Manifold Definition and 319 Threads

Does anyone know why every 2D manifold is conformally flat.

I have a question. Is it true that any curve in 2-dimensional manifold which tangent vector is null at each point is null geodesic? (In 2-dimensional manifold there are only 2 null direcitions at each point).

Suppose I have a manifold. I say that it can support a certain configuration of gravity field described by metric tensor \gamma. I do not write \gamma_{\mu\nu}, because that would immediately imply a reference to a particular chart. A tensor field, however, exists on a manifold unrelated to this...

I always thought one could define a manifold as a collection of points with a distance function or metric tensor. But in a layperson's book by Penrose, he defined a manifold as a collection of points with a rule for telling you if a function defined on the manifold is smooth. He says this is...

Homework Statement The problem I am facing is 2.9 in Sean Carroll's book on general relativity (Geometry and Spacetime) I should note that I am not studying this formally and so a full solution would not be unwelcome, though I understand that forum policy understandably prohibits it. The...

Homework Statement Given S1={(x,y) in R2: x2+y2=1}. Show that S1 is a 1-dimensional manifold. Homework Equations The Attempt at a Solution Let f1:(-1,1)->S1 s.t. f1(x)=(x,(1-x2)1/2). This mapping is a diffeomorphism from (-1,1) onto the top half of the circle S1. I was...

I posted this problem on the Classical Mechanics Subforum last week but have not received many responses - hopefully someone can help here as I've spent hours racking my brain, trying to work this out! Homework Statement There is a particle of mass 'm' moving in a manifold with the following...

Hi I am trying to work through the solution to the attached problem (see attachments). Now, I can't understand several things in the solution: The Lagrangian in question is: L={\frac{m}{2}}{g_{ij}(x)}.{\dot{x^{i}}{\dot{x^{j}} 1)is g_{ij} a matrix with diag(-1,1,1,1), ie. the metric tensor...

Dear all, I've always wondered where the name "manifold" comes from? Any idea? Thanks, Goldbeetle

Say I sit at some point P in a Calabi-Yau manifold. Are there geodesics which start from P and return to P? Are there "geodesics" which start from P and return to P but may make a "side trip first"? Is the number of geodesics which start at P and end at P infinite or finite and does that...

I'm looking for a simple example of a differentiable manifold that doesn't have an associated riemann metric. thanks

I apologize for the poorly worded title. Let me try to explain my question better. A scientific theory must be predictive to be useful. Since we only know what happened in the past, the global topology of spacetime cannot be an input to the theory. Given space-like slices/"chunk" of the...

Dear all, I am so sorry for my stupid questions. Currently, I am looking for documents (lecture notes) on tubes assembly and manifold for aircraft engine. What are those tubes assembly and manifold? What are those characteristic? I did quite a lot google search, however, I only found any good...

Conventional manifold definition refers to the neighbor of every point having a Euclidean space description. http://en.wikipedia.org/wiki/Manifold" But if most manifolds have additional property of some curvature, then won't such manifold definition actually be describing a tangent space i.e...

Can someone help me out whit this I proved for the circle, but I can't prove it for this -Prove that x^4+y^4=1 (the set of points) is a manifold For the circle it was easy, but how do I take on this case? Thanks

Is there a difference between a manifold that is a result of particle interactions and say a system of elements where there is no interactions? E.g. Two particles interact with one another by exchanging force carriers and as a result they create a manifold in the form of a sphere. Isn't this...

Hi, I am struggling to understand Stephen Hawking's view of the universe as a 4D closed manifold. In a recent interview, I believe he had this to say: What I don't understand is how this theory is compatible with the scientific observation that the universe is expanding? I have 2 questions: 1)...

Question is in the title. Seems a lot of people throw that statement around as if its obvious, but it isn't obvious to me. I can kind of see how it might be true. If you take a group element, differentiate it wrt the group parameters to pull down the generators, and then evaluate this...

I don't think this is a difficult problem, but I am not sure about what is being asked in the question. I got it from Munkres' Analysis on Manifolds page 193 Q 2. Homework Statement Let A be open in R^k; let f : A-->R be of class C^r; let Y be the graph of f in R^(k+1), parametrized by...

Because of boundary points, I can sort of see intuitively why Euclidean half-space, i.e. {(x_1, ... , x_n) : x_n >= 0} is not a manifold, but is there a simple rigorous argument for why Euclidean half-space is not homeomorphic to an open set of R^n. I do not know too much topology and the...

Wikipedia gives a confusing definition of a path's length and I would like some clarity. Let M be a pseudo-Riemann manifold with metric g and let a and b be points in M.If y is a smooth function from R->M where y(0) = a and y(1) = b, then it's length is the integral \int_0^1\sqrt{\pm...

How to say a given space is a manifold? The only thing that props in my mind is to check if every open set has a euclidean coordinate chart on it. But, what if the space I am dealing with is not fully understood apriori? As in, how were the spaces of thermodynamic equilibrium states, phase and...

Homework Statement Suppose X ⊂ R^n is a k-dimensional manifold and Y ⊂ R^p is an l-dimensional manifold. Prove that: X × Y = {[x,y] ∈ R^n × R^p : x ∈ X and y ∈ Y} is a (k+l)-dimensional manifold in R^(n+p). (Hint: Recall that X is locally a graph over a k-dimensional coordinate plane...

I must demonstrate in two ways that if c(t) is an integral curve of a smooth vector field X on a smooth manifold M with c'(t_0)=0 for some t_0, then c is a constant curve. I found one way: If \theta denotes the flow of X, then because X is invariant under its own flow, we have c'(t)=X_{c(t)} =...

