Two-dimensional hyperbolic metric in polar coordinates ($d{s}^2=a^2(d{r}^2+{\sinh }^{2}rd{\varphi }^2)$)

This metric is the hyperbolic plane with curvature scale $a$. Its nonzero connection coefficients are

Every radial coordinate line $\varphi =\text{constant}$ is a geodesic. No nonconstant coordinate circle $r=r_0>0$ is a geodesic.

For the curvature conventionthe independent nonzero mixed components areand the Ricci scalar is $R=-2/a^2$.