Diffeomorphism Definition and 70 Threads

Given a topological manifold, this may or may not admit a ##C^1## atlas (i.e. starting from its maximal atlas it is or it isn't possible to rip charts from it to get an atlas of ##C^1## compatible charts). A theorem due to Whitney states that from such a topological manifold ##C^1##-atlas (if...

I just need a hint to get started, and then I reckon the rest will follow... We consider a theory where matter is a covector field ##\omega_a## which is described by a diffeomorphism-invariant action ##S_m##. Define:$$E^{a} = \frac{1}{\sqrt{-g}} \frac{\delta S_m}{\delta \omega_a}$$Also, ##T^{ab}...

Hi, ##SU(2)## group as topological space is homeomorphic to the 3-sphere ##\mathbb S^3##. Since ##SU(2)## matrices are unitary there is a natural bijection between them and points on ##\mathbb S^3##. In order to define an homeomorphism a topology is needed on both spaces involved. ##\mathbb...

Queries on Carroll's derivation of matter action ## S_M ## under a diffeomorphism: (Book B.23+4 Notes 5.35+6) ##\frac{\delta S_{M}}{\delta g_{\mu\nu}} \delta g_{\mu\nu} = \frac{\delta S_{M}} {\delta g_{\mu\nu}} \left( 2 \nabla _{(\mu}V_{\nu)} \right) =\left( 2 \right) \frac{\delta...

I need to prove that under an infinitesimal coordinate transformation ##x^{'\mu}=x^\mu-\xi^\mu(x)##, the variation of a vector ##U^\mu(x)## is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$ where ##\mathcal{L}_\xi U^\mu## is the Lie derivative of ##U^\mu## wrt the vector...

Hi Pfs Rovelli writes this in his book (Qunatum Gravity) about spin networks: Given an oriented and ordered graph there is a finite disgrete group of maps that change its order or orientation and that can be obtained as a diffeomorphism. A link is equipped the source and target functions. this...

Hi, a clarification about the following: consider a smooth curve ##γ:\mathbb R→\mathbb R^2##. It is a injective smooth map from ##\mathbb R## to ##\mathbb R^2##. The image of ##\gamma## (call it ##\Gamma##) is itself a smooth manifold with dimension 1 and a regular/embedded submanifold of...

Hi, I know there is actually no way to set up a global coordinate chart on a 2-sphere (i.e. we cannot find a family of 2-parameter curves on a 2-sphere such that two nearby points on it have nearby coordinate values on ##\mathbb R^2## and the mapping is one-to-one). So, from a formal...

Dear all, in my current week of holidays, where all the Corona-dust settles down a bit, I came across some personal notes I made a while ago about the meaning of diffeomorphism invariance, the difference between passive and active coordinate transformations, and the notion of background...

Hello, I know question surrounding this topic have been asked again before, however the more I search for the answer of my question, the more confused I become since each book, paper and thread uses its own formulation. I am trying to figure out what is a conformal transformation and as a...

https://arxiv.org/pdf/1812.06239.pdf In this paper,the authors use ricci flow to construct Lifshitz spaces. But it is known that ricci flow is limited by Riemannian manifold, which has a positive metric. but in this paper the author use ricci flow in a lorentz manifold, whose signature...

Hi Lets start off with the definition of diffeomorphism from Wolfram MathWorld: The issue is that I am learning about smooth manifolds, and in the books I've read, the map has to be smooth and have a smooth inverse. Also, the definition above doesn't say that it has to be bijective. However...

I'm trying to understand diffeomorphisms, and I thought I basically understood them, but when I tried to work out a problem I created for myself, I realized I didn't know how to answer it. So let's consider a diffeomorphism generated by a vector field ##V##. If ##X## is a point on our manifold...

I've been reading Straumann's book "General Relativity & Relativistic Astrophysics". In it, he claims that the twice contracted Bianchi identity: $$\nabla_{\mu}G^{\mu\nu}=0$$ (where ##G^{\mu\nu}=R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R##) is a consequence of the diffeomorphism (diff) invariance of the...

