  We obtain a characterization of the proximal normal cone to a prox-regular
subset of a Riemannian manifold. Moreover, some properties of Bouligand tangent
cones to prox-regular sets are described. We prove that for a prox-regular
subset S of a Riemannian manifold, the metric projection P_S to S is locally
Lipschitz on an open neighborhood of S and it is directionally differentiable
at boundary points of S. Finally, a necessary condition for a curve to be a
minimizing curve in a prox-regular set is derived.
