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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?

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up vote 3 down vote accepted

For any vector field $X$ on a smooth manifold $M$, for any point $p\in M$ (where $X_p \neq 0$, as Nils points out below), one can find (construct) a coordinate system $x^i$ on a neighborhood of $p$ so that in that neighborhood $\partial_1 = X$. See Warner, Foundations of Differentiable Manifolds and Lie Groups, Ch. 1.

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I think you have to add that $X$ is not vanishing at $p$, so that it does not vanish on a small open neighborhood of $p$. –  Nils Matthes Sep 19 '12 at 16:42
    
Thank you! Corrected. –  Neal Sep 19 '12 at 18:37
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