Sub-Riemannian distance in the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$

V. N. Berestovskii ; I. A. Zubareva

Matematičeskie trudy, Tome 18 (2015) no. 2, pp. 3-21

Résumé

We calculate distances between arbitrary elements of the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$ for special left-invariant sub-Riemannian metrics $\rho$ and $d$. In computing distances for the second metric, we substantially use the fact that the canonical two-sheeted covering epimorphism $\Omega$ of $\mathrm{SU(2)}$ onto $\mathrm{SO(3)}$ is a submetry and a local isometry in the metrics $\rho$ and $d$. Despite the fact that the proof uses previously known formulas for geodesics starting at the unity, F. Klein's formula for $\Omega$, 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.