Manifold Definition and 318 Threads

Hey, we had in the lecures the following: Let M and N be smooth manifolds, and dim(M)=dim(N)=n, while $$x^i$$ and $$ y^i$$ are coordinate functions around $$p\in M$$ respective $$F(p) \in N$$, then we get for the pullback of F Which entries has the matrix we take the determinant of? I thaught...

I'm reading a book on Lie groups and one of the first examples is on SL(2,R). It says that every element of it can be written as the product of a symmetric matrix and a rotation matrix, which I can see, but It also makes the assertion that the symmetric matrix can be parameterized by a...

Some models of gravity, inspired by the main theme of spacetime fabric of Classical GR, treat the metric of the manifold and the connection as independent entities. I want to study this theory further but I am unable to find any paper on this, on ariXiv atleast. I will be very thankful if...

Given S is a submanifold of M such that every smooth function on S can be extended to a smooth function to a neighborhood W of S in M. I want to show that S is embedded submanifold. My attempt: Suppose S is not embedded. Then there is a point p that is not contained in any slice chart. Since a...

I want to show that a closed unit ball is manifold with boundary and I attempted as uploaded. But I am not happy with the way I showed the boundary chart is injective. Am I right?

Because it is a closed space, can it make sense to fill a Calabi_Yau manifold with an ideal gas and consider waves from a point disturbance? Would the Ricci-flat condition of Calabi-Yau manifolds have anything to say about possible sound waves? Thanks!

I have a surface defined by the quadratic relation:$$0=\phi^2t^4-x^2-y^2-z^2$$Where ##\phi## is a constant with units of ##km## ##s^{-2}##, ##t## is units of ##s## (time) and x, y and z are units of ##km## (space). The surface looks like this: Since the formula depends on the absolute value of...

Hello, an engineer here that is trying to understand more in topology for some mechanics ideas. It seems to me that any closed (regular) manifold in 3d Euclidean space should be able to be describe uniquely by the combination of the following 1) volume [1 real], 2) centroid [3 reals], and 3)...

Hello! Can someone explain to me why does a parallelizable manifold implies zero Riemann tensor? In the book I read this is mentioned but not proved. This would imply that parallel-transporting a vector would be path independent. But I am not sure how to show it. Thank you!

Hello! The cohomology ring on an M-dim manifold is defined as ##H^*(M)=\oplus_{r=1}^mH^r(M)## and the product on ##H^*## is provided by the wedge product between cohomology classes i.e. ## [a]## ##\wedge## ##[c]## ##= [a \wedge c]##, where ##[a]\in H^r(M)##, ##[c]\in H^p(M)## and ##[a \wedge...

Hello. I was trying to prove that the tangent bundle TM is a smooth manifold with a differentiable structure and I wanted to do it in a different way than the one used by my professor. I used that TM=M x TpM. So, the question is: Can the tangent bundle TM be considered as the product manifold...

Part 1. The second part of my journey to the manifold ##SU(2)## deals with some representations. We start with some bases and cite the classification theorem of representations of the three-dimensional, simple Lie algebra. Continue reading ...

Can someone tell me how we know that our physical universe is geodesically complete? In response to a question I had about why we assign any meaning to the other side of a black hole’s event horizon (or its interior), I got an answer prompting me to look into the concept of geodesic...

Differentiation, Spheres, and Fiber Bundles The special unitary groups play a significant role in the standard model in physics. Why? An elaborate answer would likely involve a lot of technical terms as Lie groups, Riemannian manifolds or Hilbert spaces, wave functions, generators, Casimir...

I don't quite understand some work I'm doing creating the normal Riemann surface for the function ##f(z)=\frac{A}{\sqrt{(1-z^2)(k^2-z^2)}}##. I can use Schwarz-Christoffel transforms to map the function to a rectangular polygon in the zeta-plane then map this rectangle onto a torus. But I...

This is not homework, self study, so I'm going to post here, where people know things :P. a) Show that a topological space can't be both a 1-manifold and an n-manifold for any n>1. If a topological space were both a 1-manifold and an n-manifold, then every point would have a neighborhood...

