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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.