$S$ is an n dimensional unit sphere such that $S^n=(x\in \Bbb R^{n+1}: |x|=1)$ with some fixed orientation and $\omega$ is a volume form on $S$. Prove that $\omega$ is closed. Prove that $\omega$ is not exact
I really do not know how to do these two proofs. Can someone please please help me with this?
Thanks in advance!
I believe this is just a consequence of Stokes' Theorem. First of all, all $n$-forms on an $n$-dimensional manifold $M$ are closed, since $H^{n+1}(M) = 0$
In case this is not obvious: If $\beta$ is an $n+1$ form then for each $p \in M, \beta_p$ acts on $n+1$-elements of $T_pM$. Since $\dim T_pM = n$ then necessarily any $n+1$ vectors are linearly dependent, so $\beta_p = 0$.
On the other hand, if $M$ is a orientable closed manifold (compact without boundary), then a volume form cannot be exact. To see why this cannot be the case, if $\omega$ is a volume form then $$ \int_M \omega >0$$ On the other hand, if $\omega = d\eta$ for some $n-1$ form eta, then by Stokes' theorem $$ \int_M \omega = \int_M d\eta = \int_{\partial M} \eta = 0$$ since $\partial M = \emptyset$. This is a contradiction.