Local minimality of weak geodesics on prox-regular subsets of Riemannian manifolds

In this paper we prove that every locally minimizing curve with constant speed in a prox-regular subset of a Riemannian manifold is a weak geodesic. Moreover, it is shown that under certain assumptions, every weak geodesic is locally minimizing. Furthermore a notion of closed weak geodesics on prox-regular sets is introduced and a characterization of these curves as nonsmooth critical points of the energy functional is presented.