Characterization of strongly convex K\"ahler-Berwald metrics

TL;DR

This paper provides a geometric characterization of strongly convex K"ahler-Berwald metrics on complex manifolds, establishing conditions for their connections and proving a rigidity theorem for metrics with constant holomorphic sectional curvature.

Contribution

It introduces a new geometric characterization of strongly convex K"ahler-Berwald metrics and links their properties to the parallelism of the complex structure and connection coincidences.

Findings

Characterization of when the complex structure is horizontally parallel.

Conditions under which Cartan and Chern-Finsler connections coincide.

Rigidity theorem for metrics with constant holomorphic sectional curvature.

Abstract

Let $F: T^{1,0}M\rightarrow[0,+\infty)$ be a strongly convex complex Finsler metric on a complex manifold $M$ and $\pmb{J}$ the canonical complex structure on the complex manifold $T^{1,0}M$. We give a geometric characterization of strongly convex K\"ahler-Berwald metrics. In particular, we prove that $\pmb{J}$ is horizontally parallel with respect to the Cartan connection iff $F$ is a K\"ahler-Berwald metric. We also prove that the Cartan connection and the Chern-Finsler connection associated to $F$ coincide iff $\pmb{J}$ is both horizontal and vertical parallel with respect to the Cartan connection. Based on these results, we give a rigidity theorem of strongly convex K\"ahler-Berwald metrics with constant holomorphic sectional curvatures.