Hopf Conjecture

Hopf Conjecture

OpenAI's claim

A topological Einstein-energy gap: Assuming the manuscript's normalized classification of nonnegatively curved Einstein four-manifolds, a sufficiently small scale-invariant $L^2$ trace-free Ricci energy is claimed to force a simply connected closed nonnegatively curved four-manifold to be diffeomorphic to $S^4$, $\mathbb{CP}^2$ or $S^2\times S^2$. No additional geometric bounds are imposed.

Positive-Einstein four-manifold classification: The paper claims that closed connected positively curved Einstein four-manifolds are, up to scaling, the round sphere, Fubini--Study projective plane, or round real projective space; the last possibility accommodates nonorientability.

Zero-plane Einstein rigidity: The claim assumes a positive Einstein constant on a closed four-manifold; if sectional curvature is nonnegative and one plane is flat, the universal cover is the product of two round constant-curvature surfaces; the radii are fixed by the Einstein normalization.

Mathematical statement of the claimed result

Mathematical claim 1.

Assume the companion classification: every closed connected smooth four-manifold with $\operatorname{Ric}=3g$ and sectional curvature $\ge0$ has universal cover isometric to round $S^4$, Fubini--Study $\mathbb{CP}^2$ normalized by $\operatorname{Ric}=3g$, or $S^2(1/\sqrt3)\times S^2(1/\sqrt3)$. Under that explicit premise, the claim gives an absolute $\epsilon_0>0$ such that any closed connected simply connected smooth four-manifold with $\sec_g\ge0$ and \[\int_M\left|\operatorname{Ric}_g-\tfrac14\operatorname{Scal}_g g\right|^2\,dV_g<\epsilon_0\] is diffeomorphic, without specified orientation, to $S^4$, $\mathbb{CP}^2$ or $S^2\times S^2$. The conclusion is topological rather than isometric, and the theorem is expressly conditional on the classification premise.

Mathematical claim 2.

In real dimension four, assume $M$ is smooth, closed and connected, and its metric is Einstein with $\sec_g>0$, there is $a>0$ such that $(M,ag)$ is claimed to be isometric to standard round $S^4$, Fubini--Study $\mathbb{CP}^2$, or standard round $\mathbb{RP}^4$. Einstein means $\operatorname{Ric}=\lambda g$ for a constant $\lambda$. Orientability is not required; the Einstein hypothesis is essential to this statement and is absent from the general Hopf conjecture.

Mathematical claim 3.

For a connected smooth closed four-manifold with $\operatorname{Ric}_g=3g$ and sectional curvature $\ge0$, the existence of one zero-curvature tangent two-plane is claimed to force its universal Riemannian cover to be \[S^2(1/\sqrt3)\times S^2(1/\sqrt3).\] Its corollary classifies every such universal cover as this product, round $S^4$ of curvature $1$, or Fubini--Study $\mathbb{CP}^2$ with $\operatorname{Ric}=3g$. This is an Einstein four-dimensional assertion, not a classification of all nonnegatively curved manifolds.

Outline of the argument

A topological Einstein-energy gap: A hypothetical countersequence is normalized and shown to satisfy initial geometry and Sobolev bounds. A uniform smoothing interval controls its high-curvature region and propagates the Einstein defect and negative-sectional-curvature error. The limit is a compact nonnegatively curved positive-Einstein space, whose companion classification identifies the topology and contradicts the sequence.

Positive-Einstein four-manifold classification: After normalizing the Einstein constant, coupled inequalities track the two Weyl blocks distinguished by the Hodge star. Weighted variance, Hessian, and moment comparisons reduce the analytic step to exact polynomial signs, forcing one block to vanish. Characteristic identities and a volume argument identify the oriented models, followed by an orientation-cover analysis of the possible quotient.

Zero-plane Einstein rigidity: Bonnet--Myers reduces the analysis to the compact simply connected cover. The two Weyl blocks satisfy a sharp coupled spectral constraint at the boundary of the curvature cone. Extending the weighted comparisons to this boundary and tracking equality forces the product geometry rather than merely a weaker norm bound.

