On the Riemann-Finsler Geometry of Tangent Bundle of Lie Groups with Two-Dimensional Commutator Subgroup

TL;DR

This paper explores the Riemannian and Finsler geometry of tangent bundles of Lie groups with a two-dimensional commutator subgroup, focusing on curvature relations, Randers metrics, and geodesic properties.

Contribution

It establishes curvature relationships, characterizes Randers metrics of Berwald and Douglas types, and provides explicit curvature formulas for tangent bundles of these Lie groups.

Findings

Relationship between sectional curvatures of G and TG

Conditions for Randers metrics to be of Berwald and Douglas type

Explicit formulas for the Riemannian curvature tensor on the tangent bundle

Abstract

We begin by studying the Riemannian geometry of the tangent Lie group $TG$ associated with a Lie group $G$ whose commutator subgroup is two-dimensional, equipped with the lift of a left-invariant Riemannian metric on $G$. We establish the relationship between the sectional curvatures of $G$ and those of $TG$. Next, we define a Randers metric on $G$ from a left-invariant Riemannian metric and a left-invariant vector field, and lift it vertically and completely to $TG$. We investigate the conditions under which this Randers metric is of Berwald and Douglas type, respectively, and compute the flag curvatures in the Berwald case. In an addendum, we discuss geodesic vectors and bi-invariant Riemannian metrics on these Lie groups, highlighting the special unimodularity conditions. Finally, we provide explicit formulas for the Riemannian curvature tensor on the tangent bundle of such a Lie group.