A unifying local convergence result for Newton's method in Riemannian manifolds

Abstract

We consider the problem of nding a singularity of a vector eld X on a complete Riemannian manifold. In this regard we prove a uni ed result for local convergence of Newton's method. Inspired by previous work of Zabrejko and Nguen on Kantorovich's majorant method, our approach relies on the introduction of an abstract one-dimensional Newton's method obtained using an adequate Lipschitz-type radial function of the covariant derivative of X. The main theorem gives in particular a synthetic view of several famous results, namely the Kantorovich, Smale and Nesterov-Nemirovskii theorems. Concerning real-analytic vector elds an application of the central result leads to improvements of some recent developments in this area.