In mathematics, the Grassmannian $\mathbf{Gr}(r, V)$ is a space which parameterizes all linear subspaces of a vector space $V$ of given dimension $r$.

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Defining a metric in the tangent spaces $ T_xM $

I'm working with a metric $D$ over the manifold of grassman $G_n(\mathbb{R}^{d})$ and have difficulties to extend $D$. Let me explain: If $M$ is a submanifold of dimension $n$ in $\mathbb{R}^{d}$ ...
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Hermitian metric on line bundle over the Grassmannian

We know that the Grassmannian manifold $G(k,\mathbb{C}^n)$ can be embedded in the projective space $\mathbb{C}P^N$ for $N= {n\choose k}-1$ by the Plucker embedding $P$. On $\mathbb{C}P^N$ we have ...
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Classification of rank $\geq 2$vector bundles over Grassmanians

Are there classification results of higher rank (complex) vector bundles over (complex) Grassmanian manifolds? For example, we know that line bundles are in correspondence with the $H^2(G)$, the ...
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What is the dimension of the space of planes in $\Bbb R^3$?

What is the dimension of the space of planes in $\Bbb R^3$ and how do we reach the answer? Clarification: What I am searching for is what is the least number of parameters that I need. For example, ...
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44 views

How does the multiplicative group of a finite field, considered as a vector space, act on subspaces?

Given that a finite vector space $V = \operatorname{GF}(p)^n$ corresponds to the finite field $F = \operatorname{GF}(p^n)$, I'm wondering about how the multiplicative subgroup of $F$, $F^*$, acts on ...
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Complexity of finding M nodes in a graph to maximize the pairwise minimum distance between nodes

I want to know the complexity of finding a set of M nodes, $\{U_1,\dots,U_M\}$, in a given graph $G$, to maximize $d(U_i,U_j)$ over all pairs $i\neq j$, where $d(\cdot,\cdot)$ is the length of the ...
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The last coordinates of basis vectors are a chart: mistake in this example?

While trying to understand local chart on Grassmannians I came across this example in this book: Take $V = \mathbb R^2$ and $U,W$ two subspaces generated by linearly independent vectors. The books ...
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The last coordinates of basis vectors are a chart

Let $Gr$ denote the Grassmannian and let $Gr(2, T\mathbb R^3) = \bigcup_{x \in T\mathbb R^3} Gr(2, T_x \mathbb R^3)$. Consider one $2$-dimensional subspace of $\mathbb R^3$, that is, one element of ...
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Question about vectors in the Grassmannian in this example

Consider $f: \mathbb R^2 \to \mathbb R^3$ defined by $(t,s) \mapsto (t^2 + 2s, t^3 + 3ts, t^4 + 4t^2 s)$. Let $Gr$ denote the Grassmannian and let $Gr(2, T\mathbb R^3) = \bigcup_{x \in T\mathbb R^3} ...
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Dimensions of Grassmannians?

I'm trying to work out the dimensions of some examples of Grassmannians but I can't seem to do it. Here is what I understand: The Grassmanian $G(k,n)$ is the set of all $k$ planes in $\mathbb ...
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Looking For a Neat Proof of the Fact that the Grassmannian Manifold is Hausdorff

$\newcommand{\R}{\mathbf R}$ Let $V$ be an $n$-dimensional vector space and $k$ be an integer less than $n$. A $k$-frame in $V$ is an injective linear map $T:\R^k\to V$. Let the set of all the ...
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A Continuous Choice of $k$-Subspaces of a Vector Space Gives a Continuous Choice of Bases

$\newcommand{\R}{\mathbf R}$ The Grassmannian $G_k(\R^n)$ as a topoplogical space is defined in the following way: Let $F_k(\R^n)$ be the collection of all the linearly independent lists of size $k$ ...
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Proving that certain incidence correspondence is a projective variety.

Let $M$ be the projective space of nonzero $m\times n$ matrices up to scalars (in $\mathbb{K}$). In Joe Harris' Algebraic Geometry: A first course, in order to find the dimension of $M_{k}=\{A\in ...
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Inclusionwise maximal linear subvarieties of a projective variety

Let $X\subseteq\mathbb P^n$ be a complex, projective variety. A linear subspace $L\subseteq\mathbb P^n$ will be called a maximal linear subspace of $X$ if $L\subseteq X$ and for any linear subspace ...
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Diameter of the Grassmannian

Just an interesting question that came to my mind while studying(!): Since the Grassmannian $G(k,\mathbb{C}^n)$ is a compact manifold, what do we know about its diameter? Do we know any estimate? ...
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Motivation for Grassmannian variety

I need some information about the Grassmanian variety for my final project in algebraic geometry course that I am taking. My questions are: Why do we define the Grassmannian variety? Do we use ...
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Ricci curvature of the Grassmannian?

