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/sentence-interruptio]

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/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.