The metric of the Poincare-AdS$_3$ geometry is given in the Wikipedia article on the Poincare coordinates of AdS$_3$ geometry:
$$ds^{2} = \alpha^{2}\left(\frac{du^{2}}{u^{2}} + u^{2}g_{\alpha\beta}dx^{\alpha}dx^{\beta}\right),$$
where $\alpha$ is supposedly the AdS$_3$ radius.
It appears that $0 < r < \infty$, in which the Poincare-AdS$_3$ geometry is not a cylinder.
But isn't the Poincare-AdS$_3$ geometry a cylinder? Where have I gone wrong in my thinking?
