Perhaps the question is quite easy to answer, however, I am encountering a total blackout on it. If you have given a Killing field $ zeta $ on a Riemannian manifold, you can associate via the metric a 1-form to it, namely $ eta(X) = g(zeta, X) $. Th task is now to show that the codifferential, i.e. the adjoint to the usual differential, vanishes. I have absolutely no idea how to address the problem and would appreeciate any answers.