Hopf conjecture

About 95 years old · traced to 1931

Let $M=\mathbb{S}^{2}\times\mathbb{S}^{2}$ , the product of two $2$ -spheres, regarded as a closed smooth $4$ -manifold. For a Riemannian metric $g$ on $M$ and a point $p\in M$ , let $K_{g}(\sigma)$ denote the sectional curvature of $g$ at $p$ in the direction of a $2$ -dimensional linear subspace $\sigma\subset T_{p}M$ ; the metric $g$ has positive sectional curvature if $K_{g}(\sigma)>0$ for every $p\in M$ and every such $\sigma$ .

Then for every smooth Riemannian metric $g$ on $M$ there exist a point $p\in M$ and a $2$ -plane $\sigma\subset T_{p}M$ with

K_{g}(\sigma)\le 0,

that is, $\mathbb{S}^{2}\times\mathbb{S}^{2}$ carries no Riemannian metric of positive sectional curvature.

More generally, if $X$ is a compact Riemannian symmetric space of rank greater than one, then $X$ admits no Riemannian metric all of whose sectional curvatures are positive.

References

Primary source

Wikipedia

Additional references

Wikipedia, Hopf conjecture, the article this problem comes from.

geometry unsolved-problem

Posted 1 month ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A 2026 preprint claims to overturn the conjecture by constructing positive-curvature metrics on the relevant four-dimensional space, but the result has not been independently verified.

The Hopf conjecture asserts that $\mathbb{S}^{2}\times\mathbb{S}^{2}$ admits no metric with everywhere positive sectional curvature, and more generally rules this out for compact symmetric spaces of rank greater than one. It has long been open in the unrestricted case.

Known results

Liu and Wan (2019) gave a conditional obstruction via an inequality for harmonic $2$ -forms; verifying it generally would prove the conjecture.
Seaman ruled out metrics on $\mathbb{S}^{2}\times\mathbb{S}^{2}$ satisfying $0.1714\leq\sec_M\leq1$ .
A 2014 result rules out positive curvature under sufficiently large torus symmetry, but not for arbitrary metrics.
Circle-symmetry classifications imply that any hypothetical positively curved metric here would have at most a finite isometry group.

August 2026 claimed resolution

On August 20, 2026, Simon Brendle posted a preprint claiming a resolution; a subsequent 2026 paper reports that Brendle and Hung constructed positive-sectional-curvature metrics on $\mathbb{S}^{2}\times\mathbb{S}^{2}$ . Discussion initially identified possible flaws, while a later assessment reported no serious remaining objection; no independent mathematical verification is recorded.

Current status (as of September 2026): A claimed counterexample would settle the stated conjecture negatively, but the Brendle--Hung construction remains unverified, so the problem is not yet mathematically settled.

Sources

No solutions have been posted yet.