Sub-Riemannian geodesics on the Heisenberg 3D nil-manifold

TL;DR

This paper investigates the properties of sub-Riemannian geodesics on the 3D Heisenberg nil-manifold, including flow dynamics, bounds on distances, and cut time estimates, contributing to geometric control theory.

Contribution

It provides a detailed analysis of geodesic flow dynamics and sharp bounds on sub-Riemannian distances on the Heisenberg nil-manifold, extending understanding of geometric structures.

Findings

Characterization of periodic and dense geodesic orbits

Sharp bounds for sub-Riemannian balls and distances

Estimates of cut time for geodesics

Abstract

We study the projection of the left-invariant sub-Riemannian structure on the 3D Heisenberg group $G$ to the Heisenberg 3D nil-manifold $M$ -- the compact homogeneous space of $G$ by the discrete Heisenberg group. First we describe dynamical properties of the geodesic flow for $M$: periodic and dense orbits, and a dynamical characterization of the normal Hamiltonian flow of Pontryagin maximum principle. Then we obtain sharp twoside bounds of sub-Riemannian balls and distance in $G$, and on this basis we estimate the cut time for sub-Riemannian geodesics in $M$.