Grassmann manifolds
E911363
Grassmann manifolds are smooth parameter spaces that classify all k-dimensional linear subspaces of an n-dimensional vector space and serve as fundamental objects in topology, geometry, and the study of characteristic classes.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Grassmann manifolds canonical | 1 |
| Grassmannian Gr(k,n) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11219608 — 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: Grassmann manifolds Context triple: [Characteristic Classes, hasSubject, Grassmann manifolds]
-
A.
theory of G-structures
The theory of G-structures is a framework in differential geometry that studies geometric structures on manifolds defined by reductions of the frame bundle to a Lie group G, encompassing and unifying many classical geometries such as Riemannian, symplectic, and complex structures.
-
B.
Riemannian manifolds
Riemannian manifolds are smooth manifolds equipped with an inner product on each tangent space that allows one to measure lengths, angles, and curvature in a curved geometric setting.
-
C.
Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
-
D.
Weingarten map
The Weingarten map is a differential geometric operator on a surface that encodes how the surface’s normal vector field changes, thereby describing the surface’s extrinsic curvature.
-
E.
Fubini–Study form
The Fubini–Study form is the canonical Kähler form on complex projective space, encoding its standard Hermitian and symplectic geometry.
- 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: Grassmann manifolds Target entity description: Grassmann manifolds are smooth parameter spaces that classify all k-dimensional linear subspaces of an n-dimensional vector space and serve as fundamental objects in topology, geometry, and the study of characteristic classes.
-
A.
theory of G-structures
The theory of G-structures is a framework in differential geometry that studies geometric structures on manifolds defined by reductions of the frame bundle to a Lie group G, encompassing and unifying many classical geometries such as Riemannian, symplectic, and complex structures.
-
B.
Riemannian manifolds
Riemannian manifolds are smooth manifolds equipped with an inner product on each tangent space that allows one to measure lengths, angles, and curvature in a curved geometric setting.
-
C.
Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
-
D.
Weingarten map
The Weingarten map is a differential geometric operator on a surface that encodes how the surface’s normal vector field changes, thereby describing the surface’s extrinsic curvature.
-
E.
Fubini–Study form
The Fubini–Study form is the canonical Kähler form on complex projective space, encoding its standard Hermitian and symplectic geometry.
- F. None of above. chosen
Statements (65)
| Predicate | Object |
|---|---|
| instanceOf |
differentiable manifold
ⓘ
homogeneous space ⓘ mathematical concept ⓘ parameter space ⓘ |
| algebraicDescription | closed subvariety of projective space via Plücker embedding ⓘ |
| appearsIn |
K-theory
NERFINISHED
ⓘ
gauge theory ⓘ index theory ⓘ moduli problems in geometry ⓘ |
| classifies | k-dimensional linear subspaces of an n-dimensional vector space ⓘ |
| cohomologyRing |
generated by Chern classes of the universal bundle (complex case)
ⓘ
generated by Stiefel–Whitney or Pontryagin classes (real case) ⓘ |
| dimensionFormula | k(n − k) ⓘ |
| field |
algebraic geometry
ⓘ
algebraic topology ⓘ complex geometry ⓘ differential geometry ⓘ homotopy theory ⓘ representation theory ⓘ symplectic geometry ⓘ |
| generalizationOf |
projective space
ⓘ
space of lines in R^n ⓘ |
| hasBundle |
universal k-plane bundle
ⓘ
universal quotient bundle ⓘ |
| hasCoordinateSystem | Plücker coordinates NERFINISHED ⓘ |
| hasInvariantMetric | metric induced from Killing form on Lie group ⓘ |
| hasStructure |
Riemannian manifold
ⓘ
compact manifold ⓘ complex manifold ⓘ homogeneous space of a Lie group ⓘ projective variety ⓘ real manifold ⓘ smooth manifold ⓘ |
| hasSymmetry |
transitive action of GL(n)
ⓘ
transitive action of O(n) ⓘ transitive action of U(n) ⓘ |
| isHomogeneousSpaceOf |
general linear group
NERFINISHED
ⓘ
special orthogonal group NERFINISHED ⓘ unitary group ⓘ |
| namedAfter | Hermann Grassmann NERFINISHED ⓘ |
| notation |
G(k,n)
ⓘ
Gr(k,n) ⓘ |
| parameterizes |
k-dimensional subspaces of C^n
ⓘ
k-dimensional subspaces of R^n ⓘ |
| quotientDescription |
Gr(k,n) ≅ GL(n)/P where P is a parabolic subgroup
ⓘ
Gr(k,n) ≅ O(n)/(O(k) × O(n − k)) ⓘ Gr(k,n) ≅ U(n)/(U(k) × U(n − k)) ⓘ |
| relatedConcept |
Plücker embedding
NERFINISHED
ⓘ
Schubert calculus NERFINISHED ⓘ Schubert variety ⓘ Stiefel manifold NERFINISHED ⓘ flag manifold ⓘ projective space ⓘ |
| specialCase |
Gr(1,n) is projective space P^{n−1}
ⓘ
Gr(k,n) is diffeomorphic to Gr(n−k,n) ⓘ Gr(n−1,n) is projective space P^{n−1} ⓘ |
| studiedIn | 19th century ⓘ |
| topologicalProperty |
compact
ⓘ
connected ⓘ simply connected for complex Grassmannians ⓘ |
| usedFor |
classification of vector bundles up to isomorphism
ⓘ
computing cohomology rings via Schubert calculus ⓘ construction of universal bundles ⓘ definition of characteristic classes ⓘ modeling classifying spaces for vector bundles ⓘ |
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: Grassmann manifolds Description of subject: Grassmann manifolds are smooth parameter spaces that classify all k-dimensional linear subspaces of an n-dimensional vector space and serve as fundamental objects in topology, geometry, and the study of characteristic classes.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Grassmannian Gr(k,n)