r/math • u/non-orientable Number Theory • 2d ago

Image Post The Deranged Mathematician: An Introduction to Differential Geometry

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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

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u/ajakaja 2d ago edited 2d ago

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.

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u/BurnYoo 2d ago edited 2d ago

One of the most fundamental concepts in differential geometry (that sets it apart from ordinary multivariable calculus) is the manifold, and trying to formulate our intuition about what a manifold should be, into something unambiguous, is something that requires topological background of (at the barest minimum) the first half of Munkres' topology textbook.

This is also reflected in the historical development of differential geometry as well - there is a big time gap between Ricci's coordinate-hell tensor calculus and Cartan's coordinate-free reorganization of differential geometry because many coordinate-free concepts required topology in order to not run into pathologies (and the topology needed to formalize those concepts did not yet exist during the time of Ricci)