Hyperbolic Geometry

by James W. Anderson

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, and 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, providing the reader with a firm grasp of the concepts and techniques of this beautiful area of mathematics.

Genres: Mathematics, Geometry

230 pages, Paperback

First published October 18, 1999

Reviews

[Ezra · Rating 3 out of 5]

Overall, I think this book was a good idea and was, from a high level, structured correctly. Additionally, I also really appreciate the liberally-given commentary that the author gives in many circumstances.

Chapters 1-4 were good. However, I felt that Chapters 5-6 were a bit messy. I also thought that a lot of the proofs throughout the book were executed in a more drawn-out and case-checking way than they needed to be, perhaps because the author wished to avoid drawing on more advanced techniques for this book, or perhaps because these proofs really must be tedious. To be clear, this is not to say that the proofs included too many details: it is that the method of proof itself was sometimes inelegant. I ultimately cannot say whether there are better proofs out there or not, but I can say that this made the book significantly less enjoyable for me.