  For a class of integral operators with kernels metric functions on manifold
we find some necessary and sufficient conditions to have finite rank. The
problem we pose has a stochastic nature and boils down to the following
alternative question. For a random sample of discrete points, what will be the
probability the symmetric matrix of pairwise distances to have full rank? When
the metric is an analytic function, the question finds full and satisfactory
answer. As an important application, we consider a class of tensor systems of
equations formulating the problem of recovering a manifold distribution from
its covariance field and solve this problem for representing manifolds such as
Euclidean space and unit sphere.
