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.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
3-dimensional geometry
ⓘ
Thurston model geometry ⓘ |
| admitsCompactQuotients | true ⓘ |
| appearsIn | Thurston’s eight geometries NERFINISHED ⓘ |
| baseSpace | 2-sphere ⓘ |
| classifiedBy | William Thurston NERFINISHED ⓘ |
| dimension | 3 ⓘ |
| fiber | real line ⓘ |
| fundamentalGroupOfCompactQuotient | extension of Z by finite group ⓘ |
| hasCanonicalProjection |
projection to R factor
ⓘ
projection to S^2 factor ⓘ |
| hasConstantCurvature | no GENERATED ⓘ |
| 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) NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| sectionalCurvatureInPlanesContainingRDirection | nonpositive or zero depending on metric choice ⓘ |
| sectionalCurvatureInS2Planes | constant positive ⓘ |
| supports | Riemannian product structure ⓘ |
| supportsGeometricStructureOn | certain 3-manifolds GENERATED ⓘ |
| typicalCompactQuotient |
S^2-bundle over S^1
GENERATED
ⓘ
twisted S^2-bundle over S^1 GENERATED ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.