Let $D$ be a distribution on $M$, and $\widetilde{V}=TM/D$. We have a canonical map (it looks like a projection)
$$ \theta: TM \rightarrow \widetilde{V} $$ $$ \theta(v):= v \text{ mod } D $$Observe that $\theta \in \Omega^1(M,\widetilde{V})=\Omega^1(M)\otimes \Gamma(\widetilde{V})$, 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 $D$.
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Author of the notes: Antonio J. Pan-Collantes
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