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In mathematics, the Grassmannian is a space which parameterizes all linear subspaces of a vector space of given dimension . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than . When is a real or complex vector space, Grassmannians are compact smooth manifolds.〔, pp. 57–59.〕 In general they have the structure of a smooth algebraic variety. The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of lines in projective 3-space and parameterized them by what are now called Plücker coordinates. Grassmannians are named after Hermann Grassmann, who introduced the concept in general. Notations vary between authors, with being equivalent to , and with some authors using or to denote the Grassmannian of -dimensional subspaces of an unspecified -dimensional vector space. ==Motivation== 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 of dimension embedded in . At each point in , the tangent space to can be considered as a subspace of the tangent space of , which is just . The map assigning to its tangent space defines a map from to . (In order to do this, we have to translate the geometrical tangent space to so that it passes through the origin rather than , and hence defines a -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 , so that every vector bundle generates a continuous map from 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. But the definition of homotopic relies on a notion of continuity, and hence a topology. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Grassmannian」の詳細全文を読む スポンサード リンク
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