  We are interested in comparing probability distributions defined on
Riemannian manifold. The traditional approach to study a distribution relies on
locating its mean point and finding the dispersion about that point. On a
general manifold however, even if two distributions are sufficiently
concentrated and have unique means, a comparison of their covariances is not
possible due to the difference in local parametrizations. To circumvent the
problem we associate a covariance field with each distribution and compare them
at common points by applying a similarity invariant function on their
representing matrices. In this way we are able to define distances between
distributions. We also propose new approach for interpolating discrete
distributions and derive some criteria that assure consistent results. Finally,
we illustrate with some experimental results on the unit 2-sphere.
