Foliations on Non-metrisable Manifolds

Abstract

Following Chap. 7 we present our second method of transferring a local structure from euclidean space to a manifold: foliations. In ${\mathbb R}^n$ we have an affine structure which leads to a partition of ${\mathbb R}^n$ into $\mathfrak c$ many affine subspaces of some fixed dimension, for example parallel lines in ${\mathbb R}^n$ or parallel planes in ${\mathbb R}^3$. Transferring this to a manifold gives what is called a foliation of the manifold with the sets corresponding to the parallel lines, planes, etc. being called leaves. Just as we may partition ${\mathbb R}^n$ into sets of the form ${\mathbb R}^p\times \{y\}$, for $y\in {\mathbb R}^{n-p}$, for each $p=1,\dots , n-1$, so we may try to foliate a manifold into a collection of ‘parallel’ leaves of any (fixed) dimension from 1 to $n-1$. Of course these leaves may well extend well beyond any particular coordinate chart. We present the definition and some examples in Sect. 8.1. In Sect. 8.2 we investigate dimension 1 foliations on certain ‘long’ manifolds, especially manifolds of the form $M\times {\mathbb L}_+$, where $M$ is a ‘small’ manifold, for example metrisable: we find that if there is at least one non-metrisable leaf then from some point ${\alpha }$ on the foliation of $M\times {\mathbb L}_+$ is just the trivial foliation whose leaves are of the form $\{x\}\times ({\alpha },{\omega }_1)$ for $x\in M$. We then apply this in Sect. 8.3 to the long cylinder ${\mathbb S}^1\times {\mathbb L}_+$ where we find that the end behaves rather like a black hole with leaves either often circulating around with constant ${\mathbb L}_+$ coordinate or falling straight to the end with constant ${\mathbb S}^1$ coordinate. In the final section, Sect. 8.4, we discuss foliations of the long plane ${\mathbb L}^2$, initially with a compact subset removed. One surprise is that the long plane ${\mathbb L}^2$ supports very few distinct foliations, unlike the real plane ${\mathbb R}^2$.