Dynamics and Topology of Conformally Anosov Contact 3-Manifolds

TL;DR

This paper investigates the existence and properties of conformally Anosov Reeb flows on 3-manifolds, providing obstructions, geometric conditions, and implications for contact structure tightness.

Contribution

It introduces new obstructions to conformally Anosov Reeb flows and links geometric conditions to dynamical properties of Reeb fields.

Findings

S^3 does not admit conformally Anosov Reeb flows

Riemannian conditions imply Reeb fields are Anosov

Curvature conditions imply universal tightness of contact structures

Abstract

We provide obstructions to the existence of conformally Anosov Reeb flows on a 3-manifold that partially generalize similar obstructions to Anosov Reeb flows. In particular, we show $\mathbb{S}^3$ does not admit conformally Anosov Reeb flows. We also give a Riemannian geometric condition on a metric compatible with a contact structure implying that a Reeb field is Anosov. From this we can give curvature conditions on a metric compatible with a contact structure that implies universal tightness of the contact structure among other things.