= Two-dimensional anti-de Sitter spacetime
{title2=$ds^2=\cosh^2r\,dt^2-dr^2$}
The unit hyperboloid $(X^0)^2+(X^1)^2-(X^2)^2=1$ in ambient signature $(++-)$ has $X^0=\cosh r\cos t$, $X^1=\cosh r\sin t$, $X^2=\sinh r$. It gives the displayed metric with $r\in\mathbb R$ and periodic $t$ modulo $2\pi$. Its time circles are timelike, so the <universal cover>, with $t\in\mathbb R$, removes these <closed timelike curves>. The metric alone does not prescribe the global identification. Both versions have constant negative curvature and two spatial ends.
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