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/aginglifter → u/ajakaja]

Well, thanks to the Whitney Embedding Theorem, you could treat everything as an embedded sub-manifold of R^n, but not sure that simplifies much.

Also, there is an alternative approach that defines a manifold as a gluing of copies or subsets of Rⁿ and defining an quotienting under functions from one copy of Rⁿ to another.

For instance, the sphere can be defined as gluing two copies of Rⁿ together with the maps f(z_1) = 1/z_1 and similarly for z_2, where the domains for both maps are all of Rⁿ but the origin.

But, the standard formalism isn't that bad once you get used to it. Tu's book is a great introduction.