Huang and Wang (arXiv:2507.15237, 2025) decompose the curvature operator of a closed Riemannian manifold into Ricci and Weyl components and prove that pinching the Weyl tensor below explicit constants — relative to a sectional- or intermediate-Ricci-curvature scale, pointwise or in Ln/2L^{n/2}, or relative to the Yamabe constant — forces positivity of the curvature operator, vanishing of Betti numbers, and the Hopf sign of the Euler characteristic. The paper leaves four issues open: (1) sharpness of the Weyl thresholds kN/(N−k)\sqrt{kN/(N-k)} (with N=(n2)N=\binom{n}{2}) and classification of the extremal algebraic curvature tensors and metrics; (2) extension of the Yamabe-constant vanishing and harmonic-Weyl gap theorems, valid only for n≥8n\ge 8, to dimensions 4–7; (3) classification of the rigid cases (Wg≡0\mathcal{W}_g\equiv 0, spherical Betti numbers) without locally conformally flat or conformally Einstein hypotheses; (4) the geometric content, beyond Betti-number vanishing, of the induced k-positivity of the curvature operator. The common objective is to determine exactly how much Weyl curvature a positively curved closed manifold can carry before the curvature-operator route to the Hopf conclusion breaks.

Background

The Hopf conjecture of the 1930s predicts that the sign of sectional curvature controls the sign of the Euler characteristic χ\chi of a closed even-dimensional Riemannian manifold: χ(M)>0\chi(M)>0 when all sectional curvatures are positive, and χ(M)\chi(M) carries the sign (−1)m/2(-1)^{m/2} when all are negative (dimension mm). It is a theorem in dimensions 2 and 4, where the Gauss–Bonnet and Chern–Gauss–Bonnet integrand itself carries the sign; in higher dimensions that integrand loses pointwise sign control under positive sectional curvature, and the conjecture is open in every even dimension at least 6. Since χ(M)=∑p(−1)pbp(M)\chi(M)=\sum_p(-1)^p b_p(M), where bp(M)b_p(M) is the pp-th Betti number (the dimension of the space of harmonic pp-forms), a standard line of attack is to force middle Betti numbers to vanish under curvature conditions stronger than sectional positivity.

The curvature operator R\mathfrak{R} is the self-adjoint endomorphism of Λ2TpM\Lambda^2 T_pM determined by ⟨R(X∧Y),Z∧W⟩=Rm(X,Y,Z,W)\langle\mathfrak{R}(X\wedge Y),Z\wedge W\rangle=Rm(X,Y,Z,W); its quadratic form on decomposable 2-vectors is the sectional curvature, so positivity of R\mathfrak{R} is strictly stronger than positive sectional curvature. By the Bochner technique — the Weitzenböck formula for differential forms, in which R\mathfrak{R} enters with a definite sign — positive R\mathfrak{R} kills harmonic pp-forms for 1≤p≤n−11\le p\le n-1, and Böhm–Wilking proved that a closed manifold with positive curvature operator is a spherical space form; in particular the Hopf conclusion holds under positive curvature operator. Petersen and Wink refined the Bochner conclusion: a lower bound on the average of the lowest n−pn-p eigenvalues of R\mathfrak{R} already yields vanishing or estimation of bpb_p. The operator R\mathfrak{R} is called kk-positive if the sum of its kk smallest eigenvalues is positive, interpolating between positivity of the whole operator (k=1k=1) and a trivial trace condition. Correspondingly, the kk-th intermediate Ricci curvature is Rick(v,v)=∑i=2k+1⟨Rm(v,ei)v,ei⟩Ric_k(v,v)=\sum_{i=2}^{k+1}\langle Rm(v,e_i)v,e_i\rangle for an orthonormal frame, so that Ric1Ric_1 recovers sectional curvature and Ricn−1Ric_{n-1} the Ricci curvature; a lower bound Rick≥kaRic_k\ge ka interpolates between the two.

