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Geometry of Surfaces

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ISBN-10: 0387977430

ISBN-13: 9780387977430

Edition: 1992

Authors: John C. Stillwell

List price: $64.99
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Book details

List price: $64.99
Copyright year: 1992
Publisher: Springer New York
Publication date: 2/3/1995
Binding: Paperback
Pages: 236
Size: 6.10" wide x 9.25" long x 0.50" tall
Weight: 0.946
Language: English

Preface
The Euclidean Plane
Approaches to Euclidean Geometry
Isometries
Rotations and Reflections
The Three Reflections Theorem
Orientation-Reversing Isometries
Distinctive Features of Euclidean Geometry
Discussion
Euclidean Surfaces
Euclid on Manifolds
The Cylinder
The Twisted Cylinder
The Torus and the Klein Bottle
Quotient Surfaces
A Nondiscontinuous Group
Euclidean Surfaces
Covering a Surface by the Plane
The Covering Isometry Group
Discussion
The Sphere
The Sphere S[superscript 2] in R[superscript 3]
Rotations
Stereographic Projection
Inversion and the Complex Coordinate on the Sphere
Reflections and Rotations as Complex Functions
The Antipodal Map and the Elliptic Plane
Remarks on Groups, Spheres and Projective Spaces
The Area of a Triangle
The Regular Polyhedra
Discussion
The Hyperbolic Plane
Negative Curvature and the Half-Plane
The Half-Plane Model and the Conformal Disc Model
The Three Reflections Theorem
Isometries as Complex Functions
Geometric Description of Isometries
Classification of Isometries
The Area of a Triangle
The Projective Disc Model
Hyperbolic Space
Discussion
Hyperbolic Surfaces
Hyperbolic Surfaces and the Killing-Hopf Theorem
The Pseudosphere
The Punctured Sphere
Dense Lines on the Punctured Sphere
General Construction of Hyperbolic Surfaces from Polygons
Geometric Realization of Compact Surfaces
Completeness of Compact Geometric Surfaces
Compact Hyperbolic Surfaces
Discussion
Paths and Geodesics
Topological Classification of Surfaces
Geometric Classification of Surfaces
Paths and Homotopy
Lifting Paths and Lifting Homotopies
The Fundamental Group
Generators and Relations for the Fundamental Group
Fundamental Group and Genus
Closed Geodesic Paths
Classification of Closed Geodesic Paths
Discussion
Planar and Spherical Tessellations
Symmetric Tessellations
Conditions for a Polygon to Be a Fundamental Region
The Triangle Tessellations
Poincare's Theorem for Compact Polygons
Discussion
Tessellations of Compact Surfaces
Orbifolds and Desingularizations
From Desingularization to Symmetric Tessellation
Desingularizations as (Branched) Coverings
Some Methods of Desingularization
Reduction to a Permutation Problem
Solution of the Permutation Problem
Discussion
References
Index