Riemannian submersions, minimal immersions and cohomology class

We prove a simple optimal relationship between Riemannian submersions and minimal immersions; namely, 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 manifold. Some related results are also presented. In the last section, we introduce a cohomology class for Riemannian submersions and provide an application.