On 2-step, corank 2 nilpotent sub-Riemannian metrics

TL;DR

This paper analyzes 2-step, corank 2 nilpotent sub-Riemannian metrics, providing explicit optimal synthesis and insights into cut and conjugate times, with implications for measure smoothness in 6D cases.

Contribution

It offers explicit optimal synthesis for these metrics and reveals that cut time generally differs from the first conjugate time, with applications to measure smoothness.

Findings

Explicit optimal synthesis for 2-step, corank 2 nilpotent sub-Riemannian metrics.

Cut time often differs from the first conjugate time, with a simple explicit formula.

Smoothness properties of the spherical Hausdorff measure in 6D cases.

Abstract

In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics that are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2 sub-Riemannian metric.