Newton's method on Riemannian manifolds: covariant alpha theory

In this paper, Smale's α theory is generalized to the context of intrinsic Newton iteration on geodesically complete analytic Riemannian and Hermitian manifolds. Results are valid for analytic mappings from a manifold to a linear space of the same dimension, or for analytic vector fields on the manifold. The invariant γ is defined by means of high-order covariant derivatives. Bounds on the size of the basin of quadratic convergence are given. If the ambient manifold has negative sectional curvature, those bounds depend on the curvature. A criterion of quadratic convergence for Newton iteration from the information available at a point is also given.