For the special linear group $SL(2,\mathbb{R}) = \{A \in Mat_{2 \times 2}(\mathbb{R})|\det A = 1\}$, I'm trying to find continuous and differentiable functions $P_k : D_k \rightarrow Mat_{2 \times 2}(\mathbb{R}), k = 1,2,...$ with open domains $D_k \subset \mathbb{R}^3$, such that $SL(2,\mathbb{R})$ is the union of the images of $P_1, P_2, ...$
Now I do understand that it makes sense to look separately at matrices with entry $a_{11} = 0$ and those with entry $a_{11} \neq 0$, but once I start to come up with examples, it just doesn't work out. So does anybody have suggestions on how I could tackle this? Thanks in advance.
By Gaussian elimination in the form of the LDU decomposition, any $M \in \text{SL}(2,\mathbb{R})$, may be factored either as $$ M = \begin{bmatrix} 1 & 0 \\ z & 1 \end{bmatrix} \begin{bmatrix} x & 0 \\ 0 & \frac{1}{x} \end{bmatrix} \begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} x & xy \\ xz & xyz + \frac{1}{x} \end{bmatrix} $$ or as $$ M = \begin{bmatrix} 1 & 0 \\ z & 1 \end{bmatrix} \begin{bmatrix} x & 0 \\ 0 & \frac{1}{x} \end{bmatrix} \begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} xy & -x \\ xyz + \frac{1}{x} & -xz \\ \end{bmatrix} $$
We know the diagonal matrix has the form it does with $x\ne 0$, as $\det M = 1$. Hence, the right hand sides as a function of $(x,y,z)$, $x\ne 0$ completely parameterize $\text{SL}(2,\mathbb{R})$.