## Foliations

A -dimensional “foliation” of an -dimensional manifold is a partition of into -dimensional connected immersed submanifolds, which are called the “leaves” of the foliation. We also ask that the tangent spaces to the leaves define a -dimensional distribution on , which we say is “induced” by , and that any connected integral submanifold of should be contained in a leaf of . It makes sense, then, that we should call a leaf of a “maximal integral submanifold” of .

One obvious family of foliations arises as follows: let , and pick some -dimensional vector subspace . The quotient space consists of all the -dimensional affine spaces “parallel” to — if we pick a representative point then is one of the cosets in . The decomposition of into the collection of is a foliation. At any point the induced distribution is the subspace , which is the image of under the standard identification of with .

Now we have another theorem of Frobenius (prolific guy, wasn’t he?) about foliations: every integrable distribution of on comes from a foliation of .

Around any point we know we can find some chart so that the slices are all integrable submanifolds of . By the assumption that is second-countable we can find a countable cover of consisting of these patches.

We let be the collection of all the slices from all the patches in this cover, and define an equivalence relation on . We say that if there is a finite sequence of slices so that . Since each is a manifold, it can only intersect another chart in countably many slices, and from here it’s straightforward to show that each equivalence class of can only contain countably many slices. Taking the (countable) union of each equivalence class gives us a bunch of immersed connected integral manifolds of , and any two of these are either equal or disjoint, thus giving us a partition. And since any connected integral manifold of must align with one of the slices in any of our coordinate patches it meets, it must be contained in one of these leaves. Thus we have a foliation, which induces .