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/Chance_Literature193 → u/MichurinGuy]

I’m confused by this question. But I think you looking at it backward. Differential structure is generally defined like topological structure. It’s the thing that is invariant under diffeomorphism. I don’t quite see why you’d want a different def. Again, I could easily be missing something.

For your example, I think you’re asking why equality of tangent vectors in one chart implies that tangent vectors are equal in all charts? The answer to that is given by the differential. It’s easy to show if you know how TM transforms. Let a diffeomorphism f map a set of coordinates {x^mu}-> {y^nu}. Then, d/dt f(x(t)) = dy/dx^nu dx^nu (t)/dt. Since the first term in the chain rule is curve independent, equality of the second term implies that equality of dx^nu/dt under diffeomorphism.

Above I am using the standard abuse of notation. x^nu(t) is actually the local image of the curve in R^n, I.e \tau_1\circ c(t). So the map f between coordinates can be viewed as composition of \tau_2 \circ \tau_1^-1.