Rank-six bundles over S^3 x S^3: classification, stabilization, and soul rigidity

Definitions and mathematical statement

Let $\xi\to S^3\times S^3$ be an oriented real vector bundle of rank six, and let $E(\xi)$ denote its twelve-dimensional total space. The Euler class is \[ e(\xi)\in H^6(S^3\times S^3;\mathbb Z)\cong\mathbb Z. \] Choose the product orientation on the base. The condition $e(\xi)\ne0$ is equivalent to its evaluation on the fundamental class being a nonzero integer; it does not depend on reversing the chosen bundle orientation. The Euler class records the primary obstruction to a nowhere-vanishing section of this oriented rank-six bundle.

Does there exist such a bundle with nonzero Euler class for which $E(\xi)$ carries a complete Riemannian metric $g$ of nonnegative sectional curvature? More generally, determine which of these bundles admit such a metric. Completeness means that all geodesics extend for all real time, an essential condition because the total space is noncompact. No metric on the zero section is prescribed, and the bundle projection need not be a Riemannian submersion. In particular, one must not impose the product round metric on a soul: a soul is a compact, totally geodesic, totally convex submanifold whose normal bundle is diffeomorphic to the complete open manifold. The source identifies an obstruction when the soul has that particular product metric. The requested existence question allows every complete metric, so it remains broader than that special geometric configuration.

Short English statement

Can a rank-six real bundle over the product of two three-spheres with nonzero Euler class have a complete nonnegatively curved total space?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20003165 (AIM-TOPOLOGY-0253), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Manifolds with nonnegative sectional curvature, item 40. Original workshop question underlying AIM-TOPOLOGY-0253. https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.