  A classical theorem, mainly due to Aleksandrov and Pogorelov, states that any
Riemannian metric on $S^2$ with curvature $K>-1$ is induced on a unique convex
surface in $H^3$. A similar result holds with the induced metric replaced by
the third fundamental form. We show that the same phenomenon happens with yet
another metric on immersed surfaces, which we call the horospherical metric.
  This results extends in higher dimension, the metrics obtained are then
conformally flat. One can also study equivariant immersions of surfaces or the
metrics obtained on the boundaries of hyperbolic 3-manifolds. Some statements
which are difficult or only conjectured for the induced metric or the third
fundamental form become fairly easy when one considers the horospherical
metric.
  The results concerning the third fundamental form are obtained using a
duality between $H^3$ and the de Sitter space $S^3_1$. In the same way, the
results concerning the horospherical metric are proved through a duality
between $H^n$ and the space of its horospheres, which is naturally endowed with
a fairly rich geometrical structure.
