On strongly convex Kähler--Berwald manifolds

Abstract

For every real Berwald strongly convex complex Finsler manifold $(M,F,J,\hat\nabla)$, if the Cartan nonlinear connection satisfies the parallelism condition $\hat\nabla J=0$, we construct a good complex vertical connection $D$, prove that $D$ is the complex Rund connection, and $F$ is Kähler-Berwald. The inverse direction also can be seen from the construction and proof. As an application, we prove that on an irreducible Hermitian symmetric space $M=G/H$, $G$-invariant complex Finsler metrics are Kähler--Berwald. Besides, we obtain the characterization of strongly pseudoconvex holomorphically invariant complex Finsler metrics on the classical domains of type-IV.