Holomorphic curvature of complex Finsler submanifolds

Abstract

Let M be a complex n-dimensional manifold endowed with a strongly pseudoconvex complex Finsler metric F . Let M be a complex m-dimensional submanifold of M , which is endowed with the induced complex Finsler metric F . Let D be the complex Rund connection associated with (M, F ). We prove that (a) the holomorphic curvature of the induced complex linear connection ∇ on (M, F ) and the holomorphic curvature of the intrinsic complex Rund connection ∇ * on (M, F ) coincide; (b) the holomorphic curvature of ∇ * does not exceed the holomorphic curvature of D; (c) (M, F ) is totally geodesic in (M, F ) if and only if a suitable contraction of the second fundamental form B(·, ·) of (M, F ) vanishes, i.e., B(χ, ι) = 0. Our proofs are mainly based on the Gauss, Codazzi and Ricci equations for (M, F ).