Differential Geometry Of Curves Puns

I am currently brushing up on my differential geometry knowledge by working through Manfredo P. Do Carmo's "Differential Geometry of Curves and Surfaces" to be ready for my upcoming postgraduate classes commencing in october.

I've attemped many of the exercises in the first few chapters and am looking to confirm my answers. Is there somewhere I can search for the solutions?

I realize that this is a highly specific question but does a curve of the form (cos(t), sin(t), -cos(t), -sin(t)) in R^(4) have a name/any interesting properties? I'm writing a linear algebra exam and playing around with some toy examples in R^(4) and this parametrization (modulo some constant) appears as the set of normalized vectors which are mutually orthogonal to two given vectors, and it looks particularly nice.

I'm an instructor for an undergraduate differential geometry course about curves and surfaces. (dunno if any of them are on reddit, but if so, hi!) We're working for Ted Shifrin's notes [pdf] which are pretty similar to do Carmo's book. The students are quite good, but many of them don't have any experience with analysis or abstract algebra.

I'd like to have students write a report on a paper. Their reports should highlight the main result of the paper and outline the argument. They should be able to relate the argument of the paper to what we've done in class. I don't expect them to understand the proof of every single lemma, but the papers should be short enough they can read and digest a lot of it. I am also happy for them to include their own voice in the introduction and conclusion: what were the most interesting parts of the paper? What are your follow-up questions? What else does it make you think about?

So I need suggestions for appropriate papers. Here are some that I like:

Tabachnikov, A Four Vertex Theorem for Polygons

Solomon, Tantrices of Spherical Curves (maybe just part of this one)

Scofield, Curves of Constant Precession

Chakerian, Johnson, Vogt, Geometric Inequality for Plane Curves with restricted curvature

Do you have any suggestions? I'd also be happy to assign sections of textbooks which don't overlap with Shifrin/do Carmo. Thanks!

Suppose that f and g are smooth curves such that f(0)=0=g(0) and 0<=f(x)<=g(x) Prove that Kf(0)<=Kg(0) (K means curvature)

Our (terrible) teacher gave us 2 days to turn 2 exercises in, one of them asking the trace of a curve (along with other stuff) for the parameterized curve: c(t)=((4/5)cost, 1-sint, (-3/5)cost). The only thing we've been told about the trace of the curve in class it's that it is the image of the curve.

My wild assumption is that if we assume that t∈[0,2pi], then the trace is gonna be given by: c([0,2pi])=([-4/5, 4/5],[0,2],[-3/5,3/5]. Does this make sense?

I'm doing an independent study this semester and wanted to build on my differential geometry knowledge. Would differential forms and applications by do Carmo be an appropriate next step?

Here's the question I'm stuck on right now. The cross-product is what's throwing me off; I know how to calculate cross-product using the determinant as well as the sine formula, but I don't know how I'm supposed to use it in the context of this question. Can someone offer some insight?

Hey everyone, I have a set of Diffy Geo problems that i need help with. I can get the first few but struggle with the rest.

Here are the problems

Any help is appreciated!

So here is the problem...

Find the equation of the curve that has the property that the y-intercept of every normal line to that curve is (0,6) given that curve must also pass through the point (3,2).

Let m be a C^2, simple closed curve.

(1) One can prove that there exists a circle of minimum radius, r, which contains the interior of m. Show that such a circle is necessarily unique.

Hint: if c1 and c2 are the centers of 2 such circles, show that the curve is inscribed in a circle of smaller radius centered at (c1 + c2)/2.

(2) Let y be a subarc of this circle. Prove that, if y (strictly) contains a half-circle, then it must intersect m at least once.

Given the rich connections between differential geometry and complex analysis is there a book that teaches complex analysis armed at say a graduate student who while comfortable with complex numbers and vector spaces has not really dived into complex analysis but has done a significant amount of differential topology and geometry?

Differential geometry and algebraic topology are not encountered very frequently in mainstream machine learning. In a new series of posts, I show how tools from these fields can be used to reinterpret Graph Neural Networks and address some of their common plights in a principled way.

First post in the series - introduction:

https://towardsdatascience.com/graph-neural-networks-through-the-lens-of-differential-geometry-and-algebraic-topology-3a7c3c22d5f?sk=791d11adb17b2e0a01f5089e4d460072

Part II will discuss the expressive power of GNNs and topological message passing.

Part III will deal with geometric flows and non-euclidean diffusion PDEs on graphs.

Part IV will show how the over-squashing phenomena can be related to graph curvature, and offer a geometric approach to graph rewiring inspired by the Ricci flow.

Dear mathematics community,

I am returning to school for a masters degree in math. As it turns out one of my chosen courses I’ve enrolled into is Differential Geometry, for which I’ve purchased the textbook already.

The introduction of the book says that a good preparation for the course is a solid understanding of linear algebra and multivariable calculus. According to the author this will suffice.

