Solution

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

For the curvature convention
$$
R^\alpha{}_{\beta\gamma\delta}
=\partial_\gamma\Gamma^\alpha_{\beta\delta}
-\partial_\delta\Gamma^\alpha_{\beta\gamma}
+\Gamma^\mu_{\beta\delta}\Gamma^\alpha_{\mu\gamma}
-\Gamma^\mu_{\beta\gamma}\Gamma^\alpha_{\mu\delta},
$$
the independent nonzero mixed components are
$$
R^r{}_{\phi r\phi}=-\sinh^2r,
\qquad
R^\phi{}_{r\phi r}=-1,
$$
and the Ricci scalar is $R=-2/a^2$.


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 Incoming links: Curvature of the two-dimensional hyperbolic polar metric