Title: Hyperbolic Geometry
Authors: James W. Cannon, William J. Floyd, Richard Kenyon, Walter R. Parry
Published: 1997-01-01
Link: https://www.math.ucdavis.edu/~kapovich/RFG/cannon.pdf
Abstract
Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry. This has not been true of the mathematicians and physicists of our generation. Nevertheless with the passage of time it has become more and more apparent that the negatively curved geometries, of which hyperbolic non-Euclidean geometry is the prototype, are the generic forms of ge-ometry. They have profound applications to the study of complex variables, to the topology of two- and three-dimensional manifolds, to the study of finitely presented infinite groups, to physics, and to other disparate fields of mathemat-ics. A working knowledge of hyperbolic geometry has become a prerequisite for workers in these fields.
These notes are intended as a relatively quick introduction to hyperbolic ge-ometry. They review the wonderful history of non-Euclidean geometry. They give five different analytic models for and several combinatorial approximations to non-Euclidean geometry by means of which the reader can develop an intuition for the behavior of this geometry. They develop a number of the properties of this geometry that are particularly important in topology and group theory.
They indicate some of the fundamental problems being approached by means of non-Euclidean geometry in topology and group theory.
Volumes have been written on non-Euclidean geometry, which the reader must consult for more exhaustive information. We recommend (Iversen 1993] for starters, and Benedetti and Petronio 1992; Thurston 1997; Ratcliffe 1994] for more advanced readers. The latter has a particularly comprehensive bibliography.
On the establishment of Hyperbolic Geometry through the pursuit of the proof of Euclid’s Fifth Postulate as resulting from the previous four (p.3).
Decisive progress came in the nineteenth century, when mathematicians abandoned the effort to find a contradiction in the denial of the fifth postulate and instead worked out carefully and completely the consequences of such a denial.
It was found that a coherent theory arises it instead one assumes that
Given a line and a point not on it, there is more than one line going through the given point that is parallel to the given line.
This postulate is to hyperbolic geometry as the parallel postulate 5’ is to Euclidean geometry.
Unusual consequences of this change came to be recognized as fundamental and surprising properties of non-Euclidean geometry: equidistant curves on either side of a straight line were in fact not straight but curved; similar triangles were congruent; angle sums in a triangle were not equal to t, and so forth.
That the parallel postulate fails in the models of non-Euclidean geometry that we shall give will be apparent to the reader. The unusual properties of non-Euclidean geometry that we have mentioned will all be worked out in Section 13, entitled “Curious facts about hyperbolic space”.
We shall consider in this exposition five of the most famous of the analytic models of hyperbolic geometry. Three are conformal models associated with the name of Henri Poincaré. A conformal model is one for which the metric is a point-by-point scaling of the Euclidean metric. Poincaré discovered his models in the process of defining and understanding Fuchsian, Kleinian, and general automorphic functions of a single complex variable.
If we consider the set of points at constant squared distance from the origin, we obtain in the Euclidean case the spheres of various radii and in Minkowski space hyperboloids of one or two sheets. We may thus define the unit -dimensional sphere in Euclidean space by the formula and -dimensional hyperbolic space by the formula . Thus hyperbolic space is a hyperboloid of two sheets that may be thought of as a “sphere” of squared radius -1 or of radius ; hence the name hyperbolic geometry. See Figure 1.