Formal verification

Family 348 describes Lean coverage for the strict-positive-curvature Einstein classification; zero-plane rigidity and the small-energy gap are separate.

A topological Einstein-energy gap: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

Positive-Einstein four-manifold classification: The manuscript links EinsteinFour.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

Zero-plane Einstein rigidity: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

• OpenAI, An L${}^{2}$ Einstein Gap for Nonnegatively Curved Four-Manifolds, 2026-10-05; citation key OAI:An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:gap.
• OpenAI, Positively curved Einstein four-manifolds, 2026-09-23; citation key OAI:Positively-curved-Einstein-four-manifolds-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:classification.
• OpenAI, Zero-Plane Rigidity for Einstein Four-Manifolds, 2026-10-04; citation key OAI:Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main; Statement source: cor:nonnegative-classification.
• Family 348: Lean scope.
• Comparator EinsteinFour.json.
• Comparator EinsteinFour.lean.
• OpenAI catalogue entry.

Hopf Conjecture

Definitions and mathematical statement

Let $(M^{2m},g)$ be a closed Riemannian manifold. For a two-plane $\sigma=\operatorname{span}(u,v)\subset T_xM$, sectional curvature is $K_g(\sigma)=\langle R(u,v)v,u\rangle/(|u|^2|v|^2-\langle u,v\rangle^2)$, with the convention making the round sphere positive. The Euler characteristic is $\chi(M)=\sum_j(-1)^j\dim H_j(M,{\mathbb{Q}})$. The conjecture states \[ K_g(\sigma)>0\text{ for every }x,\sigma \quad\Longrightarrow\quad\chi(M)>0. \] No symmetry or simple-connectedness hypothesis is imposed. Positivity of Ricci curvature or scalar curvature alone is a different and weaker curvature condition.

Formulation and concept references: [S1].

Short English statement

Must every closed even-dimensional manifold with everywhere positive sectional curvature have positive Euler characteristic?

Research attempt

Current frontier.

The unrestricted assertion must be separated from the substantial symmetry results. Nienhaus proves the positive-sectional-curvature conclusion with an additional effective isometric $T^4$-action [E1]. The 2025 source [S1], Theorem A and its first corollary, instead treats positive second intermediate Ricci curvature: dimension divisible by four and $T^{10}$ symmetry suffice for positive Euler characteristic. Neither result supplies the missing symmetry in this entry. The calculations below use the classical Weitzenböck identity, in the normalization where the unit sphere has positive curvature [E2], Section 3, $3.10$. The proposed extensions, and their failure tests, are this notebook's deductions; no novelty of the general Bochner method is asserted.

Attack route.

One sufficient route is to annihilate every odd-degree harmonic form. A second is to prove a positive integral Euler-density formula without pointwise positivity. The former makes a particularly testable claim: does positive sectional curvature force a usable lower bound for the curvature endomorphism on odd forms? I first test the pointwise version exactly, then replace it by an integral small-negative-part condition. A negative eigenvalue of an algebraic curvature term is not a counterexample manifold, but it would rule out a commonly tempting shortcut before global arguments are attempted.

Attempt.

1. The topological target of the analytic argument. It is enough to treat each connected component, and then an oriented double cover if needed: Euler characteristic multiplies by the covering degree. For a closed connected oriented even-dimensional manifold, \[ H^{\mathrm{odd}}(M;{\mathbb{R}})=0\quad\Longrightarrow\quad \chi(M)=\sum_j b_{2j}(M)\ge2. \] This is a sufficient condition, not an asserted equivalent formulation of Hopf's conjecture. Positive sectional curvature implies positive Ricci curvature. The identity \[ 0=\|\nabla\omega\|_2^2+ \int_M\langle q_p(R)\omega,\omega\rangle\,\mathrm{dvol}, \qquad \Delta\omega=0, \] with $q_1(R)=\operatorname{Ric}$, gives $b_1=0$; Poincaré duality also gives $b_{n-1}=0$ [E2]. Thus in dimension four $\chi=2+b_2>0$. In dimension six the same reasoning leaves $\chi=2+2b_2-b_3$, displaying exactly the uncontrolled odd degree.

