Geometric Inequalities for Purely Real Submanifolds in Complex Space Forms

As a generalisation of Kaehlerian slant submanifolds in Kaehler manifolds (i.e., proper slant submanifolds with the canonical endomorphism P parallel, ∇P = 0) one considers purely real submanifolds with ∇P = 0. The class of purely real submanifolds with ∇P = 0 also contains totally real submanifolds, in particular Lagrangian submanifolds. We obtain Chen-like inequalities for purely real submanifolds in complex space forms, i.e., relationships between intrinsic and extrinsic invariants of such submanifolds, involving the scalar curvature and Chen first invariant, respectively, and the squared mean curvature and the holomorphic sectional curvature of the ambient space.