What are the subtleties involved in solving vector PDEs on manifolds? Can someone suggest a reference summarizing the problems involved?
Specifically I want to solve a vector Helmholtz equation with a source field on a curved manifold (e.g. a sphere, or a sphere with distortions). How do I go about solving this? Say using finite elements? Is there any such study available?
EDIT: I must clarify that I am looking for methods to solve vector PDEs on general surfaces which are topologically equivalent to the sphere (e.g., ellipsoid, sphere + spherical harmonic deformations...). The deformations away from the sphere are not infinitesimal but finite.