Let $f,g:\mathbb{R}^m \to \mathbb{R}^n$ be two smooth functions and let $k$ be a strictly positive integer. Write $f \sim_k g$ if at each point in the domain, the determinants of all $k \times k$ minors of the Jacobians of $f$ and $g$ coincide. This is clearly an equivalence relation; and more generally, one could let $f$ and $g$ be smooth maps of smooth manifolds $X \to Y$, and ask that for each $k$-form $\omega$ on $Y$ the pullbacks $f^*\omega$ and $g^*\omega$ agree as $k$-forms on $X$.

Have such equivalences been studied and given an alternate characterization? Here's the sort of thing I'm looking for: if $k=1$ and we are mapping between Euclidean spaces, $f$ and $g$ must lie in the same orbit of the translation group acting on the codomain. Perhaps there is a similar Lie-theoretic reformulation of $\sim_k$ for maps $X \to Y$ even when $k > 1$?