Foliations on non-metrisable manifolds II: contrasted behaviours

TL;DR

This paper explores complex behaviors of 1-dimensional foliations on non-metrisable manifolds, revealing properties that prevent foliations and demonstrating diverse foliation structures on various surfaces.

Contribution

It introduces new examples of non-metrisable surfaces with unique foliation properties and analyzes the conditions under which foliations can or cannot exist.

Findings

Certain non-metrisable surfaces do not admit foliations even after removing compact subsets.

A separable surface can have a foliation with all but one leaf being metrisable.

Every non-metrisable leaf on a Type I manifold has a saturated neighborhood of non-metrisable leaves.

Abstract

This paper, which is an outgrowth of a previous paper of the authors, continues the study of dimension 1 foliations on non-metrisable manifolds emphasising some anomalous behaviours. We exhibit surfaces with various extra properties like Type I, separability and simple connectedness, and a property which we call `squat,' which do not admit foliations even on removal of a compact (or even Lindel\"of) subset. We exhibit a separable surface carrying a foliation in which all leaves except one are metrisable but at the same time we prove that every non-metrisable leaf on a Type I manifold has a saturated neighbourhood consisting only of non-metrisable leaves. Minimal foliations are also considered. Finally we exhibit simply connected surfaces having infinitely many topologically distinct foliations.