The Weyl tensor Wg\mathcal{W}_g is the totally trace-free part of the Riemann curvature tensor; in dimension n≥4n\ge 4 it vanishes exactly on locally conformally flat manifolds, and it is the conformally invariant carrier of curvature. Huang–Wang split R=S+W\mathfrak{R}=\mathfrak{S}+\mathfrak{W} into a Ricci (Schouten) component and a Weyl component, both Hermitian on Λ2TpM\Lambda^2T_pM, and bound the eigenvalues of R\mathfrak{R} in terms of Ricci data and the pointwise norm ∣Wg∣|\mathcal{W}_g|. They obtain: the Hopf conjecture for all closed locally conformally flat even-dimensional manifolds; a pointwise threshold — sec⁡g≥a>0\sec_g\ge a>0 (resp. sec⁡g≤−a<0\sec_g\le -a<0) together with ∣Wg∣<n(n−1)(n+1)(n−2) a|\mathcal{W}_g|<\sqrt{\tfrac{n(n-1)}{(n+1)(n-2)}}\,a implies R\mathfrak{R} positive (resp. negative); and, for N=(n2)N=\binom{n}{2}, the general kk-threshold ∣Wg∣<kNN−k a|\mathcal{W}_g|<\sqrt{\tfrac{kN}{N-k}}\,a under Rick≥ka>0Ric_k\ge ka>0, which forces kk-positivity and hence Petersen–Wink Betti vanishings. The eigenvalue engine is a sharp zero-sum sequence inequality applied to the trace-free Weyl component, whose eigenvalues are then majorized in operator norm by ∣Wg∣|\mathcal{W}_g|.

Two further settings appear. First, the pointwise bound can be relaxed to an Ln/2L^{n/2} bound (the norm in which the Weyl tensor is conformally invariant), and the curvature scale can be replaced by the Yamabe constant λ(g)\lambda(g) — the infimum of the normalized total scalar curvature over the conformal class of gg, a conformally invariant scalar-curvature-type quantity — giving vanishing theorems for Betti numbers and, in Corollary 1.7, the full spherical pattern of Betti numbers for n≥4n\ge 4 under an additional positive-scalar-curvature hypothesis and a different threshold, while the Yamabe-constant vanishing theorem (Theorem 1.6) requires n≥8n\ge 8. Second, for metrics with harmonic Weyl curvature (the Weyl tensor is divergence-free, div⁡Wg=0\operatorname{div}\mathcal{W}_g=0; a condition satisfied in particular by every Einstein metric), Huang–Wang prove gap theorems: unless Wg≡0\mathcal{W}_g\equiv 0, suitable Ln/2L^{n/2}-norms of Wg\mathcal{W}_g, or of 1n−2∣Ric∘g∣+∣Wg∣\tfrac{1}{\sqrt{n-2}}|\overset{\circ}{Ric}_g|+|\mathcal{W}_g| (where Ric∘g\overset{\circ}{Ric}_g is the traceless Ricci tensor), are bounded below by explicit multiples of aa or λ(g)\lambda(g); the case involving λ(g)\lambda(g) again needs n≥8n\ge 8. The reasons are visible in the published computations, and the two n≥8n\ge 8 restrictions arise from different coefficient comparisons: in the harmonic-Weyl gap proof, the improved Kato inequality for harmonic Weyl tensors, ∣∇Wg∣2≥n+1n−1∣∇∣Wg∣∣2|\nabla\mathcal{W}_g|^2\ge\tfrac{n+1}{n-1}|\nabla|\mathcal{W}_g||^2, must beat the conformal Sobolev coefficient 8(n−1)n(n−2)\tfrac{8(n-1)}{n(n-2)}, and the difference n+1n−1−8(n−1)n(n−2)=n(n−2)(n−7)−8n(n−1)(n−2)\tfrac{n+1}{n-1}-\tfrac{8(n-1)}{n(n-2)}=\tfrac{n(n-2)(n-7)-8}{n(n-1)(n-2)} is negative for n≤7n\le 7; in the Yamabe-constant vanishing proof, the corresponding positivity lemma (their Lemma 4.5), obtained from the refined Kato inequality for pp-forms, compares 1+1n−p1+\tfrac{1}{n-p} with 4p(n−p)n(n−2)\tfrac{4p(n-p)}{n(n-2)} and is stated and proved only for n≥8n\ge 8. Related prior results frame the open questions: Tran's harmonic-Weyl rigidity and gap theorems (valid for n≥6n\ge 6, but involving the traceless Ricci term as well), and Hebey–Vaugon's effective LpL^p pinching of the concircular curvature tensor ZgZ_g (the combination of Wg\mathcal{W}_g with the traceless Ricci part), which classifies the manifold as a spherical quotient only under locally conformally flat or conformally Einstein hypotheses on the conformal class.

