More on the Geometry of Mκn

In this chapter we return to the study of the model spaces Mnk. We begin by describing alternative constructions of ℍn = M−1n 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 nat...

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Bibliographic Details
Published inMetric Spaces of Non-Positive Curvature pp. 81 - 96
Main Authors Bridson, Martin R., Haefliger, André
Format Book Chapter
LanguageEnglish
Published Berlin, Heidelberg Springer Berlin Heidelberg 1999
SeriesGrundlehren der mathematischen Wissenschaften
Subjects
Cross Ratio
Geodesic Line
Geodesic Segment
Hyperbolic Distance
Hyperbolic Space
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Summary:In this chapter we return to the study of the model spaces Mnk. We begin by describing alternative constructions of ℍn = M−1n 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 Mkn can be derived from a Riemannian metric and give explicit formulae for the Riemannian metric.
ISBN:3642083994
9783642083990
ISSN:0072-7830
DOI:10.1007/978-3-662-12494-9_6