Hi I have studied GR form a number of sources - my favorite being Wald - he uses Diffeomorphism Invarience, the rest General Covarience/Invarience. Wald is more mathematically sophisticated but at rock bottom is there really any difference. My suspicion is Wald does it because there is a...

Hello! I just started reading something on differential geometry and I am not sure I understand the Difference between diffeomorphism and homeomorphism. I understand that the homeomorphism means deforming the topological spaces from one to another into a continuous and bijective way (like a...

it is often stated in texts on general relativity that the theory is diffeomorphism invariant, i.e. if the universe is represented by a manifold ##\mathcal{M}## with metric ##g_{\mu\nu}## and matter fields ##\psi## and ##\phi:\mathcal{M}\rightarrow\mathcal{M}## is a diffeomorphism, then the sets...

Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes (http://www.hartmanhep.net/topics2015/) on Quantum Gravity: In gravity, local diffeomorphisms are gauge symmetries. They are redundancies. This means that local correlation functions like ##\langle...

Hi everybody, Let V(x) a vector field on a manifold ( R^2 in my case), i am looking for a condition on V(x) for which the function x^µ \rightarrow x^µ + V^µ(x) is a diffeomorphism. I read some document speaking about the flow, integral curve for ODE solving but i fail to find a generic...

How is loop quantum gravity related to the principle of relativity, in particular to the diffeomorphism invariance of general relativity?

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...

Hi. This is my first post here in PF ( :) ). I've been reading some threads on "passive" versus "active" diffeomorphisms, and I wondered: what is the physical motivation for having GR be diffeomorphic invariant? Sure, this allows us to have solutions to Einstein's equations (EFE) up to...

I am relatively new to the concept of differential geometry and my approach is from a physics background (hoping to understand general relativity at a deeper level). I have read up on the notion of diffeomorphisms and I'm a little unsure on some of the concepts. Suppose that one has a...

It is unclear to me (1) what precisely diffeomorphism means and (2) what happens when all matter is removed from the spacetime. Sean Carrol says that: "the theory is free of "prior geometry" and there is no preferred coordinate system for spacetime." http://arxiv.org/pdf/gr-qc/9712019v1.pdf page...

The Polyakov action, S=\frac{1}{4\pi\alpha^\prime}\int d^2\sigma\sqrt{-h}h^{\alpha\beta}G_{ij}(X)\partial_\alpha X^i\partial_\beta X^j has the local symmetries, diffeomorphism on world sheet and the Weyl invariance. But is diffeomorphism on the target space also a symmetry? The target space...

[SOLVED] Diffeomorphism invariance of the Polyakov action Hi, I'm struggling with something that is quite elementary. I know that the Polyakov action is diffeomorphism invariant and Weyl invariant. Denoting the world-sheet coordinates \sigma^0 = \sigma and \sigma^1 = t and the independent...

This is a very basic question, but I cannot get my head around the following: Any physical system should be invariant under changes of coordinates, because these are just a way of parametrizing the manifold/space in which my physical system is embedded. Now, let us consider a system that...

Given a smooth vector field ##V## on a smooth manifold ##M## the uniqueness of differential equations assures that there exists a unique integral curve ##\phi^{(p)}: J \to M## for some open interval ##J \subseteq \mathbb{R}## for which ##0 \in J## and ##\dot \phi^{(p)} (0) = V_{\phi^{(p)}...

From a topological point of view a homeomphism is the best notion of equality between topological spaces. I.e. homeomorphisms preserve properties such as Euler characteristic, connectedness, compactness etc. I've understood it such that diffeomorphisms are the best notion of equality between...

I suspect this is somewhat off the beaten track here, but there may be some few that could give it a go. Einstein called his concept of coordinate independent physical theory General Covariant. The mathematicians call coordinate independent differential topology, diffeomorphism invariant...

Show that TS^1 is diffeomorphic to TM×TN. (TS^1 is the tangent bundle of the 1-sphere.) We can use the theorem stating the following. If M is a smooth n-manifold with or without boundary, and M can be covered by a single smooth chart, then TM is diffeomorphic to M×ℝ^n. Clearly, I must be...

