# De Rham Cohomology Question

Let $M$ be a compact connected $n$-manifold without boundary. Let $\mu\in\Omega^{n-1}(M)$, show that there exists a point $p\in M$ such that $d\mu(p)=0$.

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You can typeset mathematics as in LaTeX on this website: enclose mathematics in $ or $$ as appropriate. But please consider phrasing your question as a question, not as an order. – Zhen Lin Aug 5 '11 at 10:15 ## 1 Answer First, assume that$M$is oriented. Then, by Stokes's theorem, we have$\int_M d\mu = \int_{\partial M} \mu = 0$since$\partial M = \emptyset$. This can't happen if$d\mu$is never$0$. If$M$is nonorientable, let$N$by any orientable cover of$M$(say, the universal cover, or the orientation covering). Let$\pi:N\rightarrow M$be the covering map. Then by the previous case,$d(\pi^*\mu)$is$0$on some point$p$of$N$. But$0 = d(\pi^*\mu) = \pi^* (d\mu)$. Since$\pi$is a local diffeomorphism,$\pi^*$induces an isomorphism from$\Omega(M)_{\pi(p)}\rightarrow \Omega(N)_{p}$. Since the image of$d\mu$under this isomorphism is$0$, we must have$d\mu(\pi(p)) = 0\$.

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On second thought, one should definitely use the orientation covering as the universal cover may not be compact. – Jason DeVito Sep 2 '11 at 3:59