1. The problem statement, all variables and given/known data Given a manifold M with metric gab and associated derivative operator ∇a, let ξ a be a Killing vector on M. Prove that ξ a is an affine collineation for ∇a. 2. Relevant equations (a) For a vector ξ a to be an affine collineation for a derivative operator ∇a, it must satisfy ∇a∇b ξ c = Rmabc ξ m. (b) We are told that ξ a is a Killing vector, so ∇(a ξ b) = 0. (c) The action of the Riemann tensor on a vector is Rabcd ξ c = –2 ∇[a∇b] ξ d. 3. The attempt at a solution I've gone down one blind alley after another. Below I reproduce what seemed like the most promising path I've explored so far. Can somebody help me with the next step, or suggest an alternative path that could be more fruitful? Start with the right-hand side of (a): Rmabc ξ m = gcp Rmabp ξ m.It's a symmetry property of the Riemann tensor that I can switch the first pair of indices with the second: gcp Rmabp ξ m = gcp Rbpma ξ m = gcp gaq Rbpmq ξ m.We've now finagled this into the form of (c): gcp gaq Rbpmq ξ m = –2 gcp gaq ∇[b∇p] ξ q = –2 gcp ∇[b∇p] ξ a.Performing the antisymmetrization, –2 gcp ∇[b∇p] ξ a = – gcp (∇b∇p ξ a – ∇p∇b ξ a) .Now it's easy to use (b) on the first of these two terms: – gcp (∇b∇p ξ a – ∇p∇b ξ a) = + gcp (∇b∇a ξ p + ∇p∇b ξ a).But I still have that second term sitting there! What do I do about it?