Curved Spaces: From Classical Geometries to Elementary Differential Geometry

P. M. H. Wilson0521713900, 9780521713900

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.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title Page……Page 5
ISBN 0521713900……Page 6
Dedication……Page 7
4 Riemannian metrics……Page 9
Postscript, References, Index……Page 10
Preface……Page 11
1.1 Euclidean space……Page 13
1.2 Isometries……Page 16
1.3 The group O(3, R)……Page 21
1.4 Curves and their lengths……Page 23
1.5 Completeness and compactness……Page 27
1.6 Polygons in the Euclidean plane……Page 29
Exercises……Page 34
2.1 Introduction……Page 37
2.2 Spherical triangles……Page 38
2.3 Curves on the sphere……Page 41
2.4 Finite groups of isometries……Page 43
2.5 Gauss–Bonnet and spherical polygons……Page 46
2.6 Möbius geometry……Page 51
2.7 The double cover of SO(3)……Page 54
2.8 Circles on S2……Page 57
Exercises……Page 59
3.1 Geometry of the torus……Page 63
3.2 Triangulations……Page 67
3.3 Polygonal decompositions……Page 71
3.4 Topology of the g-holed torus……Page 74
Exercises……Page 79
Appendix on polygonal approximations……Page 80
4.1 Revision on derivatives and the Chain Rule……Page 87
4.2 Riemannian metrics on open subsets of R2……Page 91
4.3 Lengths of curves……Page 94
4.4 Isometries and areas……Page 97
Exercises……Page 99
5.1 Poincaré models for the hyperbolic plane……Page 101
5.2 Geometry of the upper half-plane model H……Page 104
5.3 Geometry of the disc model D……Page 108
5.4 Reflections in hyperbolic lines……Page 110
5.5 Hyperbolic triangles……Page 114
5.6 Parallel and ultraparallel lines……Page 117
5.7 Hyperboloid model of the hyperbolic plane……Page 119
Exercises……Page 124
6.1 Smooth parametrizations……Page 127
6.2 Lengths and areas……Page 130
6.3 Surfaces of revolution……Page 133
6.4 Gaussian curvature of embedded surfaces……Page 135
Exercises……Page 142
7.1 Variations of smooth curves……Page 145
7.2 Geodesics on embedded surfaces……Page 150
7.3 Length and energy……Page 152
7.4 Existence of geodesics……Page 153
7.5 Geodesic polars and Gauss’s lemma……Page 156
Exercises……Page 162
8.1 Gauss’s Theorema Egregium……Page 165
8.2 Abstract smooth surfaces and isometries……Page 167
8.3 Gauss–Bonnet for geodesic triangles……Page 171
8.4 Gauss–Bonnet for general closed surfaces……Page 177
8.5 Plumbing joints and building blocks……Page 182
Exercises……Page 187
Postscript……Page 189
References……Page 191
A,B,C……Page 193
E……Page 194
J……Page 195
R……Page 196
T……Page 197
U,V,W……Page 198