Topology of SL(2,R)

Question

I am interested in visualizing the Lie group $SL(2,\mathbb R)$ topologically. Writing a matrix as $A = \begin{pmatrix} x+w & -y+z\\ y+z& x-w \end {pmatrix}$ I could identify a matrix with determinant $1$ lying on the hypersurface $x^2+y^2-z^2-w^2=1$ or on the complex curve $|a|^2-|b|^2=1$ ($a,b \in \mathbb C$). But how to see the topological form of this manifold? Is there a nice identification?

Answer 1

In general, a connected Lie group $G$ is diffeomorphic to its maximal compact subgroup $K$ times some $\mathbb{R}^n$. In this case, $SL_2(\mathbb{R})$ is diffeomorphic to $SO(2) \times \mathbb{R}^2$ (use Gram-Schmidt).

Answer 2

I think we can parameterize the hypersurface $x^2+y^2-z^2-w^2=1$, so we will get $x=\cosh t.cosr$, $y=\cosh t.\sin r$, $z=\sinh t.\cos s$, $ w=\sinh t.\sin s$ and then by this parametrization we would have a homomorphic mapping between$ S^1 ☓D^2$ and the above hypersurface. Can any one state the explicit rule of that function?