Möbius geometry......Page 50
2.7 The double cover of SO(3)......Page 53
2.8 Circles on S2......Page 56
Exercises......Page 58
3.1 Geometry of the torus......Page 62
3.2 Triangulations......Page 66
3.3 Polygonal decompositions......Page 70
3.4 Topology of the g-holed torus......Page 73
Exercises......Page 78
Appendix on polygonal approximations......Page 79
4.1 Revision on derivatives and the Chain Rule......Page 86
4.2 Riemannian metrics on open subsets of R2......Page 90
4.3 Lengths of curves......Page 93
4.4 Isometries and areas......Page 96
Exercises......Page 98
5.1 Poincaré models for the hyperbolic plane......Page 100
5.2 Geometry of the upper half-plane model H......Page 103
5.3 Geometry of the disc model D......Page 107
5.4 Reflections in hyperbolic lines......Page 109
5.5 Hyperbolic triangles......Page 113
5.6 Parallel and ultraparallel lines......Page 116
5.7 Hyperboloid model of the hyperbolic plane......Page 118
Exercises......Page 123
6.1 Smooth parametrizations......Page 126
6.2 Lengths and areas......Page 129
6.3 Surfaces of revolution......Page 132
6.4 Gaussian curvature of embedded surfaces......Page 134
Exercises......Page 141
7.1 Variations of smooth curves......Page 144
7.2 Geodesics on embedded surfaces......Page 149
7.3 Length and energy......Page 151
7.4 Existence of geodesics......Page 152
7.5 Geodesic polars and Gauss's lemma......Page 155
Exercises......Page 161
8.1 Gauss's Theorema Egregium......Page 164
8.2 Abstract smooth surfaces and isometries......Page 166
8.3 Gauss–Bonnet for geodesic triangles......Page 170
8.4 Gauss–Bonnet for general closed surfaces......Page 176
8.5 Plumbing joints and building blocks......Page 181
Exercises......Page 186
Postscript......Page 188
References......Page 190
Index......Page 192
