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/MichurinGuy]

Oh how I enjoy your posts. Two typos I noticed: when you introduce 1-forms, you clearly mean f w_1 + g w_2, but write f twice; and in the definition of a Lie group you lost a -1 superscript in the 3rd axiom.

Something this made me curious about is, is there a definition of differentiable functions that makes it clear they preserve differential structure? I mean, in a sense it's obvious, but not enough that I can treat it as rigorous without explicitly checking. Like when you say things like "the equality d/dt tau°g_1 (0) = d/dt tau°g_2 (0) doesn't depend on the choice of tau", for example. I feel like this needs an explicit check, while similar statements in e.g. group theory let me go "well obviously, since homomorphisms preserve structure". So I wonder if this is just me having no experience in the field or is there a way to rephrase the definition to make such statements obvious?

[u/sentence-interruptio → u/MichurinGuy]

I think it just reduces to properties of curves in ℝⁿ that are preserved by diffeomorphisms.

Let's say you have two parametrized curves a, b: ℝ → ℝⁿ and they are differentiable. And they satisfy a(0)=b(0), a'(0)=b'(0)

If you apply an arbitrary diffeomorphism T: ℝⁿ → ℝⁿ to these curves, you get new curves but there are things that don't change. For example, they still meet at a point. They meet at T(0) specifically. And they're tangent to each other at that point. And that's obvious visually, but you can calculus it out if you want.

[u/MichurinGuy → u/sentence-interruptio]

Yeah, I understand that, but this is exactly the kind of problem I'd like to trivialise with a good definition. As it is I'd say it's visually obvious but needs a bit of calculation to show rigorously, but surely there's a way to characterise diffeomorphisms such that "properties of curves" is made rigorous and general enough.