About this book
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.
- DOI https://doi.org/10.1007/1-84628-220-9
- Copyright Information Springer-Verlag London Limited 2005
- Publisher Name Springer, London
- eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
- Print ISBN 978-1-85233-934-0
- Online ISBN 978-1-84628-220-1
- Series Print ISSN 1615-2085
- Buy this book on publisher's site