HYPERBOLIC GEOMETRY

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, taking the approach that hyperbolic geometry consists of the study of those quantities invariant under the action of a natural group of transformations. 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, the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications. The style and level of the book, which assumes few mathematical prerequisites, make it an ideal introduction to this subject and provide the reader with a firm grasp of the concepts and techniques of this beautiful area of mathematics. The Springer Undergraduate Mathematics Series (SUMS) is a new series for undergraduates in the mathematical sciences. From core foundational material to final year topics, SUMS books take a fresh and modern approach and are ideal for self-study or for a one- or two-semester course. Each book includes numerous examples, problems and fully-worked solutions.