  In this paper we extend recent developments in computational optimal
transport to the setting of Riemannian manifolds. In particular, we show how to
learn optimal transport maps from samples that relate probability distributions
defined on manifolds. Specializing these maps for sampling conditional
probability distributions provides an ensemble approach for solving nonlinear
filtering problems defined on such geometries. The proposed computational
methodology is illustrated with examples of transport and nonlinear filtering
on Lie groups, including the circle $S^1$, the special Euclidean group $SE(2)$,
and the special orthogonal group $SO(3)$.
