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, providing the reader with a firm grasp of the concepts and techniques of this beautiful area of mathematics. Topics covered include the upper half-space model of the hyperbolic plane, Möbius transformations, the general Möbius group and the subgroup preserving path length in the upper half-space model, arc-length and distance, the Poincaré disc model, convex subsets of the hyperbolic plane, and the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications.

This updated second edition also features:

• an expanded discussion of planar models of the hyperbolic plane arising from complex analysis;

• the hyperboloid model of the hyperbolic plane;

• a brief discussion of generalizations to higher dimensions;

• many new exercises.

Klappentext

Thoroughly updated, featuring new material on important topics such as hyperbolic geometry in higher dimensions and generalizations of hyperbolicity

Includes full solutions for all exercises

Successful first edition sold over 800 copies in North America

Inhalt

1. The Basic Spaces.- 2. The General Möbius Group.- 3. Length and Distance in ?.- 4. Other Models of the Hyperbolic Plane.- 5. Convexity, Area, and Trigonometry.- 6. Groups Acting on ?.- Solutions.- Further Reading.- References.- Notation.