ON THE SPACE OF ORIENTED GEODESICS OF HYPERBOLIC 3-SPACE

We construct a kahler structure (J, Omega, G) on the space L (H-3) of oriented geodesics of hyperbolic 3-space H-3 and investigate its properties. We prove that (L(H-3), J) is biholomorphic to P-1 x P-1 - (Delta) over bar, where (Delta) over bar is the reflected diagonal, and that the Kahler metric G is of neutral signature, conformally flat and scalar flat. We establish that the identity component of the isometry group of the metric G on L (H-3) is isomorphic to the identity component of the hyperbolic isometry group. Finally, we show that the geodesics of G correspond to ruled minimal surfaces in H-3, which are totally geodesic if and only if the geodesics are null.