Has anyone explored the notion of the Newton-Raphson method on manifolds? Or to put it another way, on $\mathbb R^n$, is there a natural coordinate free way of defining an iterate of the Newton-Raphson method?
I have some ideas on how to develop such a theory: my notions will require a need for a Riemannian metric, and will probably invoke parallel transport in some manner. But I don't want to reinvent the wheel, and I'm hoping someone else already developed the theory.
I did try google, but quite likely I used the wrong combination of keywords.