Grassmannian - Wikipedia
In mathematics, the Grassmannian (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all - dimensional linear subspaces of an -dimensional vector space over a field that has a differentiable structure.
1. Basic properties of the Grassmannian The Grassmannian can be defined for a vector space over any field; the cohomology of the Grassmannian is the best understood for the complex case, and this is our focus. Following [MS], the complex Grass-mannian G n(Cm+n) is the set of n-dimensional complex linear spaces,
We define a topology on Gr(m, n) be declaring the a subset U ⊂ Gr(m, n) is open if and only if π−1(U) is open in FR(n, m), and further declare that a function f : U → k is regular if and only if f π is a regular function on π−1(U). This gives Gr(m, n) the structure of. a …
The Grassmannian Gn(Rk) is the manifold of n-planes in Rk. As a set it consists of all n-dimensional subspaces of Rk. To describe it in more detail we must first define the Steifel manifold. Definition 2.1. The Stiefel manifold Vn(Rk) is the set of orthogonal n-frames of Rk. Thus the points of it are n-tuples of orthonormal vectors in Rk.
1. The Grassmannian Grassmannians are the prototypical examples of homogeneous varieties and pa-rameter spaces. Many of the constructions in the theory are motivated by analogous constructions for Grassmannians, hence we will develop the theory for the Grass-mannian in detail. In this section, we begin by reviewing the basic facts about ...
invertible matrix on the left. In other words the Grassmannian is the set of equivalence classes of k nmatrices under the action of GL k(K) by multiplication on the left. It is not hard to connect the two constructions. Given the matrix A, then form all possible k …
What is the topology on the Grassmannian …
Therefore, we can define a topology on the Grassmannian as the strong topology generated by these maps $\mathbb{R}^{kn} \to G_n(\mathbb{R}^{n+k})$, i.e. $U \subseteq G_n(\mathbb{R}^{n+k})$ is open if and only if the corresponding subsets of …
Grassmannian -- from Wolfram MathWorld
2024年12月18日 · The Grassmannian Gr(n,k) is the set of k-dimensional subspaces in an n-dimensional vector space. For example, the set of lines Gr(n+1,1) is projective space. The real Grassmannian (as well as the complex Grassmannian) are examples of manifolds.
The complex Grassmannian Grk(Cn) is the set of complex k-dimensional linear subspaces of Cn. It is a com-pact complex manifold of dimension k(n k) and it is a homogeneous space of the unitary group, given by U(n)=(U(k) U(n k)).
The Real Grassmannian Gr(2;4) We discuss the topology of the real Grassmannian Gr(2;4) of 2-planes in R 4 and its double cover Gr + (2;4) by the Grassmannian of oriented 2-planes.