Structure 1-form of a distribution $D$

Let $D$ be a distribution on $M$, and $\tilde{V}=TM/D$. We have a canonical map (it looks like a projection)

Observe that $\theta \in {\mathrm{\Omega }}^1(M,\tilde{V})={\mathrm{\Omega }}^1(M)\otimes \mathrm{\Gamma }(\tilde{V})$, and it is called structure 1-form of $D$.

In other words, is the projection over, in some sense, the vertical bundle. When $D$ is a connection on a fiber bundle (if $M$ has itself an internal bundle structure), we have a well defined projection of $T_{p}M$ on a vertical subspace, and the structure 1-form corresponds to the connection 1-form.

This is related to the dual description of the distribution, since in a local chart $\theta$ can be seen as $n-k$ forms such that their annihilator describe $D$.