Hyperbolic Geometry / Edition 1

Hyperbolic Geometry / Edition 1

by James W. Anderson
     
 

ISBN-10: 1852331569

ISBN-13: 9781852331566

Pub. Date: 10/28/1999

Publisher: Springer-Verlag New York, LLC

The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries. This book provides a self-contained introduction to the subject, suitable for third or fourth year undergraduates. The basic approach taken is to define hyperbolic lines and develop a natural group of transformations preserving

Overview

The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries. This book provides a self-contained introduction to the subject, suitable for third or fourth year undergraduates. The basic approach taken is to define hyperbolic lines and develop a natural group of transformations preserving hyperbolic lines, and then study hyperbolic geometry as those quantities invariant under this group of transformations.

Topics covered include the upper half-plane model of the hyperbolic plane, Mobius transformations, the general Mobius group, and their subgroups preserving the upper half-plane, hyperbolic arc-length and distance as quantities invariant under these subgroups, the Poincare disc model, convex subsets of the hyperbolic plane, the hyperbolic area, and the Gauss-Bonnet formula and its applications.

The style and level of the book, which assumes few mathematical prerequisites, make it an ideal introduction to this subject and provides the reader with a firm grasp of the concepts and techniques of this beautiful part of the mathematical landscape.

Product Details

ISBN-13:
9781852331566
Publisher:
Springer-Verlag New York, LLC
Publication date:
10/28/1999
Series:
Undergraduate Mathematics Series
Edition description:
Older Edition
Pages:
248
Product dimensions:
6.70(w) x 9.22(h) x 0.59(d)

Table of Contents

Preamble to the Second Editionvii
Preamble to the First Editionix
1The Basic Spaces1
1.1A Model for the Hyperbolic Plane1
1.2The Riemann Sphere [characters not reproducible]8
1.3The Boundary at Infinity of H18
2The General Mobius Group23
2.1The Group of Mobius Transformations23
2.3Transitivity Properties of Mob[superscript +]30
2.3The Cross Ratio36
2.4Classification of Mobius Transformations39
2.5A Matrix Representation42
2.6Reflections48
2.7The Conformality of Elements of Mob53
2.8Preserving H56
2.9Transitivity Properties of Mob(H)62
2.10The Geometry of the Action of Mob(H)65
3Length and Distance in H73
3.1Paths and Elements of Arc-length73
3.2The Element of Arc-length on H80
3.3Path Metric Spaces88
3.4From Arc-length to Metric92
3.5Formulae for Hyperbolic Distance in H99
3.6Isometries103
3.7Metric Properties of (H,d[subscript H])108
4Planar Models of the Hyperbolic Plane117
4.1The Poincare Disc Model117
4.2A General Construction130
5Convexity, Area, and Trigonometry145
5.1Convexity145
5.2Hyperbolic Polygons154
5.3The Definition of Hyperbolic Area164
5.4Area and the Gauss-Bonnet Formula169
5.5Applications of the Gauss-Bonnet Formula174
5.6Trigonometry in the Hyperbolic Plane181
6Nonplanar models189
6.1The Hyperboloid Model of the Hyperbolic Plane189
6.2Higher Dimensional Hyperbolic Spaces209
Solutions to Exercises217
Referfences265
List of Notation269
Index273

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