Explicit stable homogeneous representatives over S^2 x S^2

Definitions and mathematical statement

Let $M$ be a closed connected simply connected smooth manifold that admits a Riemannian metric of nonnegative sectional curvature. Let $\xi\to M$ be an arbitrary smooth real vector bundle of finite rank $r$. For an integer $k\geq0$, its stabilization is the fiberwise direct sum \[ \xi\oplus\underline{\mathbb R}^{k}, \qquad \underline{\mathbb R}^{k}=M\times\mathbb R^k. \] The total space of this bundle is naturally diffeomorphic to $E(\xi)\times\mathbb R^k$.

The stable converse soul question asks whether for every pair $(M,\xi)$ as above there exists some $k\geq0$ for which this total space admits a complete Riemannian metric with sectional curvature at least zero on every tangent two-plane. The integer $k$ may depend on the bundle and base; no uniform rank bound is required. The sought metric need not be a product metric, need not restrict to the originally given metric on the zero section, and need not make the bundle projection a Riemannian submersion. Completeness on the noncompact total space is part of the conclusion. Once a single stabilization works, all larger stabilizations work by taking products with Euclidean space, so existence of one $k$ is equivalent to existence for all sufficiently large $k$. The underlying converse question concerns which vector bundles can arise geometrically as normal bundles of souls; this version permits trivial additional directions before seeking the curvature realization.

Short English statement

Does adding sufficiently many trivial directions make every real vector bundle over a simply connected nonnegatively curved closed manifold admit a complete nonnegative-curvature metric?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20003168 (AIM-TOPOLOGY-0256), 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 43. Original workshop question underlying AIM-TOPOLOGY-0256. https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.
• S3 David González-Álvaro and Marcus Zibrowius, The stable converse soul question for positively curved homogeneous spaces. Formulation and special cases. https://arxiv.org/abs/1707.04711.