P-36 An inequality on Riemannian submersion invariant and theta-slant isometric immersion

It has been known that if a Riemannian manifold admits a non-trivial Riemannian submersion with totally geodesic fibers, then it cannot be isometrically immersed in any Riemannian manifold of non-positive sectional curvature as a minimal submanifold. B.Y. Chen proved this using an inequality involving the submersion invariant and his inequality shows the upper bound of the invariant Ăπ if a manifold is Lagrangian submanifold. Recently, the lower bound was found and furthermore, another inequality can be derived if we consider a θ-slant submanifold in complex space forms