I'm reading Gilmore's Lie Group, Physics and Geometry and I'm trying to solve the exercise 1 at page 8, in which you have to find the full parametrization of a 2X2 $SL(2,R)$ matrix as a product of a symmetric matrix and a rotation one.
While retrieving the rotation matrix is really simple, I'm struggling to find a suitable parametrization for the symmetric part.
The book is claiming this is a 2-Manifold which can be parametrized by the two-sheeted hyperboloid $$z^2 – x^2 -y^2 = 1$$
Initially I thought to simply write the symmetric matrix as: $$\begin{bmatrix}z+x & y\\y & z-x\end{bmatrix}$$
which determinant is 1, but this is not a 2-Manifold, since showing 3 degree of freedom.
I've also tried the hyperbolic parametrization, i.e.:
$$\begin{align} x=sinh(u)cos(v) \\ y=sinh(u)sin(v) \\ z=\pm cosh(u)\end{align}$$
, which is better but I cannot see how to put this explicitly in matrix form, to get the determinant = 1.
I'm sure it's related to the KAN decomposition of a $SL(2,R)$ matrix, somehow.
Any help is appreciated. Thanks in advance.
EDIT: I just realised that combining the 2 claims above we could write keep my symmetric 3-Manifold symmetric matrix replacing the hyperbolic relations and getting a symmetric matrix un u and v, but which solution to keep for the z?
Choosing the + may work:
$$\begin{bmatrix}cosh(u)+sinh(u)cos(v) & sinh(u)sin(v)\\sinh(u)sin(v) & cosh(u)-sinh(u)cos(v)\end{bmatrix}$$
What do you think about this, please?
Best Answer
The matrix you added in your edit is correct. The choice of $z=+cosh(u)$ and $z=-cosh(u)$ is exactly what makes the hyperboloid two-sheeted, since $cosh(u)$ is strictly positive.