On the space of oriented geodesics of hyperbolic 3-space

Abstract

We construct a Kähler 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 Kähler 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.

25 pages, AMS-LATEX, 4 figures, Version 2: new references, typos corrected, results on geodesics added

References in corpus (1)

• An Indefinite Kaehler Metric on the Space of Oriented Lines