I'm having trouble going to different coordinates for the hyperbolic geometry metric:
$$dl^2=d\chi^2\, - \, \frac{1}{\kappa}\sinh^2(\sqrt{-\,\kappa}\,\,\chi)\,\big(\,d\theta^2 \, +\, \sin(\theta)^2 \,d\phi^2 \, \big), \,\,\,\,\,\, \kappa <0$$
How does one go to the hyperbolic embedding space with:
$$dL^2=(dX^1)^2+(dX^2)^2+(dX^3)^2-(dX^4)^2$$
\begin{align} &X_1 = \frac{1}{\sqrt{-\,\kappa}}\, \sinh(\sqrt{-\,\kappa}\,\,\chi) \sin{\theta}\cos{\phi}\\ &X_2 = \frac{1}{\sqrt{-\,\kappa}}\, \sinh(\sqrt{-\,\kappa}\,\,\chi) \sin{\theta}\sin{\phi}\\ &X_3 = \frac{1}{\sqrt{-\,\kappa}}\, \sinh(\sqrt{-\,\kappa}\,\,\chi) \cos{\theta}\\ &X_4 = \frac{1}{\sqrt{-\,\kappa}}\, \cosh(\sqrt{-\,\kappa}\,\,\chi) \end{align}