Generalizing homogeneous spaces, it was proved by Grove–Ziller that cohomogeneity one manifolds under certain restrictions provide an important class of spaces which admit metrics of non-negative sectional curvature.

On these manifolds we identify conditions under which vector bundles over them (up to suitable stabilizations) admit metrics of non-negative sectional curvature as well—thus providing a certain converse to the soul theorem. We achieve this by relating the bundles to equivariant ones up to stabilization.

Beside constructions of bundle metrics, we essentially draw on K-theory computations to obtain the result. Moreover, we use the connection between (rational) K-theory and cohomology in order to link equivariant K-theory to equivariant (singular) cohomology—investigating the latter via rational homotopy theory. These methods are also applied to answer related open questions concerning the (equivariant) K-theory of homogeneous spaces.

This talk reports on joint work with David González-Álvaro and Marcus Zibrowius.