Consider the smooth manifold of all real $n\times m$ matrices $M_{n\times m}(\mathbb R)$ and a smooth action of the orthogonal group $O(n)$: $$g\cdot A := gA.$$
I would like to understand the quotient space $X := M_{n\times m}(\mathbb R) / O(n)$, especially in the case when $n \le m$.
This is what I know:
- $X$ is Hausdorff, as the action is proper.
- The action is not free (as the stabilizer of the $\mathbf{0}$ matrix is non-trivial).
- It is also non-transitive (as the $\mathbf 0$ matrix has trivial orbit).
- As there exists the QR decomposition, I imagine that $X$ may be related to the space of upper-triangular matrices.
- This lead me to asking whether the quotient space of non-zero matrices modulo $O(n)$ would have better properties. However, I think that it won't (if $m > n$ one can consider the block matrix consisting of the identity matrix $\mathbf 1_{n}$ and the zero matrix $\mathbf 0_{n\times (m-n)}$).
This leads to the following questions:
- Do you know a reference studying this space? (Or a similar one, like the quotient of the space of non-zero matrices, or spaces of square matrices, like $M_{n\times n}(\mathbb R)/O(n)$ or $GL(n)/O(n)$)?)
- Is $X$ even a topological manifold? I doubt it, but I don't have a proof.
- Can $X$ be the quotient of the space of upper-triangular matrices modulo some action of a discrete group?
In particular, I'd be interested in a reference, if there is a paper studying this problem.