  The Fr\'echet mean generalizes the concept of a mean to a metric space
setting. In this work we consider equivariant estimation of Fr\'echet means for
parametric models on metric spaces that are Riemannian manifolds. The geometry
and symmetry of such a space is encoded by its isometry group. Estimators that
are equivariant under the isometry group take into account the symmetry of the
metric space. For some models there exists an optimal equivariant estimator,
which necessarily will perform as well or better than other common equivariant
estimators, such as the maximum likelihood estimator or the sample Fr\'echet
mean. We derive the general form of this minimum risk equivariant estimator and
in a few cases provide explicit expressions for it. In other models the
isometry group is not large enough relative to the parametric family of
distributions for there to exist a minimum risk equivariant estimator. In such
cases, we introduce an adaptive equivariant estimator that uses the data to
select a submodel for which there is an MRE. Simulations results show that the
adaptive equivariant estimator performs favorably relative to alternative
estimators.
