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Stream: learning: questions

Topic: Exterior Derivative

Sam Tenka (May 03 2020 at 23:08):

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${\color{red}^\star}$ 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 ${\color{red}^\circ}$, 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!

${\color{red}^\star}$ Actually, the uniqueness statement has some caveats:

• we must of course permit scaling by constants
• there are also natural zero maps and natural identity maps for appropriate pairs of functors
• the tensor of two natural maps gives a natural map (between tensored versions of the original functors), so really the statement is that the exterior derivative (and identity and zero) generate the natural maps.
• and for some reason, the result doesn't seem to include the lowest dimension: that of going from functions to one-forms

${\color{red}^\circ}$ e.g. general bundles, not just vector bundles; non-tensorial transformations under actions of jet groups, not just tensorial transformations under GLn

John Baez (May 04 2020 at 05:01):

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.

John Baez (May 04 2020 at 05:01):

That sort of condition is the sort of thing a differential geometer can sink their teeth into and start to prove things about.

John Baez (May 04 2020 at 05:03):

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 $\mathbb{R}$-bundle over to the functor sending any manifold to its cotangent bundle.

John Baez (May 04 2020 at 05:04):

That seems like a somewhat manageable thing to prove.

John Baez (May 04 2020 at 15:34):

The easy part is to see that the exterior derivative of a function is natural - as Rongmin points out, this is just the

$\phi^* (df) = d (\phi^* f)$

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.

Sam Tenka (May 04 2020 at 16:30):

Thanks for the hints! Yes, I think these are enough clues for me to start working on it as an exercise :slight_smile: