I have a question about the directional derivative of the Ricci scalar along a Killing Vector Field. What conditions are necessary on the connection such that [itex]K^\alpha \nabla_\alpha R=0[/itex]. Is the Levi-Civita connection necessary? I'm not sure about it but I believe since the Lie derivative is more 'fundamental' than the covariant derivative it might be not necessary to have a Levi-Civita connection, but maybe I'm just conjecturing nonsense. Hope anyone can help me find an answer.