Let’s say that we have a one-dimensional distribution on a manifold . Around any point we can find a patch and an everywhere-nonzero vector field on so that spans for every . Then we know we can find a chart around such that on . Then the curve with coordinates and for all other is a one-dimensional integral submanifold of through .
Now, this doesn’t always work for every distribution . It turns out that the key ingredient is that all one-dimensional distributions are integrable; we can show that any integrable -dimensional distribution has an integrable submanifold through any point .
To give an even more detailed statement, let be a -dimensional integrable distribution on . for every there is a chart with , , and so that for any the set is an integral submanifold of . Further, any connected integral submanifold of comes in this way.
Since this statement is purely local we can get away with working in some set of local coordinates to start with, although obviously not the one we’re trying to find. As such, we will assume that , that , and that is spanned by the for . We’ll also let be the projection onto the first components, so is an isomorphism. By parallel translation, for all in a neighborhood of .
Now, like we did yesterday we can use these isomorphisms to build vector fields on belonging to that are -related to the on . Then , but since is integrable we know that . Since is an isomorphism on , we conclude that . Now we can find a coordinate patch around with on , just as in the one-dimensional case. It’s no loss of generality to tweak it until we have . This gives us an integral submanifold through the origin.
But our assertion goes further! Let , where is the complementary projection to . This map has maximal rank everywhere, so we know that for each the preimage is an -dimensional submanifold . The tangent space to consists of exactly those vectors in the kernel of , but since is a diffeomorphism these are exactly those vectors such that is in the kernel of — those .
Conversely, if is a connected integral manifold of contained in . If , then is in , which is spanned by the for . Thus . And so for all . Since is connected, is constant, and thus comes from picking a value for each .
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 .