A direct Proof for Quadratic Convergence of the Geometric Newton Method

We consider the problem of numerically computing a critical point of a functional $J\colon M\rightarrow R$ where $M$ is a Riemannian manifold. Due to local quadratic convergence a popular choice to solve this problem is the geometric Newton method. The proofs for quadratic convergence either use computations in a chart or require additional geometric quantities such as parallel translation. In this short note we provide a direct proof for quadratic convergence.