  Examples of almost-positively and quasi-positively curved spaces of the form
M=H((G,h)xF) were discovered recently. Here, h is a left-invariant metric on a
compact Lie group G, F is a compact Riemannian manifold on which the subgroup H
of G acts isometrically on the left, and M is the orbit space of the diagonal
left action of H on (G,h)xF 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.
