The differentiable diameter sphere problem: a three-gate obstruction

Definitions and mathematical statement

Let $(M,g)$ be a closed connected smooth Riemannian manifold of dimension $n\geq2$. Closed means compact without boundary. For a two-dimensional linear subspace $\sigma\subseteq T_xM$, let $\sec_g(\sigma)$ be its sectional curvature. Write $d_g$ for geodesic distance and $\operatorname{diam}(M,g)=\sup_{x,y\in M}d_g(x,y)$. The curvature normalization $\sec_g\geq1$ means that every tangent two-plane at every point has curvature at least one.

The differentiable diameter question asks whether \[ \sec_g\geq1,\qquad \operatorname{diam}(M,g)>\frac\pi2 \quad\Longrightarrow\quad M\text{ is diffeomorphic to }S^n. \] Here $S^n$ carries its standard smooth structure, and a diffeomorphism is a bijective smooth map with smooth inverse. The conclusion concerns the smooth type of the manifold, not whether its given metric is round. No upper sectional-curvature bound or pinching ratio is part of the hypothesis. The strict diameter inequality is retained; the endpoint $\pi/2$ allows different familiar topological types and is a separate rigidity problem. The classical diameter theorem provides the topological sphere conclusion under these hypotheses; the issue is whether an exotic smooth sphere could still occur. The assertion is quantified over every dimension and every metric satisfying the displayed normalization, with no symmetry or simple-connectivity assumption added. Equivalently, it asks whether any nonstandard smooth sphere is excluded from this entire normalized diameter range.

Short English statement

Does curvature at least one and diameter greater than $\pi/2$ force the standard smooth sphere, rather than only its topological type?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20003127 (AIM-TOPOLOGY-0215), 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 2. Original workshop question underlying AIM-TOPOLOGY-0215. https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.