# Is correct this proposition of Warner's Foundations of differentiable Manifolds and Lie Groups?

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 $$r distribution $$\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.

• 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? Aug 20, 2021 at 22:23
• 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 $\{f_{\alpha \beta}\}$. Aug 21, 2021 at 4:37
• 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. Aug 21, 2021 at 4:57
• 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. Aug 21, 2021 at 5:05