Let $M$ be an $m$-dimensional oriented manifold with boundary $\partial M$, that is connected and oriented. I now have to show that if $M$ is compact then if $\alpha$ is a volume form on $\partial M$, there does not exist a $\beta \in \Omega^{m-1}(M)$ such that $d\beta = 0$ and $\iota^*\beta = \alpha$ where $\iota$ is the inclusion map.
I think this proof is correct, but I would like to get some confirmation:
Assume that there is a $\beta \in \Omega^{m-1}(M)$ that has these properties. Because $M$ is compact, we now that $\beta$ has compact support and hence, we may apply Stokes's Theorem, which says that
\begin{equation*}
0 = \int_M d\beta = \int_{\partial M} \iota^*\beta = \int_{\partial M } \alpha
\end{equation*}
But $\alpha$ is a volume form on so $\int_{\partial M } \alpha \neq 0$, which concludes the proof.
I also have to check whether this holds when $M$ is not compact but I don't really have an idea on how to start with this statement.
Best Answer
Your proof is fine.
For the noncompact case, what's one of the simplest non-compact manifolds $M$ with boundary you can think of? (You can give easy examples with $\partial M$ non-compact or compact.)