Contract the isotropic-curvature formula on $a,c$. In dimension $n$ it gives
$$
R_{bd}=(n-1)K g_{bd},
\qquad
R=n(n-1)K.
$$
Substitution into the <contracted Bianchi identity> gives
$$
(n-1)\nabla_bK=\frac12n(n-1)\nabla_bK.
$$
Since $n>2$, $\nabla_bK=0$, so $K$ is constant on each connected component. Thus this is <constant sectional curvature>, with
$$
\boxed{K=\frac{R}{n(n-1)}}.
$$
This argument is <Schur theorem in pseudo-Riemannian geometry>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-309/1/b/solution