Dual approach to distributions

In general, Warner's construction

(see preliminaries of my thesis)
distributions can be approached from a dual point of view.
In @warner, it is shown that a distribution $\mathcal{Z}$, in the sense of a submodule of $TM$, can be described by its annihilator,

where a $k$-form $\omega$ is said to annihilate $\mathcal{Z}$ if $\omega (Y_1,\dots ,Y_k)=0$ on $U$ whenever $Y_1,\cdots ,Y_k\in \mathcal{Z}$. The set $\text{Ann}(\mathcal{Z})$ is an algebraic ideal of the ring ${\mathrm{\Omega }}^{\ast }(M)$. Then, it is proven that this ideal is locally generated by 1-forms (@warner, Proposition 2.28):

Around each point $m\in M$, there exists a neighborhood $U$ and $d-p$ independent 1-forms ${\omega }_1,\dots ,{\omega }_{d-p}$ on $U$ such that:

• Every 1-form in $\text{Ann}(\mathcal{Z})$ on $U$ is a linear combination (with smooth functions as coefficients) of these ${\omega }_i$.
• A global 1-form belongs to $\text{Ann}(\mathcal{Z})$ if and only if, when restricted to every such neighborhood $U$, it lies in the ideal generated by the corresponding ${\omega }_1,\dots ,{\omega }_{d-p}$.

So, locally, we have a Pfaffian system (i.e., a submodule of ${\mathrm{\Omega }}^1(U)$ for certain $U\subseteq M$) denoted by ${\mathcal{Z}}^{\circ }$, and which is generated by pointwise linearly independent 1-forms

In a local setup

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 $X_1,\dots ,X_r$. Since we have a contractible open set $U$ we can complete to a frame $X_1,\dots ,X_r,Y_1,\dots ,Y_{n-r}$ and take the associated coframe ${\zeta }_1,\dots ,{\zeta }_r,{\omega }_1,\dots ,{\omega }_{n-r}$. This way, if we define

it is easy to show that

After that, we define

which is the same as $\text{Ann}(\mathcal{Z})$.

The Pfaffian system ${\mathcal{Z}}^{\circ }$, the associated subbundle of $T^{\ast }M$ 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])

Relation to the structure 1-form

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