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 \({\mathbb R}^n\) 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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© Springer Science+Business Media Singapore 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AucklandAucklandNew Zealand

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