Minimizing Curves in Prox-regular Subsets of Riemannian Manifolds

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