  In this paper we use the relationship between conformal metrics on the sphere
and horospherically convex hypersurfaces in the hyperbolic space for giving
sufficient conditions on a conformal metric to be radial under some constrain
on the eigenvalues of its Schouten tensor. Also, we study conformal metrics on
the sphere which are invariant by a $k-$parameter subgroup of conformal
diffeomorphisms of the sphere, giving a bound on its maximum dimension.
  Moreover, we classify conformal metrics on the sphere whose eigenvalues of
the Shouten tensor are all constant (we call them \emph{isoparametric conformal
metrics}), and we use a classification result for radial conformal metrics
which are solution of some $\sigma _k -$Yamabe type problem for obtaining
existence of rotational spheres and Delaunay-type hypersurfaces for some
classes of Weingarten hypersurfaces in $\h ^{n+1}$.
