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.

翻译：我们研究了从流形$\mathcal{X}$到向量丛$\mathcal{E}$的映射求零点的牛顿法。在该设定下，需要向量丛$\mathcal{E}$上的联络来确保牛顿方程良定义，同时需要流形$\mathcal{X}$上的收缩映射来计算牛顿更新。我们利用黎曼距离的巴拿赫空间变体，基于适当的可微性概念讨论了局部收敛性。此外，我们将仿射协变阻尼策略推广至当前设定。最后，讨论了该方法的两项简单应用，即向量场不动点与泛函驻点的求解。