Manifold Definition and 336 Threads

DO WE ALREADY HAVE EVIDENCE OF DARK ENERGY - OUR MANIFOLD? If LIGO I with it’s 10^-21 sensitivity, VIRGO etc. don’t detect gravity waves, might this then be interpreted as indicating that C_R pseudo-Riemanian spacetime continuum (i.e. manifold’s) stiffness is not INsignificant; rather...

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...

Abstract: [Moderator's note: no responsibility for content. LM] This is an attempt to define a bulk which corresponds to both a string cosmological and brane cosmological formalism which disallows other branes that exist outside this universe. Our bulk is homogenous manifold similar to a...

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...

Abstract: [Moderator's note: no responsibility for content. LM] This is an attempt to define a bulk which corresponds to both a string cosmological and brane cosmological formalism which disallows other branes that exist outside this universe. Our bulk is homogenous manifold similar to a...

Abstract: [Moderator's note: no responsibility for content. LM] This is an attempt to define a bulk which corresponds to both a string cosmological and brane cosmological formalism which disallows other branes that exist outside this universe. Our bulk is homogenous manifold similar to a...

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?

I only learned the meaning of a manifold recently and in the most elementary terms but I thought that I might link it with this example. It seems that the universe is made out of 3D objects. So put all of them together (stars, black holes, galaxies etc) and you have the whole universe. It...

I'm having trouble understanding exactly what this manifold is. Let me draw an analogy: Say I have a flat map of the world. The map is a two-dimensional surface with a coordinate chart on it. However, its embedded in a higher three-dimensional space. So by analogy, is the four dimensional...

In general relativity, curved spacetime is described by a manifold and a metric or frame on top of it. Can the manifold coordinates carry units of, say, meters and seconds, or do the metric components have those units?

Hi people, I'm learning differential geometry in a book (Intro to smooth manifolds, by John Lee) and I have some difficulties with the tangent distributions. Actually, I don't know what to do if, given a distribution spanned by some vectors fields, I want to find its integral manifolds. Can...

Is it correct that the definition of a smooth manifold is an equivalence class (under diffeomorphism) of atlasses ? (this discussion is related to a discussion I try to start in general relativity concerning the hole argument).

We have a subset X, which is contained in R^4 (i.e., it is contained in the reals in 4 dimensions). (a) We must prove that the following two equations represent a manifold in the neighborhood of the point a = (1,0,1,0): (x_1)^2+(x_2)^2-(x_3)^2-(x_4)^2=0 and x_1+2x_2+3x_3+4x_4=4. (b) Also we...

I was thinking about something yesterday and I couldn't quite figure it out. It's about the question if an atlas is a countable set. Because we know that every manifold is second countable, so it has a countable basis. But does every element of the basis fit inside a chart domain? If that's the...

How does the orientation on M induce an orientation on the boundary of M? I follow the book Lectures on Differential Geometry by Chern, do not understand the proof. The proof is the Jacobian Matrix of the transformation between coordinates of two charts has positive determinant (oriented...

Hello, I am working on building some intake manifolds for an I4 2.0L street car project. The engine is being built for maximum power output, using a larger turbocharger. My basic plan was a log style, tapered plenum manifold, as you can see here...

Hi guys, Im thinking about designing an intake manifold for my car for fun. I don't necessarily want to make it or anything but I think it would be a good project to do. Can you guys recommend me some books or software that would help me out? I don't have much of an idea with CAD, but...

According to my text, a manifold should be 1) Hausdorff (that is t-2 separable, so there are disjoint open sets which are neighborhoods for any two points x and y), 2) locally euclidian (that there is a neighborhood U of a point x that is homeomorphic to an open subset U' of Rn (the RxR...xR...

The main problem I have with this question is just the wording: If M is an oriented manifold by means of the restriction of the form dx \wedge dy, describe explicitly the induced orientation on \partial M -- i.e. clockwise or counterclockwise in the plane z = 1. I don't understand the...

Is the manifold a space defined by the metric tensor or is it a completetly different thing. I'm new to tensor analysis though. Thanks.

So the equations of QM give eigenfunctions and eigenvalues. The eigenfunctions form a complete set with which any state is a combination of such. When measuring, the superposition of states collapse to one of the eigenfunctions. And the probability that some state with be measured in a...

From this reference: titled From Classical to Quantum Mechanics, I quote the following: ( \xi^i are coordinate functions) Let M be a manifold of dimension n. If we consider a non-degenerate Poisson bracket, i.e. such that \{\xi^i,\xi^j\} \equiv \omega^i^j is an inversible...

OK. Suppose you have a surface with a closed curve as a boundary. Then suppose that surface grows like a soap bubble but the boundary is stationary like the orifice through which air passes to make the bubble grow. It would seem that the 2D surface grows in both dimensions in the middle of the...

nash proved that any manifold can be embedded in R^3 in which the higher dimensional manifold gets crumpled and smoothness is lost. is it possible that 11 dimensional space has already crumpled into our three dimensional universe and that wormholes exist precisely as a direct result of the...

Hi, here are a pair of questions that I can't find the answer: What's the difference between an orbifold and a Calabi-Yau manifold? How many Calabi-Yau manifolds there exist for cubic meter of space?

what's the notation of a manifold?