Curvature Bounds and Casorati Pinching for Submanifolds in Kähler Product Manifolds

In this paper, we establish sharp pinching inequalities that relate the generalized δ-Casorati curvatures to the normalized scalar curvature of submanifolds immersed in Kähler product manifolds endowed with a quarter-symmetric metric connection. The results are obtained for a broad range of geometric configurations, encompassing several important classes of submanifolds. Moreover, we prove that the derived inequalities are optimal by completely characterizing the submanifolds for which equality holds, showing that these cases correspond precisely to invariantly quasi-umbilical submanifolds with trivial normal connection.