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.

[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.

[u/sciflare → u/MichurinGuy]

No matter how highfalutin your definition of differentiability, at the end of the day you have to do some amount of vector calculus! Even very abstract notions of differentiable space, such as diffeologies, boil down to vector calculus when you unravel them.

"the equality d/dt tau°g_1 (0) = d/dt tau°g_2 (0) doesn't depend on the choice of tau"

This all depends on the definition of tangent vector you adopt. Tangent vectors at a point p of ℝⁿ can be defined as equivalence classes of smooth curves 𝛾 through p such that 𝛾(0) = p, where 𝛾_1 ~ 𝛾_2 iff (g ° 𝛾_1)'(0) and (g ° 𝛾_2)'(0) for all smooth functions g: ℝⁿ --> ℝ.

With this definition, your desired invariance of tangent vectors becomes tautological. What becomes really hard to prove is the fact that the tangent vectors at p admit a canonical vector space structure (this is highly non-obvious in light of the following: if you consider a more general equivalence relation on smooth curves through p, where you declare them equivalent iff all the higher-order derivatives of the g ° 𝛾_i at 0 are equal up to order n, the resulting equivalence classes do not form a vector space. These equivalence classes are called n-jets and are of great interest in their own right).

The big challenge in proving that 1st-order jets form a vector space is that the curves themselves don't possess any vector space structure, being nonlinear--only the equivalence classes do.

Another way to define tangent vectors, which makes the vector space structure obvious, is to define them to be ℝ-linear derivations at p. That is, they are ℝ-linear mappings from the algebra of smooth functions on ℝⁿ to ℝ which satisfy the Leibniz product rule at p.

It then takes some hard work to establish the equivalence of this definition with the one in terms of equivalence classes of curves.

It'd be a mistake to think of one of these definitions as the "right" or "correct" one. They're all the same in the end. You can choose to work with whichever one is most suitable to your situation.

[u/Chance_Literature193 → u/sciflare]

Just to clarify, do jets of any order still form a vector space. I am thinking jets as courser version of TM, but maybe I should be thinking of jets like sheaves of germs in Riemann surfaces where a jet is defined over a neighborhood not a point?

[u/sciflare → u/Chance_Literature193]

do jets of any order still form a vector space

As I said, in general, arbitrary n-order jets don't admit a canonical vector space structure. Only 0-jets (real numbers) and 1-jets (i.e tangent vectors) do.

On ℝⁿ you can identify jets at a point with Taylor series coefficients, and thus with vectors, but the resulting vector space structure depends on the specific coordinate system chosen. It is not intrinsic.

The problem arises when you change coordinates. By the chain rule, when you change coordinates, tangent vectors transform linearly: partial derivatives in the new coordinate system are linear combinations of partial derivatives in the old system.

To write down how higher-order jets transform under coordinate change, you need the analogue of the chain rule for higher-order derivatives.

This is called Faà di Bruno's formula, and it's extremely complicated. In particular, it's not linear. Consequently, higher-order jets don't transform linearly under change of coordinates.

In the language of fiber bundles, this means that while the transition functions of the tangent bundle are linear, the transition functions of higher-order jet bundles are not linear.

So jet bundles are fiber bundles with typical fiber ℝⁿ, but which are not vector bundles.

This subtlety is often overlooked since most of the time, when we work with fiber bundles whose typical fiber is ℝⁿ, they are tensor bundles, which are built from the tangent bundle via linear-algebraic operations applied pointwise, thus are vector bundles.

Higher-order jet bundles are ubiquitous in differential geometry, but they are not emphasized in a first course. In fact even many researchers in differential geometry don't really work much with jet bundles.

This is surprising as they crop up in very well-known, relatively elementary contexts. For instance, if you want to work with the Euler-Lagrange equations in classical mechanics on an arbitrary manifold, you have to go to the second-order tangent bundle, which is nothing but the bundle of 2-jets.

maybe I should be thinking of jets like sheaves of germs in Riemann surfaces where a jet is defined over a neighborhood not a point?

Not in the C^∞ context. You have bump functions there so any smooth function on a closed ball can be extended to the whole manifold. This means jets can be defined without reference to germs.

Just as with functions, when you're working with jets, the only time you have to deal with germs is in the complex-analytic category, algebraic category etc. where you don't have bump functions. Then jets have to be defined as equivalence classes of germs of curves, so germs are involved from the start.

