Constructing an Analytic Mapping for SL(2;R) using Quadratic Forms

[r16]

Homework Statement

Construct the analytic mapping $$\phi(x,y)$$ for the $$H^{2+} \times S^1$$ representation of $$SL(2;R)$$

Homework Equations

$$g(x) \circ g(y) = g(\phi(x,y))$$

The Attempt at a Solution

So, all points in SL(2;R) lie on the manifold $$H^{2+} \times S^1$$. I also know that SL(2;R) is 3 dimensional, so I will parametrize it as x=[x y $$\theta$$].

For a point to lie on $$H^{2+}$$ it has to satisfy the quadratic form $$z^2 -x^2 -y^2=1$$ in $$R^3$$. I calculated the 3x3 matrix, H, for this quadratic form, which has diagonal [-1 -1 1] and zeros everywhere else.

My goal is to calculate the matrix rep for $$g(x) \in SL(2;R)$$ by multiplying H and the rotation matrix for $$S^1$$ which is well known, and then using this information end up solving for $$\phi$$

My problem is that H, my matrix for the quadratic form, is not paramaterized by x and y, its elements are just constants. How do I find a quadratic form for $$H^2$$ that is paramaterized by x and y?

 

[arkajad]

The mapping $\phi$ is supposed to be from where to where? Real analytic or complex analytic?