The geometry of surfaces is an ideal starting point for learning geometry, for, among other reasons, the theory of surfaces of constant curvature has maximal connectivity with the rest of mathematics. This text provides the student with the knowledge of a geometry of greater scope than the classical geometry taught today, which is no longer an adequate basis for mathematics or physics, both of which are becoming increasingly geometric. It includes exercises and informal discussions.
Author(s): John Stillwell
Edition: Corrected
Publisher: Springer
Year: 1992
Language: English
Pages: 228
Cover......Page 1
Geometry of Surfaces......Page 2
Copyright - ISBN: 0387977430......Page 3
Preface......Page 6
Contents......Page 8
1.1 Approaches to Euclidean Geometry ......Page 12
1.2 Isometries ......Page 13
1.3 Rotations and Reflections ......Page 16
1.4 The Three Reflections Theorem ......Page 20
1.5 Orientation-Reversing Isometries ......Page 22
1.6 Distinctive Features of Euclidean Geometry ......Page 25
1.7 Discussion ......Page 29
2.1 Euclid on Manifolds ......Page 32
2.2 The Cylinder ......Page 33
2.3 The Twisted Cylinder ......Page 36
2.4 The Torus and the Klein Bottle ......Page 37
2.5 Quotient Surfaces ......Page 40
2.6 A Nondiscontinuous Group ......Page 44
2.7 Euclidean Surfaces ......Page 45
2.8 Covering a Surface by the Plane ......Page 47
2.9 The Covering Isometry Group ......Page 50
2.10 Discussion ......Page 52
3.1 The Sphere S^2 in E^3......Page 56
3.2 Rotations ......Page 59
3.3 Stereographic Projection ......Page 61
3.4 Inversion and the Complex Coordinate on the Sphere ......Page 63
3.5 Reflections and Rotations as Complex Functions ......Page 67
3.6 The Antipodal Map and the Elliptic Plane ......Page 71
3.7 Remarks on Groups, Spheres and Projective Spaces ......Page 74
3.8 The Area of a Triangle ......Page 76
3.9 The Regular Polyhedra ......Page 78
3.10 Discussion ......Page 80
4.1 Negative Curvature and the Half-Plane ......Page 86
4.2 The Half-Plane Model and the Conformal Disc Model ......Page 91
4.3 The Three Reflections Theorem ......Page 96
4.4 Isometries as Complex Functions ......Page 99
4.5 Geometric Description of Isometries ......Page 103
4.6 Classification of Isometries ......Page 107
4.7 The Area of a Triangle ......Page 110
4.8 The Projective Disc Model ......Page 112
4.9 Hyperbolic Space ......Page 116
4.10 Discussion ......Page 119
5.1 Hyperbolic Surfaces and the Killing-Hopf Theorem ......Page 122
5.2 The Pseudosphere ......Page 123
5.3 The Punctured Sphere ......Page 124
5.4 Dense Lines on the Punctured Sphere ......Page 129
5.5 General Construction of Hyperbolic Surfaces from Polygons ......Page 133
5.6 Geometric Realization of Compact Surfaces ......Page 137
5.7 Completeness of Compact Geometric Surfaces ......Page 140
5.8 Compact Hyperbolic Surfaces ......Page 141
5.9 Discussion ......Page 143
6.1 Topological Classification of Surfaces ......Page 146
6.2 Geometric Classification of Surfaces ......Page 149
6.3 Paths and Homotopy ......Page 151
6.4 Lifting Paths and Lifting Homotopies ......Page 154
6.5 The Fundamental Group ......Page 156
6.6 Generators and Relations for the Fundamental Group ......Page 158
6.7 Fundamental Group and Genus ......Page 164
6.8 Closed Geodesic Paths ......Page 165
6.9 Classification of Closed Geodesic Paths ......Page 167
6.10 Discussion ......Page 171
7.1 Symmetric Tessellations ......Page 174
7.2 Conditions for a Polygon to Be a Fundamental Region ......Page 178
7.3 The Triangle Tessellations ......Page 183
7.4 Poincare's Theorem for Compact Polygons ......Page 189
7.5 Discussion ......Page 193
8.1 Orbifolds and Desingularizations ......Page 196
8.2 From Desingularization to Symmetric Tessellation ......Page 200
8.3 Desingularizations as (Branched) Coverings ......Page 201
8.4 Some Methods of Desingularization ......Page 205
8.5 Reduction to a Permutation Problem ......Page 207
8.6 Solution of the Permutation Problem ......Page 209
8.7 Discussion ......Page 212
References ......Page 214
Index ......Page 218
