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.

翻译：我们研究用于寻找从流形$\mathcal{X}$到向量丛$\mathcal{E}$的映射零点的牛顿法。在此框架下，需要$\mathcal{E}$上的一个联络使牛顿方程良定，同时需要$\mathcal{X}$上的一个回缩来计算牛顿迭代步。我们基于适当的可微性概念，利用黎曼距离的巴拿赫空间变体来讨论局部收敛性。我们还将仿射协变阻尼策略推广到当前框架。最后，我们将探讨该方法的一些应用场景，包括向量场不动点的求解、流形上的变分问题以及泛函临界点的计算。