On the superlinear convergence of Newton's method on Riemannian manifolds

TL;DR

This paper proves that Newton's method converges superlinearly when finding singularities of vector fields on Riemannian manifolds, under certain invertibility conditions, extending classical convergence results to a geometric setting.

Contribution

It establishes the well-definedness and superlinear convergence of Newton's method on Riemannian manifolds, a novel extension of classical Euclidean results.

Findings

Newton's method is well-defined near the singularity under invertibility assumptions.

The sequence generated by Newton's method converges superlinearly to the singularity.

The results extend convergence theory to Riemannian geometric contexts.

Abstract

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.