Left invariant complex Finsler metrics on a complex Lie group

Abstract

In this paper, we consider a left invariant complex Finsler metric $F$ on a complex Lie group. Using the technique of invariant frames, we prove the following properties for $(G,F)$. First, the metric $F$ must be a complex Berwald metric. Second, its complex spray $\chi=w^i\delta_{z^i}$ on $T^{1,0}G\backslash0$ can be extended to a holomorphic tangent field on $T^{1,0}G$. If we view $\chi$ as a real tangent field on $TG$, it coincides with the canonical bi-invariant spray structure on $G$. Third, we prove that the strongly K\"{a}hler, K\"{a}hler, and weakly K\"{a}hler properties for $F$ are equivalent. More over, $F$ is K\"{a}hler if and only if $G$ has an Abelian Lie algebra. Finally, we prove that the holomorphic sectional curvature vanishes.