Sub-Riemannian geometry on some step-two Carnot groups

TL;DR

This paper advances the understanding of sub-Riemannian geometry on step-two Carnot groups by characterizing GM-groups, analyzing their geometric properties, and providing new proofs for classical control problems and distance formulas.

Contribution

It introduces a characterization of GM-groups via geometric properties, extends results to all step-two groups, and offers new proofs for known optimal control and distance problems.

Findings

Characterization of GM-groups through sub-Riemannian properties

Extension of main results to all step-two groups of corank 2

New proof of the Gaveau-Brockett optimal control problem and distance formulas

Abstract

This paper is a continuation of the previous work of the first author. We characterize a class of step-two groups introduced in \cite{Li19}, saying GM-groups, via some basic sub-Riemannian geometric properties, including the squared Carnot-Carath\'{e}odory distance, the cut locus, the classical cut locus, the optimal synthesis, etc. Also, the shortest abnormal set can be exhibited easily in such situation. Some examples of such groups are step-two groups of corank $2$, of Kolmogorov type, or those associated to quadratic CR manifolds. As a byproduct, the main goal in \cite{BBG12} is achieved from the setting of step-two groups of corank $2$ to all possible step-two groups, via a completely different method. A partial answer to the open questions \cite[(29)-(30)]{BR19} is provided in this paper as well. Moreover, we provide a entirely different proof, based yet on \cite{Li19}, for the Gaveau-Brockett optimal control problem on the free step-two Carnot group with three generators. As a byproduct, we provide a new and independent proof for the main results obtained in \cite{MM17}, namely, the exact expression of $d(g)^2$ for $g$ belonging to the classical cut locus of the identity element $o$, as well as the determination of all shortest geodesics joining $o$ to such $g$.