Algebraic characterizations of sub-Riemannian geodesics in semi-simple, connected, compact Lie groups

András Domokos, Matthew Krauel, Vincent Pigno, Corey Shanbrom, Michael VanValkenburgh

Published 2017-04-16Version 1

In this paper we will use the algebraic information encoded in the root system of a semi-simple, connected, compact Lie group to describe properties of sub-Riemannian geodesics. First we give an algebraic proof that all sub-Riemannian geodesics are normal. We then find characterizations and lengths of the Riemannian and sub-Riemannian geodesic loops in simple, simply connected, compact Lie groups. We provide specific calculations for $SU (2)$ and $SU (3)$.