Let $M$ be a smooth manifold. Let $\omega\in \Omega^k(M)$ be the closed k-form on $M$.If we assume $\int_S\omega \ne 0$ on $S$(Assume $S $ is an compact oriented k-dimensional submanifold of $M$)
Prove $S$ can not be the boundary of some oriented compact smooth submanifold with boundary in $M$ (denote it $N$).
My attempt if exist such $N$ then we can apply stokes's theorem $$\int_{\partial N} \omega = \int_S \omega \ne 0$$
while $$\int_{\partial N}\omega = \int_N d\omega = 0$$ since $\omega$ is closed.Hence we are done.
The question is why compactness condition is necessary in the proof above?
One possible reason is Stokes's theorem holds only for smooth form with compact support. Hence compactness of $S$ and $N$ makes $i^*_S\omega$,and $i^*_N (d\omega)$ has well defined integral is my interpretation correct?
(related question here)
Best Answer
Indeed, Stokes' theorem does not apply if $N$ is not compact.
For a very simple example, let $M=\mathbb{R}$ and $S=\{0\}$. Then any constant function $\omega$ is a closed $0$-form and $\int_S\omega\neq 0$ if $\omega$ is nonzero, but $S$ is the boundary of the oriented submanifold $[0,\infty)$ of $M$. Note that in this example, the integrals $\int_S \omega$ and $\int_N d\omega$ are still perfectly well-defined; they just aren't equal. Compact support is needed to apply Stokes' theorem and conclude that they are equal, not that they are well-defined.