Hi, I've been reading through Yau's proof of the Calabi conjecture (1) and I was quite intrigued by the relation R_{i\bar{j}} = - \frac{\partial^2}{\partial z^i \partial \bar{z}^j } [\log \det (g_{s\bar{t}}) ] derived therein. g_{s \bar{t} } is a Kahler metric on a Kahler manifold (I'm...

I got this book here that mentions en passant that the connected components of a (topological) manifold are open in the manifold. That's not true in a general topological space, so why does Hausdorff + locally euclidean implies it? I don't see it.

This is a problem many of the grad students have probably encountered, it's in Chapter 0 of Riemannian Geometry by Do Carmo. Do Carmo proved that the tangent bundle of a differentiable manifold is itself a differentiable manifold by constructing a differentiable structure on TM, where M is a...

Can a 4-dimensional manifold with the Schwarzschild metric be embedded into a flat manifold of 5 (or more if necessary) dimensions? In other words, are there functions of t,r,\theta , \phi and M such that if x_1 = f_1 (t,r,\theta ,\phi ,M) x_2 = f_2 (t,r,\theta ,\phi ,M) . . etc...

in atomic physics, sometimes one would encounter the termilogy ''ground state manifold'' my question is, the ground state of an atom is usually unique How come the ''ground state manifold''? It means several nearly degenerate level? are these level stable?

Dear all, I can formally understand one of the many definitions of tangent vectors to a manifold, but what are they in reality? It should depend on the nature of the points of the manifold, for example, if M={set of events of general relativity}, then vectors are velocities. Other examples...

I have a intake manifold temperature that is way to high (105 by mid day normally 90) that is killing me.I have replaced the aftercooler and rebuilt the aux water pump that supplys the cooling water to the aftercooler as well as cleaned out the secondary cooling marely tower,all to no avail is...

Hello all, this is my first post on this site, so I'll try not to look stupid. I am currently trying to design a new intake manifold for a turbo 1.6L engine. I have never attempted to this before, but I have a few ideas in mind, and would like some feedback. First, I could use a D-shape pipe...

Does a manifold necessarily have a metric? Does a manifold without metric exist? If it exists, what is its name?

Does a manifold necessarily have a metric? Does a manifold without metric exist? If it exists, what is its name?

Hi all: I have just met a problem. If say there is a triangle ijk on a manifold, D(i), D(j), D(k) are the geodesic distances from a far point to i,j,k respectively. Then g = [D(i) - D(k); D(j) - D(k)], what does g describe? Does is describe the gradient of the vertex k? If u = Vi-Vk, v =...

Hi, I am new to manifold and having a hard time on it. :frown: Could anyone please help me on the following problem. Please write down your thoughts. Thanks alot. Prove that (S^n) X R is parallelizable for all n.

Let w be a 1-form on smooth manifold M. Then is there a vector field X such that locally w(X)=f where f:M-->R continuous? How can I prove it? Thanks.

Homework Statement My book ("General Relativity" by Hobson on page 32) says that an N-dimensional manifold has 1/2 * N * (N-1) independent metric functions. I am confused about why there is a limit at all to the number of independent metric functions g_{\mu \nu} . It probably has to do with...

Homework Statement I don't have great expectation that this will get a reply but here goes, because this is bugging me. I will assume that you are familiar with the notation used by Spivak. In the last section of chapter 4, he shows how to integrate a k-form on R^m over a singular k-cube...

Is there any difference between a vector subspace and a linear manifold. Paul Halmos in Finite Dimensional Vector Spaces calls them the same thing. Hamburger and Grimshaw in Linear Trasforms in n Dimensional Vector Space does not use the word subspce at all. Planet Math says a Linear...

I've been reading about the abstract formulation of dynamics in terms of symplectic manifolds, and it's amazing how naturally everything falls out of it. But one thing I can't see is why the generalized momenta should be cotangent vectors. I can see why generalized velocities are tangent...

can you be given a suitable smooth atlas to the subset M of plane that M to be a differentiable manifold? M={(x,y);y=absolute value of (x)}

This cute mechanical engineer mentioned in a message that he had designed some manifolds, so that's why I'm asking this (in addition to my general interest in physics, of course). I'm not sure if they were intake or exhaust manifolds, but that's not the problem now. I figured that...

So I was wondering about this... if \omega is a k-form and \eta is a l-form, and m is a k+l+1 manifold in \mathbb{R}^n, what's the relationship between \int_M \omega\wedge d\eta and \int_M d\omega\wedge \eta given the usual niceness of things being defined where they should be, etc. etc. The...

How can a smooth deformation of a Lorentzian manifold possibly create one or more singularities?

My text defines differentiability of f:M\rightarrow \mathbb{R} at a point p on a manifold M as the differentiability of f\circ \phi^{-1}:\phi(V) \rightarrow \mathbb{R} on the whole of phi(V) for any chart (U,\phi ) containing p, where V is an open neighbourhood of p contained in U. Is this...

Hi there, hope someone is able to help me. I am trying to find some info on early intake manifold designs, for Ford 4 cylinder engines, otherwise known as the T-, A- and B- engines. These engines came standard with two intake manifold inlets, and 4 exhaust manifold outlets cast into the block...

Hello, I wish to show that on 3-dimensional manifolds, the weyl tensor vanishes. In other words, I want to show that the curvature tensor, the ricci tensor and curvature scalar hold the relation Please, if anyone knows how I can prove this relation or refer to a place which proves the...

I've been reading the book "Geometrical Methods for Mathematical Physics" by Schutz. I can't understand/visualize the definition of contituity given on this page 7. I.e. where it states in the 3rd paragraph I don't understand/can't vizualize this definition and reconcile it with the normal...

A question on a General Relativity exam that I have asks how many linearly independent Killing fields there can be in an n-dimensional manifold. I'm sure I've seen this question before and I think that the answer is n(n+1)/2, but I can't remember why! Any help?