Complex Hypersurfaces in Kählerian Manifolds with Constant Holomorphic Sectional Curvature: Geometric Identities and Curvature Constraints

Abstract

This study introduces a new framework for analyzing complex hypersurfaces embedded in Kähler manifolds characterized by vanishing Bochner curvature tensors. Diverging from traditional approaches, we examine the intricate interplay between the second fundamental form and the shape operator under these specific curvature conditions. Our investigation reveals unique curvature identities and scalar invariants that emerge solely due to the vanishing Bochner curvature. Notably, we establish precise conditions under which such hypersurfaces become totally geodesic and constant holomorphic sectional curvature. The findings offer fresh insights into the intrinsic and extrinsic geometry of complex hypersurfaces, potentially influencing future research in complex differential geometry and theoretical physics.