Using <Cartan's second structure equation> gives
$$
\boxed{\Theta_{01}=\frac12e^0\wedge e^1,
\qquad \Theta_{02}=\frac12e^0\wedge e^2,
\qquad \Theta_{12}=\frac12e^1\wedge e^2}.
$$
Since $\Theta_{\mu\nu}=R_{\mu\nu\rho\sigma}e^\rho\wedge e^\sigma/2$, the independent nonzero lowered <Riemann curvature tensor> components are
$$
\boxed{R_{0101}=R_{0202}=R_{1212}=\frac12},
$$
with all others determined by the symmetries of the <Riemann curvature tensor>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 309 / 4 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-309/4/c/solution