On the Space of Oriented Geodesics of Hyperbolic 3-Space.

Authors

Nikos Georgiou, Department of Mathematics, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland.
BRENDAN GUILFOYLE, Department of Mathematics, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland.Follow

Document Type

Article

Disciplines

Mathematics

Publication Details

Rocky Mountain Journal of Mathematics

© 2010 Rocky Mountain Mathematics Consortium.

Abstract

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.

Recommended Citation

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.