Metabelian distributions and sub-Riemannian geodesics

TL;DR

This paper characterizes metabelian distributions in sub-Riemannian geometry, showing that geodesics in rank-2 cases are continuously differentiable and that singular trajectories project within analytic varieties.

Contribution

It introduces a characterization of metabelian distributions via principal bundle structures and analyzes the regularity of geodesics in these contexts.

Findings

Geodesics in rank-2 metabelian distributions are of class C^1.

Strictly singular trajectories project into analytic varieties.

Metabelian distributions are characterized through principal bundle structures.

Abstract

We begin by characterizing metabelian distributions in terms of principal bundle structures. Then, we prove that in sub-Riemannian manifolds with metabelian distributions of rank $r$, the projection of strictly singular trajectories to some $r$-dimensional manifold must remain within an analytic variety. As a consequence, for rank-2 metabelian distributions, geodesics are of class $C^1$.