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Let $V$ be a vector field on a smooth manifold $M$.

Are there nice conditions under which there exists a (Riemannian) metric on $M$ such that $V$ is the gradient of some smooth function on $M$?

One obstruction is that gradient vector fields have no closed integral curves (since a function is increasing on integral curves of its gradient).

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    Just a comment about the local version of the question for non-vanishing vector fields. About any point where your vector field does not vanish, there is an open neighborhood with coordinates that straighten out the integral curves, i.e. $X=\partial_z$ with $z$ one of the coordinates. $\partial_z=\nabla z$ if we take the gradient w.r.t. the pullback of the standard Euclidean metric tensor along the coordinate chart. Thus, locally around non-singular points there is always such a metric.2015-05-13

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