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On page 8 of Kobayashi & Nomizu, Volume I, it is stated that $d(f^{*} \omega) = f^{*}(d \omega)$, that is, the exterior derivative on forms is a natural operation. Of course, one can show this simply by writing out the definition in terms of coordinates and simply computing the pullback etc. (see the question here).

My question is: is there a more conceptual way to see this from the characterization of the exterior derivative given in K&N? I feel I am missing something here.

More generally, the same applies for the Lie derivative, Lie bracket etc. - is there a more conceptual way of seeing that such operations are natural?

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TJ2
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1 Answers1

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I will stick in my answer to the question of naturality under local differemorphisms. (The natruality condition for the exterior derivative is stronger, since it hods for arbitrary smooth functions, but this is a different story.) Naturality under local diffemorphisms is equivalent to the fact that the operation is independent of the choice of local coordinates. This is because both local parametrizations (i.e. the inverses of local charts) and chart changes are local diffeomorphisms, and conversely any local diffeomorphism can be interprested as a local change of coordinates.

Hence from my point of view a conceptual explanation of the invariance of an operation is that it can be described in a way which is manifestly independent of coordinates. For Lie derivatives, such a description can be given via flows of vector fields (which are manifestly independent of coordinates, since they can just be described by integral curves). Denoting the flow of a vector field $\xi$ by $\phi_t$ and the pullback of a tensor field $s$ by $(\phi_t)^*s$, one can consider $(\phi_t)^*s(x)$ as a smooth curve in the fiber at $x$ of the tensor bundle in question, and its derivative in $t=0$ equals $(\mathcal L_\xi s)(x)$. Applied to vector fields, one gets $\mathcal L_{\xi}\eta=[\xi,\eta]$ and naturality of the Lie bracket follows. Naturality of the exterior derivative can then be deduces from the global formula of for the exterior derivative (which only uses the Lie derivative of smooth functions and the Lie bracket of vector fields) or from the Cartan formula (since naturality of insertion operators is obvious). This point of view is developed in detail in the book by Kolar, Michor and slovak, which is available here.

Andreas Cap
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