Spiraling of sub-Riemannian geodesics around the Reeb flow in the 3D contact case

TL;DR

This paper describes how sub-Riemannian geodesics in a 3D contact manifold spiral around Reeb orbits, providing a detailed geometric understanding of their behavior for large initial momenta.

Contribution

It offers a precise description of the spiraling behavior of geodesics around Reeb orbits in 3D contact sub-Riemannian manifolds, utilizing Melrose's normal form.

Findings

Geodesics spiral around Reeb orbits in phase and configuration space

Analysis based on Melrose's normal form along Reeb orbits

Provides a detailed geometric characterization of geodesic behavior

Abstract

We consider a closed three-dimensional contact sub-Riemannian manifold. The objective of this note is to provide a precise description of the sub-Riemannian geodesics with large initial momenta: we prove that they "spiral around the Reeb orbits", not only in the phase space but also in the configuration space. Our analysis is based on a normal form along any Reeb orbit due to Melrose.