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. I know that $\text{GL}(n + k, \mathbb{R})$ is a principal $H$ bundle over the Grassmann manifold $G_n(\mathbb{R}^{n+k})$. Let $E$ denote the total space of this principal bundle. Let $\rho: H \to \text{GL}(n, \mathbb{R})$ be the surjective homomorphism obtained by letting $H$ act on the first $n$ coordinates of $\mathbb{R}^{n+k}$, and let $E \times_H \mathbb{R}^n$ be the total space of the $\mathbb{R}^n$ bundle over $G_n(\mathbb{R}^{n+k})$ obtained from this action. How do I see that $E \times_H \mathbb{R}^n$ is isomorphic to the total space of the tautological bundle $\gamma^n$ over $G_n(\mathbb{R}^{n+k})$?

  • $\begingroup$ It seems the bundles are isomorphic as bundles, so why just consider their total spaces? $\endgroup$ Dec 16, 2015 at 21:11

1 Answer 1


Write an element of $GL(n+k,\mathbb R)$ as $\begin{pmatrix}C & D\\ E& F\end{pmatrix}$, then the principal $H$-bundle sends this element to $\rm{Im}\begin{pmatrix}C \\ E\end{pmatrix}$ in the Grassmannian.

Write elements of $E\times_H\mathbb R^n$ as $\left[ \begin{pmatrix}C & D\\ E& F\end{pmatrix}, x\right]$.

Write elements in the total space of the tautological bundle as $(V,w)$, where $V$ is the $n$-dimensional subspace and $w\in V$.

The iso is then given as $\left[ \begin{pmatrix}C & D\\ E& F\end{pmatrix}, x\right]\mapsto \left( \rm{Im}\begin{pmatrix}C \\ E\end{pmatrix} ,\begin{pmatrix}C \\ E\end{pmatrix}\cdot x\right)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.