A best possible inequality for curvature-like tensor fields

We give an inequality for curvature-like tensor fields and apply this to Lagrangian submanifolds of complex space forms and to centroaffine hypersurfaces. In both settings we investigate the equality case and give a classification theorem if equality is attained at every point of the submanifold. We also provide an example showing that this inequality is best possible, in a sense explained in the paper.