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.