Let $\bar{M}$ be a manifold with boundary and let $M$ be its interior. Is this statement correct?
A smooth k-form $\alpha$ on $\bar{M}$ is closed (exact) if and only if its restriction to $M$, i.e. $\alpha|_{M}$, is closed (exact).
I can also assume that $\bar{M}$ is a compact subset of $\mathbb{R}^{n}$ if necessary.
Thanks!
Arzhang