Vector bundles of non-negative curvature over cohomogeneity one manifolds

TL;DR

This paper investigates conditions under which vector bundles over cohomogeneity one manifolds admit metrics of non-negative sectional curvature, using equivariant K-theory and rational homotopy theory to identify suitable structures.

Contribution

It introduces new methods to determine the existence of non-negative curvature metrics on vector bundles via equivariant K-theory and explicit constructions.

Findings

Explicit metrics of non-negative curvature constructed

Equivariant structures identified through K-theory comparison

Surjectivity properties analyzed via rational homotopy theory

Abstract

We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization. Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vector bundles via a comparison of their equivariant and non-equivariant K-theory. For this, in particular, we transcribe equivariant K-theory to equivariant rational cohomology and investigate surjectivity properties of induced maps in the Borel fibration via rational homotopy theory.