Cut locus of a left invariant Riemannian metric on \(\mathrm{SO}_3\) in the axisymmetric case. (English) Zbl 1352.53044

Summary: We consider a left invariant Riemannian metric on \(\mathrm{SO}_3\) with two equal eigenvalues. We find the cut locus and the equation for the cut time. We find the diameter of such metric and describe the set of all most distant points from the identity. Also we prove that the cut locus and the cut time converge to the cut locus and the cut time in the sub-Riemannian problem on \(\mathrm{SO}_3\) as one of the metric eigenvalues tends to infinity.


53C30 Differential geometry of homogeneous manifolds
53C20 Global Riemannian geometry, including pinching
53C17 Sub-Riemannian geometry
53C22 Geodesics in global differential geometry
49J15 Existence theories for optimal control problems involving ordinary differential equations
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