Clifford-Wolf homogeneous left invariant $(\alpha,\beta)$-metrics on compact semi-simple Lie groups

TL;DR

This paper classifies left invariant $(eta,eta)$-metrics on compact semi-simple Lie groups that are restrictively Clifford-Wolf homogeneous, showing they must be of Randers type, thus providing a complete classification.

Contribution

It introduces the concept of good normalized datum for homogeneous $(eta,eta)$-spaces and proves that restrictively CW-homogeneous metrics are necessarily Randers type on compact semi-simple Lie groups.

Findings

Restrictively CW-homogeneous $(eta,eta)$-metrics are of Randers type.

Complete classification of such metrics on compact semi-simple Lie groups.

Provides a new framework using good normalized datum for studying homogeneous Finsler spaces.

Abstract

Let $(M,F)$ be a connected Finsler space. An isometry of $(M,F)$ is called a Clifford-Wolf translation (or simply CW-translation) if it moves all points the same distance. The compact Finsler space $(M,F)$ is called restrictively Clifford-Wolf homogeneous (restrictively CW-homogeneous) if for any two sufficiently close points $x_1,x_2\in M$, there exists a CW-translation $\sigma$ such that $\sigma(x_1)=x_2$. In this paper, we define the good normalized datum for a homogeneous non-Riemannian $(\alpha,\beta)$-space, and use it to study the restrictive CW-homogeneity of left invariant $(\alpha,\beta)$-metrics on a compact connected semisimple Lie group. We prove that a left invariant restrictively CW-homogeneous $(\alpha,\beta)$-metric on a compact semisimple Lie group must be of the Randers type. This gives a complete classification of left invariant $(\alpha,\beta)$-metrics on compact semi-simple Lie groups which are restrictively Clifford-Wolf homogeneous.