Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 309 / 2 / b / iv / Solution

Choose any tangent vector $V^a\in T_p\Sigma$ and let the affinely parametrized geodesic with initial tangent $V^a$ start at $p$. If $\Sigma$ is totally geodesic, then $U^a{n}_a$ remains zero. Its initial derivative is therefore zero. Part iii, with $U^a{n}_a=0$ at $p$, givesThis holds for every tangent $V$. Since $K$ is symmetric, the polarization identity implies