Let $G(k, \mathbb{C}^n)$ be the Grassmannian of $k-$dimensional complex linear subspaces of $\mathbb{C}^n.$ We know that the Grassmannian can be embedded to the projective space ...
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When is a wedge decomposible?

Shaferevich (in AG book 1) tells me that a wedge $x \in \bigwedge ^r V$ is decompsible iff $(y . x) \wedge x = 0$ for all $y \in \bigwedge^{r-1} V$, where $.$ is the convolution pairing. I see in ...
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Rules for integrating a Grassmann quantity

So I've read that for Grassmann numbers, integration is the same as differentiation. Indeed, on Wikipedia, integration of Grassmann quantities are defined so this is true: $$\int 1 \, d \theta = 0$$ ...
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Universal Property of the Universal Line

In "An Invitation to Quantum Cohomology" by Kock and Vainsencher, they talk about "the universal line", which is defined as the variety $U=\{ (L,p)\in Gr(1,\mathbb{P}^r)\times \mathbb{P}^r | p\in L ...
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Grassmannian a Fano manifold?

I am interested in answering the following: Is it true that the Grassmannian $G(k,\mathbb{C}^n)$ is a Fano manifold? How can I see if its anticanonical bundle is ample?
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Is the determinant bundle the pullback of the $\mathcal O(1)$ on $\mathbb P^n$ under the Plucker embedding?

Let $V$ be a $n$-dimensional complex vector space and consider the Grassmannian of complex $k$-planes $Gr(k,V)$. The Plucker embedding is an embedding $p:Gr(k,V) \to \mathbb P^M$ where $M = ...
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Is it possible to continuously choose one-dimensional subspace in each k-dimensional subspace?

Does there exist a continuous map from Grassmann manifold to projective space $Gr^n(V) \to \mathbb P(V)$, such that image of every n-dimensional subspace lies (1-dimensional subspace) in this ...
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Path-connectedness of Grassmannian

Let $\mathrm{Gr}(k,X^n)$ be Grassmannian. How to prove that $\mathrm{Gr}(k,X^n)$ is path-connected (without using homology theorem) ?
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How to think Grassmannian as a projective variety?

I'm just looking for some explanation for the grassmannian as a projective variety and plücker embedding.
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Corresponding a vector subspace to a point of the space.

In this article http://en.wikipedia.org/wiki/Tautological_bundle we read: Grassmannians by definition are the parameter spaces for linear subspaces, of a given dimension, in a given vector ...
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A question on Grassmannian manifolds and universal line bundles

Problem. Fix an $ n \in \mathbb{N} $. Is it true that there exists a $ k \in \mathbb{N} $ such that every smooth complex line bundle over a smooth $ n $-dimensional real manifold is the pullback of ...
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Hypermatrices, hyperdeterminants and Grassmannians.

Let $Gr(k,n)$ the Grassmannian manifold of the $k$-planes in $\mathbb{C}^n$ and consider the Plucker embedding $\pi: Gr(k,n) \to \mathbb{P}(\Lambda^k \mathbb{C}^n)$. Let $A$ be the set of $n \times n$ ...
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Is the Cone over Grassmannian manifold a determinantal variety?

Let consider the Grassmann manifold $Gr(k,n)$ in the Plucker embedding and the Cone over $Gr(k,n)$, say $C(Gr(k,n))$. On the other hand consider $M$ the set of $n \times n$ skew-symmetric matrices. ...
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The space of minimal geodesics on $SU(2m)$

In the proof of Bott periodicity for the unitary group in Milnor's Morse theory (Lemma 23.1, page 128), it is asserted that the space of minimal geodesics from $I$ to $-I$ in the special unitary group ...
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Does the Tangent Space Vary Continuously with The Points On a Manifold?

I recently read about Grassmannian manifolds. The following question naturally comes to mind. Let $GR_k(\mathbf R^n)$ is the grassmannian manifold of $k$ dimensional linear subspaces of $\mathbf ...
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cohomology of complex grassmannians and quaternion grassmannians (the finite dimensional case)

Let $G_k(\mathbb{C}^N)$ be the complex grassmannian manifold. Let $G_k(\mathbb{H}^N)$ be the quaternion grassmannian manifold. Let $p$ be a prime integer (we may also impose extra conditions, such as ...
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What is the principal bundle structure of $O(n)$?

Consider the map $\pi:O(n)\rightarrow G(k,n)$ which maps $A\in O(n)$ to the subspace of $\mathtt{R}^n$ spanned by the first $k$ columns of $A$. Here $G(k,n)$ is the Grassmannian manifold. My question ...
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Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
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A question about a universal family over a Grassmannian.

