What is Diffeomorphism: Definition and 74 Discussions

I am terribly confused on the issue of trivial tangent bundles. I understand intuitively why some tangent bundles are trivial and others are not, but I'm having trouble figuring out how to show it. Even the most trivial example, show that T\mathbb{R}^n is diffeomorphic to \mathbb{R}^{2n} I...

I've read that GR is diffeomorphism invariant, I asked a math buddy of mine and I have a VERY BASIC idea of what that means in this case - the theory is the same regardless of your choice of coordinates? Noether's theorem states that for every symmetry there's a corresponding conservation...

I am posting my question in this forum because it is about a basic conceptual aspect of LQG discussed in Rovelli's book Quantum Gravity. He makes the following statement on page 67 (here, "e" refers to the vierbein): I do not understand the part in boldface. First, he means that the...

While reading C.C.Pugh's "Real Mathematical Analysis" I've encountered a following statement: "A starlike set U \subset \mathbb{R}^n contains a point p such that the line segment from each q\in U to p lies in U. It is not hard to construct a diffeomorphism from U to \mathbb{R}^n." It's little...

What is the relationship between being globally diffeomorphic and the Jacobian of the diffeomorphism? All I can think of is that if the Jacobian at a point is non-zero, then the map is bijective around that point. For example, if: f(x)=x_0+J(x_0)(x-x_0) where J(x0) is the Jacobian...

Having some generic curved spacetime, what are the Noether currents that are guaranteed to exist by diffeomorphism invariance? Is the energy-momentum tensor such a current?

I have seen it mentioned in various places that the Lie algebra of the diffeomorphism group of a manifold M is identifiable with the Lie algebra of all vector fields on M, but I have not found a demonstration of this. I can show that the map \rho: Lie(Diff(M)) \to Vect(M), ~~~ \rho(X)_p =...

Hi; I am pretty sure that sqrt(-g) is diffeomorphism-invariant. I am wondering if all powers of this are diffeo-invariant too. For example, are -g, g^2, etc. all invariants too?

In the definition of F-related vector fields, F must be a diffeomorphism. Why must it be a diffeomorphism? What if F is smooth and bijective, but not a diffeo?

when I first learned about homeomorphic sets, I was given the example of a doughnut and a coffee cup as being homeomorphic since they could be continuously deformed into each other. fair enough. Recently I heard another such example being given about diffeomorphisms: "Take a rubber cube...

Why is a non zero jacobian the necessary condition for a diffeomorphism? How to prove it?

I read about Einstein's hole argument and about diffeomorphism. Pasive diffeomorphism is changing of coordinate system. I do not understand, what is active diffeomorphism. I hear an example with rubber, which is streched. But I imagine this as passive diffeomorphism. Regards

I'm trying to figure out what the surface x4 + y6 + z2 = 1 looks like. I want to say that it is diffeomorphic to the sphere because (x2)2 + (y3)2 + (z)2 = 1 but i can't seems to actually construct the diffeomorphism (I am having problems with the x2 being invertible). Please let me know if...

Hi: I am trying to show that if we have a diffeomorphism f:M-->N and C is a positively-oriented Jordan curve in M ( so that., the winding number of C about any point in its interior is 1 ) , then f(C) is also positively-oriented in the same sense. It seems like something...

Is it fair to think of a diffeomorphism as being a "stronger" condition then a homeomorphism? I know this is probably a dumb question, but I'm trying to teach myself some topology, and still waiting for Munkres to come in the mail. :)

I initially posted this question in the Beyond the Standard Model forum since diffeomorphism invariance is a key ingredient of loop quantum gravity but it was suggested that I post the question here. Why is Einstein's theory diffeomorphism invariant? A diffeomorphism is basically a map of...

I was watching one of Smolin's online lecture (the link was provided by Marcus in the thread "What's new that's happening in quantum gravity" or something to that effect) and Smolin makes a big deal on the difference between diffeomorphism invariance and invariance under general coordinate...

Homework Statement M is a smooth manifold, U \subset M is a proper open set. Show that there exists a smooth non-trivial diffeomorphism from M onto itself which restriction on M - U is identity ("identity outside U"). Homework Equations The Attempt at a Solution If there exists a non-zero...

q1. Homework Statement Let f : X ->Y, g : Y->Z be smooth. Show the composite is smooth. If f, g are diffeomorphisms, so is the composite. q2.Let B= {x : |x|^2 < a^2}. Show that x -> ax/[(a^2 − |x|^2)^1/2] is a diffeomorphism. Homework Equations The Attempt at a Solution For q1...

Are the following functions \mathbb{R}^2\rightarrow\mathbb{R}^2 diffeomorphisms. If not is there an open set containing the origin on which the function is a diffeomorphism to its image? 1. (x,y)\mapsto(x+y^3, y) 2. (x,y)\mapsto(x+x^3,x) I have the definition of a diffeomorphism...

The "hole argument" and diffeomorphism invariance First, let me give a summary of my understanding of the "hole argument": Consider a space-time completely filled with matter with exception of a finite space-time volume that contains no matter (a hole). The hole is located between two spatial...

I know what a diffeomorphism is. But what is diffeomorphism invariance? And why is it important in physics? Thanks.

Rovelli has introduced an extended idea of diffeomorphism which is smooth except at a finite number of points. The group of extended (or 'almost smooth') diffeomorphisms plays a role in the Fairbairn/Rovelli paper that just came out and also in the key chapter 6 of the new book "Quantum...

There has been a lot of internet discussion of a paper by Ashtekar, Fairhurst, and Willis called "Quantum gravity, shadow states, and quantum mechanics" gr-qc/0207106, and one of the authors, Josh Willis, replied on SPR today (in the thread "LQG and diffeomorphism group cocycles") Here is a...