(Locally) shortest arcs of special sub-Riemannian metric on the Lie   group SO(3)
  The authors find geodesics, shortest arcs, diameter, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group SO(3), under condition that the metric is right-invariant relative to the Lie subgroup $SO(2)\subset SO(3)$. 
Valera Berestovskii and Irina Zubareva
math.DG