Clifford-Wolf translations of homogeneous Randers spheres

Shaoqiang Deng,Ming Xu

doi:10.1007/s11856-013-0037-4

Shaoqiang Deng, Ming Xu

Open Access

https://doi.org/10.1007/s11856-013-0037-4

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Abstract

In this paper, we study Clifford-Wolf translations of homogeneous Randers metrics on spheres. It turns out that we can present a complete description of all the local one-parameter groups of Clifford-Wolf translations for homogeneous Randers metrics on spheres. The most important point of this paper is a new phenomenon in Finsler geometry. Namely, we find that there are some CW-homogeneous Randers spaces which are essentially not symmetric. This is a great difference compared to Riemannian geometry, where any CW-homogeneous Riemannian manifold must be locally symmetric.

Highlights

In this paper we continue our study concerning Clifford-Wolf translations of Finsler spaces in our previous article ([DXP])
Our main goal here is to give a complete description of Clifford-Wolf translations of homogeneous Randers metrics on spheres
Recall that a Clifford-Wolf of a locally compact connected metric space is an isometry of the space which moves all the point in the same distance

Summary

Introduction

In this paper we continue our study concerning Clifford-Wolf translations of Finsler spaces in our previous article ([DXP]). Our main goal here is to give a complete description of Clifford-Wolf translations of homogeneous Randers metrics on spheres. It was proved in [BN09] that any restrictively CW-homogeneous Riemannian manifold must be locally symmetric. The notion of CW-homogeneous and restrictively CW-homogeneous Riemannian manifold can be generalized to the Finsler case (see Definition 2.5 below). It is a natural problem to find out whether the above conclusions still hold for Finsler spaces and to give a complete classification of all the CW-homogeneous Finsler spaces This problem is much more difficult compared to the Riemannian case. We will give a complete list of all the Clifford-Wolf translations of the homogeneous Randers metrics on spheres

Preliminary

Homogeneous Randers metrics on spheres

Findings

Randers Spheres with unitary isometry groups