Group actions and closed geodesics for some hyperbolic manifolds Peter Kramer (peter.kramer@uni-tuebingen.de)
Abstract Hyperbolic spaces $H^n \sim SO(n,1,R)/SO(n,R)$ carry a pseudoriemannian metric of constant negative curvature [1]. As universal covering spaces of a closed hyperbolic manifolds $M$ they provide particular models for a cosmos with closed geodesics, compare M Lachieze-Rey and J-P Luminet [2]. Shortest closed geodesics and the autocorrelation may be compared with astrophysical evidence. We extend the continuous group analysis from $M=$ the double torus on $H^2$ given in [3] to $M=$ the hyperbolic dodecahedral space, Weber and Seifert (1932), Best (1971). The first homotopy groups $\pi(M)$ are (normal) subgroups of hyperbolic Coxeter groups. We describe closed geodesics starting at general points, their length and direction on $M$ by actions of the homotopy and related continuous groups on $H^2,H^3$. We analyze the first homology groups, the autocorrelation function and the notion of elementary systems on the manifolds. \vspace{0.2cm} [1] R G Ratcliffe, {\em Foundations of Hyperbolic Manifolds}, Springer, New York 1994 [2] M Lachieze-Rey and J-P Luminet, {\em Cosmic Topology}, Phys. Rep. {\bf 254} (1995) 135-214 [3] P Kramer and M Lorente, {\em The double torus as a 2D cosmos: groups, geometry and closed geodesics}, J Phys {\bf A 35} (2002) 1961-1981 |