On δ-Casorati curvature invariants of Lagrangian submanifolds in quaternionic Kähler manifolds of constant q-sectional curvature

Lagrangian submanifolds, a class of Riemannian submanifolds that arose naturally in the context of Hamiltonian mechanics, play an important role in some modern theories of physics. In this paper, using an optimization technique on submanifolds immersed in Riemannian manifolds, we first obtain some sharp inequalities for δ -Casorati curvature invariants of Lagrangian submanifolds in quaternionic space forms, i.e. quaternionic Kähler manifolds of constant q -sectional curvature. Then we show that in the class of Lagrangian submanifolds in quaternionic space forms, there are only two subclasses of ideal Casorati submanifolds, namely the family of totally geodesic submanifolds and a particular subfamily of H -umbilical submanifolds. Finally, we provide some examples to illustrate the obtained results. In particular, we point out that an entire family of ideal Casorati Lagrangian submanifolds can be constructed using the concept of quaternionic extensor introduced by Oh and Kang in the early 2000s.