Classification of the geodesic curves of the sub-Riemannian LR system on the Heisenberg group in dimension 5

TL;DR

This paper classifies geodesic curves in a specific sub-Riemannian system on the 5D Heisenberg group, providing explicit solutions and quadrature computations for the Hamiltonian system.

Contribution

It offers a complete classification of geodesic curves and explicit solution methods for the sub-Riemannian LR system on the 5D Heisenberg group.

Findings

Complete classification of geodesic curves.

Explicit quadrature for solutions.

Hamiltonian system shown to be completely integrable.

Abstract

We study the geodesic flow corresponding to the left-invariant sub-Riemannian metric and the right-invariant distribution on the second Heisenberg group. The corresponding Hamiltonian system is completely integrable and in this paper we study its solutions. We obtain a complete classification of the geodesic curves. Moreover, we compute $r(t)$ as the first step of the quadrature.