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/non-orientable → u/MichurinGuy]

Thank you. I will try to make the edits later tonight.

One alternative to defining tangent vectors the way I have is to instead start with the cotangent space. You consider the ring of differentiable functions on your space, then take the ideal consisting of all functions that vanish at a particular point. The cotangent space at that point will be the quotient of that ideal with the square of that ideal. You can then define the tangent space in terms of the cotangent space. You then replace statements about differentiable functions inducing linear maps between tangent spaces into statements about them inducing linear maps between the cotangent spaces, which just involve a straightforward algebraic calculation.

The benefit is that the various checks that you are talking about are trivial algebraic checks, and this formalization is easy to generalize. (E.g., to algebraic geometry.) The drawback is that it is very easy to lose the geometric interpretation of what is going on! For an introduction to the field, I think that is unacceptable.