Sub-Riemannian distance in the Lie groups SU(2) and SO(3)

Abstract

We calculate distances between arbitrary elements of the Lie groups SU(2) and SO(3) for special left-invariant sub-Riemannian metrics ρ and d. In computing distances for the second metric, we substantially use the fact that the canonical two-sheeted covering epimorphism Ω of SU(2) onto SO(3) is a submetry and a local isometry in the metrics ρ and d. Despite the fact that the proof uses previously known formulas for geodesics starting at the unity, F. Klein’s formula for Ω, trigonometric functions, and the conventional differential calculus of functions of one real variable, we focus attention on a careful application of these simple tools in order to avoid the mistakes made in previously published mathematical works in this area.