Characterizations of Complex Finsler Metrics

Munteanu (Complex spaces in Finsler, Lagrange and Hamilton Geometries, Kluwer Academic Publishers, Dordrecht, 2004) defined the canonical connection associated to a strongly pseudoconvex complex Finsler manifold (M, F). We first prove that the holomorphic sectional curvature tensors of the canonical connection coincide with those of the Chern-Finsler connection associated to F if and only if F is a Kahler-Finsler metric. We also investigate the relationship of the Ricci curvatures (resp. scalar curvatures) of these two connections when M is compact. As an application, two characterizations of balanced complex Finsler metrics are given. Next, we obtain a sufficient and necessary condition for a balanced complex Finsler metric to be Kahler-Finsler. Finally, we investigate conformal transformations of a balanced complex Finsler metric.