Lie sphere group
E581258
The Lie sphere group is the continuous symmetry group that preserves the incidence and contact relations of spheres, planes, and points in Lie sphere geometry.
Statements (40)
| Predicate | Object |
|---|---|
| instanceOf | mathematical group ⓘ |
| actsOn |
space of oriented planes
ⓘ
space of oriented spheres ⓘ space of points in Euclidean space ⓘ |
| appearsIn |
classical differential geometry of surfaces
ⓘ
modern geometric analysis ⓘ |
| characterizedBy | preservation of oriented contact between hyperspheres ⓘ |
| definedOn | space of contact elements of Euclidean space ⓘ |
| definedVia | orthogonal transformations of a space with signature (n+1,2) ⓘ |
| describedAs | group of transformations preserving Lie contact structure ⓘ |
| dimension | (n+2)(n+3)/2 for n-dimensional Euclidean space ⓘ |
| field |
Lie sphere geometry
ⓘ
differential geometry ⓘ |
| generalizes | conformal group of the sphere ⓘ |
| hasConcept | Lie sphere transformations NERFINISHED ⓘ |
| hasProperty |
acts by projective transformations on Lie quadric
ⓘ
continuous symmetry group ⓘ encodes geometry of oriented spheres as null lines ⓘ non-compact (for standard signatures) ⓘ real Lie group ⓘ transitive on oriented contact elements ⓘ |
| hasStructure | Lie algebra isomorphic to so(n+1,2) ⓘ |
| hasSubgroup | Möbius group of the n-sphere NERFINISHED ⓘ |
| namedAfter | Sophus Lie NERFINISHED ⓘ |
| preserves |
Lie quadric in projective space
ⓘ
contact relations of spheres ⓘ incidence relations of spheres ⓘ oriented contact between planes ⓘ oriented contact between points and spheres ⓘ oriented contact between spheres ⓘ |
| relatedTo |
Laguerre geometry
NERFINISHED
ⓘ
Möbius group NERFINISHED ⓘ conformal geometry ⓘ contact geometry ⓘ cyclidic geometry ⓘ orthogonal group O(n+1,2) NERFINISHED ⓘ projective geometry ⓘ |
| usedIn |
integrable systems in differential geometry
ⓘ
theory of Dupin cyclides ⓘ theory of sphere congruences ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.