Locally symmetric homogeneous Finsler spaces

TL;DR

This paper characterizes when quotient spaces of simply connected symmetric Finsler spaces are homogeneous, showing they occur precisely when the acting group consists of Clifford translations, and classifies these translations.

Contribution

It provides a characterization of homogeneous quotient Finsler spaces via Clifford translations and classifies all such translations in symmetric Finsler spaces.

Findings

Quotient of symmetric Finsler space by a discrete group is homogeneous iff the group consists of Clifford translations.

Complete classification of Clifford translations in symmetric Finsler spaces.

Established conditions under which the quotient space inherits homogeneity.

Abstract

Let $(M,F)$ be a connected Finsler space and $d$ the distance function of $(M,F)$. A Clifford translation is an isometry $\rho$ of $(M,F)$ of constant displacement, in other words such that $d(x,\rho(x))$ is a constant function on $M$. In this paper we consider a connected simply connected symmetric Finsler space and a discrete subgroup $\Gamma$ of the full group of isometries. We prove that the quotient manifold $(M, F)/\Gamma$ is a homogeneous Finsler space if and only if $\Gamma$ consists of Clifford translations of $(M,F)$. In the process of the proof of the main theorem, we classify all the Clifford translations of symmetric Finsler spaces.