  We derive a relationship between the eigenvalues of the Weyl-Schouten tensor
of a conformal representative of the conformal infinity of a hyperbolic
Poincar\'e manifold and the principal curvatures on the level sets of its
uniquely associated defining function with calculations based on [9] [10]. This
relationship generalizes the result for hypersurfaces in ${\H}^{n+1}$ and their
connection to the conformal geometry of ${\SS}^n$ as exhibited in [7] and gives
a correspondence between Weingarten hypersurfaces in hyperbolic Poincar\'e
manifolds and conformally invariant equations on the conformal infinity. In
particular, we generalize an equivalence exhibited in [7] between
Christoffel-type problems for hypersurfaces in ${\H}^{n+1}$ and scalar
curvature problems on the conformal infinity ${\SS}^n$ to hyperbolic Poincar\'e
manifolds.
