Characterizations of complex Finsler connections and weakly complex Berwald metrics

Let M be a complex manifold endowed with a strongly pseudoconvex complex Finsler metric F. In this paper, characterizations of the complex Rund connection, complex Berwald connection and complex Hashiguchi connection that associated to F are given. The precise relationship of holomorphic sectional curvature, holomorphic bisectional curvature and Ricci scalar curvature of F with respect to these connections are obtained. Moreover, it is proved that the conformal change F˜=eσ(z)F of F is a weakly complex Berwald metric on M if and only if F is a weakly complex Berwald metric on M.