Left Invariant Randers Metrics of Berwald type on Tangent Lie Groups

TL;DR

This paper investigates the relationship between flag curvature of tangent bundles with Randers metrics of Berwald type and the sectional curvature of the base Lie group, providing classifications for 3D cases.

Contribution

It establishes the connection between flag and sectional curvatures in tangent Lie groups with Berwald Randers metrics and classifies 3D Lie groups with such structures.

Findings

Relation between flag curvature of tangent bundle and sectional curvature of Lie group

Classification of 3D Lie groups with Berwald Randers metrics on tangent bundles

Characterization of geodesic vectors in these structures

Abstract

Let $G$ be a Lie group equipped with a left invariant Randers metric of Berward type $F$, with underlying left invariant Riemannian metric $g$. Suppose that $\widetilde{F}$ and $\widetilde{g}$ are lifted Randers and Riemannian metrics arising from $F$ and $g$ on the tangent Lie group $TG$ by vertical and complete lifts. In this article we study the relations between the flag curvature of the Randers manifold $(TG,\widetilde{F})$ and the sectional curvature of the Riemannian manifold $(G,g)$ when $\widetilde{F}$ is of Berwald type. Then we give all simply connected $3$-dimentional Lie groups such that their tangent bundles admit Randers metrics of Berwarld type and their geodesics vectors.