Nil geometry
E888037
Thurston geometry
homogeneous geometry
left-invariant Riemannian geometry
non-Euclidean geometry
three-dimensional model geometry
Nil geometry is one of Thurston’s eight three-dimensional model geometries, characterized by a non-Euclidean, nilpotent Lie group structure that appears in the classification of 3-manifolds.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Nil geometry canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10807810 — 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: Nil geometry Context triple: [geometrization conjecture, hasCanonicalGeometry, Nil geometry]
-
A.
Geometry
Geometry is René Descartes’ foundational work that introduced analytic geometry, uniting algebra and Euclidean geometry through the use of coordinates.
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B.
Inventional Geometry
Inventional Geometry is an educational work by William George Spencer that introduces geometric concepts through intuitive, discovery-based learning rather than formal proofs.
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C.
Grundlagen der Geometrie
Grundlagen der Geometrie is David Hilbert’s foundational 1899 treatise that rigorously axiomatizes Euclidean geometry and helped shape modern mathematical logic and the axiomatic method.
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D.
Non-Euclidean Geometry
Non-Euclidean Geometry is a branch of mathematics that studies geometrical systems in which Euclid’s parallel postulate does not hold, leading to alternative models of space such as hyperbolic and elliptic geometry.
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E.
Mogul geometry library
Mogul geometry library is a crystallographic software tool developed by the Cambridge Crystallographic Data Centre for analyzing and validating molecular geometries using data from the Cambridge Structural Database.
- 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: Nil geometry Target entity description: Nil geometry is one of Thurston’s eight three-dimensional model geometries, characterized by a non-Euclidean, nilpotent Lie group structure that appears in the classification of 3-manifolds.
-
A.
Geometry
Geometry is René Descartes’ foundational work that introduced analytic geometry, uniting algebra and Euclidean geometry through the use of coordinates.
-
B.
Inventional Geometry
Inventional Geometry is an educational work by William George Spencer that introduces geometric concepts through intuitive, discovery-based learning rather than formal proofs.
-
C.
Grundlagen der Geometrie
Grundlagen der Geometrie is David Hilbert’s foundational 1899 treatise that rigorously axiomatizes Euclidean geometry and helped shape modern mathematical logic and the axiomatic method.
-
D.
Non-Euclidean Geometry
Non-Euclidean Geometry is a branch of mathematics that studies geometrical systems in which Euclid’s parallel postulate does not hold, leading to alternative models of space such as hyperbolic and elliptic geometry.
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E.
Mogul geometry library
Mogul geometry library is a crystallographic software tool developed by the Cambridge Crystallographic Data Centre for analyzing and validating molecular geometries using data from the Cambridge Structural Database.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Thurston geometry
ⓘ
homogeneous geometry ⓘ left-invariant Riemannian geometry ⓘ non-Euclidean geometry ⓘ three-dimensional model geometry ⓘ |
| admits |
Nil manifolds as quotients
ⓘ
compact quotients ⓘ left-invariant Riemannian metrics ⓘ |
| appearsIn | Thurston’s geometrization program NERFINISHED ⓘ |
| basedOn | Heisenberg group NERFINISHED ⓘ |
| contrastsWith |
Euclidean geometry E^3
NERFINISHED
ⓘ
Sol geometry ⓘ hyperbolic geometry H^3 ⓘ spherical geometry S^3 ⓘ |
| hasAlgebraicStructure | step-2 nilpotent Lie algebra ⓘ |
| hasBianchiType | Bianchi type II NERFINISHED ⓘ |
| hasCanonicalMetric | standard left-invariant metric on Heisenberg group ⓘ |
| hasCurvatureProperty |
non-constant sectional curvature
ⓘ
non-positive Ricci curvature in some directions ⓘ |
| hasDimension | 3 ⓘ |
| hasFundamentalExample | upper triangular 3x3 real matrices with ones on the diagonal ⓘ |
| hasGeodesicProperty | geodesics are not straight lines in coordinates ⓘ |
| hasGroupOperation | non-commutative group law on R^3 ⓘ |
| hasIsometryGroup | semidirect product of Heisenberg group with automorphisms preserving metric ⓘ |
| hasIsometryGroupProperty | acts transitively ⓘ |
| hasLieGroupProperty |
nilpotent
ⓘ
non-abelian ⓘ simply connected ⓘ |
| hasNameOrigin | named from nilpotent Lie group structure ⓘ |
| hasStructure | Lie group with left-invariant metric ⓘ |
| hasSymmetryType | anisotropic ⓘ |
| hasTopology | R^3 as underlying manifold ⓘ |
| hasTypicalQuotient |
Heisenberg nilmanifold
NERFINISHED
ⓘ
Nil manifold ⓘ |
| hasUnderlyingLieGroup | three-dimensional Heisenberg group NERFINISHED ⓘ |
| hasVolumeGrowth | polynomial volume growth ⓘ |
| isNot |
Euclidean geometry
NERFINISHED
ⓘ
hyperbolic geometry ⓘ space of constant curvature ⓘ spherical geometry ⓘ |
| isOneOf | Thurston’s eight geometries ⓘ |
| occursAsGeometryOf | some Seifert fibered spaces ⓘ |
| relatedTo | Seifert fibered 3-manifolds ⓘ |
| studiedIn |
3-manifold topology
ⓘ
Riemannian geometry NERFINISHED ⓘ geometric group theory ⓘ |
| usedIn | classification of 3-manifolds ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Nil geometry Description of subject: Nil geometry is one of Thurston’s eight three-dimensional model geometries, characterized by a non-Euclidean, nilpotent Lie group structure that appears in the classification of 3-manifolds.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.