  We 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 k\"ahler metric on the tangent space $TM$.
  The equivariant map has a simple geometric interpretation.
