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I am trying to find a differentiable structure on the Grassmannian, which is the set of all $k$-planes in $\mathbb{R}^{n} $. To do this, I have to show that for any given $\alpha$, $\beta$, the set $$\left\{ \left(AA_{\alpha}^{-1}\right)_{\alpha'}:A\in\mbox{M}_{k}\left(n\times k,\mathbb{R}\right),A_{\alpha},A_{\beta}\in\mbox{GL}\left(k,\mathbb{R}\right)\right\} $$ is open in the set of $(n-k)\times k$ matrices.

Here is the notation:

-$\alpha$ is just a $k$-tuple $\left(\alpha_{1},\ldots,\alpha_{k}\right)$ such that $1\leq\alpha_{1}<\cdots<\alpha_{k}\leq n$, and $\beta$ is another $k$-tuple $\left(\beta_{1},\ldots,\beta_{k}\right)$ such that $1\leq\beta_{1}<\cdots<\beta_{k}\leq n$.

-$\mbox{M}_{k}\left(n\times k,\mathbb{R}\right)$ is the set of all $n\times k$ matrices with rank $k$, which is an open subset of the set of all $n\times k$ matrices.

-Given $A\in\mbox{M}_{k}\left(n\times k,\mathbb{R}\right)$, $A_{\alpha}$ is defined to be the $k\times k$ submatrix of $A$ whose $i$th row is the $\alpha _{i}$th row of $A$. We define $A_{\alpha '}$ to be the $(n-k)\times k$ submatrix of $A$ which consists of the remaining rows.

I am unsure how to proceed.

Eigenfield
  • 1,532
  • $M_k(n\times k)$ is not open in $M(n\times k)$. For example, when $n=2,k=1$, $M_k(n\times k)$ is the line $x=y$ in $M(n\times k) = \mathbb{R}^2$. – Quang Hoang Aug 23 '17 at 07:28
  • Actually $\mbox{M}_{1}\left(2\times 1,\mathbb{R}\right)$ is equal to $\mathbb{R}^{2}-{0} $, which is open. – Eigenfield Aug 23 '17 at 08:57

1 Answers1

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Fix some $k$-tuples $\alpha$, $\beta$ and let $S = \{ A \in M_k(n \times k, \mathbb{R}) \mid A_\alpha, A_\beta \in \mathrm{GL}(k,\mathbb{R}) \}$. We want to show that the image of the map $$ \begin{align} \phi : S &\to M((n-k)\times k,\mathbb{R}) \\ A &\mapsto (A{A_\alpha}^{-1})_{\alpha'} \end{align} $$ is open. Hence, it is enough to prove that $\phi$ is an open map.

We can write $\phi$ as a composition $$\phi = p_{\alpha'} \circ m \circ (i,\mathrm{inv}\circ \pi_\alpha)$$ where

  • $i : S \to M(n \times k)$ is the inclusion map;
  • $\pi_\alpha : S \to \mathrm{GL}(k)$ is defined by $\pi_\alpha(A) = A_\alpha$;
  • $\mathrm{inv} : \mathrm{GL}(k) \to \mathrm{GL}(k)$ is matrix inversion;
  • $m : M(n\times k) \times \mathrm{GL}(k) \to M(n \times k)$ is given by matrix multiplication;
  • $p_{\alpha'} : M(n\times k) \to M((n-k) \times k)$ is defined by $p_{\alpha'}(A) = A_{\alpha'}$.

Note that these maps are all open. Indeed,

  • $i$ is open because $S$ is an open subset of $M(n\times k)$;
  • $p_{\alpha'} : M(n\times k) \to M((n-k) \times k)$ is a surjective linear map;
  • $\pi_\alpha$ is the restriction of a surjective linear map $M(n\times k) \to M(k \times k)$ to the open subset $S \subset M(n \times k)$;
  • $\mathrm{inv} : \mathrm{GL}(k) \to \mathrm{GL}(k)$ is a homeomorphism.

Since a composition of open maps is open, this shows that $\phi$ is open, as desired.

Math536
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