Chern-lattice obstructions to positively curved principal two-torus bundles

Definitions and mathematical statement

Let $T^2=S^1\times S^1$ be the two-dimensional torus, with its usual Lie-group structure. A smooth principal $T^2$-bundle $\pi:E\to B$ consists of a smooth free right $T^2$-action on $E$ whose orbit space is $B$, with local equivariant trivializations $\pi^{-1}(U)\cong U\times T^2$. Free means that the only torus element fixing any point is the identity. Consider bundles for which the total space, and hence the base, is a closed connected smooth manifold.

Does there exist such a bundle whose total space admits a Riemannian metric $g$ with \[ \sec_g(\sigma)>0 \qquad\text{for every two-plane }\sigma\subseteq T_xE \text{ and every }x\in E? \] The question is existential over the dimension, the base, the bundle, and the metric. No metric is prescribed on the base, and the positively curved metric on $E$ is not required to make the torus action isometric. In particular, the target is not restricted to connection metrics or Riemannian submersions. Equivalently, can a closed manifold carrying a free smooth two-torus action also carry some metric of positive sectional curvature? A merely locally free action, with finite nontrivial stabilizers, would give a different question involving an orbifold quotient and does not supply the required principal bundle. Curvature is required in every tangent two-plane, not only in planes transverse to the torus orbits.

Short English statement

Can the total space of a principal two-torus bundle carry a metric of strictly positive sectional curvature?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20003148 (AIM-TOPOLOGY-0236), 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 23. Original workshop question underlying AIM-TOPOLOGY-0236. https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.