Obstruction to Positive Curvature on Homogeneous Bundles

Examples of almost-positively and quasi-positively curved spaces of the form M = H\((G, h) × F ) were discovered recently [9],[8]. Here h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup H ⊂ G acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G, h)×F with the induced Riemannian submersion metric. We prove that no new examples of strictly positive sectional curvature exist in this class of metrics. This result generalizes the case F = {point} proven by Geroch [5].