Hello! I am reading how to integrate on an orientable manifold. So we have ##f:M \to R## and an m-form (m is the dimension of M): ##\omega = h(p)dx^1 \wedge ... \wedge dx^m##, where ##h(p)## is another function on the manifold which is always positive as the manifold is orientable. The way...

Hello! So I have 2 vector fields on a manifold ##X=X^\mu\frac{\partial}{\partial x^\mu}## and ##Y=Y^\mu\frac{\partial}{\partial x^\mu}## and this statement: "Neither XY nor YX is a vector field since they are second-order derivatives, however ##[X, Y]## is a vector field". Intuitively makes...

Hello! I am not sure I understand the idea of vector field on a manifold. The book I read is Geometry, Topology and Physics by Mikio Nakahara. The way this is defined there is: "If a vector is assigned smoothly to each point on M, it is called a vector field over M". Thinking about the 2D...

The answer to the question of the thread title is yes, according to what I found on web. Now a manifold is by definition a topological space that (aside from other conditions) is locally Euclidean. What does such condition means? Is it the same as saying that each of its points must have a...

What's the mathematical expression for manifold duplication in Many Worlds? In other words, how do the manifolds duplicate themselves endlessly?

Hi. I've been thinking about vectors, coordinate systems and all things associated for a long time. I'd like to know if (at least in the context of General Relativity) my interpretation of these subjects is correct. I will try to summarize my thoughts as follows: - We start with a general...

Why is i that two cones connected at their vertices is not a manifold? I know that it has to do with the intersection point, but I don't know why. At that point, the manifold should look like R or R2?

I understand the definition of continuity on a manifold based on open sets. I was questionning myself about what is the corresponding physical meaning of an open set of a manifold (M, Power-set-of-M, Atlas). Is it a simple (maybe simplest) assumption in order to define mathematically continuity...

So if I pick any 2 points on a 2d manifold, say x1 and x2, then the distance between these two points is a secant line that passes through 3 space that isn't part of the manifold. So no matter what, there doesn't exist an point epsilon, e , where ||e ||>||0 || and ||x2-x1||<|| e || No matter...

How do I find the surface area of a f(x) rotated around the y axis?

Homework Statement A parameterized ##p##-subset ##(U,F)## of a manifold ##M^{n}## is ''irregular'' at ##u_0## if rank ##F<p## at ##u_0##. Show that if ##\alpha^{p}## is a form at such a ##u_0## then ##F^{*}\alpha^{p}=0##. Homework Equations The Attempt at a Solution A parameterized...

Source: Basically the video talk about how moving from A to A'(which is basically A) in an anticlockwise manner will give a vector that is different from when the vector is originally in A in curved space. $$[(v_C-v_D)-(v_B-v_A)]$$ will equal zero in flat space...

I am really struggling with one concept in my study of differential geometry where there seems to be a conflict among different textbooks. To set up the question, let M be a manifold and let (U, φ) be a chart. Now suppose we have a curve γ:(-ε,+ε) → M such that γ(t)=0 at a ∈ M. Suppose further...

Suppose we have an n-dimensional manifold Mn and take a coordinate neighborhood U with associated coordinate map φ: U → V where V is an open subset of ℝn. So far I'm clear on this. However, where I become confused is when some books say that φ-1 is called a parameterization of U and basically...

Let $U$ be a compact set in $\mathbb{R}^k$ and let $f:U\to\mathbb{R}^n$ be a $C^1$ bijection. Consider the manifold $M=f(U)$. Its volume distortion is defined as $G=det(DftDf).$ If $n=1$, one can deduce that $G=1+|\nabla f|^2$. What happens for $n>1$? Can one bound from below this $G$? If...

I firstly learned about duality in context of differentiable manifolds. Here, we have tangent vectors populating the tangent space and differential forms in its co-tangent counterpart. Acting upon each other a vector and a form produce a scalar (contraction operation). Later, I run into the...

