Canonical frames in contact 3-manifolds and applications

TL;DR

This paper explores special global frames in contact 3-manifolds, linking them to Finsler structures, and derives curvature-related estimates, including a classical systole bound on positively curved spheres.

Contribution

It introduces a new class of global frames inspired by Cartan and Bryant, providing rigidity results and sharp curvature estimates in contact 3-manifolds.

Findings

Established a link between the special frames and curvature functions.

Derived sharp estimates for Reeb orbit actions.

Reproved Toponogov's systole bound for positively curved S^2.

Abstract

We study contact 3-manifolds $Y$ with a special global frame inspired by Cartan's structure equations. This frame is dual to a generalized Finsler structure defined by Bryant. We present some examples and rigidity results on the class of manifolds whose frame satisfies certain natural conditions on a scalar function $K\colon Y\to \mathbb{R}$, related to the frame. This function realizes the curvature when $Y$ is the unit tangent bundle with respect to a metric on a surface. As applications, we obtain sharp estimates for the action of a Reeb orbit in terms of this scalar function, under the assumption that the frame satisfies specific conditions. In particular, we recover a classical upper bound on the systole of positively curved metrics on $S^2$ due to Toponogov.