Left invariant complex Finsler metrics on a complex Lie group

TL;DR

This paper studies left invariant complex Finsler metrics on complex Lie groups, proving they are complex Berwald metrics with specific curvature properties and characterizing when they are K"ahler.

Contribution

It establishes that such metrics are necessarily complex Berwald, extends their spray to a holomorphic field, and characterizes K"ahler conditions in terms of the Lie algebra.

Findings

Metrics are complex Berwald.

Holomorphic spray extends to a global field.

K"ahler condition equivalent to Abelian Lie algebra.

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.