Codifferential of 1-form associated to a Killing field vanishes?

[dongo]

Suppose you are given a Killing field $\xi$ on a manifold $(M, g)$. We can associate to this vector field a 1-form via the metric. Denote this 1-form by $\eta$. Show that the codifferential of the 1-form vanishes, i.e., $\delta\eta = 0.$



As far as I know, a Killing field is a vector field on a manifold whose local flow consists of local isometries. The more pragmatic to check wether $\eta$ is Killing or not is to calculate the Live derivate of the metric of the manifold with respect to the vector field in question. It has to be zero, i.e. $L_{\eta}(g) = 0$. The codifferential of the usual differential operator is defined as the formally adjoint operator to the differential, where one can explicitly calculate it to be $\delta = (-1)^{nk}\ast d \ast = -\sum_{i=1}^{n}\iota_{e_{i}}\nabla_{e_i}$. Here, $n = dim(M)$ and the $e_{i}$ form an orthonormal basis with respect to the metric $g$. The codifferential acts on the space of k + 1 -forms and sends them to k - forms (on M), i.e., $\delta : \Omega^{k+1} \rightarrow \Omega^{k}.$[itex] <br /> <br /> <br /> <b>Since the question arose in a book on Kähler geometry, more precisely in asection where holomorphic vector field on Kähler-Einstein manifolds, I assume, it suffices to treat this case. However, in the general case, we know that the 1-form, associated to a real holomorphic vector field, can be (some sort of) hodge decomposed in three parts, namely $/ix = dh + d^{c}f + /ix^{H}$ where the first term on the right-hand side involves the usual differential. the second is the twisted differential, i.e., [itex]d^{c}=$\sum_{i=1}^{n}Je_{i}\wedge \nable_{e_{i}}$, and the last component is the harmonic part. Since we deal with a Kähler manifolds, and we know, that Killing fields are (real) holomorphic, we can use that the Laplace-operator on M is equal to: $\Delta = 2\Delta^{\bar{\partial}}$. Note that this is only the case on Kähler-manifolds. And this is basically the point from which on I ahve no clue any longer. I'd appreciate hints rather solutions as I would ultimately like to come (with a little help of course) to a solution on my own.[/itex]</b>$<br /> By the way, this is my first post, so if you have any recommendations concerning typesetting etc. you're welcome to utter them. Especially TEX help would be valuable because I have almost no experinece when it comes to that.<br /> <br /> <br /> dongo$[/itex]

 

[dongo]

No need to reply, got a hang on it. We know that Killing fields are holomorphic, and we know that when it comes to the Dolbeaut decomposition, a (p,0) form is holomorphic if and only if it is Dolbeaut-harmonic. But this actually implies that the codifferential vanishes. (This is a complex analogon to the Hodge-decomposition). As we have on Kähle manifolds that being Dolbeaut-harmonic implies being harmonic w.r.t to the usual Laplace operator, we have the solution (at least for Kähler manifolds, but I am satisfied with that ;) )