  In this paper we give a natural condition for when a volumorphism on a
Riemannian manifold $(M,g)$ is actually an isometry with respect to some other,
optimal, Riemannian metric $h$. We consider the natural action of volumorphisms
on the space $\M_\mu^s$ of all Riemannian metrics of Sobolev class $H^s$,
$s>n/2$, with a fixed volume form $\mu$. An optimal Riemannian metric, for a
given volumorphism, is a fixed point of this action in a certain complete
metric space containing $\M_\mu^s$ as an isometrically embedded subset. We show
that a fixed point exists if the orbit of the action is bounded. We also
generalize a mean ergodic theorem and a fixed point theorem to the nonlinear
setting of complete global Alexandrov nonpositive curvature spaces.
