# Codifferential of 1-form associated to a Killing vector field

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.

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The codifferential of the 1-form dual to a vector field is the divergence of the vector field. Now, if $\zeta$ is Killing, then it obeys $$\nabla_a \zeta_b + \nabla_b \zeta_a = 0$$ Now just take the trace. – José Figueroa-O'Farrill Jul 28 '11 at 15:02
Thanks a lot. This was exactly the kind of answer I was looking for. – gggg gggg Jul 28 '11 at 16:55
You're welcome. I found it easier to answer the question than pointing you to the literature. I have the feeling that the question is not quite appropriate for MO, though. In fact, rereading the question makes me somewhat suspicious that this may have been homework, in which case I'd like to apologise to the MO community for this lapse in judgement :( – José Figueroa-O'Farrill Jul 28 '11 at 22:46
No, this is not homework. I do not study, so I teach myself things out of the literature. If I had people to discuss exercises with, it would be much easier, but I haven't, so I have to ask. Normally, I would not have asked here, but in the other forums, such questions are far to specific. If I am no longer allowed to post questions that can boost my understanding of the theory, then I will have to accept it even though I will certainly not be happy about it. – gggg gggg Nov 28 '11 at 16:54