Suppose $M$ is an $n$-dimensional smooth manifold. Under what conditions is the following statement true?
If $\omega$ is a closed $k$-form on $M$ that is non-zero at every point, then the de Rham cohomology class of $\omega$ is non-trivial, i.e. $\omega$ is not exact.
We can phrase it equivalently in terms of the contrapositive:
If $\omega$ is an exact $k$-form on $M$, then it vanishes at some point of $M$.
It is trivially true for the case $k=0$. If we assume that $M$ is compact, then it becomes true for $k=1$ and $k=n$, for different reasons:
- If $\omega$ is an exact $1$-form, then $\omega=df$ for a smooth function $f:M\to\mathbb R$. Since $M$ is compact, the range of $f$ is a closed interval $[a,b]$. Then $df$ vanishes at any point $p$ with $f(p)=a$ or $f(p)=b$.
- If $\omega$ is a non-vanishing $n$-form, then its integral is non-zero and so it is not exact.
The proofs here are quite specific to $1$ and $n$, so I see no reason to believe the statement will hold for other $k$. Does it hold for $k$ other than $1$ and $n$? If not, I'd be curious about both general classes of counterexamples, and/or stricter conditions on $M$ that would make the statement hold.
Here is one counterexample for $n=3,k=2$, which should generalize to arbitrary $1<k<n$.
Let $\mathbb{T}^3=\mathbb{R}^3/(2\pi\mathbb{Z})^3$ be the $3$-torus, with local coordinates $x,y,z$ induced from $\mathbb{R}^3$ (whose differentials $dx,dy,dz$ are globally well defined). Consider the $1$-form $$ \omega=\sin(z)dx-\cos(z)dy $$ whose exterior derivative is $$ d\omega=\cos(z)dz\wedge dx+\sin(z)dz\wedge dy $$ which, evidently, is nonvanishing.
I don't have any insights to offer as to why this is only possible for $1<k<n$, but it seems to have to do with the extra local degrees of freedom that these intermediate $k$-forms have, allowing the differential equations to be solved without running into global boundary-matching problems.
(Also, given the form of $\omega$, I suspect that some cases can be related to the (non)existence of contact structures, but I have very little knowledge of this subject.)