Some rigid results on doubly twisted product complex Finsler manifolds

Let ( \(M_{1}\times_{(\lambda_{1},\lambda_{2})}M_{2},F\) $$M_1{\times }_{({\lambda }_1,{\lambda }_2)}{M}_2,F$$ ) be a doubly twisted product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds (M₁, F₁) and (M₂, F₂). Denote by M the product manifold of M₁ and M₂. In this paper, we first give the characterization for (M, F) to be of constant holomorphic curvature, and prove that (M, F) is of constant holomorphic curvature if and only if (M, F), (M₁, F₁), and (M₂, F₂) all have vanishing holomorphic curvature under the condition that ln λ₁ and ln λ₂ are both pluriharmonic functions. Secondly, we obtain the classification for (M, F) to be a complex Landsberg manifold. Finally, we prove that a strongly convex doubly twisted product complex Finsler manifold (M, F) is projectively flat (resp. dually flat) if and only if λ₁F₁ and λ₂F₂ are both complex Minkowski metrics.