Let $w$ and $u$ be nowhere-vanishing smooth differential forms fields of degree $n$ on a smooth manifold $M$ (aka smooth sections of $\Omega^n(M)$).
When does there exist an automorphism $f: M \to M$ such that $f^*u = w$?
Intuitively - can one always massage the base space $M$ so that $u$ gets pulled back to $w$? If not, can this hold by placing some additional constraints on the forms? Are there simple counter examples for this?