A dynamical characterization of universally tight lens spaces

TL;DR

This paper characterizes when a contact 3-manifold is a standard lens space using Reeb flow conditions, and constructs rational global surfaces of section for certain Reeb flows on lens spaces.

Contribution

It provides necessary and sufficient conditions for lens spaces to be standard based on Reeb flow properties and introduces methods for constructing global surfaces of section.

Findings

Characterization of standard lens spaces via Reeb flow conditions

Construction of rational global surfaces of section for Reeb flows

Conditions for nondegenerate Reeb flows on lens spaces

Abstract

We give necessary and sufficient conditions for a closed connected co-orientable contact $3$-manifold $(M,\xi)$ to be a standard lens space based on assumptions on the Reeb flow associated to a defining contact form. Our methods also provide rational global surfaces of section for nondegenerate Reeb flows on $(L(p,q),\xi_{\rm std})$ with prescribed binding orbits.