Stellar foliation structures on surfaces

TL;DR

This paper introduces zebra structures on surfaces, a generalization of translation and dilation structures, and investigates conditions for canonical representatives of homotopy classes of loops and arcs.

Contribution

It defines zebra structures and establishes conditions for the existence of canonical representatives in homotopy classes on surfaces.

Findings

Existence of canonical representatives linked to triangulations with edges joining singularities.

Equivalent geometric conditions for closed surfaces to have canonical representatives in all homotopy classes.

Zebra structures generalize known geometric structures like translation and dilation structures.

Abstract

We introduce the notion of a zebra structure on a surface, which is a more general geometric structure than a translation structure or a dilation structure that still gives a directional foliation of every slope. We are concerned with the question of when a free homotopy class of loops (or a homotopy class of arcs relative to endpoints) has a canonical representative or family of representatives, either as closed leaves or chains of leaves joining singularities. We prove that such representations exist if the surface has a triangulation with edges joining singularities (in the zebra structure sense). In the special case when the surface is closed, we describe several geometric conditions that are equivalent to the existence of canonical representations in every homotopy class of closed curves.