Under the assumption that' is a strongly convex weakly Khler 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.
Let （E, F） be a complex Finsler vector bundle over a compact Kahler manifold （M, g） with Kahler form φ We prove that if （E, F） is a weakly complex Einstein-Finsler vector bundle in the sense of Aikou （1997）, then it is modeled on a complex Minkowski space. Consequently, a complex Einstein-Finsler vector bundle （E, F） over a compact Kahler manifold （M, g） is necessarily φ-semistable and （E,F）=（E1,F1）……（Ek,Fk）,where Fj ：= F｜Ej, and each （Ej, Fj） is modeled on a complex Minkowski space whose associated Hermitian vector bundle is a φ-stable Einstein-Hermitian vector bundle with the same factor c as （E, F）.
Let （M1, F1） and （M2, F2） be two strongly pseudoconvex complex Finsler man- ifolds. The doubly wraped product complex Finsler manifold （f2 M1 x h M2, F） of （M1, F1） and （M2, F2） is the product manifold M1 x M2 endowed with the warped product complex 2 2 Finsler metric F2 = f2F1 ＋ fl F2, where fl and f2 are positive smooth functions on M1 and M2, respectively. In this paper, the most often used complex Finsler connections, holomorphic curvature, Ricci scalar curvature, and real geodesics of the DWP-complex Finsler manifold are derived in terms of the corresponding objects of its components. Necessary and sufficient conditions for the DWP-complex Finsler manifold to be K/ihler Finsler （resp., weakly K/ihler Finsler, complex Berwald, weakly complex Berwald, complex Landsberg） manifold are ob- tained, respectively. It is proved that if （M1, F1） and （M2,F2） are projectively flat, then the DWP-complex Finsler manifold is projectively flat if and only if fl and f2 are positive constants.
