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

Ming Xu, Shaoqiang Deng|arXiv (Cornell University)|Jan 9, 2013

Advanced Differential Geometry Research11 references3 citations

TL;DR

This paper classifies left-invariant restrictively Clifford-Wolf homogeneous (α, β)-metrics on compact semi-simple Lie groups by proving that such metrics must be of Randers type. Using a good normalized datum and analyzing Killing vector fields of constant length, the authors establish that non-Riemannian (α, β)-metrics satisfying restrictive CW-homogeneity are necessarily Randers metrics, providing a complete classification for compact semi-simple Lie groups via induction on the group's simple factors.

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.

Motivation & Objective

• To classify left-invariant restrictively Clifford-Wolf homogeneous (α, β)-metrics on compact semi-simple Lie groups.
• To determine whether such metrics must be of Randers type.
• To extend previous classifications of CW-homogeneous Finsler metrics to the broader class of (α, β)-metrics.
• To establish a complete classification of left-invariant (α, β)-metrics on compact semi-simple Lie groups that are restrictively CW-homogeneous.

Proposed method

• Introduce the concept of a 'good normalized datum' for homogeneous non-Riemannian (α, β)-spaces to standardize the metric representation.
• Use the correspondence between local one-parameter groups of CW-translations and Killing vector fields of constant length (KVFCLs) in the Finsler setting.
• Analyze the structure of KVFCLs on compact semi-simple Lie groups using direct sum decompositions of the Lie algebra.
• Prove that the function φ in F = αφ(β/α) is real analytic on (−1, 1) under non-degeneracy assumptions on the β-form.
• Apply mathematical induction on the number of simple factors in the semi-simple Lie group decomposition.
• Use the fact that real analyticity of φ combined with Randers structure on subgroups forces the full metric to be Randers.

Experimental results

Research questions

• RQ1 Are there any non-Randers (α, β)-metrics that are restrictively Clifford-Wolf homogeneous on compact semi-simple Lie groups?
• RQ2 What conditions on the function φ in F = αφ(β/α) ensure restrictive CW-homogeneity?
• RQ3 Does restrictive CW-homogeneity imply full CW-homogeneity for left-invariant Finsler metrics on compact semi-simple Lie groups?
• RQ4 Can the space of Killing vector fields of constant length be used to classify (α, β)-metrics on semi-simple Lie groups?
• RQ5 Is the function φ in a good normalized datum for a restrictively CW-homogeneous (α, β)-metric necessarily real analytic on (−1, 1)?
• RQ6 Does the restriction of a restrictively CW-homogeneous (α, β)-metric to a simple factor remain restrictively CW-homogeneous?

Key findings

• Any left-invariant restrictively Clifford-Wolf homogeneous (α, β)-metric on a compact connected semi-simple Lie group must be a Randers metric.
• The classification of such metrics reduces completely to the Randers case, providing a full classification for compact semi-simple Lie groups.
• The function φ in the metric F = αφ(β/α) is real analytic on (−1, 1) when the α-dual of β has non-zero components in all simple factors of the Lie algebra.
• Restrictive CW-homogeneity on a semi-simple group implies the same property on each simple factor, enabling inductive classification.
• The restriction of a restrictively CW-homogeneous (α, β)-metric to a simple subgroup is also restrictively CW-homogeneous.
• If the restriction of F to a simple factor is Randers and φ is real analytic, then the full metric F is Randers.