Diameter of $SU_2$ for a left-invariant axisymmetric Riemannian metric

TL;DR

This paper derives an explicit formula for the diameter of $SU_2$ equipped with a left-invariant axisymmetric Riemannian metric, effectively computing the diameter of Berger's sphere.

Contribution

It provides the first explicit formula for the diameter of $SU_2$ with axisymmetric metrics, generalizing previous results on Berger's spheres.

Findings

Explicit diameter formula for $SU_2$ with axisymmetric metrics

Extension of diameter computations to Berger's spheres

Enhanced understanding of geometric properties of left-invariant metrics

Abstract

We consider the Lie group $SU_2$ endowed with a left-invariant axisymmetric Riemannian metric. This means that a metric has eigenvalues $I_1 = I_2, I_3 > 0$. We give an explicit formula for the diameter of such metric. Other words, we compute the diameter of Berger's sphere.