What the paper does not do: it never claims its constants are optimal, never identifies the algebraic curvature tensors or metrics that attain the thresholds, leaves dimensions 4≤n≤74\le n\le 7 outside its Yamabe-based and harmonic-Weyl gap statements, classifies no rigid case beyond citing results under extra hypotheses, and draws no geometric conclusion from the induced kk-positivity beyond Betti vanishing. These four gaps constitute the problem below.

Problem Statement

Setup. Let (M,g)(M,g) be a closed (compact, without boundary) Riemannian manifold of dimension n≥4n\ge 4, let N=(n2)N=\binom{n}{2}, let R\mathfrak{R} be its curvature operator, Wg\mathcal{W}_g its Weyl tensor with pointwise norm ∣Wg∣|\mathcal{W}_g|, RickRic_k its kk-th intermediate Ricci curvature, λ(g)\lambda(g) its Yamabe constant, bp(M)b_p(M) its Betti numbers and χ(M)\chi(M) its Euler characteristic. Huang–Wang prove, among other results:

(P1) if sec⁡g≥a>0\sec_g\ge a>0 (resp. sec⁡g≤−a<0\sec_g\le -a<0) and ∣Wg∣<n(n−1)(n+1)(n−2) a=NN−1 a|\mathcal{W}_g|<\sqrt{\frac{n(n-1)}{(n+1)(n-2)}}\,a=\sqrt{\frac{N}{N-1}}\,a, then R\mathfrak{R} is positive (resp. negative), so for even nn the Euler characteristic carries the Hopf sign;

(P2) more generally, if Rick≥ka>0Ric_k\ge ka>0 and ∣Wg∣<kNN−k a|\mathcal{W}_g|<\sqrt{\frac{kN}{N-k}}\,a for some 1≤k≤n−11\le k\le n-1, then R\mathfrak{R} is kk-positive (their Proposition 3.3(1); the printed Corollary 3.4 draws the same conclusion under the stronger hypothesis sec⁡g≥a\sec_g\ge a, which implies Rick≥kaRic_k\ge ka), and the Petersen–Wink vanishings of Betti numbers follow;

(P3) integral Ln/2L^{n/2} and Yamabe-constant versions of these vanishing theorems (their Theorems 1.4, 1.6 and Corollary 1.7), with the Yamabe-constant statement of Theorem 1.6 restricted to n≥8n\ge 8 (Corollary 1.7 covers n≥4n\ge 4 only under an additional positive-scalar-curvature hypothesis and a different threshold);

(P4) gap theorems under harmonic Weyl curvature (their Theorem 1.9): if Wg≢0\mathcal{W}_g\not\equiv 0, then the normalized Ln/2L^{n/2}-norm of Wg\mathcal{W}_g (case 1, under Rick≥ka>0Ric_k\ge ka>0 for some 1≤k≤⌊n−12⌋1\le k\le\lfloor\frac{n-1}{2}\rfloor), or of 1n−2∣Ric∘g∣+∣Wg∣\frac{1}{\sqrt{n-2}}|\overset{\circ}{Ric}_g|+|\mathcal{W}_g| (case 2, requiring n≥8n\ge 8 and λ(g)>0\lambda(g)>0), is bounded below by explicit multiples of aa or λ(g)\lambda(g).

The four questions below share one objective: determine exactly how much Weyl curvature a closed positively curved manifold can carry before the curvature-operator/Bochner route to the Hopf conclusion breaks, and what happens at the breaking point. Each question is posed independently: a complete resolution settles all four, while a solution of any single question is a partial result.

(Q1) Sharp constants and extremals. For each 1≤k≤n−11\le k\le n-1, determine the optimal constant cn,kc_{n,k} such that Rick≥ka>0Ric_k\ge ka>0 together with ∣Wg∣<cn,k a|\mathcal{W}_g|<c_{n,k}\,a still forces R\mathfrak{R} to be kk-positive; in particular, decide whether cn,1=n(n−1)(n+1)(n−2)c_{n,1}=\sqrt{\frac{n(n-1)}{(n+1)(n-2)}} is optimal. Prove optimality or improve the constant — the published values come from a sharp inequality for arbitrary zero-sum eigenvalue sequences (using only the trace-freeness of the Weyl component) plus a norm majorization of that component, and therefore ignore the further algebraic constraints, notably the first Bianchi identity, satisfied by actual Weyl curvature operators, so neither direction is automatic. Classify the algebraic curvature tensors, and geometrically the metrics on closed manifolds, that attain or approach the sharp thresholds, and decide whether the corresponding kNN−k\sqrt{\frac{kN}{N-k}}-scaled factors in the Ln/2L^{n/2} inequalities of (P3) and (P4) are sharp.

