Curvature and Riemannian Submersions

We study Riemannian submersions from positively curved manifolds and from Einstein manifolds. We first prove a diameter rigidity theorem for Riemannian submersions.Secondly we show that there is no nontrivial Riemannian submersion from positively curved four manifolds such that either the mean curvature vector field or the norm of the O'Neill tensor is basic. We also classify Riemannian submersions from compact four-dimensional Einstein manifolds with totally geodesic fibers.