Geometric Inequalities for Purely Real Submanifolds in Complex Space Forms

摘要

As a generalisation of Kaehlerian slant submanifolds in Kaehler manifolds (i.e., proper slant submanifolds with the canonical endomorphism P parallel, a double dagger P = 0) one considers purely real submanifolds with a double dagger P = 0. The class of purely real submanifolds with a double dagger 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.