Everything about Grassmannian totally explained
In
mathematics, a
Grassmannian is a space which parameterizes all
linear subspaces of a
vector space V of a given
dimension. For example, the Grassmannian
Gr1(
V) is the space of lines through the origin in
V, so it's the same as the
projective space PV. Grassmannians are named in honor of
Hermann Grassmann.
Motivation
By giving subspaces a
topological structure it's 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. Though such concepts may seem strangely out of place they can coincide with things that one is interested in, and can describe ideas that couldn't be considered otherwise—or at least describe them more economically.
A natural example comes from
tangent bundles of smooth manifolds embedded in
Euclidean space. Suppose we've a manifold
M of dimension
r embedded in
is a
Radon measure with respect to the metric space topology and is uniform in the sense that every ball of the same radius (with respect to this metric) is of the same measure.
Further Information
Get more info on 'Grassmannian'.
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