We define the curvature of the distribution as the $\mathbb{R}$-bilinear skew-symmetric map
$$ \omega: \Gamma(D)\times \Gamma(D) \rightarrow \Gamma(\widetilde{V}) $$ $$ \omega(X,Y):=\theta([X,Y])=[X,Y] \text{ mod } \Gamma(D) $$where $\theta$ is the structure 1-form of the distribution.
Indeed, $\omega$ is $\mathcal{C}^{\infty}(M)$-linear, so it comes from a bundle map
$$ \omega:\Lambda^2M\rightarrow \widetilde{V} $$([Vitagliano 2017] exercise 3.6)
When $\omega=0$ we have that $D$ is involutive distribution. If it has full rank it is called completely non-integrable.
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Author of the notes: Antonio J. Pan-Collantes
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