  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.

General Mathematics