The ICM Conjectures record reports the following assessment; its claims and bibliography are retained as source material.

Verdict: open

Precise problem

The problem asks the classical Hopf product problem: whether the smooth manifold $S^2 \times S^2$ admits any Riemannian metric of strictly positive sectional curvature.

The expected conjectural answer in the literature is negative, but I found no published or otherwise citable theorem that proves nonexistence for all Riemannian metrics on $S^2 \times S^2$, and no construction of such a positive-curvature metric.

Current status

The exact unrestricted problem remains open. Recent survey literature still describes it as open in general, while noting that it is settled in special metric classes. The correct status is therefore Open.

Original source and status references

• Title: Differential Geometry; Its Past and Its Future

• Authors: Shiing-Shen Chern; records a question asked by H. Hopf

• Venue/year: Actes, Congrès international des mathématiciens, Nice 1970, Tome 1, pp. 41-53

• DOI/arXiv/URL: https://www.hiroleetanaka.com/pdfs/2014-fall-230a-icm1970-chern-differential-geometry.pdf

• Relevance: Problem I asks exactly whether the product of two 2-dimensional spheres admits a Riemannian metric of strictly positive curvature.

• Title: Bounds on Sectional Curvature and Interactions with Topology

• Author: Manuel Amann

• Venue/year: Jahresbericht der Deutschen Mathematiker-Vereinigung 123, 27-55, 2021; published online 2020

• DOI/arXiv/URL: https://doi.org/10.1365/s13291-020-00225-x

• Relevance: States the Hopf conjecture for $S^2 \times S^2$, says it is settled only in special metric classes, and says it is widely open in general.

• Title: Harmonic 2-forms and positively curved 4-manifolds

• Authors: Kefeng Liu and Jianming Wan

• Venue/year: Science China Mathematics 64, 1613-1620, 2021; published online 2020

• DOI/arXiv/URL: https://doi.org/10.1007/s11425-019-1641-9; arXiv:1909.00110

• Relevance: The introduction says the Hopf conjecture $S^2 \times S^2$ has no positive sectional curvature metric is still widely open except for special cases.

Special cases progress

• Title: On the topology of positively curved 4-manifolds with symmetry

• Authors: Wu-Yi Hsiang and Bruce Kleiner

• Venue/year: Journal of Differential Geometry 29(3), 615-621, 1989

• DOI/arXiv/URL: https://doi.org/10.4310/jdg/1214443064

• Result and limitation: A compact simply connected positively curved 4-manifold with an effective isometric circle action is homeomorphic to $S^4$ or $\mathbb{CP}^2$. Since $S^2 \times S^2$ is neither, this rules out positive-curvature metrics on $S^2 \times S^2$ with nontrivial circle symmetry. It does not rule out metrics with finite or trivial isometry group.

• Title: A Remark On Positively Curved 4-Manifolds

• Author: Christian Bär

• Venue/year: Differential Geometry and its Applications 4(1), 71-75, 1994

• DOI/arXiv/URL: https://doi.org/10.1016/0926-2245(94)90009-4; PDF: https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Publikationen/hopf.pdf

• Result and limitation: If a compact oriented connected 4-manifold has indefinite intersection form, then it admits no metric with $K \ge 1$ and $\|\nabla R\|_{L^\infty} \le 2/\pi$. This applies to $S^2 \times S^2$, but only under the extra derivative bound.

• Title: Geometrically formal 4-manifolds with nonnegative sectional curvature

• Author: Christian Bär

• Venue/year: Communications in Analysis and Geometry 23, 479-497, 2015

• DOI/arXiv/URL: https://doi.org/10.4310/CAG.2015.v23.n3.a2; arXiv:1212.1325

• Result and limitation: Shows that the Hopf product conjecture - namely, the nonexistence of a positive-sectional-curvature metric on $S^2\times S^2$ - holds for formal metrics and related harmonic-form length conditions. This is a restriction on the metric class and does not cover arbitrary metrics on $S^2 \times S^2$.

• Title: Positive biorthogonal curvature on $S^2 \times S^2$

• Author: Renato G. Bettiol

• Venue/year: Proceedings of the American Mathematical Society 142(12), 4341-4353, 2014

• DOI/arXiv/URL: https://doi.org/10.1090/S0002-9939-2014-12173-3; arXiv:1210.0043

• Result and limitation: Constructs metrics on $S^2\times S^2$ with positive biorthogonal curvature (and in fact positive Ricci curvature and positive distance curvature $\sec_\theta$ for all $\theta>0$), a package Bettiol describes as intermediate between admitting Ric > 0 and admitting sec > 0.

• Title: Harmonic 2-forms and positively curved 4-manifolds

• Authors: Kefeng Liu and Jianming Wan

• Venue/year: Science China Mathematics 64, 1613-1620, 2021; published online 2020

• DOI/arXiv/URL: https://doi.org/10.1007/s11425-019-1641-9; arXiv:1909.00110

• Result and limitation: If a compact oriented 4-manifold has positive sectional curvature and all harmonic 2-forms satisfy $|\nabla\phi|^2 \ge 2|d|\phi||^2$, a Kato-type inequality, then its intersection form is definite. Since $S^2 \times S^2$ is indefinite, this would rule it out under the assumption of the Kato-type inequality. The inequality is not known for arbitrary positive-curvature metrics.

• Title: Conformal Metrics to a Product or Doubly Warped Product on $S^2 \times S^2$ and the Hopf Conjecture

• Authors: Thierno Seck and Athoumane Niang

• Venue/year: Journal of Mathematics 2022, Article ID 6268017, 5 pp., 2022

• DOI/arXiv/URL: https://doi.org/10.1155/2022/6268017

• Result and limitation: Proves no positive sectional curvature within two metric classes: conformal metrics to the product metric and doubly warped product metrics. These classes do not include all Riemannian metrics on $S^2 \times S^2$.

• Title: Metrics Induced by $\mathbb{R}^{6+k}$ on the Graph of k-Smooth Functions and the Hopf Conjecture

• Authors: Thierno Seck, Athoumane Niang, and Adama Thiandoum

• Venue/year: In Nonlinear Analysis, Geometry and Applications, Trends in Mathematics, Birkhäuser, pp. 285-298, 2024

• DOI/arXiv/URL: https://doi.org/10.1007/978-3-031-52681-7_13

• Result and limitation: By Perdomo's Theorem A (Isr. J. Math. 156 (2006)), every Riemannian metric on S²×S² is, up to homothety, isometric to a graph metric over the standard embedding S²×S² ⊂ ℝ⁶ in sufficiently high codimension; since homothety preserves the sign of sectional curvature, the preprint's claimed theorem — that no graph of an ℝᵏ-valued function over S²×S² has positive sectional curvature — would, if correct, resolve the conjecture in full. The claim is equivalent in strength to the conjecture itself; it appears only in an unrefereed preprint, has not been validated, and the refereed literature through 2026 continues to treat the conjecture as open. We therefore do not count it. (Perdomo's published Theorem B establishes the graph statement only for scalar-valued F over the product metric.)

What remains unresolved

It remains unknown whether an arbitrary Riemannian metric on $S^2 \times S^2$ can have strictly positive sectional curvature everywhere, or whether such metrics are impossible without extra hypotheses such as symmetry, pinching/derivative bounds, formality-type conditions, Kato inequalities, conformal/warped-product form, or graph-induced form.