On product complex Finsler manifolds

Abstract

Let (M, F) be the product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds $(M_{1,;}F_1)$ and $(M_{2,;}F_2)$ with $F=sqrt{f(K,H)}$ and $K=F_1^2$, $H=F_2^2$. In this paper, we prove that (M; F) is a weaklyKähler–Finsler (resp. weakly complex Berwald) manifold if and only if $(M_{1,;}F_1)$ and $(M_{2,;}F_2)$ are both weakly Kähler–Finsler (resp. weakly complex Berwald) manifolds, which is independent of the choice of function f . Meanwhile, weprove that (M; F) is a complex Landsberg manifold if and only if either $(M_{1,;}F_1)$ and $(M_{2,;}F_2)$ are both complex Landsberg manifolds and $f=c_1K+c_2H$ with $c_1,;c_2$ or $(M_{1,;}F_1)$ and $(M_{2,;}F_2)$ are both Kähler–Finsler manifolds.