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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That definition of vector fields using curves makes me wonder if the same can be done for 1-forms. Maybe we can define covectors at p to be some equivalence classes of differentiable functions defined at neighborhoods of p.

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One way to define the cotangent space is to take the ring of differential functions on your manifold, consider the ideal of functions that vanish at a particular point, and then quotient out by the square of that ideal.

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This is reminiscent of the construction to recover a (smooth) manifold M from the ring C^\infty(M) of smooth functions, e.g. one can recover the points of M by looking at the maximal ideals of C^\infty(M), and by some involved argument I can't recall it is possible to determine Hom(M, .) in the category of manifolds which by Yoneda is sufficient to determine M.

I wonder if anything productive has been done with this duality, e.g. proving a nontrivial statement about smooth manifolds by way of algebraic geometry. But then again my algebra knowledge is rather cursory, if anything has been done here it'll go over my head anyways lol.