In Carrol's text, he shows that the covariant derivative of the Ricci scalar is zero along a Killing vector. He then goes on to say something about how this intuitively justifies our notion of geometry not changing along a Killing vector. This same informal reasoning would seem to imply that the Ricci tensor (and Riemann tensor for that matter) is covariantly constant along a Killing vector. However, Carroll has no discussion of this, nor can I find it in any other source (which leads me to think it's probably not true). My question is: Is the covariant derivative of the Ricci tensor zero along a Killing vector? If so, how does one show this? If not, is there a conceptual way of understanding this and/or what is a counterexample? Thanks.