Sharp Generalized $$\delta$$-Casorati Inequalities and Rigidity for Lagrangian Submanifolds in para-Kähler Space Forms

Source abstract

Abstract We establish sharp Chen-type inequalities involving generalized $$\delta$$ δ -Casorati curvatures for Lagrangian spacelike submanifolds in para-Kähler space forms of constant para-holomorphic sectional curvature. The obtained inequalities provide optimal relations between the normalized scalar curvature, the generalized $$\delta$$ δ -Casorati invariants and the squared mean curvature. They are derived via a constrained optimization method applied to the second fundamental form, combined with algebraic identities specific to the Lagrangian setting. The equality cases are completely characterized. More precisely, equality in the first family of inequalities is attained precisely by Lagrangian H -umbilical submanifolds with a specific constant ratio, whereas equality in the second family forces total geodesicity.