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I learned something cool, but I don't know why it's true. My question is about how to learn why it's true.
Specifically: the exterior derivative is the only natural map between (contravariant) functors from Manifolds to Vector Bundles! At least, according to the first page of the preface of "Natural Operations in Differential Geometry" by Kolar, Michor, and Slovak, which seems to treat it as folklore.
It seems that this result appears as a corollary after about 200 pages of material presented in much greater generality , which I alas feel insufficiently sophisticated to absorb. I didn't find a clear paper citation, either, although I might just have missed it in such a large text. Are there papers or blogs etc that prove this result directly?
Thanks!
Actually, the uniqueness statement has some caveats:
e.g. general bundles, not just vector bundles; non-tensorial transformations under actions of jet groups, not just tensorial transformations under GLn
That sort of result is cool. I've looked at that book. If I had to prove it, I'd use the fact that naturality implies such operations must commute with the Lie derivative w.r.t. any vector field - this is an "infinitesimal" statement of naturality.
That sort of condition is the sort of thing a differential geometer can sink their teeth into and start to prove things about.
If I were you, and I really wanted to understand this sort of result, I'd classify all natural transformations from the functor sending any manifold to the trivial -bundle over to the functor sending any manifold to its cotangent bundle.
That seems like a somewhat manageable thing to prove.
The easy part is to see that the exterior derivative of a function is natural - as Rongmin points out, this is just the
equation one sees in most treatments of 1-forms. The harder, more fun part is to figure out all the natural transformations from functions to 1-forms. I really think this is better to figure out than to look up.
Thanks for the hints! Yes, I think these are enough clues for me to start working on it as an exercise :slight_smile: