Let $X$ be a vector field on a manifold $M$. Is there a necessary and sufficient condition on $X$ for it to be locally equal to the coordinate vector $\partial_j$ for some coordinate system?
For any Riemannian metric on $M$, the 1-forms $dx_j$ corresponding to the coordinate vector fields are closed forms. So, a necessary condition is that the 1-form corresponding to $X$ under any metric must be closed. Is this also sufficient? If not, what is a necessary and sufficient condition?