Grassmann manifolds

E911363

differentiable manifold homogeneous space mathematical concept parameter space

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.

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Observed surface forms (1)

Surface form	Occurrences
Grassmannian Gr(k,n)	1

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 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Characteristic Classes → hasSubject → Grassmann manifolds ⓘ

Plücker coordinates → relatedTo → Grassmann manifolds ⓘ

this entity surface form: Grassmannian Gr(k,n)