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 (\( \mathcal{M},\mathcal{F} \)).