2. An exact obstruction in dimension six. On oriented ${\mathbb{R}}^4$, write $\Lambda^2=\Lambda^2_+\oplus\Lambda^2_-$ in the orthonormal bases \[ \frac{12+34}{\sqrt2},\ \frac{13-24}{\sqrt2},\ \frac{14+23}{\sqrt2};\qquad \frac{12-34}{\sqrt2},\ \frac{13+24}{\sqrt2},\ \frac{14-23}{\sqrt2}, \] where $ij=e_i\wedge e_j$. Consider the symmetric curvature operator \[ R_4=\begin{pmatrix}A&0\\0&C\end{pmatrix},\qquad A=\operatorname{diag}(-1/2,-1/2,4),\quad C=I_3. \] It satisfies the first Bianchi identity because $\operatorname{tr}A= \operatorname{tr}C=3$. This can also be checked directly: in the lexicographic wedge basis its only potentially nonzero Bianchi expression with four distinct indices is \[ (R_4)_{12,34}-(R_4)_{13,24}+(R_4)_{14,23} =-3/4-3/4+3/2=0. \] For a unit decomposable two-form $\eta$, the identity $\eta\wedge\eta=0$ implies $\|\eta_+\|^2=\|\eta_-\|^2=1/2$. Consequently \[ \langle R_4\eta,\eta\rangle\ge\tfrac12(-\tfrac12)+\tfrac12=\tfrac14. \] All sectional curvatures are positive, although the curvature operator itself has two negative eigenvalues.

Here $\operatorname{Ric}=3I$. On two-forms the Weitzenböck endomorphism is $6I-2R_4$, so \[ \alpha=(14+23)/\sqrt2,\qquad q_2(R_4)\alpha=-2\alpha. \] Extend this algebraic tensor by zero to ${\mathbb{R}}^4\oplus{\mathbb{R}}^2$ and put \[ R_6=R_4\oplus0+\tfrac1{10}I_{\Lambda^2{\mathbb{R}}^6}. \] Projection of a decomposable two-form onto $\Lambda^2{\mathbb{R}}^4$ is still decomposable. The extended first term is therefore nonnegative on planes, and $R_6$ has sectional curvature at least $1/10$ on every plane. On $\psi=\alpha\wedge e_5$, however, the product representation and the constant curvature identity $q_p(\kappa I)=\kappa p(n-p)I$ give \[ q_3(R_6)\psi=(-2+9/10)\psi=-\tfrac{11}{10}\psi. \tag{211.1} \] These normalizations can be verified without a curvature-sign convention imported from another text: if $L_{ij}$ is the infinitesimal orthogonal rotation acting on exterior forms, set \[ q_p(R)=-\sum_{i<j,\,k<\ell}R_{ij,k\ell}L_{ij}L_{k\ell}. \] Then $q_p(I)=p(n-p)I$. The accompanying exact rational computation constructs this $20$-by-$20$ matrix for $211.1$. Its eigenvalues, with multiplicities, are \[ -11/10\ (2),\quad39/10\ (8),\quad49/10\ (6),\quad79/10\ (4). \] The displayed eigenpair and Bianchi identity are checked symbolically, not inferred from floating-point eigenvalues. Thus the implication "positive sectional curvature gives positive $q_p$ in every odd degree" is false already for an algebraic tensor in dimension six.

