S^2 × R geometry
E888036
S² × R geometry is one of Thurston’s eight model geometries, describing 3-manifolds that locally look like the product of a 2-sphere with a line and have isometry groups reflecting this product structure.
All labels observed (1)
| Label | Occurrences |
|---|---|
| S^2 × R geometry canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10807808 — 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: S^2 × R geometry Context triple: [geometrization conjecture, hasCanonicalGeometry, S^2 × R geometry]
-
A.
Lorentzian geometry
Lorentzian geometry is the branch of differential geometry that studies manifolds equipped with metrics of Lorentzian signature, providing the mathematical framework for general relativity and spacetime physics.
-
B.
Möbius geometry
Möbius geometry is a branch of geometry that studies properties of figures invariant under Möbius (conformal) transformations of the extended complex plane or higher-dimensional spheres.
-
C.
Lie sphere geometry
Lie sphere geometry is a branch of differential geometry that studies the properties and transformations of spheres (and related objects like planes and points) using the methods of Lie groups and projective geometry.
-
D.
Weyl geometry
Weyl geometry is a generalization of Riemannian geometry that allows the length of vectors to vary under parallel transport, forming the geometric framework for Weyl’s original gauge theory.
-
E.
SL(2,R)
SL(2,R) is the Lie group of 2×2 real matrices with determinant 1, fundamental in representation theory, geometry, and the study of symmetries in mathematics and physics.
- 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: S^2 × R geometry Target entity description: S² × R geometry is one of Thurston’s eight model geometries, describing 3-manifolds that locally look like the product of a 2-sphere with a line and have isometry groups reflecting this product structure.
-
A.
Lorentzian geometry
Lorentzian geometry is the branch of differential geometry that studies manifolds equipped with metrics of Lorentzian signature, providing the mathematical framework for general relativity and spacetime physics.
-
B.
Möbius geometry
Möbius geometry is a branch of geometry that studies properties of figures invariant under Möbius (conformal) transformations of the extended complex plane or higher-dimensional spheres.
-
C.
Lie sphere geometry
Lie sphere geometry is a branch of differential geometry that studies the properties and transformations of spheres (and related objects like planes and points) using the methods of Lie groups and projective geometry.
-
D.
Weyl geometry
Weyl geometry is a generalization of Riemannian geometry that allows the length of vectors to vary under parallel transport, forming the geometric framework for Weyl’s original gauge theory.
-
E.
SL(2,R)
SL(2,R) is the Lie group of 2×2 real matrices with determinant 1, fundamental in representation theory, geometry, and the study of symmetries in mathematics and physics.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
3-dimensional geometry
ⓘ
Thurston model geometry ⓘ |
| admitsCompactQuotients | true ⓘ |
| appearsIn |
Thurston’s eight geometries
ⓘ
linked to:
geometrization conjecture
|
| baseSpace | 2-sphere ⓘ |
| classifiedBy | William Thurston ⓘ |
| dimension | 3 ⓘ |
| fiber | real line ⓘ |
| fundamentalGroupOfCompactQuotient | extension of Z by finite group ⓘ |
| hasCanonicalProjection |
projection to R factor
ⓘ
projection to S^2 factor ⓘ |
| hasConstantCurvature | no ⓘ |
| hasCurvatureInRDirection | zero ⓘ |
| hasCurvatureInS2Direction | positive ⓘ |
| hasFactor |
R
ⓘ
S^2 ⓘ |
| hasFiniteVolumeCompactModels | true ⓘ |
| hasModelMetric | product of round metric on S^2 and Euclidean metric on R ⓘ |
| hasOrientationReversingIsometries | true ⓘ |
| hasSymmetryType | product of spherical and Euclidean symmetries ⓘ |
| hasUnderlyingManifold | S^2 × R ⓘ |
| isDistinctFrom |
E^3 geometry
ⓘ
H^2 × R geometry ⓘ S^3 geometry ⓘ |
| isGeodesicallyComplete | true ⓘ |
| isHomogeneous | true ⓘ |
| isIsotropic | false ⓘ |
| isLocallyIsometricTo | S^2 × R with product metric ⓘ |
| isNotSimplyConnectedCompactly | compact quotients have infinite fundamental group ⓘ |
| isometryGroup | Isom(S^2) × Isom(R) ⓘ |
| isometryGroupContains |
O(3)
ⓘ
R (translations) ⓘ SO(3) ⓘ
linked to:
rotation group SO(3)
|
| isotropyInRDirection | translational symmetry ⓘ |
| isotropyInS2Direction | full rotational symmetry ⓘ |
| isProductGeometry | true ⓘ |
| isProductOfConstantCurvatureSpaces | true ⓘ |
| isUniversalCoverOf | geometric S^2 × R 3-manifolds ⓘ |
| isUsedIn | geometrization of 3-manifolds ⓘ |
| localModelFor | 3-manifolds locally isometric to S^2 × R ⓘ |
| occursAsGeometryOf | Seifert fibered spaces with spherical base and zero Euler number ⓘ |
| relatedTo | Seifert fibered spaces ⓘ |
| sectionalCurvatureInPlanesContainingRDirection | nonpositive or zero depending on metric choice ⓘ |
| sectionalCurvatureInS2Planes | constant positive ⓘ |
| supports | Riemannian product structure ⓘ |
| supportsGeometricStructureOn | certain 3-manifolds ⓘ |
| typicalCompactQuotient |
S^2-bundle over S^1
ⓘ
twisted S^2-bundle over S^1 ⓘ |
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: S^2 × R geometry Description of subject: S² × R geometry is one of Thurston’s eight model geometries, describing 3-manifolds that locally look like the product of a 2-sphere with a line and have isometry groups reflecting this product structure.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.