# Length spectra of sub-Riemannian metrics on compact Lie groups

## TL;DR

This paper computes the length spectra for a canonical sub-Riemannian structure on any compact Lie group, revealing that shortest loops are identical to those in the Riemannian case, with explicit results for SU(2) and SU(3).

## Contribution

It provides the first explicit length spectrum calculations for sub-Riemannian structures on compact Lie groups, extending understanding beyond Riemannian cases.

## Key findings

- Shortest loops are the same in Riemannian and sub-Riemannian cases.
- Explicit length spectra for SU(2) and SU(3).
- Sub-Riemannian length spectra can be explicitly determined for compact Lie groups.

## Abstract

Length spectra for Riemannian metrics are well studied, while sub-Riemannian length spectra have been 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).