Lifting Group Actions and Nonnegative Curvature

TL;DR

This paper proves that certain vector bundles over CP^2 admit complete nonnegative curvature metrics by constructing specific group actions, and it addresses lifting problems of group actions in principal bundles.

Contribution

It introduces a method to construct nonnegative curvature metrics on vector bundles over CP^2 using cohomogeneity one actions and solves a lifting problem for SO(k) bundles over 4D bases.

Findings

All non-spin vector bundles over CP^2 admit nonnegative curvature metrics.

Constructed a cohomogeneity one action with singular orbits of codimension 2.

Solved the lifting problem for SO(k) principal bundles over 4D simply connected bases.

Abstract

We show that all vector bundles over CP^2 which are not spin admit a complete metric with nonnegative sectional curvature. In the proof we construct a nonnegatively curved metric on the corresponding principle bundle by showing that it admits a cohomogeneity one action with singular orbits of codimension 2. This is closely related to the problem of when an action of G on the base of an L principle bundle lifts to the total space, such that the lift commutes with L. We solve this lifting problem for all SO(k) principle bundles over a 4-dimensional simply connected base B with G a cohomogeneity one action on B.