Hopf conjectured in the 1930s that every closed, even-dimensional Riemannian manifold of positive sectional curvature has positive Euler characteristic. Kennard, Mouillé, and Nienhaus (2025) proved the first dimension-independent symmetry-rank thresholds for this conclusion under the weaker condition of positive second intermediate Ricci curvature (Ric2>0\mathrm{Ric}_2>0): torus symmetry of rank 10 forces positive Euler characteristic in dimensions divisible by four, rank 8 controls the odd rational cohomology of fixed-point components, and rank 9 (with connected isotropy) forces partial four-periodicity of rational cohomology. They explicitly ask whether, for each higher intermediate Ricci curvature condition Rick>0\mathrm{Ric}_k>0 with k≥3k\ge 3, some dimension-independent integer d(k)d(k) exists such that torus symmetry of rank at least d(k)d(k) again forces positive Euler characteristic, noting that their key tools fail in that range. A linked calibration question, posed in the same source, is whether the proven k=2k=2 thresholds (8, 9, 10, 11) are sharp.

Background

The kk-th intermediate Ricci curvature of an nn-dimensional Riemannian manifold is the condition ∑i=1ksec⁡(x,yi)>0\sum_{i=1}^{k}\sec(x,y_i)>0 for every orthonormal (k+1)(k+1)-frame (x,y1,…,yk)(x,y_1,\dots,y_k), where sec⁡\sec denotes sectional curvature; it is written Rick>0\mathrm{Ric}_k>0. It interpolates between positive sectional curvature (k=1k=1) and positive Ricci curvature (k=n−1k=n-1), and Rick>0\mathrm{Ric}_k>0 implies Ricℓ>0\mathrm{Ric}_\ell>0 for all ℓ≥k\ell\ge k, so smaller kk is the stronger hypothesis and Ric2>0\mathrm{Ric}_2>0 is the strongest condition weaker than positive sectional curvature. Hopf's conjecture asserts that every closed, even-dimensional manifold with sec⁡>0\sec>0 has positive Euler characteristic χ\chi; it remains one of the central open problems of global Riemannian geometry. The conclusion fails if the curvature hypothesis is relaxed: for example, S3×S3\mathbb{S}^3\times\mathbb{S}^3 carries a metric with Ric2>0\mathrm{Ric}_2>0 (due to Wilking) and S7×S7\mathbb{S}^7\times\mathbb{S}^7 carries such a metric (DeVito–Domínguez-Vázquez–González-Álvaro–Rodríguez-Vázquez), both with Euler characteristic zero, so some auxiliary hypothesis is needed. The Grove symmetry program supplies one: the symmetry rank is the rank of a torus acting effectively by isometries. For sec⁡>0\sec>0, the Hopf conclusion holds under torus symmetry of rank at least a logarithm of the dimension (Kennard 2013), then at least 5 (Kennard–Wiemeler–Wilking 2021), then at least 4 (Nienhaus 2022). For Ric2>0\mathrm{Ric}_2>0, Mouillé (2022) proved the Hopf conclusion for closed simply connected manifolds in even dimensions other than 6 under large, dimension-dependent symmetry rank, and Kennard–Mouillé (2024) classified such manifolds at maximal symmetry rank; Kennard–Mouillé–Nienhaus (2025) then proved the first dimension-independent thresholds: with Ric2>0\mathrm{Ric}_2>0 and dimension divisible by four, a T8\mathbb{T}^8-action forces every fixed-point component to have vanishing odd rational cohomology, and a T10\mathbb{T}^{10}-action forces χ(M)>0\chi(M)>0; with connected isotropy groups, T9\mathbb{T}^9 forces four-periodicity of rational cohomology in degrees 1 through n−1n-1, with T10\mathbb{T}^{10} and T11\mathbb{T}^{11} corollaries in dimensions 4n4n and 4n+14n+1. (Four-periodicity means multiplication by some degree-4 class gives isomorphisms in a range of degrees; it implies vanishing of odd Betti numbers, the Betti numbers being the dimensions of the rational cohomology groups.) The engine of these results combines Myers' theorem (which still applies for every k≤n−1k\le n-1, since Rick>0\mathrm{Ric}_k>0 implies positive Ricci curvature and hence finite fundamental group) with a Connectedness Lemma specific to Ric2>0\mathrm{Ric}_2>0 — fixed-point components of circle actions are highly connected inclusions — which feeds Wilking's Periodicity Lemma and a new Partial Four Periodicity Theorem of Kennard–Mouillé–Nienhaus that lifts the periodicity to rational cohomology under H1H^1-vanishing hypotheses. The Connectedness Lemma is where the method breaks for k≥3k\ge 3: the hypothesis Rick>0\mathrm{Ric}_k>0 is then too weak to force high connectivity of the relevant inclusions, so no periodic cohomology input is currently available at all. For k≥3k\ge 3 the only known topological rigidity (Kennard–Mouillé 2024, Theorems B and C) assumes strong topological hypotheses — (k−1)(k-1)-connectedness or the integral cohomology of CPn\mathbb{C}\mathrm{P}^n up to degree k+2k+2 — together with near-maximal, dimension-dependent symmetry rank r=⌊(n+k)/2⌋−1r=\lfloor (n+k)/2\rfloor-1. Related open threads in the same paper include whether the k=2k=2 thresholds are optimal (the known Euler-characteristic-zero examples with Ric2>0\mathrm{Ric}_2>0 carry only T3\mathbb{T}^3- or T4\mathbb{T}^4-symmetry) and realizability constraints on partially periodic cohomology rings, where an Adams-type theorem for periodic cohomology is conjectured by Harper–Kennard.

