Closed Form and Pullback compatibility

Question

given:

$U,V \subset \mathbb{R}^N, f\in C^1(V,U)$ a diffeomorphism

Let $\omega$ be a k-Form on U and $f^*\omega$ a closed Form.

Then with $ 0 = df^*(\omega) = d \omega(df)$ we have ,that $\omega$ is a closed Form. Is this correct?

Answer 1

If $f$ is a diffeomorphism, $f^*$ is an isomorphism of vector-spaces of $(k+1)$-forms, hence $0=df^*\omega = f^*d\omega$ implies $d\omega=0$.