<Cartan's second structure equation> gives
$$
\boxed{\Theta^i{}_0=\Theta^0{}_i
=\frac{p_i(p_i-1)}{t^2}e^0\wedge e^i},
$$
and, for $i\neq j$,
$$
\boxed{\Theta^i{}_j=\frac{p_ip_j}{t^2}e^i\wedge e^j}.
$$
With all indices lowered and the curvature convention in the question, the independent nonzero orthonormal <Riemann curvature tensor> components are
$$
\boxed{R_{0i0i}=\frac{p_i(1-p_i)}{t^2},
\qquad
R_{ijij}=\frac{p_ip_j}{t^2}\quad(i\neq j)},
$$
together with those obtained from the Riemann symmetries.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 3 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-309/3/d/solution