P-30 A note on an inequality of Riemannian submersion invariant

Buller Hallway

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 maximum value of the invariant. I could find another inequality that gives the minimum of the submersion invariant under a certain assumption.