Its easy to check that every matrix in $SL_2(\mathbb{R})$ can be wrtten uniquely as $$\begin{pmatrix} \begin{pmatrix} \lambda & \alpha \\ 0 & \lambda^{-1}\end{pmatrix}\begin{pmatrix} \cos{\theta} & -\sin{\theta} \ \\sin{\theta} & \cos{\theta} \end{pmatrix}$$end{pmatrix}$where$\lambda>0$. This is exactly written then as a coset representative of the quotient you wrote down above. You can arrive at the hyperbolic metric by following the definition of the pushforward metric from a left invariant metric on$SL_2(\mathbb{R})$. Notice$(\alpha,\lambda)$is a point in the upperhalf plane. The latex misbehaved, those should be$2\times 2$matrices. 3 Fix a tex display Its easy to check that every matrix in$SL_2(\mathbb{R})$can be wrtten uniquely as $$\begin{pmatrix} \lambda & \alpha \\ 0 & \lambda^{-1}\end{pmatrix}\begin{pmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{pmatrix}$$ where$\lambda>0$. This is exactly written then as a coset representative of the quotient you wrote down above. You can arrive at the hyperbolic metric by following the definition of the pushforward metric from a left invariant metric on$SL_2(\mathbb{R})$. Notice$(\alpha,\lambda)$is a point in the upperhalf plane. The latex misbehaved, those should be$2\times 2$matrices. 2 Typesetting Its easy to check that every matrix in$SL_2(\mathbb{R})$can be wrtten uniquely as $$\begin{pmatrix} \lambda & \alpha \\ 0 & \lambda^{-1}\end{pmatrix}\begin{pmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{pmatrix}$$ where$\lambda>0$. This is exactly written then as a coset representative of the quotient you wrote down above. You can arrive at the hyperbolic metric by following the definition of the pushforward metric from a left invariant metric on$SL_2(\mathbb{R})$. Notice$(\alpha,\lambda)$is a point in the upperhalf plane. The latex misbehaved, those should be$2\times 2\$ matrices.