Meanwhile, we prove 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 $c_1, c_2$ positive constants, or $(M_1, F_1)$ and $(M_2, F_2)$ are both Kähler-Finsler manifolds.
📌 3 Key Takeaways
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Objective & Hypothesis: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 $K=F_1^2$, $H=F_2^2$.
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Key Findings & Data:In this paper, we prove that $(M, F)$ is a weakly Käahler-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 $f$.
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Conclusion & Impact:Meanwhile, we prove 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 $c_1, c_2$ positive constants, or $(M_1, F_1)$ and $(M_2, F_2)$ are both Kähler-Finsler manifolds.
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 weakly Käahler-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, we prove 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 $c_1, c_2$ positive constants, or $(M_1, F_1)$ and $(M_2, F_2)$ are both Kähler-Finsler manifolds.