The diameter sphere theorem up to diffeomorphism

A closed Riemannian manifold with sectional curvature at least 1 and diameter greater than $\pi/2$ is known to be homeomorphic to a sphere. Is it also diffeomorphic to the standard sphere, or could such a metric live on an exotic sphere? For the stronger pinching hypothesis of the quarter-pinched sphere theorem the smooth answer is known, but for the diameter hypothesis it is open.

Problem

Let $M$ be a closed connected Riemannian manifold of dimension $n$ with sectional curvature $\sec_M \ge 1$ and diameter $\operatorname{diam}(M) > \pi/2$. Must $M$ be diffeomorphic to the standard sphere $S^n$?

Equivalently: is there an exotic sphere (a smooth manifold homeomorphic but not diffeomorphic to $S^n$) that admits a metric with $\sec \ge 1$ and $\operatorname{diam} > \pi/2$?

Background

The question comes from the AIM workshop on manifolds with non-negative sectional curvature. The diameter sphere theorem of Berger and of Grove and Shiohama (1977) says such an $M$ is homeomorphic to $S^n$. Its proof uses critical point theory for distance functions, which gives topological rather than smooth conclusions. Exotic spheres exist in many dimensions, and it is a central question in comparison geometry which smooth structures carry positive or strongly constrained curvature.

What is known

• Grove and Shiohama (1977): $\sec \ge 1$ and $\operatorname{diam} > \pi/2$ imply that $M$ is homeomorphic to $S^n$.
• Brendle and Schoen proved the differentiable sphere theorem: a manifold with pointwise $1/4$-pinched positive sectional curvature is diffeomorphic to a spherical space form. As Petersen and Wilhelm point out, globally $1/4$-pinched simply connected manifolds form only a tiny part of the class with $\sec \ge 1$ and $\operatorname{diam} > \pi/2$, which contains manifolds that are Gromov–Hausdorff close to an interval.
• For the Gromoll–Meyer exotic 7-sphere, Weiss showed that it cannot carry $\sec \ge 1$ with radius greater than $\pi/2$, and Grove and Wilhelm (1997) that it cannot carry $\sec \ge 1$ with four points at pairwise distance greater than $\pi/2$. Whether some exotic sphere admits $\sec \ge 1$ with diameter greater than $\pi/2$ is open (Petersen and Wilhelm).

What would count as progress

• A proof that $\sec \ge 1$ and $\operatorname{diam} > \pi/2$ imply diffeomorphism with $S^n$, in all dimensions or in a dimension where exotic spheres exist (such as 7).
• A construction of such a metric on an exotic sphere. Curvature bounds for an explicit metric can be checked symbolically or with rigorous numerics.
• Intermediate results: smooth rigidity under extra hypotheses, for instance with a lower bound on volume or injectivity radius, or with diameter close to $\pi$.

References

1. K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math. 106 (1977), 201–211.
2. K. Grove and F. Wilhelm, Metric constraints on exotic spheres via Alexandrov geometry, J. Reine Angew. Math. 487 (1997), 201–217.
3. S. Brendle and R. Schoen, Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc. 22 (2009), 287–307.
4. P. Petersen and F. Wilhelm, An exotic sphere with positive sectional curvature, arXiv:0805.0812 (2008).

Source

Imported from AIM Problem Lists nnsectcurvature 1.2, curated by American Institute of Mathematics. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.