Which geodesic flows are left-handed ?

TL;DR

This paper characterizes when geodesic flows on hyperbolic 2-orbifolds are left-handed, showing it occurs only for spheres with three conic points, and explores implications for open book decompositions.

Contribution

It provides a complete classification of left-handed geodesic flows on hyperbolic 2-orbifolds and links this to open book decompositions on 3-conic spheres.

Findings

Geodesic flow on hyperbolic 2-orbifolds is left-handed only for spheres with three conic points.

On 3-conic spheres, certain collections of geodesics form open book decompositions.

The lift of zero-homology geodesics on these spheres serve as bindings for open books.

Abstract

We prove that the geodesic flow on the unit tangent bundle to a hyperbolic 2-orbifold is left-handed if and only if the orbifold is a sphere with three conic points. As a consequence, on the unit tangent bundle to a 3-conic sphere, the lift of every finite collection of closed geodesics that is zero in integral homology is the binding of an open book decomposition.