= Solution
Using the stated convention for the <Riemann curvature tensor>, the first independent component is
$$
\begin{aligned}
R^r{}_{\phi r\phi}
&=\partial_r\Gamma^r_{\phi\phi}
-\Gamma^\phi_{\phi r}\Gamma^r_{\phi\phi}\\
&=-(\cosh^2r+\sinh^2r)+\cosh^2r\\
&=\boxed{-\sinh^2r}.
\end{aligned}
$$
The other is
$$
\begin{aligned}
R^\phi{}_{r\phi r}
&=-\partial_r\Gamma^\phi_{r\phi}
-\Gamma^\phi_{r\phi}\Gamma^\phi_{\phi r}\\
&=\operatorname{csch}^2r-\coth^2r\\
&=\boxed{-1},
\end{aligned}
$$
using the <hyperbolic Pythagorean identity> $\coth^2r-\operatorname{csch}^2r=1$. Antisymmetry in the final two indices supplies the components with those indices reversed. These are the <curvature of the two-dimensional hyperbolic polar metric> components.
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