On symmetries of a sub--Riemannian structure with growth vector $(4,7)$

TL;DR

This paper investigates the symmetries of a specific sub-Riemannian structure with growth vector (4,7), revealing two distinct types of geodesics and utilizing symmetry reduction to analyze their properties.

Contribution

It characterizes the symmetries of a (4,7) sub-Riemannian structure and analyzes their effects on geodesic behavior using symmetry reduction techniques.

Findings

Two types of geodesics identified: intersecting and contained in the symmetry fixed point set.

Symmetry reduction helps in understanding geodesic properties.

Geodesic behavior is significantly influenced by the structure's symmetries.

Abstract

We study symmetries of specific left--invariant sub--Riemannian structure with filtration $(4,7)$ and their impact on sub--Riemannian geodesics of corresponding control problem. We show that there are two very different types of geodesics, they either do not intersect the fixed point set of symmetries or are contained in this set for all times. We use the symmetry reduction to study properties of geodesics.