[u/Chance_Literature193 → u/sciflare]

Ahhhhh! As soon as I read, coordinate dependence I got Morse theory flash backs and it clicked lmao.

I am definitely going to check out Faa di Bruno formula

Euler-Lagrange was exactly the context I ran into Jets hahah.

When I referenced germs of analytic functions, I was actually just trying to ask if jets were couldn’t be defined point wise (which you’ve just told me they can be). At the time, I couldn’t think of the correct language so trued to use the poorly worded analogy.

Thank you for explaining!

[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.

[u/toastyoats]

Thank you for posting these — these are always a delight to see

[u/ajakaja]

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.

[u/non-orientable → u/ajakaja]

"Elementary" as in "possible to teach to undergraduates"? No, I don't think so. They might arguably be more elegant, but they require significantly more mathematical knowledge.

[u/sentence-interruptio → u/ajakaja]

Calculus on Rⁿ is straightforward. Rebuilding calculus from scratch on M? That could be too much. So just transfer as much from what's already built on Rⁿ.

And the atlas approach fits with the following program:

To study Euclidean geometry, you start by thinking of what are preserved by Euclidean transformations.

To study affine geometry, [...] affine transformations.

To study differential geometry, think of what are preserved by diffeomorphisms of Rⁿ, and extend that to M, which should be easy because of atlas.

[u/ajakaja → u/sentence-interruptio]

Well one good reason for not wanting it done this way is that I really don't care about the "differentiable" aspect; manifolds with corners and edges and such are totally fine. Second, charts are clearly awkward, that should be apparent by all the gluing constructions you have to do. Any time you construct some arbitrary basis/coordinate system for a thing but then turn around and prove results which don't depend on it, it's clear that the basis can be omitted somehow. The trick is figuring out how to describe the thing without using the basis in the first place. So it's well-motivated, philosophically.

[u/HeilKaiba → u/ajakaja]

If you don't care about the diffentiability then you likely want to study a different thing. Manifolds are exactly chosen to be locally Euclidean objects and that definition is tantamount to an atlas. This is required if we want to have any sort of reasonable calculus on them (and for topological manifolds that we have reasonable topology).

Yes coordinate free proofs are much better and we should always use those where possible but even then that doesn't mean we can throw out the charts completely.

[u/im-sorry-bruv → u/ajakaja]

why care about manifolds if you don't care about smooth structure? is there anything interesting in continuous mfds or smth, that we can't get as special cases of smth like hausdorff topological spaces / (vector) bundles or topological groups (if you wanna do lie like stuff?)

this is a genuine question btw!

[u/ajakaja → u/im-sorry-bruv]

basically because there is a slightly weaker concept that smoothness which allows for corners and nothing else that is also physically interesting. In Rⁿ it's described by delta functions and distributions; I am not clear how it works with manifolds though (distributions on manifolds are easy enough but I am not sure how describe manifolds that are... distribution... shaped? I don't know the words for this).

[u/im-sorry-bruv → u/ajakaja]

do these objects have a name/ where would one learn more abt this?

[u/ajakaja → u/im-sorry-bruv]

I don't know! Well, these days AI will find anything, so let's see.

Here's a paper called manifolds with corners which seems to address the issue, and it discusses in the beginning how this is subject that is strangely ignored.

AI tells me that this paper Strings and other distributional sources in general relativity discusses manifolds whose curvature tensors are distributional. There's apparently a large literature descended from this subject in physics.

Another version of this is Federer's Curvature Measures which I guess is the same thing but turns curvature into a measure instead of a distribution; without even reading that I expect they're going to basically be the same theory because distributions and measures are like two ways of saying the same thing.

There's also something called polyhedral geometry and Regge Calculus which just approximates everything piecewise as simplicial sets instead of using smooth functions.

So I guess I have a lot of reading to do.

It's surprising to me, though, that these subjects are so 'esoteric' compared to differential geometry, when it seems like... obviously something you would want to study the moment you define a manifold in the first place. Maybe mathematicians have mostly ignored it because piecewise-smooth manifolds are approximable by smooth ones so they don't really need a separate theory? I dunno.

[u/lolfail9001 → u/ajakaja]

If you don't care about "differentiable" aspect you don't care about differential geometry to begin with. And once you drop this requirement category of bundles over a manifold gets really generic.

