Motivated by the following RG question we ask a related question as follows:

We identify $\mathbb{R}^{n} \otimes \mathbb{R}^{m}$ with $\mathbb{R}^{mn}$. We define $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$ as a subgroup of $GL(\mathbb{R}^{n} \otimes \mathbb{R}^{m})$ which consist simple tensors $T\otimes S\;\; \text{where}\;\;T\in GL(\mathbb{R}^{n}),\;S\in GL(\mathbb{R}^{m})$.

Is the action of $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$ on $\mathbb{R}P^{(mn-1)}$ a transitive action? If not, what is the topological description of the quotion $\mathbb{R}P^{mn-1}/GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$?

The above action is the restriction of the natural action of $GL(\mathbb{R}^{n} \otimes \mathbb{R}^{m})$

Note: "$GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$" is just a notation so we do not confuse it with the tensor product of two groups or some thing else. However, as I heard from Todd Trimble, this group is isomorphic to $\frac{GL(\mathbb{R}^{n}) \times GL(\mathbb{R}^{m})} {\mathbb{R}^{+}}$ with obvious action $\lambda.(A,B)=(\lambda A, \lambda^{-1}B)$