I refer to this paper on Moduli Spaces by Ravi Vakil. I am uploading a screenshot: What can possibly be a universal family over $G(k,n)$? For example, let us take the set of all linear subspaces ...
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Find ideal defining $Gr_2(\mathbb{C}^5)$ in Pluker embedding

Let $Gr_k(\mathbb{C}^n)$ the Grassmannian variety of $k$-planes in the complex space $\mathbb{C}^n$. We can consider the Pluker embedding $$ \mathcal{P}: Gr_k(\mathbb{C}^n) \to \mathbb{P}(\Lambda^k ...
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Finding Grassmann coefficients

If we for example represent the Grassmann variables as $4\times4$ matrices, then is there some procedure to find the matrices $f_i$, so that we will get the following decomposition of some given ...
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Stabilizer of an element in the Grassmannian

Let $Gr_{n,k} = \{W\leq \mathbb{C}^n:\mbox{dim(W)}=k \}$ be the set of $k$ dimensional subspaces of $\mathbb{C}^n$. We have a group action: $$GL_n(\mathbb{C})\times Gr_{n,k} \to Gr_{n,k} \ , \ ...
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Identities for the Hilbert–Schmidt norm of products of projections.

I've been studying different metrics on the Grassmannian $Gr(k,n)$ of k-dimensional linear subspaces of $\mathbb{R}^n$ and found myself needing some identities for the norm of a product of orthogonal ...
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82 views

Is a ruled surface of degree>2 always singular?

Let $X=\mathbb{C}\mathbb{P}^3$ and let $V\subset X$ be a closed algebraic sub variety. By V-is ruled, I mean that for every point in $V$ there is a line passing through it which also lies in V. ...
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Isomorphism between projective space and grassmanian $\mathrm{Grass}(1,n+1)$

I have to show that the projective space $\mathbb{P}^n$ is isomorphic to the grassmanian $\mathrm{Grass}(1,n+1)=\{V\subseteq\mathbb{R}^{n+1}:V\,\text{linear subspace,}\,\dim\,V\,=1\}$ as well as ...
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De Rahm Cohomology of Complex Grassmannian

Since the complex Grassmannian $G_k(\mathbb{C}^n)\cong SU(n)/S(U(k)\times U(n-k))$ is connected and simply connected, the first two de Rahm cohomology groups are given by $$ ...
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Induced measurable subbundle

Let $G_k(\mathbb{R}^m)=\{ W: W$ is subspace of $\mathbb{R}^m, \dim W=k \}$ measurable and suppose that the application $$ \displaystyle{\begin{array}{rccl} h:&Z&\longrightarrow& ...
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In search for a topology

I'm looking for a way to convergence on subspaces. If $G_k(\mathbb{R}^m)=\{ W: W$ is subspace of $\mathbb{R}^m, \dim W=k \}$ and consider in $G_k(\mathbb{R}^m)$ one topology $\tau$. I would like to ...
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Convergence equivalence

If $G_k(\mathbb{R}^m)=\{ W: W$ is subspace of $\mathbb{R}^m, \dim W=k \}$ and consider in $G_k(\mathbb{R}^m)$ one topology $\tau$ where $U\in \tau$ is open iff the set $\widehat{U}=\lbrace v: v\in ...
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Is there an embedding of projective varieties $\mathrm{Grass}(r,n)\hookrightarrow(\mathbb{P}^{n-1})^{\times r}$?

Let $k$ be an algebraically closed field, and let $r\le n$ be positive integers. Let $\mathrm{Grass}(r,n)$ be the projective variety of all $r$-dimensional planes in $k^n$. Notice that ...
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How to understand following expression for gaussian integral of grassmannian function?

Let's have grassmannian numbers $\theta_{i}$: $$ \theta_{i}\theta_{j} = -\theta_{j}\theta_{i}, \quad \theta_{i} x = x\theta_{i}, $$ where x is just ordinary number. How to understand (or how to show ...
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Grassmannian as a submanifold of the exterior product

I'm looking for a proof of the fact that if $V$ is a finitely dimensional vector space, then $G_p(V) \setminus \{0\}$ is a submanifold of $\Lambda_pV$. Here $G_p(V) = \{ v_1 \wedge ... \wedge v_p \ ...
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Isotropy group for the subset of Grassmannian

Consider a complex Grassmannian $Gr_{k}(C^{n})$, which is a symmetric space with symmetry group $U(n)$ (i.e. unitary group). Consider a subspace $S_{0}$ of the Grassmannian determined by the canonical ...
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Grassmannian bundle: any good reference?

I have met the notion of Grassmannian bundle of a vector bundle over a variety in intersection theory, but anywhere I look I just find a brief recall of how the stalks look like (my references so far ...