| In this paper we study the global geometric properties of an open manifold with nonnegative sectional curvature. Cheeger and Gromoll's well-known Soul Theorem states that any such manifold, M, contains a compact submanifold, S⊂M , called the "soul of M", whose normal bundle is diffeomorphic to M. In 1994, Perelman proved that the metric projection p:M→S is a well defined Riemannian submersion. The main purpose of this paper is to explore consequences of Perelman's result. Along the way we develop some general theory for Riemannian submersions which is of interest independent of its application to nonnegative curvature. For example, we study "bounded Riemannian submersions" (submersions whose A and T tensors are both bounded in norm). When applied to the metric projection onto a soul, p:M→S , our results imply that M is quasi-isometric to any single fiber of p . Additionally, Perelman's theorem enables us to bound the volume growth of M from above and below. We also address the converse of the Soul Theorem; that is, the question of which vector bundles over spheres (or more general souls) admit metrics of nonnegative curvature. For example, we prove that only finitely many vector bundles over a given soul admit a nonnegatively curved metric satisfying a fixed upper bound for the vertical curvatures at the soul. Also, we translate the question of whether a bundle admits nonnegative curvature into the question of whether it admits a connection and a tensor which together satisfy a certain differential equation. |
