  In this paper, we generalize Magnanini-Sakaguchi's result [MS3] from
Euclidean space to spaces of constant curvature. More precisely, we show that
if a conductor satisfying the exterior geodesic sphere condition in the space
of constant curvature has initial temperature 0 and its boundary is kept at
temperature 1 (at all times), if the thermal conductivity of the conductor is
inverse of its metric, and if the conductor contains a proper sub-domain,
satisfying the interior geodesic cone condition and having constant boundary
temperature at each given time, then the conductor must be a geodesic ball.
Moreover, we show similar result for the wave equations and the Schr\"{o}dinger
equations in spaces of constant curvature.
