Mathematics

Questions tagged [grassmannian]

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

In mathematics, the Grassmannian $\mathbf{Gr}(r, V)$ is a space which parameterizes all linear subspaces of a vector space $V$ of given dimension $r$. For example, the Grassmannian $\mathbf{Gr}(1, V)$ is the space of lines through the origin in $V$, so it is the same as the projective space of one dimension lower than $V$.

When $V$ is a real or complex vector space, Grassmannians are compact smooth manifolds. In general they have the structure of a smooth algebraic variety (Wikipedia).

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Compactness of the Grassmannian

Let $V$ be a finite-dimensional inner product space. For $0 \leq d \leq \text{dim}(V)$, define the Grassmannian $G(V, d)$ to be the set of all $d$-dimensional linear subspaces of $V$, equipped with the metric $d(X, Y) = \Vert P_X - P_Y \Vert$, where…
Tom Jonathan
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Fundamental groups of Grassmann and Stiefel manifolds

Could someone provide details on how to compute fundamental groups of real and complex Grassmann and Stiefel manifolds?
google
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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 $k$-frames in $V$ be denoted by $F_k(V)$. It is clear…
caffeinemachine
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What are the attaching maps for the real Grassmannian?

The Grassmannian $G_n(\mathbb{R}^k)$ of n-planes in $\mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition. The study of characteristic classes tells us that these Schubert cells generate the cohomology of the…
wckronholm
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Topology on the general linear group of a topological vector space

Let $K$ be a topological field . Let $V$ be a topological vector space over $K$ (if it makes things convenient, you may assume it is finite dimensional). Naive Question: Is there a canonical way of defining a topology on $\text{GL}(V)$? Attempted…
Zev Chonoles
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When does variété mean manifold?

Following advice from this post , I am in the process of translating Ehresmann's 1934 paper "Sur la Topologie de Certains Espaces Homogènes" from French to English. French-English dictionaries online and Google translate are helping me out quite a…
wckronholm
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Difference between Grassmann and Stiefel manifolds

I'm working on an optimization problem on manifolds and I'm having a bit of a conceptual issue with choosing between the Grassmann and Stiefel manifolds. Grassmann(2, 3) is the linear subspace of dimension 2 within the space $\mathbb{R}^3$, so all…
bibliolytic
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intrinsic proof that the grassmannian is a manifold

I was trying to prove that the grassmannian is a manifold without picking bases, is that possible? Here's what I've got, let's start from projective space. Take $V$ a vector space of dimension n, and $P(V)$ its projective space. To imitate the…
John Salvatierrez
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Avoid the planes - the geometry of grassmannians

Suppose we have $n$ planes $H_1, \ldots, H_n$ in $\mathbb{R}^m$ of codimension $q$ , or equivalently of dimension $d=m-q$ . I want to choose a vector which does not belong to the planes in a continuous way. There are two versions of this problem,…
Andrea Marino
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Cohomology of Grassmannian

Let $G_r$ the infinite complex Grassmannian manifold. We know that $H^{*}(G_r)=\mathbb{C}[x_{1}, \cdots, x_{n}]$ where $x_i$ are the Chern classes of tautological bundle. But $H^{*}(G_r)$ is also isomorphic to the ring $\mathbb{C}[c_{1}, \cdots,…
ArthurStuart
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A Proof of the Hausdorffness of the Grassmannian Using the Basics

$\DeclareMathOperator{\Span}{span} \newcommand{\R}{\mathbf R} \newcommand{\mc}{\mathcal} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\grassman}{GR} \newcommand{\set}[1]{\{#1\}} \DeclareMathOperator{\id}{Id}$ I had previously asked here if…
caffeinemachine
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Chern classes of tangent bundle over the Grassmannian G(2,4)

What are the Chern classes of the tangent bundle $\tau_G$ of the Grassmannian $G=G(2,4)$ of lines in $\mathbb{P}^3$? This is Exercise 5.37 on page 191 of 3264 & All That by Eisenbud and Harris. By Theorem 3.5, the tangent bundle $\tau_G$ is…
user571305
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The Plucker relations are sufficient

Consider the Grassmannian of codimension-$d$ subspaces of a given vector space $E$ (over an arbitrary field), which I will define as $$ \operatorname{Gr}^d(E) = \{\text{linear surjections } \sigma: E \to F \mid \text{$F$ is any $d$-dimensional…
Joppy
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Second homotopy group of real Grassmannians $\textrm{Gr}(n,m)$, special case $n=m=2$ not clear.

I have been considering real Grassmanians $$\textrm{Gr}(n,m)=O(n+m)/O(n)\times O(m)$$ appearing in certain condensed matter physics context (space of real flat-band Hamiltonians $Q(k)$ with $n$ occupied and $m$ unoccupied bands, the reality comes…
Tomáš Bzdušek
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Quantifying the angle metric on the Grassmannian in terms of the norm on the exterior power

Let $V$ be a finite-dimensional Hilbert space and $Gr_k(V)$ the Grassmannian of $k$ -dimensional subspaces of $V$ . The $k$ th exterior power $\bigwedge^k(V)$ can be equipped with a scalar product by extending $$ \langle v_1\wedge \cdots \wedge v_k,…
Sven Pistre
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