The Relation Between Automorphism Group and Isometry Group of Left Invariant $ (\alpha,\beta)$-metrics

TL;DR

This paper explores the relationship between automorphism and isometry groups of left invariant $(eta)$-metrics on Lie groups, extending previous results from Randers to more general $(eta)$-metrics.

Contribution

It generalizes earlier findings by establishing conditions under which a subgroup of automorphisms can be realized as a subgroup of the isometry group for $(eta)$-metrics on Lie groups.

Findings

Automorphism and isometry groups of left invariant $(eta)$-metrics are related.

Existence of $(eta)$-metrics with prescribed automorphism subgroups as isometry subgroups.

Extension of previous Randers metric results to general $(eta)$-metrics.

Abstract

This work generalizes the results of an earlier paper by the second author, from Randers metrics to $(\alpha,\beta)$-metrics. Let $F$ be an $(\alpha,\beta)$-metric which is defined by a left invariant vector field and a left invariant Riemannian metric on a simply connected real Lie group $G$. We consider the automorphism and isometry groups of the Finsler manifold $(G,F)$ and their intersection. We prove that for an arbitrary left invariant vector field $X$ and any compact subgroup $K$ of automorphisms which $X$ is invariant under them, there exists an $(\alpha,\beta)$-metric such that $K$ is a subgroup of its isometry group.