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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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  • $\begingroup$ 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. $\endgroup$ Sep 21, 2012 at 8:23
  • $\begingroup$ @MarcoGolla: there, it was assumed $X$ never zero. But, ok, it may be a good hypothesis to add to my question. $\endgroup$
    – Qfwfq
    Sep 21, 2012 at 8:31
  • $\begingroup$ Have you looked at Michael Farber's "Topology of closed one-forms" book? I believe he discusses related problems in the last chapter. $\endgroup$
    – Mark Grant
    Sep 21, 2012 at 8:42

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