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Geometry of Surfaces : Universitext - John Stillwell

Geometry of Surfaces



Published: 3rd February 1995
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Geometry used to be the basis of a mathematical education; today it is not even a standard undergraduate topic. Much as I deplore this situation, I welcome the opportunity to make a fresh start. Classical geometry is no longer an adequate basis for mathematics or physics-both of which are becoming increasingly geometric-and geometry can no longer be divorced from algebra, topology, and analysis. Students need a geometry of greater scope, and the fact that there is no room for geometry in the curriculum un- til the third or fourth year at least allows us to assume some mathematical background. What geometry should be taught? I believe that the geometry of surfaces of constant curvature is an ideal choice, for the following reasons: 1. It is basically simple and traditional. We are not forgetting euclidean geometry but extending it enough to be interesting and useful. The extensions offer the simplest possible introduction to fundamentals of modem geometry: curvature, group actions, and covering spaces. 2. The prerequisites are modest and standard. A little linear algebra (mostly 2 x 2 matrices), calculus as far as hyperbolic functions, ba- sic group theory (subgroups and cosets), and basic topology (open, closed, and compact sets).

The Euclidean Plane
Approaches to Euclidean Geometryp. 1
Isometriesp. 2
Rotations and Reflectionsp. 5
The Three Reflections Theoremp. 9
Orientation-Reversing Isometriesp. 11
Distinctive Features of Euclidean Geometryp. 14
Discussionp. 18
Euclidean Surfaces
Euclid on Manifoldsp. 21
The Cylinderp. 22
The Twisted Cylinderp. 25
The Torus and the Klein Bottlep. 26
Quotient Surfacesp. 29
A Nondiscontinuous Groupp. 33
Euclidean Surfacesp. 34
Covering a Surface by the Planep. 36
The Covering Isometry Groupp. 39
Discussionp. 41
The Sphere
The Sphere S[superscript 2] in R[superscript 3]p. 45
Rotationsp. 48
Stereographic Projectionp. 50
Inversion and the Complex Coordinate on the Spherep. 52
Reflections and Rotations as Complex Functionsp. 56
The Antipodal Map and the Elliptic Planep. 60
Remarks on Groups, Spheres and Projective Spacesp. 63
The Area of a Trianglep. 65
The Regular Polyhedrap. 67
Discussionp. 69
The Hyperbolic Plane
Negative Curvature and the Half-Planep. 75
The Half-Plane Model and the Conformal Disc Modelp. 80
The Three Reflections Theoremp. 85
Isometries as Complex Functionsp. 88
Geometric Description of Isometriesp. 92
Classification of Isometriesp. 96
The Area of a Trianglep. 99
The Projective Disc Modelp. 101
Hyperbolic Spacep. 105
Discussionp. 108
Hyperbolic Surfaces
Hyperbolic Surfaces and the Killing-Hopf Theoremp. 111
The Pseudospherep. 112
The Punctured Spherep. 113
Dense Lines on the Punctured Spherep. 118
General Construction of Hyperbolic Surfaces from Polygonsp. 122
Geometric Realization of Compact Surfacesp. 126
Completeness of Compact Geometric Surfacesp. 129
Compact Hyperbolic Surfacesp. 130
Discussionp. 132
Paths and Geodesics
Topological Classification of Surfacesp. 135
Geometric Classification of Surfacesp. 138
Paths and Homotopyp. 140
Lifting Paths and Lifting Homotopiesp. 143
The Fundamental Groupp. 145
Generators and Relations for the Fundamental Groupp. 147
Fundamental Group and Genusp. 153
Closed Geodesic Pathsp. 154
Classification of Closed Geodesic Pathsp. 156
Discussionp. 160
Planar and Spherical Tessellations
Symmetric Tessellationsp. 163
Conditions for a Polygon to Be a Fundamental Regionp. 167
The Triangle Tessellationsp. 172
Poincare's Theorem for Compact Polygonsp. 178
Discussionp. 182
Tessellations of Compact Surfaces
Orbifolds and Desingularizationsp. 185
From Desingularization to Symmetric Tessellationp. 189
Desingularizations as (Branched) Coveringsp. 190
Some Methods of Desingularizationp. 194
Reduction to a Permutation Problemp. 196
Solution of the Permutation Problemp. 198
Discussionp. 201
Referencesp. 203
Indexp. 207
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387977430
ISBN-10: 0387977430
Series: Universitext
Audience: General
Format: Paperback
Language: English
Number Of Pages: 236
Published: 3rd February 1995
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.37 x 16.71  x 1.27
Weight (kg): 0.33