A delta-Invariant for QR-Submanifolds in Quaternion Space Forms

Starting from an inequality involving the invariant delta(D) for an anti-holomorphic submanifold of a complex space form [1] and using optimization methods on Riemannian manifolds, we establish a corresponding inequality for the invariant delta(D-perpendicular to) defined on QR-submanifolds in quaternion space forms, in terms of the squared mean curvature. We obtain a relationship between intrinsic and extrinsic invariants for QR-submanifolds of quaternion space forms.