  In this paper we study the Newton's method for finding a singularity of a
differentiable vector field defined on a Riemannian manifold. Under the
assumption of invertibility of covariant derivative of the vector field at its
singularity, we establish the well definition of the method in a suitable
neighborhood of this singularity. Moreover, we also show that the generated
sequence by Newton method converges for the solution with superlinear rate.
