Characterizations of complex Finsler Metrics

TL;DR

This paper explores the relationships between different curvature tensors in complex Finsler geometry, characterizes balanced metrics, and examines conditions for K"ahler-Finsler metrics and conformal transformations.

Contribution

It provides new characterizations of balanced complex Finsler metrics and establishes conditions under which these metrics are K"ahler-Finsler, advancing understanding of complex Finsler geometry.

Findings

Holomorphic sectional curvature tensors coincide iff the metric is K"ahler-Finsler.

Characterizations of balanced complex Finsler metrics are given.

Conditions for conformal transformations of balanced metrics are established.

Abstract

Munteanu defined the canonical connection associated to a strongly pseudoconvex complex Finsler manifold $(M,F)$. We first prove that the holomorphic sectional curvature tensors of the canonical connection coincide with those of the Chern-Finsler connection associated to $F$ if and only if $F$ is a K\"ahler-Finsler metric. We also investigate the relationship of the Ricci curvatures (resp. scalar curvatures) of these two connections when $M$ is compact. As an application, two characterizations of balanced complex Finsler metrics are given. Next, we obtain a sufficient and necessary condition for a balanced complex Finsler metric to be K\"ahler-Finsler. Finally, we investigate conformal transformations of a balanced complex Finsler metric.