2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119854We equip the whole tangent space $TM$ to a hyperbolic manifold $M$ (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of $M$ extend to isometries of $TM$ by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In the case of hyperbolic space, we exhibit an equivariant diffeomorphism between $TM$ and the fourth symmetric complex domain of E. Cartan, also known as the Lie ball. The closure of the Lie ball appears as a horospheric compactification of the tangent bundle to hyperbolic space, and its Bergmann metric gives an intrinsic natural kähler metric on the tangent space $TM$. The equivariant map has a simple geometric interpretation.Geometric TopologyDifferential Geometry51M10, 58A05text