Conditions for nonnegative curvature on vector bundles and sphere bundles

Abstract

This paper addresses J. Cheeger and D. Gromoll's question about which vector bundles admit a complete metric of nonnegative curvature, and it relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that any vector bundle that admits a metric of nonnegative curvature must admit a connection, a tensor, and a metric on the base space, which together satisfy a certain differential inequality. On the other hand, a slight sharpening of this condition is sufficient for the associated sphere bundle to admit a metric of positive curvature. Our results sharpen and generalize M. Strake and G. Walschap's conditions under which a vector bundle admits a connection metric of nonnegative curvature. IntroductionA well-known question in Riemannian geometry is to what extent the converse of Cheeger and Gromoll's soul theorem holds. Their theorem states that any complete noncompact manifold, M, with nonnegative sectional curvature is diffeomorphic to the normal bundle of a compact totally geodesic submanifold, ⊂ M, called the soul of M (see [5]). The converse question is the classification problem: Which vector bundles over compact nonnegatively curved base spaces can admit complete metrics of nonnegative curvature? There are vector bundles that are known not to admit nonnegative curvature, but in all such examples the base space has an infinite fundamental group (see [13], [17], [1], [2]). Trivial positive results include all vector bundles over S 1 , S 2 , and S 3 , T S n for any n, and more generally all homogeneous vector bundles over homogeneous spaces. As for nontrivial positive results, D. Yang obtained nonnegatively curved metrics on rank 2 vector bundles over C P n #C P n (see [20]). More recently, K. Grove and W. Ziller constructed nonnegatively curved metrics on all vector bundles over S 4 and S 5 (see [7]).Our first result is a necessary condition for a vector bundle to admit a metric of nonnegative curvature. Suppose that M is an open (i.e., complete and noncompact)

Examples of Manifolds with Non-negative Sectional Curvature

“…It induces a horizontal distribution on the sphere bundle and the fibers are endowed with a metric of constant curvature. An analogue of 6.2 for sphere bundles was proved in [75]: (2) is hyperfat as well. If the base is a manifold, the condition ∇R = 0 is rather restrictive.…”

Metrics with nonnegative curvature on $${S^2 \times \mathbb{R}^4}$$

“…2, we prove rigidity results for connection metric with nonnegative curvature, including: Proposition 1. 5 For any connection metric with nonnegative curvature on an R 4 bundle over S 2 , the holonomy group of the normal bundle of the soul must lie in a maximal torus of S O (4). In other words, ν( ) globally decomposes as two orthogonal ∇-invariant R 2 -bundles over S 2 .…”

Nonnegatively and positively curved invariant metrics on circle bundles

“…where Part (2) of the theorem was proven in [11] by constructing a metric on the associated vector bundle in such a way that derivative considerations force the boundary of a small metric tube about the zero-section to have positive curvature. For the positively curved metric guaranteed by part (2) of the theorem, note that the function φ : B → R is not the fiber-length function.…”