Let $u\epsilon C^{2}\left(\overline{\Omega}\right)$be a positive
solution of the differential equation $\Delta u+f\left(u\right)=0$
in $\Omega$ with boundary condition u=0 on $\partial\Omega$ where
f is a C$^{1}$ function and $\Omega$ is a geodesic ball in the hyperbolic
space $\mathbf{H}^{\mathbf{n}}$ $\left(\textrm{respectively}\:\textrm{sphere}\:\mathbf{S^{\mathbf{n}}}\right)$.
Further in case of sphere we assume that $\overline{\Omega}$ is contained
in a hemisphere. Then we prove that u is radially symmetric.
