Differential Manifold Puns

Of course we're talking about XY not being a vector field on our manifold M, and [X,Y] being a vector field on our manifold M

I am interested in using differential geometry for a project-based ML course that I’m doing next semester. I need to know the place to start learning about this type of ML and whether or not that place is GNN’s.

I think differential geometry is a really cool topic in math. I want to do an ML project that involves this math because of the interest and it also seems like geometry is becoming more prevalent in ML so I want to start learning more about it during this course.

What I’m wondering is if GNN’s are the right method to start looking in to if I’m interested in using manifolds in ML. Is a GNN the best tool to allow me to analyze information embedded in a manifold? (don’t know if this is the right terminology)

If you think using manifolds and differential geometry in a project is too advanced for someone that doesn’t know much about it yet then please tell me and any suggestions on where to start are appreciated.

Thank you in advance for any help.

I have recently started learning about neural representations for a potential undergrad research project this summer and came across the manifold hypothesis and started learning more. The subject seems very interesting to me.

I’m wondering if knowledge of this kind of math would be useful for ML or if there is really no need to get this deep of an understanding? Is the knowledge of manifolds, topology and differential geometry used enough to justify taking a series of courses in university based on these topics?

Thanks in advance!

Generous compensation for any graduate/PhD-level person in math who can take my midterm.

So I am taking a class officially titled "Calculus on Manifolds", this is my first semester of grad school and of my 3 classes this is the one giving me the most trouble. (I am also in graduate topology and linear algebra classes, but I have at least taken these topics in undergrad, this manifold/differential stuff is pretty new to me.) We are using Manfredo do Camo's "Differential Forms and Applications". It isn't my favorite text book, but it seems to cover most of our material in a relatively concise way. For supplement I have been using Fortney's "A Visual Introduction to Differential Forms and Calculus on Manifolds" which is slightly easier to read (and has some nice graphics to help intuit some of the concepts".

Here is the course description " Integration of functions of several variables. Differential forms and differential manifolds, Line integrals, integration on manifolds, Stoke’s Theorem and Poincare’s Lemma. "

I think we might have skipped past the integration of several variables as that is taught in the pre-requisite class for this course so I don't really need a textbook that includes this.

I was hoping some of you might be able to recommend me some more introductory books on these topics that maybe either you were able to learn out of, or that you have heard people use with decent success.

Thanks a lot in advance everyone!

I am currently a junior and finished Calc 3 and linear algebra this year. I'm not super interested in DEs and I just found out I only have a few days and I need to develop a curriculum for an independent study in mathematics for next year. I want to generally design a curriculum on advanced calculus- tensors, manifolds, differential forms i.e. there are so many texts online and so many different approaches to learning the material so can someone help me create a structured order to learning tensor calculus, differential forms and calculus on manifolds, that could include some preliminary info from other sources at the beginning to cover up any holes I might have? Thanks!

Hey guys! So yeah, the title kinda says ir all, I’m trying to peove that for smooth manifolds M, N the tangent bundle T(M\times N) is diffeomorphic to TM \times TN. I don’t know how to describe the elements of the tangent bundles to make it convenient to see the diffeomorphism... Also, I supposed the tangent bundles have to be endowed with some sort of differentiable structure, how to do it? I don’t know if this helps bur I already proved that for all (p, q) in M\times N, the tangent space T_(p,q) (M\timesN) is isomorphic to the product of tangent spaces T_p M \times T_q N. Anyways, please help me out, any help is appreciated.

I want to prove that the definition of exterior derivative doesn't depend on the coordinates chosen (on the chart). We define the exterior derivative of a covector field (u_i dx^(i)) as:

d(u_i dx^(i)) = du_i ∧ dx^i

But if I use different coordinates I should still obtain the same exterior derivative. Say dx^(i) = A^(i)_j dy^(j), then: u_i dx^(i)= u_i A^(i)_j dy^(j). Now to calculate d in these coordinates:

d( u_i A^(i)_j dy^(j))

= d(u_i A^(i)_j )∧dy^(j)

= (du_i) A^(i)_j ∧dy^(j) + u_i d(A^(i)_j )∧dy^(j) , (by the product rule, although the mistake is probably here)

= (du_i) ∧( A^(i)_j dy^(j)) + u_i d(A^(i)_j )∧dy^(j)

= du_i ∧ dx^(i) + u_i d(A^(i)_j )∧dy^(j)

To conclude we check if the derivatives calculated in different coordinates give the same form:

du_i ∧ dx^(i) = du_i ∧ dx^(i) + u_i (dA^(i)_j) ∧ dy^(j)

Nope, not equal. But they should be. What did I do wrong?

