Vol. 296, No. 2, 2018

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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
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(2) and SU(3).

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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
Authors
András Domokos
Department of Mathematics and Statistics
California State University Sacramento
6000 J Street
Sacramento, CA 95819
United States
Matthew Krauel
Department of Mathematics and Statistics
California State University Sacramento
6000 J Street
Sacramento, CA 95819
United States
Vincent Pigno
Department of Mathematics and Statistics
California State University Sacramento
6000 J Street
Sacramento, CA 95819
United States
Corey Shanbrom
Department of Mathematics and Statistics
California State University Sacramento
6000 J Street
Sacramento, CA 95819
United States
Michael VanValkenburgh
Department of Mathematics and Statistics
California State University, Sacramento
6000 J Street
Sacramento, CA 95819
United States