Differential and Hyperbolic Geometry in More Dimensions

Abstract

The differential geometry of Chapter 5 is extended to spaces with more than two dimensions. General manifolds are defined, and a space is determined by a manifold equipped with a line element or metric tensor. In particular, the line element for three-dimensional hyperbolic space is given in terms of the spherical coordinates, and a second proof that the geometry of the horosphere is Euclidean is given. Various models of three-dimensional hyperbolic space are described. Lastly, spaces with indefinite metric, in which the metric tensor is not required to be a positive definite matrix, are discussed. A particular case of this is the Minkowski geometry used in the theory of special relativity. The Minkowski geometry appears in the next chapter in connection with the relation between hyperbolic isometries and Lorentz transformations of relativity theory.