(I use Einstein's convention.)

Edit: I DIDN'T MEAN DISCRETE. I meant Exterior derivative.

L_V V = [V,V]=0

Ofc, but I'd like to avoid brackets and try to understand Lie transport. That is induced by the flow of V which I don't understand very well.

Imagine M=(the real line) and V points on the positive direction and is always increasing. Then since V is increasing as you in the direction of V, it sounds reasonable to say L_V V is not 0. But this is wrong! Why?

>!(Potential answer: Maybe it's better to visualize a reparametrized real line R, where you stretch and clump it in such a way that the integral curves all have constant speed 1. If we assume the velocity of the curve is V in this reparametrized R, then V doesn't change, so it's Intuition obvious L_V V=0. But this suggests it's probably better to think of L_V as happening in a reparametrized neighborhood such that V has norm 1.)!<

I think I somewhat understand the formalism.

There's one valid way i know if how to think about the Parallel Transport. If the manifold (eg a sphere) is embedded in R^(n) then: To transport u from p along v to p', I simply move u while a force normal to sphere acts on it to keep it tangent to the sphere throughout. The resulting u' at p' is what we wanted.

But this is 3D, the picture is complicated. How do I draw Parallel Transport of the sphere in 2D? >!I want to do something similar to this: 1. Chose an orthogonal basis that includes v. Draw some geodesics creating a grid that will serve as coordinates. If u at p is an arrow pointing 2 squares to the left and 1 up of p in the grid. Then the transported u' at p' will just be the arrow with the same coordinates p'. I say similar because this doesn't work.!<

ok so i have a couple of basic questions about the riemannian gradient. i have a chart \Phi from a subset of R^m to a submanifold of R^n. (let's just say the subset of R^m is a tangent space of the submanifold) I want to use it to relate the gradient of a function f in the chart (\nabla f) to its gradient (grad f) on the manifold. I found the metric tensor G by doing d \Phi ^T d \Phi and the inverse G^-1. I can find the Riemannian gradient by multiplying G^-1 * \nabla f at a point p in the tangent space. However, when i do this i get an m-dimensional vector. what basis is this with respect to? is it the standard basis for R^m except rotated to the tangent space at Phi(p)? im confused because i read the columsn of d \Phi are the basis of the tangent space.

also by the chain rule, i have \nabla f = d \Phi ^T grad f. this should give me grad f if i multiply \nabla f by the generalized inverse of d \Phi^T, right? but it gives me something totally different ):

my second question is just how do you compare grad f at different points on the submanifold aka how does parallel transport work. if im in the same chart, can i compare grad f at two different points by comparing G^-1 * \nabla f?

I have been thinking about function spaces, and was wondering if there is enough overlap of differential geometry and Banach Manifolds to give any useful structure to some function spaces.

After perusing wikipedia and google, I have found nothing about differential geometry ideas on Banach manifolds, especially tangent, cotangent, or tensor bundles. These seems useful for formalizing the derivative of a functional and parallel transport in function spaces.

Are there any good resources on this topic? It seems that there should be enough interest in the topic to make it worthwhile for investigation, and there are no apparent problems with extending the ideas above to Banach manifolds.

I'm looking for some graduate texts on the subjects in the title and preferably some including relevant material on multilinear algebra and tensor analysis. I'm aware of do Carmo's Differential Geometry of Curves and Surfaces as well as Gravitation by Thorne et al. I'd like to supplant the former (whose language is a bit uncanonical) and be equipped to comprehend the latter. I did my undergrad in physics and math, have a good fundamental understanding of general topology, manifolds, and abstract algebra and have done work in special relativity, if that helps.

This is HW. I don't know how to do it. What topology should Y have? I would think it'd be the subset topology from R^2. However then Y isn't locally Euclidean at the middle point of Y. Any help?

I have done the basic Real Analysis, Linear Algebra, Metric Spaces and Group Theory. I have started doing Topology and Measure Theory. Now, I need to know about Manifolds and have to start Differential Geometry but I am having trouble finding a proper book, I have tried a few books but all of them are disorganised and talk only about one kind of manifold euclidean or rimannian manifold. Is there a proper book where it starts from a general concept of manifold and talk about the different types?