Problem Statement

For each integer k≥3k\ge 3, does there exist an integer d(k)d(k) such that every closed, even-dimensional Riemannian manifold MM with positive kk-th intermediate Ricci curvature (Rick>0\mathrm{Ric}_k>0) admitting an effective isometric torus action of rank at least d(k)d(k) has positive Euler characteristic χ(M)>0\chi(M)>0? Either there is, for every k≥3k\ge 3, a proof that such a threshold exists, with an explicit value of d(k)d(k) — in the k≤2k\le 2 theory the corresponding results additionally give the companion conclusion that the odd Betti numbers vanish — or there is a construction showing that for some k≥3k\ge 3 no such threshold exists, i.e., for arbitrarily large rr an even-dimensional closed manifold with Rick>0\mathrm{Ric}_k>0, an effective isometric Tr\mathbb{T}^r-action, and χ≤0\chi\le 0. A further question posed in the same source for k=2k=2 is whether the proven thresholds are sharp — in particular, can the rank-8 statement (in dimensions divisible by four, vanishing of the odd rational cohomology of fixed-point components), the rank-9 statement (with connected isotropy groups and a non-empty fixed-point set, partial four-periodicity of rational cohomology in degrees 11 through n−1n-1), the rank-10 statement (positive Euler characteristic in dimensions divisible by four), or the rank-11 statement (with connected isotropy groups, rational homology of a sphere in dimensions 4n+14n+1) be lowered? This question is linked to, but separate from, the question above. These four statements are independent, so establishing or lowering any one of them is a self-contained result that does not settle the others. The known Euler-characteristic-zero examples with Ric2>0\mathrm{Ric}_2>0 carry only T3\mathbb{T}^3- or T4\mathbb{T}^4-symmetry, so the optimal thresholds, where they differ from the proven ones, lie in the intermediate symmetry range.

References

  1. 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
  2. Lee Kennard, "On the Hopf Conjecture with Symmetry", Geometry & Topology 17 (2013) 563–593. DOI: 10.2140/gt.2013.17.563, https://arxiv.org/abs/1203.3808
  3. 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
  4. 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
  5. Lawrence Mouillé, "Torus actions on manifolds with positive intermediate Ricci curvature", Journal of the London Mathematical Society 106 (2022) 3792–3821. DOI: 10.1112/jlms.12674, https://arxiv.org/abs/2101.05826
  6. Lee Kennard, Lawrence Mouillé, "Positive intermediate Ricci curvature with maximal symmetry rank", Journal of Geometric Analysis 34 (2024), article 129. DOI: 10.1007/s12220-024-01575-z, https://arxiv.org/abs/2303.00776
  7. Philipp Reiser, David J. Wraith, "Intermediate Ricci curvatures and Gromov's Betti number bound", Journal of Geometric Analysis 33 (2023), article 364. DOI: 10.1007/s12220-023-01423-6, https://arxiv.org/abs/2208.13438
  8. Jason DeVito, Miguel Domínguez-Vázquez, David González-Álvaro, Alberto Rodríguez-Vázquez, "Positive Ric2\mathrm{Ric}_2 curvature on products of spheres and their quotients via intermediate fatness", arXiv:2410.18846, 2024. DOI: 10.48550/arxiv.2410.18846, https://arxiv.org/abs/2410.18846
  9. Elahe Khalili Samani, Lawrence Mouillé, "Positive intermediate Ricci curvature on cohomogeneity one manifolds in low dimensions", arXiv:2507.17920, 2025. DOI: 10.48550/arxiv.2507.17920, https://arxiv.org/abs/2507.17920
  10. Hasan M. El-Hasan, Russell Phelan, Frederick Wilhelm, "Do Riemannian Submersions Preserve Positive Intermediate Ricci Curvature?", arXiv:2507.17839, 2025. DOI: 10.48550/arxiv.2507.17839, https://arxiv.org/abs/2507.17839
  11. Yujie Wu, "Comparison Theorems and the Intermediate Ricci Curvature Assumption", arXiv:2510.11205, 2025. DOI: 10.48550/arxiv.2510.11205, https://arxiv.org/abs/2510.11205
  12. John R. Harper, Lee Kennard, "Extending Adams' theorem from singly generated to periodic cohomology", arXiv:2412.16340, 2024. DOI: 10.48550/arxiv.2412.16340, https://arxiv.org/abs/2412.16340