If mathematics is the queen of the sciences, then hyperbolic geometry surely must be one of her crown jewels. Gauss, Lobachevski, and Bolyai first discovered hyperbolic geometry by denying Euclid's parallel postulate. Klein and Poincaré then polished the subject into a shining gem by crafting connections with complex analysis and group theory. These origins suggest two possible approaches to studying or teaching hyperbolic geometry: the synthetic and the analytic.
The synthetic approach is appealing since the subject remains grounded in its history; the analytic approach, however, makes it easy to exploit connections with other areas of mathematics and to progress rapidly toward questions of current research interest. But, connecting hyperbolic geometry to other areas of mathematics requires some knowledge of those other areas and making progress on current research questions usually requires a fair dose of "mathematical maturity". For a professor, graduate student, or sophisticated undergraduate with this background, the analytic path to hyperbolic geometry is a thoroughly enjoyable route.
Hyperbolic Geometry from a Local Viewpoint is a compact introduction to two-dimensional hyperbolic geometry with a distinctly analytic flavor. For those accustomed to the dense writing of the professional mathematical world, this book is a pleasure to read. It is well written, with only a few paragraphs of less than stellar quality. There are remarkably few typographical mistakes and effective use is made of choice examples. Indeed, one of the major strengths of the text is how a few well-chosen examples are used to demonstrate multiple points and to motivate new definitions or criteria. For the typical undergraduate mathematics major, however, there are not nearly enough such examples. Caution should be used if you are considering this book for such a student.
The mathematics is reasonably self-contained; the one major prerequisite is familiarity with the theorems and techniques of an undergraduate complex analysis course. The necessary theorems are stated in the text and a few of them are given proofs or sketches of proofs. The required topological concepts, such as "2-dimensional manifold," are briefly defined but some prior exposure to point set topology will make the reading easier.
As should be expected, Möbius transformations of the Riemann sphere, upper half plane, and unit disc are the central tools for studying hyperbolic geometry. The first two chapters introduce these functions and give the standard classification of them in terms of their fixed points. Also in chapter two, hyperbolic distance is introduced. This is done by defining a "hyperbolic density" on the unit disc and defining the length of a path to be the path integral of the hyperbolic density. The hyperbolic distance between two points in the unit disc is the infimum of the lengths of all paths joining the two points. The group of isometries of the hyperbolic plane (i.e. the unit disc with hyperbolic distance) is the group of Möbius functions which preserve the unit disc setwise. A discrete subgroup of this group is called a Fuchsian group. Fuchsian groups are used throughout the text to study hyperbolic structures on surfaces. The first two chapters of this book are well worth reading for this introduction to analytic hyperbolic geometry.
Chapter three summarizes important results from complex analysis, giving special attention to the Riemann mapping theorem and the Schwarz lemma, for which proofs are given. Chapter four defines the concept of "Riemann surface" (i.e. a surface together with an atlas having holomorphic overlap maps). It also succinctly defines the concepts of "fundamental group" and "covering space" (the latter in a way which will cause a topologist to blink twice). The proof that surfaces have universal covering spaces is outlined, done by piecing together fundamental domains rather than by a method which generalizes to other topological spaces. This has the advantage, however, of motivating a discussion of fundamental domains for the action of a Fuchsian group on the unit disc. Covering spaces lead naturally to a discussion of the uniformization theorem and the uniformization theorem motivates much of the remaining work in the book.
The quotient of the unit disc by a Fuchsian group is a surface and the hyperbolic density, being locally defined on the unit disc and invariant under Möbius transformations, descends to a hyperbolic density on the surface. The uniformization theorem guarantees that every orientable surface which is not a sphere or a torus has such a hyperbolic structure. This is the most natural way of defining a hyperbolic metric on a surface. However, the uniformization theorem, relying as it does on the Riemann mapping theorem, is not constructive and so an important question is how to compute, or at least approximate, the hyperbolic length of a given path on a given Riemann surface. This is equivalent to approximating the hyperbolic density function on a surface. Considerable portions of the book are devoted to developing the necessary tools for finding such approximations for paths on surfaces which are subsets of the complex plane. This emphasis on the hyperbolic density function is what provides the book's title.
For a topologist such as myself the amount of analysis in the book is wearying. Happily, this weariness is lightened by three chapters of applications which highlight the authors' own research on iterated function systems. The inequalities in these chapters are daunting, but the results are interesting and several open problems are stated.
As has surely been gathered from the previous comments, the authors have taken an approach to hyperbolic geometry which highlights its analytic aspects and which allows them to progress rapidly towards the frontier of a certain type of research into hyperbolic geometry. For those who are more topologically inclined, the first few chapters are a concise introduction to the major ideas of two dimensional hyperbolic geometry and could serve as a useful basis for seeking out other books. From these beginning chapters one might take several possible routes. One would be to learn more about the "global" properties of hyperbolic geometry which have played an important role in group theory and topology. Two notions especially worth mentioning due to their prominence in group theory are "delta-hyperbolic" and "CAT(k)" spaces. Both concepts generalize certain aspects of hyperbolic geometry. Not surprisingly, Hyperbolic Geometry from a Local Viewpoint doesn't proceed very far down this path. Indeed, the constant negative curvature of the hyperbolic plane is mentioned only in passing. One might also learn more about higher dimensional hyperbolic geometry. 3-dimensional hyperbolic geometry is currently a very exciting and active area of research and has close connections to the recently solved Poincaré and Geometrization conjectures. Another possible project would be to explore the applications of Möbius functions to dynamical systems, such as in . That book is particularly accessible and contains many beautiful images. It has surprisingly little overlap with the present volume. The present text could also provide a source of interesting examples or projects for a complex analysis course. It might even motivate for such a course. Finally, if one has, or wants to have, research interests aligned with those of the authors, then Hyperbolic Geometry from a Local Viewpoint would be an excellent introduction to the tools and ideas necessary for such research.
 Mumford, Series, and Wright. Indra's Pearls: The Vision of Felix Klein. Cambridge, 2002.
Scott Taylor is a 3-manifold topologist and a recent immigrant to Maine from southern California. His non-mathematical interests include playing the (French) horn and entertaining his 7 month old son.
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.