3. A replacement allowing a negative part. Let $n\ge3$, $V=\operatorname{vol}M$, and suppose a scalar Sobolev inequality has been established with a constant $S$: \[ \|u\|_{2n/(n-2)}^2\le S\left(\|du\|_2^2+V^{-2/n}\|u\|_2^2\right). \tag{211.2} \] Fix a degree $p$ and suppose there are $a_p>0$ and a nonnegative function $W_p\in L^{n/2}$ with \[ q_p(R)(x)\ge(a_p-W_p(x))I. \] Define $b_p=S\|W_p\|_{n/2}$. If \[ b_p<1,\qquad b_p V^{-2/n}<a_p, \tag{211.3} \] then $H^p(M;{\mathbb{R}})=0$.

Indeed, for a harmonic $p$-form the Bochner identity gives \[ \|\nabla\omega\|_2^2+a_p\|\omega\|_2^2 \le\int W_p|\omega|^2 \le\|W_p\|_{n/2}\|\omega\|_{2n/(n-2)}^2. \] Apply $211.2$ to $|\omega|$, using the weak Kato inequality $|d|\omega||\le|\nabla\omega|$. It follows that \[ (1-b_p)\|\nabla\omega\|_2^2+ (a_p-b_pV^{-2/n})\|\omega\|_2^2\le0. \] Both coefficients are strictly positive, proving the claim. The application to the nonsmooth norm of a form follows by replacing it by $(|\omega|^2+\delta^2)^{1/2}$ and passing to the limit; compactness of $M$ and smoothness of $\omega$ justify the passage. If these hypotheses hold in every odd degree, the argument in part 1 proves positive Euler characteristic. The quantities in $211.3$ are compatible with metric rescaling: $a_p$ scales as curvature, while the $L^{n/2}$ curvature norm is scale invariant.

Stress test.

The matrix is not a metric on a closed manifold with $\chi\le0$. Realizing an algebraic tensor locally does not supply a global metric, harmonic form, or topology. It only eliminates the pointwise Bochner positivity lemma. Conversely, having some negative eigenvalues does not by itself disprove the integral criterion; harmonic forms need not concentrate on their negative eigenspaces.

Condition $211.3$ is a genuine additional hypothesis. One may formally set $W_p=(a_p-\lambda_{\min}(q_p))_+$, but positive sectional curvature has not been shown here to make its norm small relative to the Sobolev constant. Neither scaling nor compactness repairs this: scaling leaves the critical norm unchanged, and compactness of the individual manifold supplies no uniform control over this class of metrics. At the equality threshold in $211.3$, the proof only gives nonnegativity and does not force vanishing. In addition, vanishing of all odd cohomology could be unnecessarily strong; a successful Euler-characteristic argument might instead control the alternating sum without this vanishing. The present work does not establish such a cancellation or an everywhere positive Euler density.

Outcome.

Outcome: Exploratory approach with unresolved gap.

The attempt gives an explicit exact obstruction to a pointwise odd-form Bochner proof, together with a fully proved integral criterion that would replace that false premise. It recovers the elementary four-dimensional case and pinpoints the first uncontrolled degree in dimension six. The missing step is a global estimate of the negative curvature contribution, relative to the Sobolev inequality, derived from sectional positivity alone, or a different argument controlling Euler characteristic without odd-degree vanishing. No counterexample to Hopf's conjecture, and no unrestricted proof, is obtained.

Sources

• S1 Lee Kennard, Lawrence Mouille, Jan Nienhaus. On Hopf's conjecture and positive second intermediate Ricci curvature. 2025. https://arxiv.org/abs/2507.16936. Catalogue formulation source.
• E1 Jan Nienhaus, An improved four-periodicity theorem and a conjecture of Hopf with symmetry, arXiv:2211.13151v1 (23 November 2022), abstract and main Hopf-with-$T^4$ theorem. https://arxiv.org/abs/2211.13151.
• E2 Uwe Semmelmann and Gregor Weingart, The Weitzenböck machine, Compositio Mathematica 146 $2010$, 507--540; arXiv:math/0702031, Section 3, especially $3.9$--$3.10$. https://arxiv.org/pdf/math/0702031.