  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 OG_n 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 OG_n and H. Moreover, we show that OG_3 is K\"{a}hler
and find an orthogonal almost complex structure on OG_7.
