On the fundamental formulas of complex Finsler submanifolds

Let M be a complex manifold endowed with a strongly pseudoconvex Finsler metric F, and M be a complex submanifold of M endowed with the induced complex Finsler metric F. In this paper, the Gauss, Codazzi and Ricci equations are obtained with respect to the Chern-Finsler connection on (M,F), the relationship between the torsion of the induced Chern-Finsler connection and the torsion of the Chern-Finsler connection on the ambient Finsler manifold are obtained. As applications of the fundamental formulas, we first prove that the holomorphic curvature of the induced complex Finsler metric F does not exceed the holomorphic curvature of F, and then give a characterization of the totally geodesic complex Finsler submanifold in terms of the horizontal components of the second fundamental form of (M,F).