Newton method for finding a singularity of a special class of locally Lipschitz continuous vector fields on Riemannian manifolds

In this paper, we extend some results of nonsmooth analysis from Euclidean context to the Riemannian setting. In particular, we discuss the concept and some properties of locally Lipschitz continuous vector fields on Riemannian settings, such as Clarke generalized covariant derivative, upper semicontinuity and Rademacher theorem. We also present a version of Newton method for finding a singularity of a special class of locally Lipschitz continuous vector fields. Under mild conditions, we establish the well-definedness and local convergence of the sequence generated by the method in a neighborhood of a singularity. In particular, a local convergence result for semismooth vector fields is presented. Furthermore, under Kantorovich-type assumptions the convergence of the sequence generated by the Newton method to a solution is established, and its uniqueness in a suitable neighborhood of the starting point is verified.