Given a -dimensional distribution on an -dimensional manifold , we say that a -dimensional submanifold is an “integral submanifold” of if for every . That is, if the subspace of spanned by the images of vectors from is exactly .
This is a lot like an integral curve, with one slight distinction: in the case on an integral curve we also demand that the length of match that of , not just the direction (up to sign).
Now, if for every there exists an integral submanifold of with , then is integrable. Indeed, let and belong to . Since is an isomorphism of vector spaces at every point, we can find and that are -related to and , respectively. That is, for all , and similarly for and . But then we know that , and so .