Left Invariant Randers Metrics of Berwald type on Tangent Lie Groups

Farhad Asgari, Hamid Reza Salimi Moghaddam

Published 2016-10-05Version 1

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.