Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & B\end{pmatrix},$$where $A \in \text{GL}(n, \mathbb{R}$), $B \in \text{GL}(k, \mathbb{R})$, and $*$ is arbitrary. How do I see that $\text{GL}(n + k, \mathbb{R})$ is a principal $H$ bundle over the Grassmann manifold $G_n(\mathbb{R}^{n+k})$?

## 1 Answer

The first thing to note is that the Grassman manifold is a homogeneous manifold, since the group $G=GL(n+k,\mathbb R)$ naturally acts on it transitively. This implies that $G_n(\mathbb R^{n+k})=G/H$, where $H$ is a stabilizer of (any) point $x\in G_n(\mathbb R^{n+k})$.

Now take $x=\mathrm{span}(e_1,\dots e_n)$, where $e_1,\dots,e_{n+k}$ is a basis of $\mathbb R^{n+k}$. Clearly, an element $M\in G$ stabilizes $x$ iff it has the form $$M = \begin{pmatrix} A & * \\ 0 & B\end{pmatrix}.$$

Hence $G$ is a principle $H$-bundle over $G/H=G_n(\mathbb R^{n+k})$.