Completely integrable distribution
(also, flat coordinate chart)
Completely integrable imply, obviously, having integral submanifolds. Therefore, it is an involutive distribution.
Moreover, completely integrable distributions are also characterized by the fact that every maximal integral manifold has just dimension $\operatorname{dim}(D)$ ([Lychagin 1991] page 4). The manifold is then decomposed in leaves of a foliation.
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Author of the notes: Antonio J. Pan-Collantes
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