Regular foliations and integrable distributions correspond

In a regular foliation chart $(x,y)$, declare the tangent distribution to [given] be the span of ${\mathrm{\partial }}_{x^1},\dots ,{\mathrm{\partial }}_{x^k}$. Because overlap maps send plaque directions to plaque directions, these local $k$-planes patch to a smooth rank-$k$ distribution. Plaques are local integral manifolds, so the distribution is integrable.

Conversely, let $\mathcal{D}$ be an integrable rank-$k$ distribution. By the [given] local Frobenius theorem, every point has a flat chart $(x,y)$ for $\mathcal{D}$. Choose a covering by sufficiently small restrictions of these charts, refining overlap domains into plaque-coordinate neighborhoods. On each such overlap the new transverse coordinate has differential zero in every old plaque direction, so it is locally a function only of the old transverse coordinate. The refined charts therefore have transitions of the form required by a regular foliation atlas. This asserts existence of a compatible refined atlas; it does not claim that every unrestricted flat chart belongs to one common atlas.

In that atlas the plaques are precisely the local integral pieces of [given] $\mathcal{D}$, so the global leaves are exactly the maximal connected integral manifolds. The two constructions therefore recover one another.

Thus regular foliations and integrable distributions correspond. [given] ∎