Metrics with Nonnegative Curvature on S^2xR^4

TL;DR

This paper investigates nonnegatively curved metrics on S^2xR^4, establishing rigidity theorems and extending Wilking's almost-positive curvature metric to a nonnegative curvature metric on the entire space.

Contribution

It proves new rigidity results for connection metrics and extends Wilking's almost-positive curvature metric to a complete nonnegatively curved metric on S^2xR^4.

Findings

Holonomy group of the normal bundle lies in a maximal torus of SO(4)

Wilking's metric extends to a nonnegatively curved metric on S^2xR^4

Detailed geometric description of the extended metric

Abstract

We study nonnegatively curved metrics on S^2xR^4. First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking's almost-positively curved metric on S2xS3 extends to a nonnegatively curved metric on S^2xR^4 (so that Wilking's space becomes the distance sphere of radius 1 about the soul). We describe in detail the geometry of this extended metric.