2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/123904Let 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 Kähler and find an orthogonal almost complex structure on OG_7.Communicated at ICM 2006. Submitted to Glasgow Math. J. on November 3, 2006Differential Geometrytext