Unflat connections in 3-sphere bundles over 𝑆⁴

Abstract

The paper concerns connections in 3 3 -sphere bundles over 4 4 -manifolds having the property of unflatness, which is a necessary condition in order that a natural construction give a Riemannian metric of positive sectional curvature in the total space. It is shown that, as conjectured by A. Weinstein, the only 3 3 -sphere bundle over S 4 {S^4} with an unflat connection is the Hopf bundle.