Proposition 2.28 in Warner's book goes against my intuition. I guess that I'm missing something, but I think it is not totally correct. I paste it here:
where $\mathcal{I}(\mathcal{D})=\left\{\omega \in E^{*}(M): \omega \text { annihilates } \mathcal{D}\right\}$.
Specifically, I disagree with part c), the uniqueness of the distribution $\mathcal{D}$. At a first glance, I can not "feel" how the ideal $\mathcal{I}(\mathcal{D})$, which is a "global" data of the manifold (it is made of differential forms defined on the whole $M$), is enough to codify the distribution $\mathcal{D}$, that is given by local vector fields defined maybe only on open sets $U$. Perhaps we can have local differential forms that codify the distribution but they don't assure us the existence of globally defined 1-forms...
If we go inside the proof, it says that if $\mathcal{D}\neq\mathcal{D_1}$ then $\mathcal{I}(\mathcal{D})\neq \mathcal{I}(\mathcal{D_1})$, and I think that this is incorrect. For example, what if a rank $r<dim(M)$ distribution $\mathcal{D}$ is such that $\mathcal{I}(\mathcal{D})=\{0\}$ (because there is no global 1-form annihilating the distribution)? In this case the trivial distribution given by $TM$ share the same "annihilator".
I would hope that the distribution could be expressed by global sections but not of $T^*M$ but of some $T^*M\otimes E$ being $E$ a rank 1 bundle that let us to glue the locally defined 1-forms.
I don't know if I am being clear enough.
