On the geometry of the space of oriented lines of the hyperbolic space
Marcos Salvai
Abstract
Let H be the n-dimensional hyperbolic space of constant sectional curvature -1 and let G be the identity component of the isometry group of H. We find all the G-invariant pseudo-Riemannian metrics on the space OGn of oriented geodesics of H (modulo orientation preserving reparametrizations). We characterize the null, time- and space-like curves, providing a relationship between the geometries of OGn and H. Moreover, we show that OG3 is Kähler and find an orthogonal almost complex structure on OG7.
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