Information AboutGrassmannian |
| CATEGORIES ABOUT GRASSMANNIAN | |
| differential geometry | |
| projective geometry | |
| algebraic homogeneous spaces | |
| algebraic geometry | |
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MOTIVATION By giving subspaces a Topological structure it is possibly 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. Though such concepts may seem strangely out of place they can coincide with things that one is interested in, and can describe ideas that could not be considered otherwise - or at least describe them more economically. A natural example comes from Tangent Bundle s of smooth manifolds embedded in Euclidean Space . Suppose we have a manifold ''M'' of dimension ''k'' embedded in . At each point ''x'' in ''M'', the tangent space to ''M'' can be considered as a subspace of the tangent space of , which is just . 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 geometrical tangent space to ''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 with maps that are Homotopic are isomorphic. But the definition of Homotopic relies on a notion of continuity, and hence a topology. These ideas are used in the definition of R-structures LOW DIMENSIONS When ''k'' = 2, the Grassmannian is the space of all planes through the origin. In Euclidean 3-space, a plane is completely characterized by the one and only line Perpendicular to it (and vice-versa); hence ''Gr''2(3) is isomorphic to ''Gr''1(3) (both of which are isomorphic to the Real Projective Plane ). THE STRUCTURE CARRIED BY A GRASSMANNIAN Grassmannians often carry a natural geometrical structure derived from ''V''. For example, when ''V'' is a real vector space the Grassmannian ''Gr''''k''(''n'') can be given the structure of a Smooth Manifold of dimension ''k''(''n'' − ''k''). For a fixed Field ''K'', we can consider for an ''n''-dimensional vector space ''V'', the set of subspaces with appropriate extra structure (e.g. a Topological Space , Homogeneous Space , Differential Manifold or Algebraic Variety ), and notice that Up To appropriate Isomorphism s, we have a Well-defined geometric object for the given pair (''n'',''k''). Supposing first that ''K'' is the Real Number or Complex Number field, the easiest approach to Grassmannians is probably to consider them as homogeneous spaces. That is, the Group Action of GL(''V'') on the ''k''-dimensional subspaces has a single orbit, as is shown in Linear Algebra . The stabilizer ''H'' of ''K''''k'' in ''K''''n'', embedded using the first ''k'' co-ordinates, can be identified quickly as the Block Matrices defined by the condition a (the upper right-hand block is 0). We can therefore identify ''Gr''''k''(''n'') as the coset space GL This then provides a Topological Space structure on the Grassmannian, and, more than that, a Smooth Structure . If ''V'' is an Inner Product Space , then the Grassmannian is in fact a Metric Space with the metric | ||
| For ''k''-dimensional Subspaces ''V'' And ''W'', Where ''P''<sub>''V''</sub> Is The | "http://wwwinformationdelightinfo/information/entry/orthogonal_projection" class="copylinks">Orthogonal Projection onto ''V'' and <math>\cdot</math> denotes the Operator Norm |
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