Hyperbolic Geometry from a Local Viewpoint
Hyperbolic Geometry from a Local Viewpoint
Written for graduate students, this book presents topics in 2-dimensional hyperbolic geometry. The authors begin with rigid motions in the plane which are used as motivation for a full development of hyperbolic geometry in the unit disk. The approach is to define metrics from an infinitesimal point of view; first the density is defined and then the metric via integration. The study of hyperbolic geometry in arbitrary domains requires the concepts of surfaces and covering spaces as well as uniformization and Fuchsian groups. These ideas are developed in the context of what is used later. The authors then provide a detailed discussion of hyperbolic geometry for arbitrary plane domains. New material on hyperbolic and hyperbolic-like metrics is presented. These are generalizations of the Kobayashi and Caratheodory metrics for plane domains. The book concludes with applications to holomorphic dynamics including new results and accessible open problems.
- Entirely self-contained text on hyperbolic geometry for plane domains, accessible to upper-level undergraduate and graduate students yet also suitable for academic researchers
- Contains over 250 exercises and worked examples
- Presents brand new material on hyperbolic and hyperbolic-like matrices
Reviews & endorsements
'Here new and interesting results are collected and presented for a target audience of graduate students and researchers, but the first half of the book is well accessible also for undergraduate students, and indeed everyone who is interested in an introduction to hyperbolic geometry.' Internationale Mathematische Nachrichten
Product details
March 2007Paperback
9780521682244
282 pages
229 × 152 × 16 mm
0.402kg
32 b/w illus. 236 exercises
Available
Table of Contents
- Introduction
- 1. Elementary transformations
- 2 Hyperbolic metric in the unit disk
- 3. Holomorphic functions
- 4. Topology and uniformization
- 5. Discontinuous groups
- 6 Fuchsian groups
- 7. General hyperbolic metric
- 8. The Kobayashi metric
- 9. The Caratheodory pseudo metric
- 10. Contraction properties
- 11. Applications
- 12 Applications II
- 13. Applications III
- 14. Estimating hyperbolic densities
- 15. Uniformly perfect domains
- 16 Appendix: Elliptic functions
- Bibliography.
