Cut time in the sub-Riemannian problem on the Cartan group

TL;DR

This paper proves the conjectured cut times for the sub-Riemannian problem on the Cartan group, providing insights into optimal trajectories and their uniqueness in a complex geometric setting.

Contribution

It confirms Sachkov's conjectured cut times for the sub-Riemannian Cartan problem and introduces a reduction to elliptic function systems for optimal synthesis.

Findings

Confirmed Sachkov's conjectured cut times.

Established a comparison with the Engel group case.

Provided a generic condition for the uniqueness of length minimizers.

Abstract

We study the sub-Riemannian structure determined by a left-invariant distribution of rank 2 on a step 3 Carnot group of dimension 5. We prove the conjectured cut times of Y. Sachkov for the sub-Riemannian Cartan problem. Along the proof, we obtain a comparison with the known cut times in the sub-Riemannian Engel group, and a sufficient (generic) condition for the uniqueness of the length minimizer between two points. Hence we reduce the optimal synthesis to solving a certain system of equations in elliptic functions.