# An extra condition on Frobenius theorem for $1$-forms

Consinder a smooth manifold $$M$$ and $$\omega$$ a smooth $$1$$-form on $$M$$. Assume that there is an open set $$U\subset M$$ such that $$\omega$$ never vanishes on $$U$$. One can define a smooth distribution $$\cal F$$ on $$TU$$ via the kernel of $$\omega$$. Frobenius theorem tells us that such distribution promotes a foliation on $$M$$ provided if $$d\omega\wedge\omega =0$$, that happens to be equivalent to the existence of a smooth $$1$$-form $$\alpha$$ such that $$d\omega = \alpha\wedge \omega.$$

My questions are, can one assume that $$\alpha$$ is closed in some case? If yes, what can one conclude something else about the smooth distribution? How restrictive is this condition?

I am particularly interested on this question because this can provide simple proofs for Calabi-Honda theorems of intrinsically harmonic 1-forms.

Locally, yes. By the Frobenius theorem, if $$\omega\wedge d\omega=0$$ then there is a local coordinate system, near each point, in which $$\omega=f \, dx$$ for some function $$f$$, and then we can take $$\alpha=df/f$$.
The existence of a global closed $$\alpha$$ imposes strong restrictions on the foliation. The foliation would transversely affine, in the sense that there would exist an open covering of the ambient manifold and a first integral over each open set of the covering such that, over non-empty intersections, the first integrals differ by an element of the affine group.
The first integrals are of the form $$\int \exp \left( \pm \int \alpha \right) \cdot \omega .$$