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 $\ddot{I}‡=w^i\hat{I}{´}_{z^i}$ on $T^{1,0}G\backslash 0$ can be extended to a holomorphic tangent field on $T^{1,0}G$. If we view $\ddot{I}‡$ as a real tangent field on $TG$, it coincides with the canonical bi-invariant spray structure on $G$. Third, we prove that the strongly Kähler, Kähler, and weakly Kähler properties for $F$ are equivalent. More over, $F$ is Kähler if and only if $G$ has an Abelian Lie algebra. Finally, we prove that the holomorphic sectional curvature vanishes.

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