The Unapologetic Mathematician

Mathematics for the interested outsider

Foliations

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

One obvious family of foliations arises as follows: let M=\mathbb{R}^n, and pick some k-dimensional vector subspace N\subseteq M. The quotient space M/N consists of all the k-dimensional affine spaces “parallel” to N — if we pick a representative point a\in M then a+N = \{a+n\vert n\in N\} is one of the cosets in M/N. The decomposition of M into the collection of M/N is a foliation. At any point a\in M the induced distribution \Delta is the subspace \Delta_a\subseteq\mathcal{T}_aM, which is the image of N under the standard identification of M with \mathcal{T}_aM.

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

Around any point we know we can find some chart (U,x) so that the slices \{q\in U\vert x^{k+j}(q)=a_{k+j}\} are all integrable submanifolds of \Delta. By the assumption that M is second-countable we can find a countable cover of M consisting of these patches.

We let \mathcal{S} be the collection of all the slices from all the patches in this cover, and define an equivalence relation \sim on \mathcal{S}. We say that S\sim S' if there is a finite sequence S=S_0,S_1,\dots,S_l=S' of slices so that S_i\cap S_{i+1}\neq\emptyset. Since each S\subseteq U is a manifold, it can only intersect another chart (V,y) in countably many slices, and from here it’s straightforward to show that each equivalence class of \mathcal{S}/\sim can only contain countably many slices. Taking the (countable) union of each equivalence class gives us a bunch of immersed connected integral manifolds of \Delta, and any two of these are either equal or disjoint, thus giving us a partition. And since any connected integral manifold of \Delta 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 \Delta.

July 1, 2011 Posted by | Differential Topology, Topology | 1 Comment

   

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