I'm given he following closed form on $\Bbb R^2$ $$\omega = (\sin^4\pi x+\sin^2\pi(x+y))dx - \cos^2\pi(x+y)dy$$
and let $\eta$ be the unique $1$-form on the torus such that $p^*\eta=\omega$, where $p$ is the usual covering $\Bbb R^2\to T^2$.
Show whether $\eta$ is closed/exact or no.
I showed that $\eta$ is not exact since there is a loop s.t. $\int_C \eta \neq 0$. But I've no idea how to deal with closeness. Any suggestion?