It is trivial to prove that the integration of a $(n-1)$ exact form on the boundary of a $n$-manifold is 0. What about the contraposative ? If the integration of a $(n-1)$-form on the boundary of a $n$-manifold is 0, is this form exact ? If not, are there particular conditions to satisfy for this to be the case ?
For example, in the case of $S^1$, any $1$-form can be written as $f(\theta)d\theta=c d\theta+dg(\theta)$, $c$ being the integral around $S^1$, and $g$ a differentiable function on $S^1$. Now in this case if the integral is 0, it implies that the form is exact. I wondered in the general case if such decomposition is always possible, because if so, it can be proven that the integral being 0 implies that the form is exact. Is this correct ?