The action of the orthogonal group $O_n(\mathbb{R})$ on the Stiefel manifold $V_{k,n}(\mathbb{R}) $ .

Question

I'm trying to prove that $O_n(\mathbb{R}) / O_{(n-k)}(\mathbb{R}) \cong V_{k,n}(\mathbb{R})$ where $ O_n(\mathbb{R}) = \left\lbrace A \in M_n(\mathbb{R}) / A A^t = I_{n}\right\rbrace $ is the orthogonal group, and $V_{k,n} (\mathbb{R}) = \left\lbrace (a_1, ...a_k) / a_i \in \mathbb{R^n} and \left\langle a_i , a_j\right\rangle = \delta _ {i,j}\right\rbrace $ is the Stiefel manifold (the space of the $k-frames$). I already proved that $V_{k,n} (\mathbb{R})$ is compact and Hausdorff and that $O_n(\mathbb{R})$ is compact. So if I can also prove that $O_n(\mathbb{R})$ acts transitively on $V_{k,n}(\mathbb{R})$ and that $O_{(n-k)}(\mathbb{R})$ is the stabilizer of the action, I can conclude. What I need is the explicit expression of the action's map : $O_n(\mathbb{R}) \times V_{k,n}(\mathbb{R}) \rightarrow V_{k,n}(\mathbb{R}) $, Or the equivalence relation associated to the action so I can check the transitivity of the action. Thanks in advance fo the help.

Answer 1

$V_{k, n}(\mathbb{R})$ is the space of isometric linear embeddings $\mathbb{R}^k \to \mathbb{R}^n$. The action of the orthogonal group is by postcomposition.

Answer 2

I think that $\phi \colon O_n \times V_k(\mathbb{R}^{n}) \to V_k(\mathbb{R}^{n})$ defined by $\phi(M,(A_1,A_2,...,A_k)) = (MA_1, MA_2, ..., MA_k)$ is the action that you need.