The flow of a <Killing vector field> consists of local isometries. Isometries preserve both the metric and its unique torsion-free metric-compatible <Levi-Civita connection>, hence $\mathcal L_K\nabla=0$. Set $X=K$ and $Y=Z=U$ in part (b). Since $\nabla_UU=0$,
$$
0=R(K,U)U+\nabla_U\nabla_UK,
$$
and antisymmetry of the first two curvature arguments gives
$$
\nabla_U\nabla_UK=R(U,K)U.
$$
This is the <geodesic deviation> equation with connecting field $K$: applying the isometry to the original geodesic produces a neighboring geodesic, and curvature determines their relative acceleration.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 309 / 1 / c / Solution

{wiki}

For a family of affinely parametrized geodesics with tangent $U$ and connecting field $J$, the Jacobi equation in the convention used here is
$$
\nabla_U\nabla_UJ=R(U,J)U.
$$
It describes the relative acceleration produced by spacetime curvature.


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