(see preliminaries of my thesis)
Distributions can be approached from a dual point of view.
In Warner_1983, it is shown that a distribution $\mathcal{Z}$, in the sense of a submodule of $TM$, can be described by its annihilator,
$$ \mbox{Ann}(\mathcal{Z})=\{\omega\in \Omega^*(U): \omega \quad \mbox{annihilates}\quad \mathcal{Z}\}, $$where a $k$-form $\omega$ is said to annihilate $\mathcal{Z}$ if $\omega(Y_1,\ldots,Y_k)=0$ on $U$ whenever $Y_1,\cdots,Y_k\in \mathcal{Z}$. The set $\mbox{Ann}(\mathcal{Z})$ is an algebraic ideal of the ring $\Omega^*(M)$. Then, in Warner_1983 it is proven that this ideal is locally generated by 1-forms (Proposition 2.28). So, locally, we have a Pfaffian system (i.e., a submodule of $\Omega^1(U)$ for certain $U\subseteq M$) denoted by $\mathcal{Z}^{\circ}$, and which is generated by pointwise linearly independent 1-forms
$$ \mathcal{Z}^{\circ}:=\mathcal{S}(\{\omega_1,\cdots,\omega_{n-r}\}). $$But if we are concerned with a local context, that is, we are allowed to reduce $M$ to arbitrary small open sets $U$, the construction can be simplified. Or if we suppose that $M$ is contractible space (and therefore any vector bundle is trivial).
The distribution $\mathcal{Z}$ is given by the submodule generated by $X_1, \ldots, X_r$. Since we have a contractible open set $U$ we can complete to a frame $X_1, \ldots, X_r,Y_1,\ldots,Y_{n-r}$ and take the associated coframe $\zeta_1,\ldots,\zeta_r,\omega_1,\ldots,\omega_{n-r}$. This way, if we define
$$ \mathcal{Z}^{\circ}=\{\omega \in \Omega^1(U): \omega(Z)=0, Z\in \mathcal{Z}\} $$it is easy to show that
$$ \mathcal{Z}^{\circ}:=\mathcal{S}(\{\omega_1,\cdots,\omega_{n-r}\}). $$After that, we define
$$ \mathcal{I}(\mathcal{Z}):=\mathcal{I}(\{\omega_1,\cdots,\omega_{n-r}\}). $$which is the same as $\mbox{Ann}(\mathcal{Z})$.
The Pfaffian system $\mathcal{Z}^{\circ}$, the associated subbundle of $T^* M$ and the ideal $\mathcal{I}(D)$, all will be called dual description of the distribution. The equivalence of the first two is given by the Serre-Swan theorem, and the equivalence of the first one and the third one is given by the lemma of belongingness to ideal.
It turns out that every distribution can be given by its dual description of the distribution.
(It is implicitly proved in [Lychagin_2021])
This collection of 1-forms constitutes the structure 1-form or, at least, is related to it. See an example in [xournal 113], although I have needed a vertical distribution
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Author of the notes: Antonio J. Pan-Collantes
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