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We start with the general case of $AdS_{p+2}$ i.e AdS space in $p+2$ dimension. \begin{equation} X_{0}^{2}+X_{p+2}^{2}-\sum_{i=1}^{p+1}X_{i}^{2} = R^2 \end{equation} This space has an isometry $SO(2,p+1)$ and is homogeneous and isotropic. The Poincare Patch is given by \begin{equation} ds^2 = R^{2}\left(\frac{du^2}{u^2}+u^2(-dt^2 +d\mathbf{x}^{2})\right) \end{equation} According to Equation (2.27) of the article, The second metric covers only half of the hyperboloid. Firstly, how do I show this. Secondly, when I go to the asymptotic limit (small radial distance), should the topology of the two spaces be different?

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It can be shown by conformal compactification of the spacetime, i.e. using coordinates which allow you to draw a penrose diagram. After you have done so, you can analytically continue the geometry and discover the other half (see chapter 2 of this) . Regarding differences in topology: I don't see any reason for this to be the case.

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