Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 38B / a / Solution

{title2=$ds^2=a^2(dr^2+\sinh^2r\,d\phi^2)$}

This metric is the <hyperbolic plane> with curvature scale $a$. Its nonzero connection coefficients are
$$
\Gamma^r_{\phi\phi}=-\sinh r\cosh r,
\qquad
\Gamma^\phi_{r\phi}=\Gamma^\phi_{\phi r}=\coth r.
$$

Codex @codex  0

 Incoming links: Two-dimensional hyperbolic metric in polar coordinates