In the paper Natural Operations on Differential Forms, the author R. Palais shows that the exterior derivative $d$ is characterized as the unique "natural" linear map from $\Phi^p$ to $\Phi^{p+1}$ (Palais' $\Phi^p$ is what is perhaps more commonly written as $\Omega^p$, and "commutes with all diffeomorphisms", I believe, means $f^*(d\omega) = d(f^*\omega)$):
the exterior derivative on $p$-forms is determined to within a scalar factor by the condition that it be a linear mapping into $p+1$ forms which commutes with all diffeomorphisms.
I've tried to read the proof in the paper, but I'm struggling to follow the details and missing a sense of the big picture of the proof.
I'm interested in Palais' claim because this characterization is the most compelling one I have seen $-$ it seems far more appropriate as an axiomatic definition of $d$ than the definitions found in many textbooks, which often define $d$ based on properties such as
$d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta$ (where $\alpha$ is a $k$-form),
$d^2=0$,
or $d(\sum \omega_{i_1…i_k}dx_{i_1} \wedge \cdots \wedge dx_{i_k}) = \sum d\omega_{i_1…i_k} \wedge dx_{i_1} \wedge \cdots \wedge dx_{i_k}$.
While these are indeed quite basic properties of $d$, they are more appropriate as theorems than as a priori assumptions. (Of course, what is more "natural" is a matter of opinion, so please don't belabor the issue.)
Since it's such an innocent-looking and natural characterization, I would like to see a clear, motivated, reasonably elementary proof of it. Why ought it to be true that there is only one natural linear map from $\Omega^p$ to $\Omega^{p+1}$, up to a constant multiple? What are the key steps of a proof?
Best Answer
I think this might be a fairly intuitive way of understanding this statement. You are defining the operator $d$ by the property $f^*(d\omega)=df^*(\omega)$ for all forms $\omega$ and all diffeomorphisms $f$. A more general linear map from $\Lambda^p$ to $\Lambda^{p+1}$ (I like this notation for the exterior p forms) would be a general affine connection $\nabla$. Now, using your defining identity twice, we find \begin{equation} f^*(d^2\omega)=d^2f^*(\omega)=0 \end{equation} which is clearly true of the exterior derivative. For a more general connection, we have \begin{equation} f^*(\nabla^2\omega)=f^*(\Omega \omega)=\tilde{\nabla}^2 f^*(\omega)=\tilde{\Omega}f^*(\omega), \end{equation} where $\Omega$ is the curvature of the connection, and $\tilde{\nabla}$ and $\tilde{\Omega}$ denote the induced connection and curvature under the diffeomorphism. Clearly, in general the form of the induced curvature of the connection depends on the diffeomorphism. However, in the case where the curvature is zero, the curvature remains form invariant under all diffeomorphisms.
So we have established a relationship between your definition and the more usual requirement that $d^2=0$. As $d$ is the unique affine connection that satisfies this requirement, we have established the result.