(Q2) Low dimensions. Decide whether the Yamabe-constant vanishing theorem (their Theorem 1.6) and the harmonic-Weyl gap theorem in its case (2) (their Theorem 1.9(2)) extend to 4≤n≤74\le n\le 7, possibly with different constants. The published proofs fail there: for Theorem 1.9(2), the coefficient n+1n−1−8(n−1)n(n−2)=n(n−2)(n−7)−8n(n−1)(n−2)\frac{n+1}{n-1}-\frac{8(n-1)}{n(n-2)}=\frac{n(n-2)(n-7)-8}{n(n-1)(n-2)}, comparing the improved Kato inequality for harmonic Weyl tensors with the conformal Sobolev constant, is negative for n≤7n\le 7, while for Theorem 1.6 the corresponding comparison 1+1n−p>4p(n−p)n(n−2)1+\frac{1}{n-p}>\frac{4p(n-p)}{n(n-2)}, obtained from the refined Kato inequality for pp-forms, is established (their Lemma 4.5) only for n≥8n\ge 8; determine whether these barriers are defects of the method or of the statements.

(Q3) Rigidity and classification. In the regimes where the gap and vanishing theorems force Wg≡0\mathcal{W}_g\equiv 0 or the spherical pattern of Betti numbers, classify the manifolds and metrics: without locally conformally flat or conformally Einstein hypotheses, are they spherical space forms or conformal quotients of the round sphere, or do other examples exist at and near the thresholds?

(Q4) Geometric content of the induced kk-positivity. Determine, for each of the following candidate conclusions — holonomy reduction, local symmetry, fundamental group, homeomorphism or diffeomorphism type — whether it follows from the kk-positivity of R\mathfrak{R} produced by the Weyl thresholds in (P2), beyond the Betti-number vanishings obtained from Petersen–Wink's theorem; a resolution of this question is a proof that the conclusion holds, or a counterexample showing that it does not.

References

  1. Teng Huang, Weiwei Wang, Decomposition of the Curvature Operator and Applications to the Hopf Conjecture, arXiv:2507.15237 (2025), https://arxiv.org/abs/2507.15237 (source of the open questions)
  2. Peter Petersen, Matthias Wink, New Curvature Conditions for the Bochner Technique, Inventiones mathematicae 224 (2021), 33–54, doi:10.1007/s00222-020-01003-3, https://arxiv.org/abs/1908.09958
  3. Hung Tran, On Closed Manifolds with Harmonic Weyl Curvature, Advances in Mathematics 322 (2017), 861–891, doi:10.1016/j.aim.2017.10.030, https://arxiv.org/abs/1602.01429
  4. Christoph Böhm, Burkhard Wilking, Manifolds with positive curvature operators are space forms, Annals of Mathematics 167 (2008), no. 3, 1079–1097, doi:10.4007/annals.2008.167.1079, https://arxiv.org/abs/math/0606187
  5. Emmanuel Hebey, Michel Vaugon, Effective L^p pinching for the concircular curvature, Journal of Geometric Analysis 6 (1996), no. 4, 531–553, doi:10.1007/bf02921622, https://doi.org/10.1007/bf02921622
  6. Lee Kennard, Lawrence Mouillé, Jan Nienhaus, On Hopf's conjecture and positive second intermediate Ricci curvature, arXiv:2507.16936 (2025), https://arxiv.org/abs/2507.16936
  7. Teng Huang, Qiang Tan, Curvature operator and Euler number, Calculus of Variations and Partial Differential Equations 64 (2025), doi:10.1007/s00526-025-03065-8, https://arxiv.org/abs/2507.22091
  8. Teng Huang, Pan Zhang, Schrödinger Operators, Integral Curvature, and the Euler Characteristic of Riemannian Manifolds, arXiv:2601.12440 (2026), https://arxiv.org/abs/2601.12440
  9. Haiping Fu, Yao Lu, Manifolds with harmonic Weyl curvature and curvature operator of the second kind, arXiv:2602.07313 (2026), https://arxiv.org/abs/2602.07313