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Concept
Grassmannian
Summary
In mathematics, the Grassmannian Gr(k, V) is a space that parameterizes all k-dimensional linear subspaces of the n-dimensional vector space V. For example, the Grassmannian Gr(1, V) is the space of lines through the origin in V, so it is the same as the projective space of one dimension lower than V.
When V is a real or complex vector space, Grassmannians are compact smooth manifolds. In general they have the structure of a smooth algebraic variety, of dimensions
The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in projective 3-space, equivalent to Gr(2, R4) and parameterized them by what are now called Plücker coordinates. Hermann Grassmann later introduced the concept in general.
Notations for the Grassmannian vary between authors; notations include Gr_k(V), Gr(k, V), Gr_k(n), or Gr(k, n) to denote the Grassmannian of k-dimensional subspaces of an n-dimensional vector space V.
By giving a collection of subspaces of some vector space a topological structure, it is possible to talk about a continuous choice of subspace or open and closed collections of subspaces; by giving them the structure of a differential manifold one can talk about smooth choices of subspace.
A natural example comes from tangent bundles of smooth manifolds embedded in Euclidean space. Suppose we have a manifold M of dimension k embedded in Rn. At each point x in M, the tangent space to M can be considered as a subspace of the tangent space of Rn, which is just Rn. The map assigning to x its tangent space defines a map from M to Gr(k, n). (In order to do this, we have to translate the tangent space at each x ∈ M so that it passes through the origin rather than x, and hence defines a k-dimensional vector subspace. This idea is very similar to the Gauss map for surfaces in a 3-dimensional space.)
This idea can with some effort be extended to all vector bundles over a manifold M, so that every vector bundle generates a continuous map from M to a suitably generalised Grassmannian—although various embedding theorems must be proved to show this.
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