  We establish a link between rational homotopy theory and the problem which
vector bundles admit complete Riemannian metric of nonnegative sectional
curvature. As an application, we show for a large class of simply-connected
nonnegatively curved manifolds that, if C lies in the class and T is a torus of
positive dimension, then "most" vector bundles over the product of C and T
admit no complete nonnegatively curved metric.
