Motivated by the [following RG question](https://www.researchgate.net/post/Does_anyone_know_that_it_is_possible_to_add_a_row_to_set_of_convex_and_compact_full_rank_matrices_such_that_rank_of_any_matrix_increases) 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 $GL(\mathbb{R}^{n} \otimes \mathbb{R}^{m})$ 

>Is really $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$  a  proper subgroup of $GL(\mathbb{R}^{n} \otimes \mathbb{R}^{m})$? If not, what is the structure of its Lie algebra?


The later question is somehow similar to the following posts(very indirectly):

http://mathoverflow.net/questions/58186/positive-element-in-c-tensor-product?lq=1


http://mathoverflow.net/questions/43138/positive-elements-in-tensor-products?rq=1


**Note:** "$GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$" is just  a terminology and notation. We do not confuse it with the tensor product of two groups or some thing else.