Curvature of left-invariant complex Finsler metric on Lie groups

TL;DR

This paper derives explicit curvature formulas for left-invariant complex Finsler metrics on Lie groups, providing conditions for Kähler properties and showing rigidity results linking curvature to the group's structure.

Contribution

It offers explicit curvature formulas and characterizes when such metrics are Kähler or weakly Kähler, revealing rigidity and classification results for complex Lie groups.

Findings

Curvature formulas expressed in terms of complex Lie algebra

Necessary and sufficient conditions for Kähler-Finsler metrics

Rigidity: such metrics are Berwald with vanishing holomorphic bisectional curvature

Abstract

Let $ G $ be a connected Lie group with real Lie algebra $ \mathfrak{g}$. Suppose $G$ is also a complex manifold. We obtain explicit holomorphic sectional and bisectional curvature formulas of left-invariant strongly pseudoconvex complex Finsler metrics $F$ on $G$ in terms of the complex Lie algebra $\mathfrak{g}^{1,0}$; we also obtain a necessary and sufficient condition for $F$ to be a K\"ahler-Finsler metric and a weakly K\"ahler-Finsler metric, respectively. As an application, we obtain the rigidity result: if $F$ is a left-invariant strongly pseudoconvex complex Finsler metric on a complex Lie group $G$, then $F$ must be a complex Berwald metric with vanishing holomorphic bisectional curvature; moreover, $F$ is a K\"ahler-Berwald metric iff $G$ is an Abelian complex Lie group.