Clifford-Wolf translations of left invariant Randers metrics on compact Lie groups

TL;DR

This paper characterizes when left invariant Randers metrics on compact Lie groups are Clifford-Wolf homogeneous, showing it occurs precisely when the indicatrix is a round sphere under a bi-invariant Riemannian metric, providing many examples.

Contribution

It provides a complete characterization of Clifford-Wolf homogeneous left invariant Randers metrics on compact simple Lie groups, linking homogeneity to the shape of the indicatrix.

Findings

Clifford-Wolf homogeneous metrics have indicatrices that are round spheres.

Characterization applies specifically to simple Lie groups.

Many non-reversible Finsler metrics are Clifford-Wolf homogeneous.

Abstract

A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the sam distance. A Finsler space $(M, F)$ is called Clifford-Wolf homogeneous if for any two point $x_1, x_2\in M$ there is a Clifford-Wolf translation $\rho$ such that $\rho(x_1)=x_2$. In this paper, we study Clifford-Wolf translations of left invariant Randers metrics on compact Lie groups. The mian result is that a left invariant Randers metric on a connected compact simple Lie group is Clifford-Wolf homogeneous if and only if the indicatrix of the metric is a round sphere with respect to a bi-invariant Riemannian metric. This presents a large number of examples of non-reversible Finsler metrics which are Clifford-Wolf homogeneous.