An obstructions to Non-negative curvature on Bundles with Pull-back connections

Abstract

Motivated to study the geometry of the exotic spheres constructed in [5], we derive a necessary condition for non-negative sectional curvature in certain total spaces of Riemannian submersions with totally geodesic fibers. In particular, we prove that the bundles in [5] and [1] have sections of negative curvature.