We construct a Kahler structure (J, Ω, G) on the space L(H~3) of oriented geodesics of hyperbolic 3-space H3 and investigate its properties. We prove that (L(H~3), J) is biholomorphic to P~1 x P~1 — Δ, where Δ is the reflected diagonal, and that the Kahler 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~3) 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~3, which are totally geodesic if and only if the geodesics are null.
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机译：我们在双曲3空间H3定向测地线的空间L（H〜3）上构造Kahler结构（J，Ω，G），并研究其性质。我们证明（L（H〜3），J）是P〜1 x P〜1-Δ的全纯形式，其中Δ是反射对角线，并且Kahler度量G是中性签名，保形平坦和标量平坦。我们建立了L（H〜3）上度量G的等距组的恒等分量与双曲等距组的恒等分量同构。最后，我们证明G的测地线对应于H〜3中的规则最小曲面，当且仅当测地线为null时，它们才完全是测地线。
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