A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

We consider the problem of finding a singularity of a differentiable vector field X defined on a complete Riemannian manifold. We prove a unified result for the existence and local uniqueness of the solution, and for the local convergence of a Riemannian version of Newton's method. Our approach relies on Kantorovich's majorant principle: under suitable conditions, we construct an auxiliary scalar equation (r) = 0 which dominates the original equation X(p) = 0 in the sense that the Riemannian Newton method for the latter inherits several features of the real Newton method applied to the former. The majorant is derived from an adequate radial parameterization of a Lipschitz-type continuity property of the covariant derivative of X, a technique inspired by previous work of Zabrejko et al. on Newton's method in Banach spaces. We show how different specializations of the main result recover Riemannian versions of Kantorovich's theorem and Smale's -theorem, and, at l...

• Type: Journal
• Year: 2008
• Where: FOCM
• Authors: Felipe Alvarez, Jérôme Bolte, Julien Munier
• Keywords: FOCM 2008, Riemannian Manifold, Riemannian Versions, Vector Field