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I'm not a geometer, but I'm currently trying to relearn some basic geometry. My question is: What is exactly the difference (if there is any!) between the codifferential simply as the pullback of a smooth map, and the codifferential of forms on an orientable manifold? By suspicion is that since the differential of a smooth map can be thought of as a special case of the exterior derivative, then the former case of the codifferential is just the lowest-degree case for the codifferential of forms.

Just to be concrete, if $X,Y$ are smooth manifolds and $f: X \rightarrow Y$, the first definition of codifferential (or pullback) is the map $\delta f: T^{\ast}_{f(x)} Y \rightarrow T^{\ast}_x X$ such that if $w \in T^{\ast}_{f(x)} Y, v \in T_x X$, then $\delta f(w)(v) = w(df(v))$.

The other definition of codifferential I've encountered is some formula involving the Hodge star (something like $\delta = \pm \star d \star$) and $\delta: \Lambda^k(X) \rightarrow \Lambda^{k-1}(X)$ lowers the degree of the form by 1. This second definition of codifferential for forms is much more mysterious to me, so I was wondering whether somebody could clear up for me what exactly the connection is here between this codifferential and the pullback of smooth maps, which in some literature is also called the codifferential. Thanks!

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There is no relation, really. The first is just named like that because it is the contravariant version of the usual differential, while the second, because it the transpose (w.r.t the star) of the exterior differential, a rather different beast. –  Mariano Suárez-Alvarez Sep 13 '12 at 22:02
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The codifferential $\delta f: T^*Y\to T^*X$ is the image of $f:X\to Y$ under the contravariant functor $T^*$. It is a map of vector bundles, makes no reference to choice of metric, and arises naturally in the smooth category.

The codifferential $d^*:\Omega^kX\to \Omega^{k-1}X$ is the adjoint of the differential $d:\Omega^{k-1}X\to\Omega^kX$ with respect to the $L^2$ inner product $\langle \omega,\eta\rangle_{L^2} = \int_X \omega\wedge *\eta$ ($*$ denotes the Hodge map). It is a map of pre-Hilbert spaces. Its definition requires a metric (one uses orthonormal frames to define $*$, for instance) and it naturally arises in the context of the de Rham complex for a smooth manifold after a choice of Riemannian metric has been made.

As Mariano Suarez-Alvarez says in the comments, there isn't really any relation between the two (at least, he's right to my knowledge). It's just a case of two different objects being called the same name.

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