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 will discuss some applications of our approach, namely, finding fixed points of vector fields, variational problems on manifolds and finding critical points of functionals.

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