Finsler geodesics, periodic Reeb orbits, and open books

TL;DR

This paper surveys the existence of periodic Reeb orbits on contact manifolds, connecting them with Finsler geometry and open book decompositions, and presents new existence results under certain conditions.

Contribution

It provides a comprehensive survey and extends existing results by establishing new existence statements for periodic Reeb orbits on contact manifolds with open book support.

Findings

Finsler geodesic flow can be viewed as a Reeb flow

Existence of periodic Reeb orbits on contact manifolds with open books

Conditions ensuring the existence or non-existence of periodic Reeb orbits

Abstract

We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore theorem that the Finsler geodesic flow can be interpreted as a Reeb flow. As a mild extension of previous results we present existence statements on periodic Reeb orbits on contact manifolds with suitable supporting open books.