The diameter sphere theorem up to diffeomorphism

Decide whether a manifold with sectional curvature at least 1 and diameter above π/2 must be diffeomorphic to the standard sphere.

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1	# The diameter sphere theorem up to diffeomorphism
2
3	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.
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5	## Problem
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7	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 $S^n$?
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9	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 $\operatorname{diam} > \pi/2$?
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11	## Background
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13	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.
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15	## What is known
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17	- Grove and Shiohama (1977): $\sec \ge 1$ and $\operatorname{diam} > \pi/2$ imply that $M$ is homeomorphic to $S^n$.
18	- 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.
19	- 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).
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21	## What would count as progress
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23	- 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).
24	- A construction of such a metric on an exotic sphere. Curvature bounds for an explicit metric can be checked symbolically or with rigorous numerics.
25	- Intermediate results: smooth rigidity under extra hypotheses, for instance with a lower bound on volume or injectivity radius, or with diameter close to $\pi$.
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27	## References
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29	1. K. Grove and K. Shiohama, [A generalized sphere theorem](https://doi.org/10.2307/1971164), Ann. of Math. 106 (1977), 201–211.
30	2. K. Grove and F. Wilhelm, [Metric constraints on exotic spheres via Alexandrov geometry](https://doi.org/10.1515/crll.1997.487.201), J. Reine Angew. Math. 487 (1997), 201–217.
31	3. S. Brendle and R. Schoen, [Manifolds with 1/4-pinched curvature are space forms](https://doi.org/10.1090/S0894-0347-08-00613-9), J. Amer. Math. Soc. 22 (2009), 287–307.
32	4. P. Petersen and F. Wilhelm, [An exotic sphere with positive sectional curvature](https://arxiv.org/abs/0805.0812), arXiv:0805.0812 (2008).
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34	## Source
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36	Imported from [AIM Problem Lists nnsectcurvature 1.2](http://aimpl.org/nnsectcurvature/1/), 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.