I've started self studying using Loring Tu's An Introduction to Manifolds, and things are going well, but I'm trying to figure out where this book fits in in the overall scheme of things.

It seems like there are so many different texts and courses with similar names at the intersection of manifolds, algebra, and topology.

• smooth manifolds
• differential manifolds
• differential Geometry
• differential topology
• geometric topology
• Riemannian manifolds
• Riemannian geometry
• algebraic topology
• algebraic geometry
• topological manifolds

I'm trying to figure out how these subjects are organized, how they fit together - is there any kind of hierarchy (whether by difficulty, the order in which they should be studied, or anything else)? Is there a lot of overlap - eg it seems to me that the study of manifolds in general should be quite similar, but apparently that's not the case since John Lee has 3 different textbooks with titles from the above list.

With "pure" analysis it was much easier to figure this out - there's a pretty standard hierarchy, something like Rudin/Apostol followed by Big Rudin or Royden, and then maybe some complex analysis on the side. And if I had wanted to go further in that direction, I would have picked up a functional analysis text like Rudin III.

But when it comes to manifolds/geometry/algebra/topology, I'm lost. Any help with making sense of this is appreciated.

I'm currently an undergraduate mathematics student learning relativity (and differential geometry because of it), and I'm having trouble understanding why the choice of the dual basis happens to look like these differentials in the form of one-forms.

As I understand it, the dual vector space to a vector space with a given basis is defined to be the one such that if e^i is a basis one-form of V* and e_i is a basis vector of V, we have that e^i (e_i) = kronecker delta. If that's the case, then why exactly is the basis of T*_p dx^i ? Are these not the "differentials" I'm used to? Are they simply defined to have the property that necessarily makes them the basis of the dual space to T_p?

Any insight would be appreciated, I don't feel like I have a good connection to what these one-forms are right now.

Additionally, I'm having trouble seeing what exactly the differential operator is and why d^2 = 0.

What I mean to ask is there a theory that addresses the equivalent of a geodesic on a surface or the curvature of a manifold in Banach or Lebesgue space assuming it has some smooth structure? How would we practically interpret these objects/properties in an abstract setting?

I don't have the mathematical development to deal with these things at all yet; I'm about to head off to grad school and I've been looking at cursory overviews of functional analysis, nonlinear PDEs, and applied dynamical systems, and this question just kind of occurred to me. Can anyone pique quench my curiosity?

Edited - Caught an error. My curiosity had already been piqued. Damn definitions.

Hey, so this is from John M Lee's Riemannian Manifolds, exercise 2.3:

> Suppose [; M^n \subset \tilde{M}^m ;] is an embedded submanifold. If [; f\in C^\infty(M);], show that [; f;] can be extended fo a smooth function in a neighborhood of [; M;] in [; \tilde{M};]

Here's what I did:
> Choose open sets [; \tilde{U}_\alpha\subset\tilde{M};] s.t. [; M\subseteq \bigcup_\alpha\tilde{U}_\alpha;]. Then the sets [; U_\alpha = \tilde{U}_\alpha\cap M;] will form an open cover of [; M;]. We use slice coordinates, i.e. if [; \tilde{U}_\alpha\leadsto(x^1,...,x^m);], then [; U_\alpha\leadsto(x^1,...,x^n);]. You'll have to excuse this abuse of notation, it's what Lee uses and it's not too bad actually.
>Now, define [; \tilde{f}_\alpha\in C^\infty(\tilde{U}_\alpha);] via [; \tilde{f}_\alpha(x^1 ,...,x^m )=f(x^1 ,...,x^n ) ;]. This is then independent of [; x^{n+1},...,x^m ;]. Let [; p_\alpha ;] be a partition of unity subordinate to [; \tilde{U}_\alpha;] (this too is an open cover of [; M;] as an embedded submanifold). Finally, set [; F=\sum_\alpha p_\alpha \tilde{f}_\alpha;]. Then [; F\in C^\infty \left(\bigcup_\alpha \tilde{U}_\alpha\right);] is an extension of [; f;].

Is this correct? I'm a bit shaky when it comes to arguments using partition of unity, how do I get more comfortable with them?

Edit: Is my mathjax botched? I can't get it rendered on my laptop.

Edit2: Thanks to /u/AFairJudgement for formatting help!

I've encountered three definitions of tangent spaces for differentiable manifolds (e.g., in Lee's Introduction to Smooth manifolds, Ch. 3), and I am trying to understand which definitions work in which contexts.

