Positive and nonnegative curvature on bundles

Bryn Mawr College, Bryn Mawr PA

Investigators

Abstract

Proposal DMS-0303326 P.I.: Kristopher Tapp (Bryn Mawr College) Title: Positive and nonnegative curvature on bundles ABSTRACT: In recent research, the PI developed tools for studying the following two general problems in Riemannian geometry: (1) For which vector bundles does the total space admit a complete Riemannian metric with nonnegative sectional curvature? (2) For which sphere bundles does the total space admit a Riemannian metric with positive curvature? The PI plans to use these tools to find new examples, obstructions, and rigidity theorems. In particular, the PI plans to construct metrics of nonnegative curvature on holomorphic vector bundles over complex projective space, and determine whether Einstein- Hermitian connections can appear as the connections in the normal bundles of their souls. Also, the PI intends to study nonnegatively curved metrics on trivial vector bundles over spheres. This problem is related to the Hopf conjecture, or more specifically to the question: how large is the family of nonnegatively curved metrics on the product of two spheres? The PI will work with undergraduate students studying the metrics on such products obtained by re-scaling the product metric by a compact Lie group which acts by isometries. Finally, the PI proposes to find obstructions to metrics of nonnegative curvature on vector bundles over spheres with prescribed soul metrics. The question of which manifolds can have nonnegative curvature is central to Riemannian geometry. Nonnegative curvature is a visually natural restriction on the way in which an object curves about in space. All known examples come from compact Lie groups with bi-invariant metrics, which are indispensable tools in diverse fields of mathematics, physics, cosmology, and other disciplines in which simplification is achieved through symmetry. The search for new examples of manifolds with nonnegative (or positive) curvature has a long history, yet very few constructions are known. Since his tools represent a construction which is substantially different form known methods, the PI believes they deserve further study.