metric transformation

The metric can be written as ## \mathrm{d}s^2 = 2\,\mathrm{d}U\,\mathrm{d}V + e^{2\beta} \begin{pmatrix} \mathrm{d}x & \mathrm{d}y \end{pmatrix} R^T(\alpha) \begin{pmatrix} e^{2\gamma} & 0 \\ 0 & e^{-2\gamma} \end{pmatrix} R(\alpha) \begin{pmatrix} \mathrm{d}x \\ \mathrm{d}y \end{pmatrix}=...