Must a closed manifold admitting a metric without conjugate points admit a metric of nonpositive curvature?
A Riemannian metric has no conjugate points if no nonzero Jacobi field along a geodesic vanishes twice. Nonpositive sectional curvature implies this, and Gulliver showed that individual metrics without conjugate points may have positive curvature somewhere, but his examples live on manifolds that already carry negatively curved metrics. On surfaces the answer is yes by uniformization. Ivanov and Kapovitch asked, and Burns and Matveev repeated for dimension three, whether every closed manifold that admits a metric without conjugate points also admits a metric of nonpositive sectional curvature. Is that true?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Riemannian geometry; manifolds without conjugate points
- Posed by
- Sergei Ivanov and Vitali Kapovitch (2014, Questions 1.1 and 8.1); repeated by Keith Burns and Vladimir Matveev (survey, Question 4.1.2)
- Year posed
- 2014
- Years open
- 12y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: there is a closed connected orientable smooth 3-manifold with a smooth metric without conjugate points that admits no smooth metric of nonpositive sectional curvature. The manifold glues two copies of (punctured torus) x by Leeb's obstructed gluing; admits no proper cocompact action on a CAT(0) space, so there is no locally CAT(0) metric, and by Ivanov-Kapovitch no metric without focal points. Not shown: higher-dimensional examples, or an answer to Ivanov-Kapovitch's other questions about fundamental groups.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. Single manuscript dated September 24, 2026. The topological obstruction is Leeb's two-piece graph-manifold example; the metric construction and its no-conjugate-points estimate are the paper's.
Verification
No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 read against Ivanov-Kapovitch's question. Lean (in lean/formalization.yaml): OAI.ThreeManifold.main_theorem in OAI/Geometry/ConjugatePoints/ThreeManifold.lean. Its statement MainStatement gives a compact connected Hausdorff second-countable smooth 3-manifold with an oriented atlas, a smooth metric whose Jacobi fields along every geodesic on every open interval vanish identically once they vanish at two distinct times, and no smooth metric with nonpositive sectional curvature. That is the headline. Not rebuilt here. The no-focal-points and CAT(0) consequences are outside the formal statement.