Second covariant derivative of a Killing vector

For a Killing vector field and the curvature convention $[{\nabla }_{\mu },{\nabla }_{\nu }]V^{\alpha }=R^{\alpha }{}_{\beta \mu \nu }{V}^{\beta }$,The identity follows by combining the Ricci identity, Killing equation, and first Bianchi identity.

Let $C^a{}_{bcd}={\partial }_d{\Gamma }_{bc}^a-{\partial }_c{\Gamma }_{bd}^a-{\Gamma }_{ec}^a{\Gamma }_{bd}^e+{\Gamma }_{ed}^a{\Gamma }_{bc}^e$, with all other index positions obtained by the metric tensor. This convention is the negative of the convention $[{\nabla }_c,{\nabla }_d]V^a=R^a{}_{bcd}{V}^b$. The Ricci identity is $k_{a;bc}-k_{a;cb}=C_a{}^d{}_{bc}{k}_d$, with $k_{a;bc}={\nabla }_c{\nabla }_b{k}_a$. For a Killing vector, the Killing equation makes this second derivative antisymmetric in $a,b$. Applying the commutator three times gives $2k_{a;bc}=(C_a{}^d{}_{bc}+C_c{}^d{}_{ba}-C_b{}^d{}_{ac})k_d=-2C_{abc}{}^d{k}_d$, where the last step uses the first Bianchi identity. Switching the sign of the Riemann curvature tensor switches the sign in this identity; moving a raised index by ordinary metric contraction does not switch it.

Every Killing vector field $K$ preserves the scalar curvature: ${\mathcal{L}}_{K}R=K^{\alpha }{\nabla }_{\alpha }R=0$. One index proof rewrites this derivative as a doubly contracted curvature commutator acting on the antisymmetric tensor ${\nabla }_{[\mu }{K}_{\lambda ]}$; both terms vanish by symmetry of the Ricci tensor.