Foliations on Non-metrisable Manifolds

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