More on the Geometry of M ⁿ _κ

Abstract

In this chapter we return to the study of the model spaces M ^k _n . We begin by describing alternative constructions of ℍⁿ = M ⁿ₋₁ attributed to Klein and Poincaré⁹. In each case we describe the metric, geodesics, hyperplanes and isometries explicitly. In the case of the Poincaré model, this leads us naturally to a discussion of the Möbius group of Sⁿ and of the one point compactification of Eⁿ. We also give an explicit description of how one passes between the various models of hyperbolic space. In the final paragraph we explain how the metric on M ⁿ _k can be derived from a Riemannian metric and give explicit formulae for the Riemannian metric.