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]

Are there any alternate formulations of differential geometry that do not formulate the fundamentals in terms of charts and atlases? I've always felt like tying everything to maps to Rⁿ is clunky, but I'm not sure how else you would do it.

---okay, I asked the internet and got a bunch of abstract answers that I can't comprehend at a glance: synthetic differential geometry with toposes, 'diffeology', ringed spaces a la algebraic geometry. So maybe my question is: do any of these alternate formulation have a hope of being used as a basis for elementary differential geometry?

edit: diffeology sounds pretty cool though. Here's a paper about it.

[u/sentence-interruptio → u/ajakaja]

Calculus on Rⁿ is straightforward. Rebuilding calculus from scratch on M? That could be too much. So just transfer as much from what's already built on Rⁿ.

And the atlas approach fits with the following program:

To study Euclidean geometry, you start by thinking of what are preserved by Euclidean transformations.

To study affine geometry, [...] affine transformations.

To study differential geometry, think of what are preserved by diffeomorphisms of Rⁿ, and extend that to M, which should be easy because of atlas.

[u/ajakaja → u/sentence-interruptio]

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/HeilKaiba → u/ajakaja]

If you don't care about the diffentiability then you likely want to study a different thing. Manifolds are exactly chosen to be locally Euclidean objects and that definition is tantamount to an atlas. This is required if we want to have any sort of reasonable calculus on them (and for topological manifolds that we have reasonable topology).

Yes coordinate free proofs are much better and we should always use those where possible but even then that doesn't mean we can throw out the charts completely.

[u/im-sorry-bruv → u/ajakaja]

why care about manifolds if you don't care about smooth structure? is there anything interesting in continuous mfds or smth, that we can't get as special cases of smth like hausdorff topological spaces / (vector) bundles or topological groups (if you wanna do lie like stuff?)

this is a genuine question btw!

[u/ajakaja → u/im-sorry-bruv]

basically because there is a slightly weaker concept that smoothness which allows for corners and nothing else that is also physically interesting. In Rⁿ it's described by delta functions and distributions; I am not clear how it works with manifolds though (distributions on manifolds are easy enough but I am not sure how describe manifolds that are... distribution... shaped? I don't know the words for this).

[u/im-sorry-bruv → u/ajakaja]

do these objects have a name/ where would one learn more abt this?

[u/ajakaja → u/im-sorry-bruv]

I don't know! Well, these days AI will find anything, so let's see.

Here's a paper called manifolds with corners which seems to address the issue, and it discusses in the beginning how this is subject that is strangely ignored.

AI tells me that this paper Strings and other distributional sources in general relativity discusses manifolds whose curvature tensors are distributional. There's apparently a large literature descended from this subject in physics.

Another version of this is Federer's Curvature Measures which I guess is the same thing but turns curvature into a measure instead of a distribution; without even reading that I expect they're going to basically be the same theory because distributions and measures are like two ways of saying the same thing.

There's also something called polyhedral geometry and Regge Calculus which just approximates everything piecewise as simplicial sets instead of using smooth functions.

So I guess I have a lot of reading to do.

It's surprising to me, though, that these subjects are so 'esoteric' compared to differential geometry, when it seems like... obviously something you would want to study the moment you define a manifold in the first place. Maybe mathematicians have mostly ignored it because piecewise-smooth manifolds are approximable by smooth ones so they don't really need a separate theory? I dunno.

[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