Sub-Riemannian distance on Lie groups $SU(2)$ and $SO(3)$

TL;DR

This paper calculates sub-Riemannian distances on Lie groups SU(2) and SO(3) using geometric and algebraic methods, leveraging the covering map between the groups to derive explicit formulas.

Contribution

It provides explicit formulas for sub-Riemannian distances on SU(2) and SO(3), utilizing the covering map and known geodesic formulas, which was not previously detailed.

Findings

Explicit distance formulas for SU(2) and SO(3)

Use of covering map to relate distances between groups

Application of classical formulas and calculus in derivations

Abstract

The authors compute distances between arbitrary elements of Lie groups SU(2) and SO(3) for special left-invariant sub-Riemannian metrics $\rho$ and $d$. To compute distances for the second metric, we essentially use the fact that canonical two-sheeted covering epimorphism $\Omega$ of the Lie group SU(2) onto the Lie group SO(3) is submetry and local isometry with respect to metrics $\rho$ and $d$. Proofs are based on previously known formulas for geodesics with origin at the unit, F. Klein's formula for $\Omega$, trigonometric functions and relations between them, usual Calculus for functions of one real variable. But in order to avoid possible mistakes, it is required sufficiently careful application of this simple tool.