Cut locus of a left invariant Riemannian metric on SO(3) in the axisymmetric case

TL;DR

This paper analyzes the cut locus and cut time for a specific class of left invariant Riemannian metrics on SO(3), providing explicit formulas, diameter, and convergence results to sub-Riemannian cases.

Contribution

It explicitly determines the cut locus and cut time for metrics with two equal eigenvalues on SO(3), and shows their convergence to sub-Riemannian limits.

Findings

Calculated the cut locus and cut time for the metric

Determined the diameter of the metric

Proved convergence to sub-Riemannian case as eigenvalue tends to infinity

Abstract

We consider a left invariant Riemannian metric on 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 SO(3) as one of the metric eigenvalues tends to infinity.