Differential geometry·OpenAI Math·Result 358·Lean formalisation listed
A three-manifold without conjugate points or nonpositive curvature
One of the 372 results in OpenAI's public mathematics catalogue, filed under differential geometry: 1 preprint, the latest dated 24 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a result titled ‘A three-manifold without conjugate points or nonpositive curvature’ (differential geometry) in a preprint dated 24 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More, is on YouTube.
Chapter 48 of 84 in Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · starts at 16:23 · watch from there on YouTube →
What OpenAI claims
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. (Lean)
Paper
A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric
OpenAI's 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.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- 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. (Lean)
- Where is the proof?
- In the paper, A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric (PDF).
- Is there a video?
- Yes, on YouTube (29:20, chapter 48).
Also in
- Chapter 8 of 9 in Calabi Flow, Affine Bernstein Problem and 7 More in Differential Geometry · starts at 2:25 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · Compilation · 29:20 · chapter 48 from 16:23 · description, sources and chapters →
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