# Anti-de Sitter space Minkowski space is the maximally symmetric solution to Einstein's equation without the cosmological constant. With a negative cosmological constant, $\Lambda<0$, the maximally symmetric space is anti-de Sitter (AdS) space. In Euclidean signature, AdS${}_n$ is just the hyperbolic space $H_n$, and the metric can be written as$\mathrm{d}s^2_{H_n}=\ell_{\rm AdS}^2(\mathrm{d}\rho^2+\sinh^2\rho \,\mathrm{d}\Omega^2_{n-1}),$where $\mathrm{d}\Omega^2_{n-1}$ is the line element of the unit $(n-1)$-sphere. It is easy to verify that it has constant negative curvature $-n(n-1)/\ell_{\rm AdS}^2$. Here $\ell_{\rm AdS}$ is called the AdS length, related to the cosmological constant via$\Lambda=-\frac{(n-1)(n-2)}{2\ell_{\rm AdS}^2}.$