On the superlinear convergence of Newton's method on Riemannian   manifolds
  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. 
Teles A. Fernandes, Orizon P. Ferreira, Yuan J. Yun
math.OC