Hi all, this might be a silly question, but I was curious. In Carroll's book, the author says that, in a manifold M , for any vector k in the tangent space T_p at a point p\in M , we can find a path x^{\mu}(\lambda) that passes through p which corresponds to the geodesic for that...

I need a reminder. What numbers or functions characterize a map from one manifold to another? More specifically, is there a continuous function that goes from one manifold to another to another to another is some parameterized way? What is that called? I'm thinking of a manifold of spacetime...

Is it possible to create a non-local manifold that co exist with the spacetime manifold? The non-local manifold being where quantum correlations took place. How do you make the two manifolds co-exist?

Hi everybody I seen in a book a quote of H. E. Winkelnkemper "Transversality unlocks the secrets of the manifold.". Can someone explain to me this citation and why transversality has all this importance in geometry? thank you in advance.

1. Homework Statement I'm taking a swing at Spivak's Differential Geometry, and a question that Spivak asks his reader to show is that if ##x\in M## for ##M## a manifold and there is a neighborhood (Note that Spivak requires neighborhoods to be sets which contain an open set containing the...

I have been working through John Lee's "Introduction to Smooth Manifolds" recently. I am having trouble visualizing the unit n-sphere as a manifold in its own right, i.e. without an ambient space. It seems impossible to visualize it without embedding it in Euclidean space. I mean, the unit...

Is there a topologist out there that wants to explain why exactly a branched line in R2 is not not a topological manifold? I know it's because there doesn't exist a chart at the point of branching, but I don't understand why not. I'm just starting to self study this, so go easy on me :).

Why are tangent spaces on a general manifold associated to single points on the manifold? I've heard that it has to do with not being able to subtract/ add one point from/to another on a manifold (ignoring the concept of a connection at the moment), but I'm not sure I fully understand this - is...

Hi, A chamber (manifold type cylinder) has 1 inlet and 8 outlets of 3 inch diameter each. 8 suction blowers are connected to the chamber's outlet. Each blower's suction flow rate is 1000 cfm. what will be the flow rate through the inlet of the chamber (diameter = 3 inches). Will the inlet...

I have what I think is a basic question. Say I have a manifold and a metric. How do I write down the most general curve for some arbitrary parameter? For example in \mathbb{R}^2 with the Euclidean metric, I think I should write \gamma(\lambda) = x(\lambda)\hat{x} + y(\lambda)\hat{y} But what...

Can one define a vector space structure on a Riemannian manifold ##(M,g)##?! By this I mean, does it make a sense to write ##x+y## where ##x,y## are arbitrary points on ##M##?

Hi all, I am not familiar with the dynamic system theory. When I was trying to understand the weakly nonlinear stability analysis, I realize the following question. It is known that the center manifold reduction can be used to study the first linear bifurcation. This lead to the...

Hello, I've just read and I think I have understood the following result : If we were to geodesically transport all points of a small 2D surface, so small that it would be flat for all purposes, in a direction vertically above it, and if this surface belongs in an arbitrary 3D manifold, then in...

Let f:p\mapsto f(p) be a diffeomorphism on a m dimensional manifold (M,g). In general this map doesn't preserve the length of a vector unless f is the isometry. g_p(V,V)\ne g_{f(p)}(f_\ast V,f_\ast V). Here, f_\ast:T_pM\to T_{f(p)}M is the induced map. In spite of this fact why...

I have recently had a lengthy discussion on this forum about coordinate charts which has started to clear up some issues in my understanding of manifolds. I have since been reading a few sets of notes (in particular referring to John Lee's "Introduction to Smooth Manifolds") and several of them...

I'm trying to work up some examples to help me understand this concept. Would the periodic flow on a solid torus be transverse to it's boundary?

I am studying Carroll's notes on GR. He defines charts as maps from an open subset in manifold M to open subset in R^n. He then writes "We therefore see the necessity of charts and atlases: many manifolds cannot be covered with a single coordinate system. (Although some can, even ones with...

So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let M be an n-dimensional manifold and let (U,\phi) and (V,\psi) be two overlapping coordinate charts (i.e. U\cap V\neq\emptyset), with U,V\subset M, covering a neighbourhood of p\in M, such that...