Let us assume that the 2-forms $dg\wedge dh$, $df \wedge dh$ and $df\wedge dg$ are exact. Is there any proof that the following identity holds
$\int f\,dg \wedge dh=\int h\, df\wedge dg=\int g\, dh\wedge df$
for periodic boundary conditions? $f,g,h$ are 0-forms.
It holds more generally than that. You have e.g.: $$ d(f g\, dh) = f \, dg \wedge dh + g \, df \wedge dh $$ So if your boundary conditions assure that the integral of $f g \, dh$ along the boundary vanishes then you arrive at $$ \int \int f \, dg \wedge dh - g \, dh \wedge df =0$$