On the Space of Oriented Geodesics of Hyperbolic 3-Space.
We construct a Kahler structure (J, Ω, G) on the space L(H³) of oriented geodesics of hyperbolic 3-space H³ and investigate its properties. We prove that (L(H³), J) is biholomorphic to P¹ × P¹ - Δ̅, where Δ̅ 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 geodesics of G correspond to ruled minimal surfaces in H³, which are totally geodesic if and only if the geodesics are null.
Georgiou, Nikos & Guilfoyle, Brendan. (2010). On the Space of Oriented Geodesics of Hyperbolic 3Space. Rocky Mountain Journal of Mathematics - ROCKY MT J MATH. 40. 10.1216/RMJ-2010-40-4-1183.