On the fundamental formulas of complex Finsler submanifolds (Q2482656)

Let \((M,F)\) be a complex manifold endowed with a strongly pseudocomplex Finsler metric and consider a complex submanifold \({\mathcal M}\) of \(M\). Taking into account the induced complex Finsler metric \({\mathcal F}\) on \({\mathcal M}\) the fundamental equations (Gauss, Codazzi, Ricci) of \(({\mathcal M},{\mathcal F})\) are derived using the Chern-Finsler connection of the ambient manifold \((M,F)\). As main applications it is proved that the holomorphic curvature of \(({\mathcal M},{\mathcal F})\) does not exceed the holomorphic curvature of \((M,F)\) and a characterization of totally geodesic complex Finsler submanifolds is obtained in terms of the horizontal components of the second fundamental form of \({\mathcal M}\).