  Our first objective in this paper is to give a natural formulation of the
Christoffel problem for hypersurfaces in $H^{n+1}$, by means of the hyperbolic
Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our
second objective is to provide an explicit equivalence of this Christoffel
problem with the famous problem of prescribing scalar curvature on $\S^n$ for
conformal metrics, posed by Nirenberg and Kazdan-Warner. This construction lets
us translate into the hyperbolic setting the known results for the scalar
curvature problem, and also provides a hypersurface theory interpretation of
such an intrinsic problem from conformal geometry. Our third objective is to
place the above result into a more general framework. Specifically, we will
show how the problem of prescribing the hyperbolic Gauss map and a given
function of the hyperbolic curvature radii in $H^{n+1}$ is strongly related to
some important problems on conformally invariant PDEs in terms of the Schouten
tensor. This provides a bridge between the theory of conformal metrics on
$\S^n$ and the theory of hypersurfaces with prescribed hyperbolic Gauss map in
$\H^{n+1}$. The fourth objective is to use the above correspondence to prove
that for a wide family of Weingarten functionals $W(\k_1,..., \k_n)$, the only
compact immersed hypersurfaces in $H^{n+1}$ on which $W$ is constant are round
spheres.
