Let (M,g) be a riemannian manifold, ∇ the Levi-Civita connection, and let X be a vector field. I define a positiv metric:
hX=∑i,jg(∇eiX,∇ejX)[e∗i⊗e∗j]
where (ei) is an orthonormal basis of the tangent space.
hX(Y,Z)=g(∇YX,∇ZX)
If hX=g, then X is called the vector potential of g.
Can we have locally a vector potential for any riemannian metric g?