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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Lie sphere group canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6282373 — 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: Lie sphere group Context triple: [Lie sphere geometry, usesGroup, Lie sphere group]
-
A.
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.
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B.
Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
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C.
Lie group
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
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D.
Lie pseudogroup
A Lie pseudogroup is a collection of local diffeomorphisms on a manifold that is closed under composition, inversion, and restriction, generalizing the concept of a Lie group to transformations defined only locally.
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E.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
- 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: Lie sphere group Target entity description: 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.
-
A.
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.
-
B.
Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
-
C.
Lie group
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
-
D.
Lie pseudogroup
A Lie pseudogroup is a collection of local diffeomorphisms on a manifold that is closed under composition, inversion, and restriction, generalizing the concept of a Lie group to transformations defined only locally.
-
E.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
- F. None of above. chosen
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
ⓘ
linked to:
Lie sphere geometry
|
| 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
ⓘ
linked to:
Möbius geometry
|
| namedAfter | Sophus Lie ⓘ |
| 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
ⓘ
linked to:
Lie sphere geometry
Möbius group ⓘ
linked to:
PSL(2,\mathbb{C})
conformal geometry ⓘ contact geometry ⓘ cyclidic geometry ⓘ orthogonal group O(n+1,2) ⓘ projective geometry ⓘ |
| usedIn |
integrable systems in differential geometry
ⓘ
theory of Dupin cyclides ⓘ theory of sphere congruences ⓘ |
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: Lie sphere group Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.