On the Space of Oriented Geodesics of Hyperbolic 3-space

We construct a Kähler structure (J,Ω, G) on the space L(H) of oriented geodesics of hyperbolic 3-space H and investigate its properties. We prove that (L(H), J) is biholomorphic to P×P−∆, where ∆ 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(H) 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 H, which are totally geodesic iff the geodesics are null. The space L(M) of oriented geodesics on a 3-manifold M of constant curvature is a 4-dimensional manifold which carries a natural complex structure J. In the case where M is Euclidean 3-space E, this complex structure can be traced back to Weierstrass [12] and Whittaker [13], with its modern re-emergence occurring in Hitchen’s study of monopoles on E [4]. More recently, this structure has been supplemented by a compatible symplectic structure, so that L(M) inherits a natural Kähler structure. This has been investigated when M = E and M = E1 [1] [2] [3] and the purpose of this paper is to study the hyperbolic 3-space case M = H. From a topological point of view L(M) is homeomorphic to S2×S2 −∆, where ∆ is the diagonal. However, from holomorphic point of view we show that: Theorem 1: The complex surface (L(H), J) is biholomorphic to P × P −∆, where ∆ is the reflected diagonal (see Definition 2). The P in the Theorem refers to the boundary of the Poincaré ball model of H, considered as the past and future infinities of the oriented geodesics, from which L(H) inherits its complex structure. We then turn to the Kähler metric G and prove: Theorem 2: The Kähler metric G is of neutral signature, conformally flat and scalar flat. We also show that, despite the (++−−) signature, this metric on L(H) faithfully reflects the hyperbolic metric g on H, in the following sense: Date: 9th February, 2007. 1991 Mathematics Subject Classification. Primary: 51M09; Secondary: 51M30.