  We are to establish necessary conditions (of the primal and dual types) for
the set of weak sharp minima of a nonconvex optimization problem on a
Riemannian manifold. Here, we are to provide a generalization of some
characterizations of weak sharp minima for convex problems on Riemannian
manifold introduced by Li et al. (SIAM J. Optim., 21 (2011), pp. 1523--1560)
for nonconvex problems. We use the theory of the Fr\'echet and limiting
subdifferentials on Riemannian manifold to give the necessary conditions of the
dual type. We also consider a theory of contingent directional derivative and a
notion of contingent cone on Riemannian manifold to give the necessary
conditions of the primal type. Several definitions have been provided for the
contingent cone on Riemannian manifold. We show that these definitions, with
some modifications, are equivalent. We establish a lemma about the local
behavior of a distance function. Using the lemma, we express the Fr\'echet
subdifferential (contingent directional derivative) of a distance function on a
Riemannian manifold in terms of normal cones (contingent cones), to establish
the necessary conditions. As an application, we show how one can use weak sharp
minima property to model a Cheeger type constant of a graph as an optimization
problem on a Stiefel manifold.
