Prove that $\omega$ is closed if $\int_{c}\omega \in \mathbb{Q}$.


Let $ \omega$ be a differentiable 1-from defined on an open subset $ U \subset \mathbb{R}^{n}$ . Suppose that for each closed differential curve $ c$ in $ U$ , $ \int_{c}\omega \in \mathbb{Q}$ . Prove that $ \omega$ is closed.

I tried some approaches but they didn’t work.

Then I found this answer https://math.stackexchange.com/a/111704/444015. The answer is:

“When a closed curve is deformed continuously with a parameter, the integral varies continuously with the parameter as well.”

But I have two questions:

  1. Why the curves are deformed continuously? Why are they homotopic?

  2. I can prove the statement and I can conclude that the integral (as a continuous function) is constant. So, $ $ f(t) = \int_{c(t)}\omega = \frac{p}{q}$ $ for each $ t$ . Thus, can I conclude that $ d\omega = f”(t) = 0$ ?