A conjecture attributed to Heinz Hopf in the early 1930s states that every closed, even-dimensional Riemannian manifold with positive sectional curvature has positive Euler characteristic. It is proved only in dimensions two and four; in every even dimension at least six it has remained open without additional hypotheses for roughly ninety years. Under Grove's symmetry program, the conclusion is known when the isometry group contains a sufficiently large torus: linear symmetry-rank bounds go back to Püttmann–Searle, a logarithmic bound to Kennard, and Kennard–Wiemeler–Wilking (2021) obtained the first dimension-independent threshold, an isometry-group rank of at least five; Nienhaus (2022) lowered this to an isometric T4T^4-action. The symmetry-free conjecture remains open, with the minimal symmetry rank (now known to be at most four) as a derived open question.

Background

For a Riemannian manifold, the sectional curvature sec⁡(x,y)\operatorname{sec}(x,y) is the Gaussian curvature of the two-dimensional surface tangent to a pair of orthonormal vectors x,yx,y; the manifold has positive sectional curvature when this quantity is positive for every such pair at every point. The Euler characteristic χ(M)\chi(M) of a closed (compact, without boundary) manifold is the alternating sum of its Betti numbers, i.e. the alternating sum of the dimensions of its rational cohomology groups. Positive sectional curvature is the strongest standard curvature condition: it implies positive Ricci curvature, and by the theorem of Bonnet–Myers a complete manifold with positive Ricci curvature has finite fundamental group, while Synge's theorem forces a closed, orientable, even-dimensional positively curved manifold to be simply connected. Hopf observed in the 1930s that all positively curved spaces known at the time — spheres, complex and quaternionic projective spaces, the Cayley plane (the projective plane over the octonions) — have positive Euler characteristic, and conjectured this is forced by the curvature: every closed, even-dimensional Riemannian manifold of positive sectional curvature has χ(M)>0\chi(M)>0. The conjecture holds in dimension two by the Gauss–Bonnet theorem and in dimension four by positivity of the Chern–Gauss–Bonnet integrand, but it remains open in every even dimension at least six. The problem sits at the center of a broader impasse: beyond the rank-one symmetric spaces (the spheres, projective spaces, and Cayley plane listed above) and a short list of exceptional examples in dimensions 6, 7, 12, 13, and 24, almost nothing is known about which closed simply connected manifolds admit positive sectional curvature, and no obstruction is known for simply connected manifolds beyond those already implied by nonnegative curvature. Since the 1990s, Karsten Grove's symmetry program has studied the conjecture under the assumption of a large isometry group. The symmetry rank of a Riemannian manifold is the rank (dimension of the largest torus) of its isometry group; if a torus Tr=(S1)rT^r=(S^1)^r acts by isometries with fixed-point set FF, then χ(M)=χ(F)\chi(M)=\chi(F), so the problem reduces to fixed-point components, which are totally geodesic submanifolds (submanifolds whose geodesics are also geodesics of the ambient manifold). In this framework the conjecture was proved with symmetry-rank bounds growing linearly in the manifold dimension nn by Püttmann–Searle, with a logarithmic bound r≥2log⁡2nr\ge 2\log_2 n (for n≡0 mod 4n\equiv 0 \bmod 4) by Kennard via four-periodicity phenomena in cohomology (a repeating period-four pattern in the rational cohomology rings of such manifolds), and by Kennard–Wiemeler–Wilking with the first rank bound independent of dimension: if the isometry group has rank at least five, then χ(M)>0\chi(M)>0 (Corollary B), derived from their structure theorem that fixed-point components of an effective isometric T5T^5-action on a closed, orientable, positively curved manifold have the rational cohomology of a sphere, complex projective space, or quaternionic projective space. Nienhaus subsequently improved the relevant machinery (an improved four-periodicity theorem) and lowered the sufficient rank to four. Related results delimit the conjecture: the conclusion fails if positive sectional curvature is weakened to positive Ricci curvature (products of spheres) or to positive second intermediate Ricci curvature Ric⁡2>0\operatorname{Ric}_2>0 (the condition that ∑i=12sec⁡(x,yi)>0\sum_{i=1}^{2}\sec(x,y_i)>0 for every orthonormal triple), so no argument can rely only on such weaker curvature information.

Problem Statement

The Hopf conjecture: does every closed, even-dimensional Riemannian manifold with positive sectional curvature have positive Euler characteristic? Equivalently, is χ(M)>0\chi(M)>0 for every closed smooth manifold M2nM^{2n}, n≥3n\ge 3, carrying a Riemannian metric whose sectional curvatures are all positive? Dimensions two and four are settled affirmatively by Gauss–Bonnet-type theorems, so the open content is every even dimension at least six, with no symmetry, finiteness, or topological hypothesis beyond closedness and even dimensionality. A derived problem, not part of the conjecture proper but its most active front: determine the minimal symmetry rank sufficient to force the conclusion — currently known to be at most four (an isometric T4T^4-action suffices by Nienhaus's improvement of Kennard–Wiemeler–Wilking's rank-five theorem), with three or fewer open and no matching lower bound known.

References

  1. Lee Kennard, Michael Wiemeler, Burkhard Wilking, Splitting of torus representations and applications in the Grove symmetry program, arXiv:2106.14723 (2021). DOI: 10.48550/arXiv.2106.14723, https://arxiv.org/abs/2106.14723
  2. Jan Nienhaus, An improved four-periodicity theorem and a conjecture of Hopf with symmetry, arXiv:2211.13151 (2022). DOI: 10.48550/arXiv.2211.13151, https://arxiv.org/abs/2211.13151
  3. Lee Kennard, Michael Wiemeler, Burkhard Wilking, Positive curvature, torus symmetry, and matroids, Journal of the European Mathematical Society (2025). DOI: 10.4171/jems/1656, https://doi.org/10.4171/jems/1656 (arXiv:2212.08152)
  4. Lee Kennard, Lawrence Mouillé, Jan Nienhaus, On Hopf's conjecture and positive second intermediate Ricci curvature, arXiv:2507.16936 (2025). DOI: 10.48550/arXiv.2507.16936, https://arxiv.org/abs/2507.16936
  5. Lee Kennard, On the Hopf conjecture with symmetry, Geometry & Topology 17 (2013), 563–593. DOI: 10.2140/gt.2013.17.563, https://doi.org/10.2140/gt.2013.17.563 (arXiv:1203.3808)
  6. Thomas Püttmann, Catherine Searle, The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank, Proceedings of the American Mathematical Society 130 (2002), 163–166. DOI: 10.1090/S0002-9939-01-06039-7, https://doi.org/10.1090/S0002-9939-01-06039-7 (arXiv:1207.4086)
  7. Manuel Amann, Lee Kennard, On a generalized conjecture of Hopf with symmetry, Compositio Mathematica 153 (2017), 313–322. DOI: 10.1112/s0010437x16008150, https://doi.org/10.1112/s0010437x16008150
  8. Wu-Yi Hsiang, Bruce Kleiner, On the topology of positively curved 4-manifolds with symmetry, Journal of Differential Geometry 29 (1989), 615–621. DOI: 10.4310/jdg/1214443064, https://doi.org/10.4310/jdg/1214443064