Characterization of strongly convex Kähler-Berwald metrics

Abstract

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

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