On the subRiemannian cut locus in a model of free two-step Carnot group

TL;DR

This paper characterizes the subRiemannian cut locus in a specific Carnot group, explicitly computes cut times and distances, and analyzes the geometric singularities at cut points using Hamiltonian methods.

Contribution

It provides a detailed description of the cut locus and explicit formulas for cut times and distances in a free two-step Carnot group with three generators.

Findings

Explicit characterization of the cut locus.

Calculation of cut times for extremal paths.

Identification of corner-like singularities at cut points.

Abstract

We characterize the subRiemannian cut locus of the origin in the free Carnot group of step two with three generators. We also calculate explicitly the cut time of any extremal path and the distance from the origin of all points of the cut locus. Finally, by using the Hamiltonian approach, we show that the cut time of strictly normal extremal paths is a smooth explicit function of the initial velocity covector. Finally, using our previous results, we show that at any cut point the distance has a corner-like singularity.