On the space of oriented geodesics of hyperbolic 3-space

Abstract

We construct a K\"ahler structure (${\mathbb{J}},\Omega,{\mathbb{G}}$) on the space ${\mathbb{L}}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space ${\mathbb{H}}^3$ and investigate its properties. We prove that (${\mathbb{L}}({\mathbb{H}}^3),{\mathbb{J}})$ is biholomorphic to ${\mathbb{P}}^1\times{\mathbb{P}}^1-\bar{\Delta}$, where $\bar{\Delta}$ is the reflected diagonal, and that the K\"ahler metric ${\mathbb{G}}$ is of neutral signature, conformally flat and scalar flat. We establish that the identity component of the isometry group of the metric ${\mathbb{G}}$ on ${\mathbb{L}}({\mathbb{H}}^3)$ is isomorphic to the identity component of the hyperbolic isometry group. Finally, we show that the geodesics of ${\mathbb{G}}$ correspond to ruled minimal surfaces in ${\mathbb{H}}^3$, which are totally geodesic iff the geodesics are null.