I need find a homeomorphism $h:SU(1,1)/U(1)\longrightarrow \mathbb{H}^2$ equivariant with the action of $SU(1,1)$ onto the disk unitary $\mathbb{D}$ where
$\mathbb{H}^2={\{(x_1,x_2,x_3)\in\mathbb{R}^3: x_1^2-x_2^2-x_3^2=1, x_1 \geq1}\}$
Idea: Remember that:
$SU(1,1)={\{\left( \begin{array}{cc} \alpha & \beta \\ \bar{\beta} & \bar{\alpha} \end{array} \right)\in\mathcal{M}_{2\times2}(\mathbb{C}):(\alpha,\beta)\in\mathbb{C}^{2},\ \alpha\bar{\alpha}-\beta\bar{\beta}=1}\} ={\{\left( \begin{array}{cc} e^{i\theta} & 0\\ 0 & e^{-i\theta} \end{array} \right)\left( \begin{array}{cc} cosh(t) & sinh(t)\\ sinh(t) & cosh(t) \end{array} \right)\left( \begin{array}{cc} e^{i\phi} & 0\\ 0 & e^{-i\phi} \end{array} \right):\phi,\theta,t\in\mathbb{R}}\}$
Then all element of $SU(1,1)/U(1)$ we can write them the way:
$\left[\left(\begin{array}{cc} e^{i\theta}cosh(t) & e^{i\theta}sinh(t) \\ e^{-i\theta}sinh(t) & e^{-i\theta}cosh(t) \end{array}\right)\right]$ where
$\left[\left(\begin{array}{cc} e^{i\theta}cosh(t) & e^{i\theta}sinh(t) \\ e^{-i\theta}sinh(t) & e^{-i\theta}cosh(t) \end{array}\right)\right]=\left(\begin{array}{cc} e^{i\theta}cosh(t) & e^{i\theta}sinh(t) \\ e^{-i\theta}sinh(t) & e^{-i\theta}cosh(t) \end{array}\right)U(1)$
Then maybe a map homeomorphism is $h:SU(1,1)/U(1)\longrightarrow \mathbb{H}^2$ defined by $h(\left[\left(\begin{array}{cc} e^{i\theta}cosh(t) & e^{i\theta}sinh(t) \\ e^{-i\theta}sinh(t) & e^{-i\theta}cosh(t) \end{array}\right)\right])=(cosh(t),cos(\theta)sinh(t),sinh(t)sin(\theta))$
But I have not been able to prove that $h$ is a homeomorphism. Is $h$ a homeomorphism?
Thanks.