A Universal Model for Conformal Geometries of Euclidean, Spherical and Double-Hyperbolic Spaces

Abstract

The study of relations among Euclidean, spherical and hyperbolic geometries dates back to the beginning of last century. The attempt to prove Euclid's fifth postulate led C. F. Gauss to discover hyperbolic geometry in the 1820's. Only a few years passed before this geometry was rediscovered independently by N. Lobachevski (1829) and J. Bolyai (1832). The strongest evidence given by the founders for its consistency is the duality between hyperbolic and spherical trigonometries. This duality was first demonstrated by Lambert in his 1770 memoir [L1770]. Some theorems, for example the law of sines, can be stated in a form that is valid in spherical, Euclidean, and hyperbolic geometries [B1832]. To prove the consistency of hyperbolic geometry, people built various analytic models of hyperbolic geometry on the Euclidean plane. E. Beltrami [B1868] constructed a Euclidean model of the hyperbolic plane, and using differential geometry, showed that his model satisfies all the axioms of hyperbolic plane geometry. In 1871, F. Klein gave an interpretation of Beltrami's model in terms of projective geometry. Because of Klein's interpretation, Beltrami's model is later called Klein's disc model of the hyperbolic plane. The generalization of this model to n-dimensional hyperbolic space is now called the Klein ball model [CFK98]. In the same paper Beltrami constructed two other Euclidean models of the hyperbolic plane, one on a disc and the other on a Euclidean half-plane. Both models are later generalized to n-dimensions by H. Poincaré [P08], and are now associated with his name. All three of the above models are built in Euclidean space, and the latter two are conformal in the sense that the metric is a point-to-point scaling of the Euclidean metric. In his 1878 paper [K1878], Killing described a hyperboloid †