Let $(M,g)$ be a Riemannian manifold and $V$ a unit Killing vector field on it. Under what condition on curvature tensor the following equation hold:
$$\nabla\_VQ=0,$$
where $Q$ is the Ricci operator defined as $g(QX,Y)=\rho(X,Y)$.


**Update1:** Einstein metrics that admit a unit Killing vector field satisfies the above equation.


**Update2:** Can anybody give an example other than Einstein manifolds that satisfy the above equation?



Under what condition the covariant derivative of Ricci operator along Killing vector field vanish?
If $V$ is a Killing vector, then necessarily its Lie derivative annihilates all curvature tensors, including the Ricci operator, namely
$$
 \mathcal{L}\_V Q = \nabla\_V Q + Q J\_V - J\_V Q = 0 ,
$$
where $J\_V$ is the ("Jacobian") endomorphism on tangent vectors defined by $\nabla\_U V = J\_V U$. Thus, the condition that you want seems to be just $Q J\_V - J\_V Q = 0$. I don't see how the unit condition on $V$ enters.


