# The Unapologetic Mathematician

## 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 $\Delta$.