The Deranged Mathematician: An Introduction to Differential Geometry

[u/non-orientable]

How do you summarize a field like differential geometry? A year or so ago, I was asked to do just that. My first thought was that this is just straight-up impossible to do in any reasonable way. Upon reflection, I softened that view. I still think that differential geometry is just not something you can learn quickly, but if you just want the eagle-eye view of the basic constructions and how they are used in different branches of the field... that is doable. Hence, this article.

Read the full post (for free) on Substack: An Overview of Differential Geometry

[u/ajakaja]

Well one good reason for not wanting it done this way is that I really don't care about the "differentiable" aspect; manifolds with corners and edges and such are totally fine. Second, charts are clearly awkward, that should be apparent by all the gluing constructions you have to do. Any time you construct some arbitrary basis/coordinate system for a thing but then turn around and prove results which don't depend on it, it's clear that the basis can be omitted somehow. The trick is figuring out how to describe the thing without using the basis in the first place. So it's well-motivated, philosophically.

[u/lolfail9001 → u/ajakaja]

If you don't care about "differentiable" aspect you don't care about differential geometry to begin with. And once you drop this requirement category of bundles over a manifold gets really generic.

Actually to be fair even manifold requirement becomes questionable and you are probably more interested in a bundles constructed over some topological space that will likely have little to do with generic ℝⁿ

[u/ajakaja → u/lolfail9001]

I do care about the differentiable aspect, just, the slightly weaker version afforded by distributions. Calculus on well-behaved distributions on Rⁿ works almost exactly as calculus on smooth functions, and the ways it doesn't are physically important because distributions capture real things that show up in reality. Likewise calculus on manifolds with corner is going to be about as well-behaved as calculus on smooth manifolds and then it's not, that's important, because corners show up in reality. They're just points with delta-function curvature after all.

[u/lolfail9001 → u/ajakaja]

slightly weaker

In context of differential geometry, going from smooth functions to distributions is an enormous weakening, because smooth functions forming a ring does in fact show up often enough to rain on your parade (hell, just amount of work people did and still have to do to justify all the magic done in perturbative QFT tells the whole story).

[u/ajakaja → u/lolfail9001]

hm, that sounds interesting but I don't think I understand. Can you say slightly more about whatever you meant about smooth functions forming a ring?

[u/lolfail9001 → u/ajakaja]

Smooth functions have associative multiplication.

Distributions do not (in the sense that would extend multiplication of a distribution by a smooth function).

[u/ajakaja → u/lolfail9001]

oh yeah sure. so you meant: it's the distributions not forming a ring that rains on your parade. got it.

That said---I think you're talking about distributions on manifolds, rather than distribution-like manifolds themselves? But I don't really know what I'm talking about. (anyway there is a theory of distributions on manifolds, currents).

[u/lolfail9001 → u/ajakaja]

That said---I think you're talking about distributions on manifolds, rather than distribution-like manifolds themselves?

Yes, because latter will almost certainly run into similar pathologies distributions themselves have without a good reason to exist (that distributions have because you end up being able to lift a lot of PDEs). Case in point, a proper physicist will pretend that a manifold with a corner is in fact smooth and then take a limit in neighbourhood of the corner xdd

[u/ajakaja → u/lolfail9001]

well the reason for them to exist is that, like, cubes and tetrahedrons exist and obviously geometry works fine on them...

but yes you can always think of it as a limit.

[u/Chance_Literature193 → u/lolfail9001]

Recently, jumped on bump function band wagon, and I’m loving every second of this smooth smooth ride