$$ d\rho^2 + \cosh^2\rho\ d\theta^2$$
Only one ? Is there any other example ?
2026-03-31 20:31:51.1774989111
$2$-dimensional Hyperbolic space with fundamental group ${\bf Z}$ and constant curvature $-1$
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You have essentially two choices: either the quotient of the hyperbolic plane by (the cyclic group generated by) a hyperbolic isometry or the quotient of the hyperbolic plane by (the cyclic group generated by) a parabolic isometry.