Manifold Definition and 318 Threads

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?