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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.

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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. – LuboŇ° Motl Nov 1 '12 at 10:27

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