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Assume that $\beta$ is a real analytic 2-form on an analytic manifold $M$ and $\alpha$ is an analytic non vanishing 1-form on $M$. Assume that $\beta \wedge \alpha=0$. Is there an analytic 1-form $\gamma$ on $M$ with $\alpha \wedge \gamma=\beta$?

This is needed in the following question:

A cohomology associated to a 1- form

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  • $\begingroup$ Probably the easiest way to verify this is to choose local co-ordinates such that $\alpha = f\,dx^1$. $\endgroup$
    – Deane Yang
    Commented Mar 29, 2014 at 21:01
  • $\begingroup$ @DeaneYang But my question is a global question. Since we search for an analytic solution we can not globalize using partition of unity $\endgroup$
    – Ali Taghavi
    Commented Mar 29, 2014 at 21:03
  • $\begingroup$ Ah, good point. $\endgroup$
    – Deane Yang
    Commented Mar 29, 2014 at 21:10
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    $\begingroup$ If there exists a global analytic vector field $v$ such that $\langle\alpha,v\rangle$ never vanishes, then $\gamma = i(v)\beta$ works. $\endgroup$
    – Deane Yang
    Commented Mar 29, 2014 at 21:16
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    $\begingroup$ A more general condition is that there exists a global vector field $v$ such that $\gamma = i(v)\beta$ never vanishes. $\endgroup$
    – Deane Yang
    Commented Mar 29, 2014 at 22:13

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