Canonical frames in contact 3-manifolds and applications

2026/03/31 by Brayan Ferreira, Marcelo Miranda, Alejandro Vicente · 1 voice
Mathematics · #math.SG #math.DG

paper · pdf

arxiv published 2026/03/31 · arxiv updated 2026/06/18

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→ 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 S² due to Toponogov.

Citations

• Finsler geodesics, periodic Reeb orbits, and open books
• An Introduction to Geometric Topology
• Generalized Finsler structures on closed 3-manifolds
• Dynamically convex Finsler metrics and J-holomorphic embedding of asymptotic cylinders
• An Introduction to Riemann-Finsler Geometry
• Projectively flat Finsler 2-spheres of constant curvature
• The Geometries of 3-Manifolds
• Dreidimensionale euklidische Raumformen

Discussions

• Brayan Ferreira, Marcelo Miranda, Alejandro Vicente: Canonical frames in contact 3-manifolds and applications https://arxiv.org/abs/2603.30027 https://arxiv.org/pdf/2603.30027 https://arxiv.org/html/2 [bsky, 0 points, 0 comments]

Related

• Tightness in contact metric 3-manifolds
• Geodesic vector fields, induced contact structures and tightness in dimension three
• Einstein Metrics Adapted to Contact Structures on 3-Manifolds
• Contact structures on 3-manifolds are deformations of foliations
• Mixed metric 3-contact manifolds and paraquaternionic Kähler manifolds