Periodic geodesics for contact sub-Riemannian 3D manifolds

TL;DR

This paper investigates the existence and properties of periodic geodesics in contact sub-Riemannian 3D manifolds, focusing on generic metrics and specific invariant metrics on quotients of SL2(R).

Contribution

It introduces new results on the existence of closed geodesics around Reeb orbits and provides a detailed analysis of periodic geodesics for right invariant metrics on SL2(R) quotients.

Findings

Existence of closed geodesics spiraling around Reeb orbits for generic metrics.

Detailed characterization of periodic geodesics on SL2(R) quotients.

Insights into the structure of geodesics in contact sub-Riemannian manifolds.

Abstract

The goal of this paper is to study periodic geodesics for sub-Riemannian metrics on a contact 3D-manifold.We develop two rather independent subjects:1) The existence of closed geodesics spiraling around periodic Reeb orbits for a generic metric.2) The precise study of the periodic geodesics for a right invariant metric on a quotient of SL2(R)