Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 54 / 1 / a / i / Solution

Use signature $(-,+,+,+)$ and the Riemann curvature tensor convention $[{\nabla }_a,{\nabla }_b]X^c=R^c{}_{dab}{X}^d$. Lower the index of the Killing vector field and set $T_{abc}={\nabla }_a{\nabla }_b{V}_c$. Differentiating the Killing equation gives $T_{abc}=-T_{acb}$, while the Ricci identity givesDefine $S_{abc}=R_{cbad}{V}^d$. The first-pair antisymmetry of the Riemann curvature tensor gives $S_{abc}=-S_{acb}$. The pair symmetry and the first Bianchi identity giveThus $D=T-S$ is symmetric in its first two indices and antisymmetric in its last two. These two symmetries force it to vanish:Consequently $T_{abc}=R_{cbad}{V}^d$. Raising $c$ using metric compatibility proves the required second covariant derivative of a Killing vector: