The Scalar Curvature on Totally Geodesic Fiberings

We give a generalization of a theorem of Llarull concerning thebehaviour of the scalar curvature while perturbing the metric. In thispaper the following is shown: let Ñ →N be a Riemannian submersion with totally geodesic fibre. IfÑ has the property that perturbingits metric towards a bigger one implies that there is a point onÑ where the perturbed scalarcurvature is less than the original one, then also the base manifoldN possesses this property. This result is applied to theprojective spaces.