Getting the AdS metric from maximally symmetric spaces

Question

I am familiar with the way we derive the form of the FRW metric by just using the fact that we have a maximally symmetric space i.e the universe is homogeneous and isotropic in spatial coordinates. Similarly, how do I get the Poincare patch of $AdS_{p+2}$ i.e $$\begin{equation} ds^2 = R^{2}\left(\frac{du^2}{u^2}+u^2(-dt^2+d\mathbf{x}^2)\right) \end{equation}$$ by using the property of maximal symmetry only.

This post imported from StackExchange Physics at 2014-07-28 11:15 (UCT), posted by SE-user Debangshu

Comments

It's the same metric written in different coordinates - well, coordinates that only cover the patch. So if you can prove the maximum symmetry in one, the appropriate coordinate transformation proves the maximum symmetry in the other coordinate system or form as well. They're locally the same geometry.

This post imported from StackExchange Physics at 2014-07-28 11:15 (UCT), posted by SE-user Luboš Motl