Normal sub-Riemannian geodesics related to filtrations of Lie algebras

TL;DR

This paper extends explicit solutions for normal sub-Riemannian geodesics from simple Lie group structures to more complex chains of subgroups and describes geodesics on associated homogeneous spaces.

Contribution

It generalizes known solutions for sub-Riemannian geodesics to arbitrary chains of Lie subgroups and characterizes geodesics on homogeneous spaces.

Findings

Extended explicit solutions to general chains of Lie subgroups.

Described normal geodesic lines on homogeneous spaces.

Connected sub-Riemannian structures to filtrations of Lie algebras.

Abstract

There is a natural way to construct sub-Riemannian structures that depend on $n$ parameters on compact Lie groups. These structures are related to the filtrations of Lie subalgebras $\mathfrak g_0 < \mathfrak g_1 < \mathfrak g_2 < \dots < \mathfrak g_{n-1}<\mathfrak g_n=\mathfrak g=Lie(G)$. In the case where $n=1$, the explicit solution for normal sub-Riemannian geodesics was provided by Agrachev, Brockett, and Jurjdevic. We extend their solution to apply to general chains of Lie subgroups. Additionally, we describe normal geodesic lines of the induced sub-Riemannian structures on homogeneous spaces $G/K$, where $\mathfrak g_0=Lie(K)$.