On Finsler geometry of tangent Lie groups

This paper is divided into two main parts. In the first part, we study left-invariant Randers metrics on Lie groups. We characterize the class of left-invariant Randers metrics with isotropic mean Berwald and isotropic Berwald curvatures on Lie groups. This yields an extension of Deng's well-known theorem for left-invariant Randers metrics with isotropic $S$-curvature. In the second part, we consider the left-invariant Randers metrics on tangent Lie groups. Let $G$ be a Lie group equipped with a left-invariant Randers metric $F$. Suppose that $F^v$ and $F^c$ denote the vertical and complete lift of $F$ on $TG$, respectively. First, we find the necessary and sufficient condition under which these metrics are weakly Berwaldian. Then, we prove that these lifting Randers metrics are isotropic Berwald metrics if and only if $F$ reduces to a Berwald metric. Finally, we give the necessary and sufficient conditions under which these metrics are of Douglas-type metrics.