OpenAI · 358-01 · 2026-09-24 · Differential geometry
A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric
preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026
Family 358. A three-manifold with no conjugate points and no nonpositively curved metric.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 358: A three-manifold without conjugate points or nonpositive curvature. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.