Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 37D / c / Solution

Lower the free index of $K$. The Killing equation makes the last two indices of ${\nabla }_{\nu }{\nabla }_{\mu }{K}_{\alpha }$ antisymmetric after the appropriate derivative is moved. Combining the three resulting Ricci identities cyclically givesApplying the covector form of the Ricci identity and then the first Bianchi identity reduces this toRaising $\alpha$ proves the second covariant derivative of a Killing vector identityContracting $\mu$ and $\nu$ and using the definition and symmetries of the Ricci tensor gives