Nonnegative curvature on stable bundles over compact rank one symmetric spaces

TL;DR

This paper proves that any vector bundle over a compact rank one symmetric space can be equipped with a nonnegative curvature metric after adding a sufficiently large trivial bundle, and explores bounds for spheres.

Contribution

It demonstrates the existence of nonnegative curvature metrics on stabilized bundles over rank one symmetric spaces and provides bounds for trivial bundle ranks over spheres.

Findings

Every vector bundle over a compact rank one symmetric space admits a nonnegative curvature metric after stabilization.

Provides upper bounds for the trivial bundle rank needed over spheres.

Extends results to complex vector bundles over other manifolds.

Abstract

In this note we show that every (real or complex) vector bundle over a compact rank one symmetric space carries, after taking the Whitney sum with a trivial bundle of sufficiently large rank, a metric with nonnegative sectional curvature. We also examine the case of complex vector bundles over other manifolds, and give upper bounds for the rank of the trivial bundle that is necessary to add when the base is a sphere.