Browse › Differential geometry › 358 A three-manifold without conjugate points or nonpositive curvature
A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric
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Abstract
We construct a closed connected orientable smooth three-manifold that admits a smooth Riemannian metric without conjugate points but admits no smooth Riemannian metric of nonpositive sectional curvature.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 358 (lean/docs/358.md, 2026). The release's Lean development covers the main theorem (a metric without conjugate points, quantified over all geodesics including constant ones and unbounded time intervals, and no smooth nonpositively curved metric); the notes put the no-focal-points and CAT(0) consequences out of scope. [family-level]
Possibly relevant literature (found by the agent, not checked)
- S. Buyalo and P. Svetlov, Topological and geometric properties of graph-manifolds (arXiv:math/0309192, 2003; St. Petersburg Math. J. 16 (2005)). This survey gives criteria for when graph manifolds admit nonpositively curved metrics, which is general context for the Leeb obstruction the paper uses for its two-piece gluing.
- S. Tchuiaga, Isometric Splitting of Metrics Without Conjugate Points on Σ × S¹ (arXiv:2504.07616, v2 Dec 2025). It claims that metrics without conjugate points on Σ × S¹, with Σ closed of genus at least 2, are Riemannian products; this is a rigidity statement on Seifert-fibered pieces, while the paper's pieces are punctured-torus products with boundary.
Digests and expositions
- Latent Seminar, No conjugate points without nonpositive curvature: a three-manifold and its geodesic neck (7 October 2026). This exposition follows the gluing obstruction, concave cometric construction, returning-geodesic estimates and countable assembly of index forms, explicitly disclaiming independent verification of the complete proof. [family-level]
- Open Mathematics Courses / Claude Opus 5.5, A three-manifold without conjugate points (undated). Seven lessons with exercises explain the manuscript's topological obstruction and neck estimates, with AI self-checking disclosed and independent review stated to be incomplete. [AI-generated per source]
- Gradient Damage, Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More (14 October 2026). The video includes a segment on this result at 16:23, with the accompanying page distinguishing its animations from proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 358
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026,
author = {{OpenAI}},
title = {{A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026/paper.pdf}{OAI:A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/358.md. The Lean statements have not been checked by people against the paper's claims.
Family 358: A three-manifold without conjugate points or nonpositive curvature
Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether the first metric-existence property implies the second.
This is the only paper in this family.