  This paper gives a new characterization of geodesic spheres in the hyperbolic
space in terms of a ``weighted'' higher order mean curvature. Precisely, we
show that a compact hypersurface $\Sigma^{n-1}$ embedded in $\H^n$ with $VH_k$
being constant for some $k=1,\cdots,n-1$ is a centered geodesic sphere. Here
$H_k$ is the $k$-th normalized mean curvature of $\Sigma$ induced from $\H^n$
and $V=\cosh r$, where $r$ is a hyperbolic distance to a fixed point in $\H^n$.
Moreover, this result can be generalized to a compact hypersurface $\Sigma$
embedded in $\H^n$ with the ratio
$V\left(\frac{H_k}{H_j}\right)\equiv\mbox{constant},\;0\leq j< k\leq n-1$ and
$H_j$ not vanishing on $\Sigma$.

Differential Geometry