Newton's method for nonlinear mappings into vector bundles

We consider Newton's method for finding zeros of mappings from a manifold $\mathcal{X}$ into a vector bundle $\mathcal{E}$. In this setting a connection on $\mathcal{E}$ is required to render the Newton equation well defined, and a retraction on $\mathcal{X}$ is needed to compute a Newton update. We discuss local convergence in terms of suitable differentiability concepts, using a Banach space variant of a Riemannian distance. We also carry over an affine covariant damping strategy to our setting. Finally, we discuss two simple applications of our approach, namely, finding fixed points of vector fields and stationary points of functionals.