Three-manifolds with positive Ricci curvature
E888034
"Three-manifolds with positive Ricci curvature" is a landmark 1982 paper by Richard S. Hamilton that introduced the Ricci flow and launched the modern geometric analysis approach to understanding the topology of three-dimensional manifolds.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Three-manifolds with positive Ricci curvature canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10807773 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Three-manifolds with positive Ricci curvature Context triple: [Richard S. Hamilton, notableWork, Three-manifolds with positive Ricci curvature]
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A.
"Finite extinction time for the solutions to the Ricci flow on certain three-manifolds"
"Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" is a landmark mathematical paper by Grigori Perelman that advances the analysis of Ricci flow in three dimensions and plays a key role in his proof of the Poincaré conjecture.
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B.
The geometry of four-manifolds
The Geometry of Four-Manifolds is a foundational monograph in differential geometry that develops the theory of smooth four-dimensional manifolds using gauge theory and Yang–Mills instantons.
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C.
Hopf conjecture (on Euler characteristic and curvature)
The Hopf conjecture on Euler characteristic and curvature is an open problem in differential geometry proposing a deep link between the sign of a manifold’s Euler characteristic and the sign of its sectional curvature, especially for even-dimensional manifolds with positive or negative curvature.
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D.
Hyperbolic Manifolds and Discrete Groups
"Hyperbolic Manifolds and Discrete Groups" is a foundational mathematical monograph that develops the theory of hyperbolic geometry and its deep connections with discrete group actions and low-dimensional topology.
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E.
Nirenberg problem in differential geometry
The Nirenberg problem in differential geometry is a classical question about prescribing Gaussian curvature on the 2-sphere via conformal deformations of the metric.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Three-manifolds with positive Ricci curvature Target entity description: "Three-manifolds with positive Ricci curvature" is a landmark 1982 paper by Richard S. Hamilton that introduced the Ricci flow and launched the modern geometric analysis approach to understanding the topology of three-dimensional manifolds.
-
A.
"Finite extinction time for the solutions to the Ricci flow on certain three-manifolds"
"Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" is a landmark mathematical paper by Grigori Perelman that advances the analysis of Ricci flow in three dimensions and plays a key role in his proof of the Poincaré conjecture.
-
B.
The geometry of four-manifolds
The Geometry of Four-Manifolds is a foundational monograph in differential geometry that develops the theory of smooth four-dimensional manifolds using gauge theory and Yang–Mills instantons.
-
C.
Hopf conjecture (on Euler characteristic and curvature)
The Hopf conjecture on Euler characteristic and curvature is an open problem in differential geometry proposing a deep link between the sign of a manifold’s Euler characteristic and the sign of its sectional curvature, especially for even-dimensional manifolds with positive or negative curvature.
-
D.
Hyperbolic Manifolds and Discrete Groups
"Hyperbolic Manifolds and Discrete Groups" is a foundational mathematical monograph that develops the theory of hyperbolic geometry and its deep connections with discrete group actions and low-dimensional topology.
-
E.
Nirenberg problem in differential geometry
The Nirenberg problem in differential geometry is a classical question about prescribing Gaussian curvature on the 2-sphere via conformal deformations of the metric.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
landmark paper in geometric analysis
ⓘ
mathematics research paper ⓘ |
| assumes | positive lower bound on Ricci curvature ⓘ |
| author | Richard S. Hamilton ⓘ |
| citedFor |
classification results for positively Ricci curved 3-manifolds
ⓘ
introduction and analysis of Ricci flow on 3-manifolds ⓘ |
| contribution |
applied Ricci flow to study topology of 3-manifolds
ⓘ
launched modern geometric analysis approach to 3-manifold topology ⓘ |
| establishes | long-time behavior of Ricci flow in dimension three under curvature conditions ⓘ |
| field |
Riemannian geometry
ⓘ
linked to:
Riemannian manifolds
differential geometry ⓘ geometric analysis ⓘ |
| focusesOn | closed three-manifolds with positive Ricci curvature ⓘ |
| hasAuthorInitials | R. S. Hamilton ⓘ |
| hasCurvatureCondition | positive Ricci curvature ⓘ |
| hasDimensionFocus | 3 ⓘ |
| historicalSignificance | first systematic use of Ricci flow in Riemannian geometry ⓘ |
| influenced |
Perelman’s work on the Poincaré conjecture
ⓘ
development of Ricci flow with surgery ⓘ |
| influencedField |
3-manifold topology
ⓘ
geometric evolution equations ⓘ global Riemannian geometry ⓘ |
| introducesConcept | Ricci flow ⓘ |
| isConsidered |
foundational work in Ricci flow theory
ⓘ
starting point of modern program to classify 3-manifolds via geometric flows ⓘ |
| language | English ⓘ |
| mainTopic |
Ricci flow
ⓘ
positive Ricci curvature ⓘ three-dimensional manifolds ⓘ |
| mathematicalSubject |
curvature conditions on manifolds
ⓘ
topology of 3-manifolds ⓘ |
| publicationYear | 1982 ⓘ |
| relatedToConcept |
curvature pinching
ⓘ
normalized Ricci flow ⓘ
linked to:
Ricci flow
spherical space forms ⓘ |
| relatedToConjecture | Poincaré conjecture ⓘ |
| result | proved that certain 3-manifolds with positive Ricci curvature are diffeomorphic to spherical space forms ⓘ |
| studies | behavior of curvature under Ricci flow ⓘ |
| studiesObject | Riemannian 3-manifolds ⓘ |
| usesMethod |
evolution equation for Riemannian metrics
ⓘ
maximum principle techniques ⓘ parabolic partial differential equations ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Three-manifolds with positive Ricci curvature Description of subject: "Three-manifolds with positive Ricci curvature" is a landmark 1982 paper by Richard S. Hamilton that introduced the Ricci flow and launched the modern geometric analysis approach to understanding the topology of three-dimensional manifolds.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.