  We define and study a family of distributions with domain complete Riemannian
manifold. They are obtained by projection onto a fixed tangent space via the
inverse exponential map. This construction is a popular choice in the
literature for it makes it easy to generalize well known multivariate Euclidean
distributions. However, most of the available solutions use coordinate specific
definition that makes them less versatile. %We propose improvements in two
directions. We define the distributions of interest in coordinate independent
way by utilizing co-variant 2-tensors. Then we study the relation of these
distributions to their Euclidean counterparts. In particular, we are interested
in relating the covariance to the tensor that controls distribution
concentration. We find approximating expression for this relation in general
and give more precise formulas in case of manifolds of constant curvature,
positive or negative. Results are confirmed by simulation studies of the
standard normal distribution on the unit-sphere and hyperbolic plane.
