# Poincare Patch covers half of the hyperboloid of AdS

We start with the general case of $AdS_{p+2}$ i.e AdS space in $p+2$ dimension. $$X_{0}^{2}+X_{p+2}^{2}-\sum_{i=1}^{p+1}X_{i}^{2} = R^2$$ This space has an isometry $SO(2,p+1)$ and is homogeneous and isotropic. The Poincare Patch is given by $$ds^2 = R^{2}\left(\frac{du^2}{u^2}+u^2(-dt^2 +d\mathbf{x}^{2})\right)$$ According to Equation (2.27) of the article http://arxiv.org/abs/hep-th/9905111, 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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