Most introductory books focus on the smooth case, a few include the C^r case for finite r, and very few touch on the analytic case. I was wondering if anyone could break down these methods and where they work and fail.

Consider a C^r manifold M (r a positive integer, infinity, or omega). The three methods I've seen of defining the tangent vectors are:

1. As derivations on the space of C^r functions
2. As derivations on the germs of C^r functions
3. As equivalence classes of curves on M

Here's what I think I've gathered so far: For smooth manifolds all three methods work equally well (they are naturally isomorphic). For analytic manifolds, method 1 might fail to be local and thus won't work, method 2 works fine, and I'm not sure about method 3. And for r finite, I know method 3 works, but I have no idea about methods 1 and 2.

Apologies if my understandings in the above paragraph are incorrect. Can anyone summarize how these three methods work in the r finite and the analytic cases? If you can include the reason for the failures of various methods, that would also be helpful. E.g., the lack of analytic bump functions may be the reason method 1 fails for analytic manifolds.

From my text:

... Then two differentiable curves c, d with initial point x are said to be tangent at x if there exists a smooth chart (U, χ) containing x, such that

[; \frac{\textup{d}}{\textup{dt}} \; \chi \; \circ \; c(t) \; |_{t=0} = \frac{\textup{d}}{\textup{dt}} \; \chi \; \circ \; d(t) \; |_{t=0} ;]

Suppose now that (V, ψ) is another smooth chart, and let τ = ψ o χ^(-1) be the associated transition map. Then by the chain rule we have:

[; \frac{\textup{d}}{\textup{dt}} \; \psi \; \circ \; c(t) \; |_{t=0} = \textup{D}(\tau)(\chi (x)) \; \frac{\textup{d}}{\textup{dt}} \; \chi \; \circ \; c(t) \; |_{t=0} ;]

From this we see that if (1) holds in one chart containing x, then it holds in any other chart containing x.

The notation in the second statement is confusing me - what happened? And how does this show that it then holds for all charts, as there is still a factor left that is not there in the first definition?

Thanks in advance.

I'm just starting to learn differential topology, and the start of the book (differential topology, Guillemin and Pollack) is some definitions to do with manifolds. Here is what I am working with:

• k-dim manifold given a subset X of Rn, X is a k-dim manifold if for any point x in X, there exists a neighbourhood V of x and an open subset U of Rk such that V and U are diffeomorphic.
• diffeomorphic given subsets X of Rn and Y of Rm, X and Y are diffeomorphic if there exists a diffeomorphism between them
• diffeomorphism given subsets X of Rn and Y of Rm, a function phi: X -> Y is a diffeomorphism if it is bijective, smooth, and its inverse is smooth

The example I am working on is showing that the unit circle is a 1-dim manifold (the text goes over it). I understand how you can map the interval (0,1) to the upper and lower semicircles of the unit circle bijectively and so on, but when I go to do it rigorously I encounter problems.

Namely, the book looks at (x,y) in the unit circle, and considers when y > 0. It then just says the function phi(x) = (x, sqrt(1 - x^2 )) maps the interval (0,1) bijectively to the upper semicircle of the unit circle. But using the definition, I need phi to map from an open subset of R1 (which would be (0,1)) to a neighbourhood of (x,y). But the upper semicircle isn't a neighbourhood (because it does not contain any open disk for any of its points)! And if I expand the semicircle (e.g. to the closed disk of length 1 with centre (0,0)), then I don't get surjectivity.

What am I doing wrong or missing? Thanks

I've always been kind of insecure with my Calculus knowledge. Once I took an undergrad intro to PDE and it was so hardcore at Calculus that I couldn't follow most of the time.

Now, the thing is, I love Analysis and I know that at some point I'll have to deal with Manifolds. A professor even told me that an analyst should be confortable with the likes of Differential Geometry.

That said, should I bother to review anything from Vector Calculus/Multivariate Analysis apart from the Inverse/Implicit Function Theorem before tackling Manifolds?

PS: I also have undergrad courses in General Topology and Abstract Algebra under my belt.

I’ve really been enjoying multivariable/vector calculus and it sounds like manifolds continue from this point?

It just occurred to me that atlases have fallen completely out of usage with the advent of smartphones and the internet. Someday I'll have students who don't know the meaning of the word in any other context but math. Makes me wonder if there are words like that in my own vocabulary for which I don't understand he original reference.