Length spectra of sub-Riemannian metrics on compact Lie groups

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

Vol. 296 (2018), No. 2, 321–340

DOI: 10.2140/pjm.2018.296.321

Abstract

Length spectra for Riemannian metrics have been well studied, while sub-Riemannian length spectra remain largely unexplored. Here we give the length spectrum for a canonical sub-Riemannian structure attached to any compact Lie group by restricting its Killing form to the sum of the root spaces. Surprisingly, the shortest loops are the same in both the Riemannian and sub-Riemannian cases. We provide specific calculations for $SU\left(2\right)$ and $SU\left(3\right)$.

Keywords

sub-Riemannian geometry, geodesics, root systems, compact Lie groups

Mathematical Subject Classification 2010

Primary: 53C17, 53C22

Secondary: 22E30, 51N30

Milestones

Received: 7 April 2017

Revised: 11 December 2017

Accepted: 2 March 2018

Published: 16 July 2018