On the space of oriented geodesics of hyperbolic 3-space

We construct a Kähler structure (J, Ω, G) on the space L(H ³) of oriented geodesies of hyperbolic 3-space H³ and investigate its properties. We prove that (L(H³), J) is biholomorphic to P¹ x P¹-Δ̄, where Δ̄ is the reflected diagonal, and that the Kähler 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(H³) is isomorphic to the identity component of the hyperbolic isometry group. Finally, we show that the geodesies of G correspond to ruled minimal surfaces in H³, which are totally geodesic if and only if the geodesies are null.