Hyperbolic Geometry
Edition: 2nd ed. 2005. Corr. 2nd printing edition
Publisher: Springer
Published: April 21, 2007
Language: eng
Format: Paperback
Number of pages: 276
ISBN-13: 9781852339340
Subjects: Non-Euclidean geometry; Mathematics; Science/Mathematics; Geometry - Non-Euclidean; Geometry; Hyperbolic geometry; Hyperbolic plane; Hyperbolicity; Mathematics / Geometry / General; Geometry, Hyperbolic; Geometry - General
First sentence: In this chapter, we set the stage for what is to come.
Description: 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, Möbius transformations, the general Möbius group, and their subgroups preserving the upper half-plane, hyperbolic arc-length and distance as quantities invariant under these subgroups, the Poincaré disc model, convex subsets of the hyperbolic plane, hyperbolic area, the Gauss-Bonnet formula 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; brief discussion of generalizations to higher dimensions; many new exercises. 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.
Notes: Springer Undergraduate Mathematics Series