Hello, we known that for each linear operator \phi:\mathbb{R}^n\rightarrow \mathbb{R}^n there exists an adjoint operator \overline{\phi} such that: <\phi(\mathbf{x}),\mathbf{y}>=<\mathbf{x},\overline{\phi}(\mathbf{y})> for all x,y in ℝn, and where <\cdot,\cdot> is the inner product. My...

Hello, the definition of diffeomorphism is: a bijection f:M\rightarrow N between two manifolds, such that both f and f-1 are smooth. Is it thus correct to say that a (admissible) change of coordinates is a diffeomorphism between two manifolds?

Hi! I'm working through this script and I'm not sure if if there is a mistake at one point, or if I'm just thinking wrong. To prove this, the group Sp(n+1) = \{A \in M(n+1, \mathbb H) | A^*A = I\} is used. Its elements operate linearly and isometrically on \mathbb{S}^{4n+3} and therefore...

I would like to study the path components (isotopy classes) of the diffeomorphism group of some compact Riemann surface. To make sense of path connectedness, I require a notion of continuity; hence, I require a notion of an open set of diffeomorphisms. What sort of topology should I put on the...

For some reason, diffeomorphism invariance seems to be treated like a second-class citizen in the land of symmetries. In nonrelativistic quantum mechanics, we consider Galilean invariance so important that we form our Hilbert space operators from irreducible representations of the Galilei...

One of the foundations of General Relativity is diffeomorphism invariance - the fact that the laws of physics are invariant under smooth coordinate transformations, and thus the laws must involve tensors. My question is, why doesn't this imply scale invariance; after all, isn't a change of...

Hagen Kleinert, in his book, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4 Edition, generalizes the Equivalence principle with what he calls a nonholonomic transformation. He transforms equations in flat space into spaces with curvature and torsion...

Let U be a non-empty open set in Rn, if f:U->Rm is a diffeomorphism onto its image, show that df(p) is injective for all p in U. How can I attack this problem?

I have trouble in showing a disk and a square is not diffeomorphic. Intuitively, I know there is smoothness problem occurs at the corner of the square if I suppose there is a diffeomorphism between the two, but how can I explicitly write down the proof? I hope someone can provide me with some...

Does anyone know of any website that has animations of what this Diffeomorphism Invariance in General Relativity can do? I read a lot of articles about it but can't seem to get the essence or visualize how it actually occurs exactly. Thanks.

I have a question... Since solutions to Einstein's field equations are diffeomorphism invariant, does that mean that solutions are metrics of constant curvature?

Homework Statement I have some problems with understanding these two things. Homeomoprhism is a function f f: M\rightarrow N is a homeomorphism if if is bijective and invertible and if both f, f^{-1} are continuous. Here comes an example, let's take function f(x) = x^{3} it is...

I'm hoping there will be some comment on this new paper of Kirill Krasnov http://arxiv.org/abs/1101.4788 Gravity as a diffeomorphism invariant gauge theory Kirill Krasnov 24 pages (Submitted on 25 Jan 2011) "A general diffeomorphism invariant SU(2) gauge theory is a gravity theory with two...

Hi, does anyone have any idea how to prove the mapping R^2->R^2 (x/(x^2+y^2), y/(x^2+y^2) is a diffeomorphism, and if it is not restrict the values so it is one I am fairly sure it is not over R^2 as it is not continuous at 0, but I don't know what values to restrict it over. I have...

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This question comes from proof of Theorem 2.47 in Folland's "real analysis: modern techniques and their applications", second edition. In particular, the question lies in the inequalities in line 7 and 8 in page 76. The first equality is an application of measure property "continuity from...

I've read that a function f given by f:U\rightarrow V is a diffeomorphism if the inverse function f^{-1} exists and is differentiable. I've also read that that function is a local diffeomorphism in a given point p\inU if it can be found a range A around p such that the function f verifies f:A ->...

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...