# Foliations on Non-metrisable Manifolds

Chapter

## Abstract

Following Chap.  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 $${\mathbb R}^2$$.

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