I am using the following representation of the $GL^+(2,\mathbb{R})$ group.
$$ \exp( a+x \sigma_1 +y \sigma_2 + b \sigma_1\sigma_2) = \exp( \begin{bmatrix}a+x & -b +y \\ b+y & a-x\end{bmatrix}) $$
If I pose $a \to 0, x\to 0$ and $y\to 0$, then I have effectively reduced the group from $GL^+(2,\mathbb{R})$ to $SO(2,\mathbb{R})$.
$$ \exp (b\sigma_1\sigma_2) = \exp( \begin{bmatrix}0 & -b \\ b & 0\end{bmatrix}) $$
What is the precise expression for $GL^+(2,\mathbb{R})/SO(2,\mathbb{R})$? If yes, what is it?
edit:
Can I find $GL^+(2,\mathbb{R})/SO(2,\mathbb{R})$ in the following way:
$$ GL^+(2,\mathbb{R})/SO(2,\mathbb{R}) = \exp( a+x \sigma_1 +y \sigma_2 + b \sigma_1\sigma_2)/\exp ( b \sigma_1\sigma_2) $$
The next step would be to factor our $ b\sigma_1\sigma_2$ from $\exp(a+ x \sigma_1 +y \sigma_2 + b\sigma_1\sigma_2)$?
But since $[x \sigma_1 +y \sigma_2, b\sigma_1\sigma_2]\neq 0$, I have to use this formula?
I end up with
$$ GL^+(2,\mathbb{R})/SO(2,\mathbb{R}) = \exp (a) \exp(x \sigma_1 +y \sigma_2)\exp( \frac{-1}{2}[ x\sigma_1 +y \sigma_2, b\sigma_1\sigma_2]) \exp (...) $$
I was previously told $GL^+(2,\mathbb{R})/SO(2,\mathbb{R})$ yields the group of all Riemmanian metrics in $\mathbb{R}^2$. But I do not see where the metrics are in my result.
This is not a quotient group but a homogeneous space, i.e. you take elements of $GL^+(2,\mathbb R)$ and identify $A$ and $B$ if and only if $B=AC$ for some $C\in SO(2)$. In general, there is no canonical way to realize a homogeneous space of a matrix group as a subset of $\mathbb R^N$ for any $N$. (This is one of the reasons why one looks at abstract manifolds.) However for this example, there is a nice explicit way to explicitly understand it: You can send a matrix $A\in GL^+(4,\mathbb R)$ and map it to $AA^t$, which is a symmetric, positive definite $2\times 2$-matrix. Now if $B=AC$ for $C\in SO(2)$, then $B^t=C^tB^t$, so $BB^t=AA^t$. One verifies that this indeed gives rise to a diffeomorphism from $GL^+(2,\mathbb R)/SO(2)$ to the space of positive definite, symmetric $2\times 2$-matrices.