Strongly convex weakly complex Berwald metrics and real Landsberg metrics

Abstract

Under the assumption that $F$ is a strongly convex weakly Kahler Finsler metric on a complex manifold $M$, we prove that $F$ is a weakly complex Berwald metric if and only if $F$ is a real Landsberg metric. This result together with Zhong (2011) implies that among the strongly convex weakly Kahler Finsler metrics there does not exist unicorn metric in the sense of Bao (2007). We also give an explicit example of strongly convex Kahler Finsler metric which is simultaneously a complex Berwald metric, a complex Landsberg metric, a real Berwald metric, and a real Landsberg metric.