The Relation Between Automorphism Group and Isometry Group of Left Invariant $ (α,β)$-metrics

Abstract

This work generalizes the results of an earlier paper by the second author, from Randers metrics to $(α,β)$-metrics. Let $F$ be an $(α,β)$-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 $(α,β)$-metric such that $K$ is a subgroup of its isometry group.

Theorems & Definitions (14)

• Definition 2.1
• Proposition 2.2
• proof
• Remark 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Corollary 2.6
• proof