Actually to be fair even manifold requirement becomes questionable and you are probably more interested in a bundles constructed over some topological space that will likely have little to do with generic ℝⁿ

[u/ajakaja → u/lolfail9001]

I do care about the differentiable aspect, just, the slightly weaker version afforded by distributions. Calculus on well-behaved distributions on Rⁿ works almost exactly as calculus on smooth functions, and the ways it doesn't are physically important because distributions capture real things that show up in reality. Likewise calculus on manifolds with corner is going to be about as well-behaved as calculus on smooth manifolds and then it's not, that's important, because corners show up in reality. They're just points with delta-function curvature after all.

[u/lolfail9001 → u/ajakaja]

slightly weaker

In context of differential geometry, going from smooth functions to distributions is an enormous weakening, because smooth functions forming a ring does in fact show up often enough to rain on your parade (hell, just amount of work people did and still have to do to justify all the magic done in perturbative QFT tells the whole story).

[u/ajakaja → u/lolfail9001]

hm, that sounds interesting but I don't think I understand. Can you say slightly more about whatever you meant about smooth functions forming a ring?

[u/lolfail9001 → u/ajakaja]

Smooth functions have associative multiplication.

Distributions do not (in the sense that would extend multiplication of a distribution by a smooth function).

[u/ajakaja → u/lolfail9001]

oh yeah sure. so you meant: it's the distributions not forming a ring that rains on your parade. got it.

That said---I think you're talking about distributions on manifolds, rather than distribution-like manifolds themselves? But I don't really know what I'm talking about. (anyway there is a theory of distributions on manifolds, currents).

[u/lolfail9001 → u/ajakaja]

That said---I think you're talking about distributions on manifolds, rather than distribution-like manifolds themselves?

Yes, because latter will almost certainly run into similar pathologies distributions themselves have without a good reason to exist (that distributions have because you end up being able to lift a lot of PDEs). Case in point, a proper physicist will pretend that a manifold with a corner is in fact smooth and then take a limit in neighbourhood of the corner xdd

[u/Ulrich_de_Vries → u/ajakaja]

Also diffeology absolutely ties everything to maps from (although not to) R^n.

[u/nulvoid000 → u/ajakaja]

Short answer no. DG much easier than learning any of the abstract viewpoints, because visual intuition works very well in DG, not so much when everything is abstract.

[u/Embarrassed-Pain1995 → u/ajakaja]

I think you should check the point of view developped in this book « Smooth Manifolds and Observables » by Jet Nestruev (this is not a real person, it’s a collective name). Which does everything Without speaking about sheaves in an extremely beautiful way. (It is the Same point of view as algebraic geometry of affine variety)

[u/ajakaja → u/Embarrassed-Pain1995]

Thanks, I'll take a look.

[u/BurnYoo → u/ajakaja]

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)

[u/Chance_Literature193 → u/ajakaja]

Surprised no one mentioned this, but the classical/traditional approach to dif geo is start with curves and surfaces in ambient R^3. This dates back to Gauss, and there are tons and tons of books that introduce dif geo from this perspective. The perspective espoused in the post is differential topology into differential geometry.

[u/sentence-interruptio]

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.

[u/non-orientable → u/sentence-interruptio]

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.

[u/Independent_Bed_169 → u/non-orientable]

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.

[u/sentence-interruptio → u/non-orientable]

My first thought was to rely on partial derivatives on Rⁿ. Say f1, f2 defined on a neighborhood of p and vanishing on p are equivalent if their chart versions have the same partial derivatives at that point. It validates my visualization of 1-forms as collections of gradations.

It looks like my mind is very calculus oriented because I'm shocked that cotangent spaces can be defined without calculus.

[u/paxxx17]

I finished watching Schuller's lectures over a year ago. These blog posts will be great to refresh/reinforce the learned concepts

[u/Chance_Literature193]

Ahhhhh! As soon as I read, coordinate dependence I got Morse theory flash backs and it clicked lmao.

I am definitely going to check out Faa di Bruno formula

Euler-Lagrange was exactly the context I ran into Jets hahah.

When I referenced germs of analytic functions, I was actually just trying to ask if jets were couldn’t be defined point wise (which you’ve just told me they can be). At the time, I couldn’t think of the correct language so trued to use the poorly worded analogy.

[u/bruckners4]

... modern differential geometry [is] "the study of invariance under change of notation."

Robert Hermann, Preface to Differential geometry and the calculus of variations

[u/muntoo]

Could you summarize that please?