  We introduce and study covariance fields of distributions on a Riemannian
manifold. At each point on the manifold, covariance is defined to be a
symmetric and positive definite (2,0)-tensor. Its product with the metric
tensor specifies a linear operator on the respected tangent space.
Collectively, these operators form a covariance operator field. We show that,
in most circumstances, covariance fields are continuous. We also solve the
inverse problem: recovering distribution from a covariance field. Surprisingly,
this is not possible on Euclidean spaces. On non-Euclidean manifolds however,
covariance fields are true distribution representations.
