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

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