Geodesics in the sub-Riemannian problem on the group $\mathrm{SO}(3)$

Geodesics of left-invariant sub-Riemannian structures are considered on the group $\mathrm{SO}(3)$. A complete description of periodic geodesics, their elementary properties, certain necessary conditions for minimality and estimates for the cut time and the diameter of the metric are presented.
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