Chapter

Hyperbolic geometry

Jacques Franchi and Yves Le Jan

in Hyperbolic Dynamics and Brownian Motion

Published in print August 2012 | ISBN: 9780199654109
Published online January 2013 | e-ISBN: 9780191745676 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199654109.003.0002

Series: Oxford Mathematical Monographs

Hyperbolic geometry

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Basic notions of hyperbolic geometry are presented: geodesics, light rays, tangent bundles etc. These are systematically derived from properties of Minkowski space: the boundary of hyperbolic space is given by the light cone, geodesics by planes intersecting the light cone in two rays, and horospheres by affine hyperplanes parallel to a light ray and intersecting the hyperbolic space. Intrinsic formulae related to the hyperbolic distance are obtained by using only elementary linear algebra within R1,d. Harmonic conjugation is also discussed in this framework. Poincaré coordinates are extended to the boundary Hd. The geodesic and horocyclic flows are defined by the right action of Ad on frames. The classical ball and upper-half-space models are presented, and the latter is related to Poincaré coordinates. A commutation relation in PSO(1, d) is established. Stable leaves and the Busemann function are introduced, and some physical interpretations are given.

Keywords: hyperbolic geometry; commutation relations; geodesic flow; horosphere; stable leaves

Chapter.  16938 words.  Illustrated.

Subjects: Mathematical and Statistical Physics

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