Are there any specifics that any of you would recommend for me to be better prepared for success in the course?

Thank you in advance :)

The differential geometry class that I'm taking focuses solely on curves and surfaces (so far we've only done curves) in R^(3). I noticed that some of the things that we've done easily generalize to R^(n), such as the tangent and normal vectors and thus the curvature, but some of the things depend on the cross product (e.g. the binormal vector, torsion, and related definitions). How much of the differential geometry of curves can be generalized to R^(n) for n > 3? As it's possible to define a nontrivial cross product in R^(7), it seems that one could define the binormal vector and torsion for a curve in R^(7) in exactly the same way as for curves in R^(3); in this case, do the curvature and torsion still uniquely determine the curve up to rigid motions? If not, is there a way to define additional functions on the curve such that all of the functions together uniquely determine a curve in R^(7) up to rigid motions? What about in R^(n) for n > 3 and n != 7, where it's impossible to define a nontrivial cross product? Is there still a way to define some analogue of the binormal vector and torsion or some other function(s) that, together with the curvature, uniquely determine a curve up to rigid motions?

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.

Recently I was reading about Tensor Calculus in order to understand some equations in thermodynamics of irreversible processes, in those lecture I found that Tensor Calculus is a subtopic in Differential Geometry and I was thinking if it is possible apply this subject to the area of Structural chemistry with the objective of predict or describe chemical properties of some compounds.

If anyone has or knows about papers talking about this topics it would be nice if you can share it, I will be appreciated.

I haven’t been able to find many sources on these fields. They are of interest to me as I do research in probably applied to control theory. Are these synonymous or is it more of a perspective difference ?

Thanks

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

How can I find the geodesics on the cylinder x^2 + y^2 = 1?

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 want to learn about DG/Topology and am looking for texts to get started. I plan on doing a review of linear algebra and calculus, but need some guidance on what books to get. I have read that An Introduction to Manifolds by Tu and Introduction to Differential Topology by Jänich and Bröcker/Basic Concepts of Algebraic Topology by Croom are good texts to get started. Are there any recommendations for these topics you could give me? And other than calc/linear algebra is there anything I should review before diving into these topics?

P.S. I'm a physics undergrad, so I don't have the same math background as most of you.

Just today, Yuchen Liu, Chenyang Xu, and Ziquan Zhuang put up a preprint solving the so-called finite generation conjecture, a conjecture in algebraic geometry that forms the last link in a long chain of conjectures in the study of the K-stability of Fano varieties, a huge topic of research in algebraic geometry over the last several decades. Since the resolution of this conjecture essentially completes this field of study, I thought it would be a good idea to post a reasonably broad discussion of it and its significance.

In this post I will summarise this research program and the significance of the paper, and where people in the field will likely turn to next.

##Introduction

Going all the way back to the beginning, the problem starts with what pure mathematicians actually want to do with themselves. The way I like to think of it is this: pure mathematicians want to find mathematical structures, understand their properties, understand the links between them, and classify them (that is, completely understand which objects can exist and hopefully what they all look like). Each of these is an important part of the pure mathematical process, but it is the last one is in some sense the "end" of a given theory, and what I will focus on.

In geometry, classification is an old and interesting problem, going back to Euclid's elements, where the Platonic solids (Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron) were completely classified. This is a fantastic classification: pick a class of geometric structures (convex regular 3-dimensional polytopes) and produce a comprehensive list (there are 5, and here is how to construct them...). Another great classification is the classification of closed oriented surfaces up to homeomorphism/diffeomorphism. For each non-negative integer g called the genus, we associated a surface with g holes in it.

##Higher dimensional classification

As you pass to more complicated geometric structures and higher dimensions, the issue becomes more complicated, for a variety of reasons. Perhaps the most obvious is that classification of all geometric structures is impossible. This is meant in a precise sense: a classification should be some kind of list or rule which can produce all possible structures of a given type. However it can be proven that every finitely presented group appears as the

Sophomore here taking AP Calc BC as a junior next year. I've been self-studying calculus in my free time and love it so far, I'm looking to take a few classes at my local community college over the summer to lay a good foundation and get into some topics I'll be learning in BC. There are two calculus-related courses at the community college - Calculus and Analytic Geometry I, and Differential Equations.

Which course would be better to take the summer before AP Calculus BC (calc I and II equivalent) for a high-schooler who is looking to lay a solid calculus foundation and get a head start on the year? Thanks!

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!

x'' + x(3+x^2) x'^2/(1-x^4) = 0

I've derived the above equation for the constant speed geodesic equation on a certain 1 dimensional Riemannian manifold. Here x(t) is in the open interval (-1,1).

Numerically, I can see that the solution looks a lot like tanh, but it is not quite.

Does anyone have advice on finding an analytic solution? Or on showing that there is no analytic solution?

Thanks!