Fritz John necessary optimality condition on Riemannian manifolds

The definitions of generalized directional derivative and generalized gradient of Lipschitz functions defined on Riemannian manifold are presented.Some properties of the directional derivative and gradient are proved by using tangent and cotangent mapping.The minimization necessary condition of nonsmooth Lipschitz functions is given.Moreover,Fritz John necessary optimality condition in mathematical programming is provided on Riemannian manifold.