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Must a closed manifold admitting a metric without conjugate points admit a metric of nonpositive curvature?
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.
Exact FrontierDelta
Prior state unknown→disproved
Scope and record
Occurred: Sep 24, 2026
Delta type: SOURCE CLAIM
Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.
Canonical aliases: Must a closed manifold admitting a metric without conjugate points admit a metric of nonpositive curvature? · No conjugate points vs nonpositive curvature
Confidence: Not scored
Registry verification: lean checked · announcement · candidate
Attribution
VibeMathed
registry · event recorded by
Unreleased internal OpenAI model
model · ai model contributor · OpenAI
Artifacts and verifiers
Compute record
No linked compute attempts recorded.
Lineage and corrections
Must a closed manifold admitting a metric without conjugate points admit a metric of nonpositive curvature? parent of this event
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. parent of this event
This event attributed to Unreleased internal OpenAI model
Must a closed manifold admitting a metric without conjugate points admit a metric of nonpositive curvature? evidence for this event
Lean formalization (ConjugatePoints three-manifold) evidence for this event
VibeMathed record: Must a closed manifold admitting a metric without conjugate points admit a metric of nonpositive curvature? evidence for this event
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