Examples of integrable sub-Riemannian geodesic flows

TL;DR

This paper presents examples of sub-Riemannian and Riemannian metrics with integrable geodesic flows, highlighting the non-holonomic case's unique features like non-compact phase space and positive topological entropy.

Contribution

It provides new explicit examples of integrable sub-Riemannian geodesic flows and explores their properties, including holonomization and entropy characteristics.

Findings

Existence of sub-Riemannian metrics with integrable geodesic flows and positive entropy

Construction of Riemannian examples via holonomization of sub-Riemannian metrics

Identification of a Liouville-integrable Hamiltonian system with positive entropy

Abstract

Motivated by a paper of Bolsinov and Taimanov DG/9911193 we consider non-holonomic situation and exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. Moreover the Riemannian examples are obtained as "holonomization" of sub-Riemannian ones. A feature of non-holonomic situation is non-compactness of the phase space. We also exhibit a Liouvulle-integrable Hamiltonian system with topological entropy of all integrals positive.