  We study power concavity of rotationally symmetric solutions to elliptic and
parabolic boundary value problems on rotationally symmetric domains in
Riemannian manifolds. As applications of our results to the hyperbolic space
${\bf H}^N$ we have:
  $\bullet$ The first Dirichlet eigenfunction on a ball in ${\bf H}^N$ is
strictly positive power concave;
  $\bullet$ Let $\Gamma$ be the heat kernel on ${\bf H}^N$. Then
$\Gamma(\cdot,y,t)$ is strictly log-concave on ${\bf H}^N$ for $y\in {\bf H}^N$
and $t>0$.
