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