Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous matrices-valued function such that $H(x)$ is full rank for every $x \in \mathbb{R}^k$. I’d like to construct a continuous function $K:\mathbb{R}^k\rightarrow {\cal M}_{k\times (k-d)}(\mathbb{R})$ such that $K(x)$ is full rank […]

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 $Gr(2, T\mathbb R^3) = \bigcup_{x \in T\mathbb R^3} Gr(2, T_x \mathbb R^3)$. Note that $$ {\partial f \over \partial t} = {\partial f \over \partial s} + […]

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 from commutation with antiunitary time-reversal squaring to $1$), and I am interested in their second homotopy group. If I understand correctly, this can be […]

$\newcommand{\R}{\mathbf R}$ 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 that $F_k(V)$ is an open subset of $L(\R^k, V)$. Define a […]

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 bit. However, I’m confused about the standard usage of the word variété in French mathematics writing. It […]

Let $V$ be a $n$-dimensional complex vector space and consider the Grassmannian of complex $k$-planes $Gr(k,V)$. The Plücker embedding is an embedding $p:Gr(k,V) \to \mathbb P^M$ where $M = \left(\begin{array}{c} n \\ k \end{array}\right)$, given by $$ \text{span}\{ u_1,\ldots,u_k\} \mapsto [u_1\wedge u_2 \wedge \cdots \wedge u_k].$$ The Grassmannian comes equipped with a tautological vector bundle […]

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 Focusing of Naive Question: Let $\mathcal{C}$ be the subcategory of $\mathsf{TVect}_K$ with the same […]

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? Thank you.

Could someone provide details on how to compute fundamental groups of real and complex Grassmann and Stiefel manifolds?

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 standard open sets when you have a basis, consider a hyperplane $H$. We […]

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