Exactness of products of forms of trivial cohomology with generic functions

Question

Say we have a some form $\omega$ of trivial cohomology. Further, we have some function $f$. Is it known what conditions $f$ needs to satisfy in order for

$\int f\cdot \omega = 0$

to hold, apart from the cases where $f=c$ or $f=0$?

Answer 1

At this level of generality it's hard to get far away from definitions. We have an exact form $\omega=d\varphi$ and would like to know when $f\omega$ is exact.

Since $d(f\omega)=df\wedge \omega+f\,d\omega=df\wedge \omega$, there is a necessary condition: $df\wedge\omega=0$.

And since $d(f\varphi)=df\wedge \varphi+f\,d\varphi=df\wedge \varphi + f\omega$, there is a sufficient condition: $df\wedge\varphi=0$.