Twisting and nonnegative curvature metrics on vector bundles over the round sphere

Abstract

A complete noncompact manifold M with nonnegative sectional curvature is diffeomorphic to the normal bundle of a compact submanifold S called the soul of M . When S is a round sphere we show that the clutching map of this bundle is restricted; this is used to deduce that there are at most finitely many isomorphism types of such bundles with sectional curvature lying in a fixed interval [0, κ]. We also examine the opposite question of how the twisting of the bundle limits the type of possible nonnegative curvature metrics on the bundle: It turns out that if the bundle does not admit a nowhere-zero section, then the normal exponential map is necessarily a diffeomorphism onto M , and the ideal boundary of M consists of a single point. In their paper [CG], Cheeger and Gromoll raised the question of which vector bundles over the round sphere admit complete metrics with nonnegative sectional curvature. The significance of this problem is that it attempts to determine to what extent a converse to the Soul theorem holds. Recall that this theorem states that every open (i.e., complete noncompact) manifold M with nonnegative curvature KM is diffeomorphic to a vector bundle over a compact totally geodesic submanifold S called a soul. A natural question then is whether all such vector bundles admit complete metrics with KM ≥ 0. In [OW], it was shown that when the soul is a Bieberbach manifold, nonnegative curvature metrics force the vanishing of the Euler class of the vector bundle. It follows that among oriented plane bundles over the torus, only the trivial one admits such a metric. The above case is fairly rigid a priori, however (since for example any bundle with curvature ≥ 0 over such a manifold must also admit a flat metric), and the corresponding question for a simply connected base remains open. An answer could provide insights on the topological structure of compact manifolds with nonnegative curvature, since any open manifold with nonnegative curvature can (after modifying its metric) be isometrically 1991 Mathematics Subject Classification. Primary 53C20.