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This question is inspired by this recent one and this one; I hope it's not too elementary.

Let $M$ be a (closed) smooth manifold and $X$ a vector field on $M$. Fix any Riemannian metric $g$ on $M$ and let $X^{\flat}=g(X,\cdot)$ be the metric-dual differential form of $X$.

If $dX^{\flat}=0$, what can be said about the trajectories of $X$?

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For example, $X$ can't have contractible (or, in fact, nullhomologous) closed orbits. This has been said in some answer to the first question you linked. –  Marco Golla Sep 21 '12 at 8:23
    
@MarcoGolla: there, it was assumed $X$ never zero. But, ok, it may be a good hypothesis to add to my question. –  Qfwfq Sep 21 '12 at 8:31
    
Have you looked at Michael Farber's "Topology of closed one-forms" book? I believe he discusses related problems in the last chapter. –  Mark Grant Sep 21 '12 at 8:42
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