On the Superlinear Convergence of Newton’s Method on Riemannian Manifolds

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