 Poincare coordinates on anti-de Sitter spacetime

For Lorentzian boundary metric $\eta$ and $z>0$, this Anti-de Sitter spacetime patch has unit curvature radius. Its orthonormal coframe has connection ${\omega }^a{}_3=-{\theta }^a$ and curvature ${\Omega }^a{}_b=-{\theta }^a\wedge {\theta }_b$, giving sectional curvature $-1$. The full local isometry algebra is $\mathfrak{so}(d-1,2)$ in spacetime dimension $d$. A patch need not support every local Killing flow for all parameter values.