Normal Sub-Riemannian Geodesics Related 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 $g_0 <g_1 < g_2 < \ldots < g_{n-1}< g_n =g=Lie(G)$. 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 $g_0=Lie(K)$.