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

In the propagated frame $T=e_0$, and the <constant sectional curvature> formula reduces <geodesic deviation> to
$$
\ddot y^0=0,
\qquad
\ddot y^i=K y^i.
$$
The stated temporal initial data give $y^0(t)=0$. Writing $\omega=\sqrt{-K}$ for $K<0$ and $\kappa=\sqrt K$ for $K>0$, the spatial displacement is
$$
\boxed{
y^i(t)=
\begin{cases}
Y^i\cos(\omega t)+(V^i/\omega)\sin(\omega t),&K<0,\\
Y^i+V^it,&K=0,\\
Y^i\cosh(\kappa t)+(V^i/\kappa)\sinh(\kappa t),&K>0.
\end{cases}}
$$
These formulas are valid to first order in the initial separation and relative velocity.

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

{title2=$R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc})$}
{wiki}

A pseudo-Riemannian manifold has constant sectional curvature $K$ when $R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc})$. In dimension $n$, its Ricci tensor and scalar are $R_{ab}=(n-1)Kg_{ab}$ and $R=n(n-1)K$.


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