Grassmannian

In mathematics, the Grassmannian Gr(k, V) is a space which parametrizes 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. In mathematics, the Grassmannian Gr(k, V) is a space which parametrizes 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 dimension k ( n − k ) . {displaystyle k(n-k).} 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 vary between authors, with G r k ( V ) {displaystyle Gr_{k}(V)} being equivalent to Gr(k, V), and some authors using G r k ( n ) {displaystyle Gr_{k}(n)} or Gr(k, n) to denote the Grassmannian of k-dimensional subspaces of an unspecified n-dimensional vector space. 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. We then find that the properties of our vector bundles are related to the properties of the corresponding maps viewed as continuous maps. In particular we find that vector bundles inducing homotopic maps to the Grassmannian are isomorphic. Here the definition of homotopic relies on a notion of continuity, and hence a topology. For k = 1, the Grassmannian Gr(1, n) is the space of lines through the origin in n-space, so it is the same as the projective space of n−1 dimensions. For k = 2, the Grassmannian is the space of all 2-dimensional planes containing the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the one and only line through the origin that is perpendicular to that plane (and vice versa); hence the spaces Gr(2, 3), Gr(1, 3), and P2 (the projective plane) may all be identified with each other.