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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: enter image description 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.

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    $\begingroup$ Your counterexample doesn't make sense; if $\mathcal{I}(\mathcal{D})=\{0\}$ then $\mathcal{D}=TM$. If $\operatorname{rank}(\mathcal{D})<\operatorname{dim}(M)$ then one can always construct a global 1-form other than $0$ which anihilates $\mathcal{D}$. Perhaps you're assuming (erroneously) that elements of $\mathcal{I}(\mathcal{D})$ must be nonvanishing? $\endgroup$
    – Kajelad
    Aug 20, 2021 at 22:23
  • $\begingroup$ I don't see how can you construct that global 1-form (for example in the codim 1 case). You would have defined local 1-forms $\omega_{\alpha}$ in an open cover $\{U_{\alpha}\}$; and it would be $\omega_{\alpha}=f_{\alpha \beta} \omega_{\beta}$ in $U_{\alpha} \cap U_{\beta}$. But this is not a global 1-form but, if anything, a global section of $T^*M \otimes E$ for a certain bundle with transition functions $\{f_{\alpha \beta}\}$. $\endgroup$ Aug 21, 2021 at 4:37
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    $\begingroup$ I'm not sure what you're trying to do with the transition functions; to obtain a global $1$-form from a local one, just multiply it by a suitably chosen bump function. $\endgroup$
    – Kajelad
    Aug 21, 2021 at 4:57
  • $\begingroup$ What I meant is that the local 1-forms are proportional. But I didn't realize that we can glue them with bump functions!! I was stuck by "holomorphic ideas", THANK YOU VERY MUCH. $\endgroup$ Aug 21, 2021 at 5:05

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