Finite-window largeness of the O'Neill tensor

Definitions and mathematical statement

Let $\pi:(M^{b+f},g)\to(B^b,h)$ be a surjective Riemannian submersion between closed connected smooth Riemannian manifolds, with $b,f\geq1$. This means that $d\pi$ is surjective and restricts to an isometry from the horizontal space $\mathcal H_x=(\ker d\pi_x)^\perp$ onto $T_{\pi(x)}B$. The vertical space $\mathcal V_x=\ker d\pi_x$ has dimension $f$, the fiber dimension. Assume all sectional curvatures of $g$ are strictly positive.

Must every such submersion satisfy \[ f<b? \] The workshop also asks the weaker question whether there exists a function $F:\mathbb N\to\mathbb N$ such that $f\leq F(b)$ for every submersion in this class, with no dependence on the metric, the total dimension, or the topology of the base. Positive-dimensional base and fiber exclude the constant-map and identity-map degeneracies. Neither totally geodesic fibers nor homogeneity of the total space is assumed.

The suggested differential-geometric approach concerns O'Neill's tensor: for horizontal vector fields $X,Y$, define $A_XY=\tfrac12[X,Y]^{\mathcal V}$. At a point it determines a linear map $\Lambda^2\mathcal H_x\to\mathcal V_x$. The accompanying structural question is what lower bounds on its image dimension positive curvature can force at some point. No unquantified assertion that this image is "large" is treated as an additional hypothesis; the two dimension inequalities above are the precise primary targets.

Short English statement

Must a Riemannian submersion from a positively curved closed manifold have smaller-dimensional fibers than base, or at least uniformly bounded fiber dimension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20003153 (AIM-TOPOLOGY-0241), 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 28. Original workshop question underlying AIM-TOPOLOGY-0241. https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.