# What is Diffeomorphism: Definition and 74 Discussions

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.

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1. ### I ##SU(2)## homeomorphic with ##\mathbb S^3##

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...
2. ### I Variation of matter action under diffeomorphism (Carroll)

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...
3. ### A Infinitesimal Coordinate Transformation and Lie Derivative

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...
4. ### A What does Rovelli mean with "oriented and ordered graph"?

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...
5. ### I Clarification about submanifold definition in ##\mathbb R^2##

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...
6. ### I Global coordinate chart on a 2-sphere

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...
7. ### I Diffeomorphism Invariance, Passive/Active Interpretations GR Insight

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...
8. ### Conformal Transformation as a diffeomorphism

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...
9. ### A Can Ricci Flow be Used in Lorentz Manifolds?

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...
10. ### A Equivalence of diffeomorphism definitions

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...
11. ### A Coordinate representation of a diffeomorphism

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...
12. ### I Diffeomorphism invariance and contracted Bianchi identity

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...
13. ### A Difference Between Covariance and Diffeomorphism Invarience

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...
14. ### I Difference between diffeomorphism and homeomorphism

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...
15. ### I Diffeomorphism invariance of GR

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...
16. ### A Diffeomorphism invariance and gauge invariance

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...
17. ### Show existence of arc-length parameterized period p'....

Homework Statement Let γ : R → Rn be a regular (smooth) closed curve with period p. Show that there exist an orientation preserving diffeomorphism ϕ: R → R, a number p' ∈ R such that ϕ(s + p') = ϕ(s) + p and γ' = γ ◦ ϕ is an arclength parametrized closed curve with period p' Homework...
18. ### Counting degrees of freedom in field theory

I'm having a bit of trouble with counting the number of physical ("propagating") degrees of freedom (dof) in field theories. In particular I've been looking at general relativity (GR) and classical electromagnetism (EM). Starting with EM: Naively, given the 4-potential ##A^{\mu}## has four...
19. ### I Condition on vector field to be a diffeomorphism.

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...
20. ### A Loop quantum gravity and diffeomorphism group

How is loop quantum gravity related to the principle of relativity, in particular to the diffeomorphism invariance of general relativity?
21. ### Why do diffeomorphic manifolds have physically identical properties?

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...
22. ### What is the physics behind GR's diffeomorphism invariance?

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...
23. ### Attempting to understand diffeomorphisms

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...
24. ### Diffeomorphism and removal of all matter from the spacetime

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...
25. ### Exploring Polyakov's Action: Diffeomorphism Symmetry?

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...
26. ### Diffeomorphism invariance of the Polyakov action

[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...
27. ### Spacetime symmetries vs. diffeomorphism invariance

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...
28. ### Local flow as a diffeomorphism.

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)}...
29. ### Diffeomorphism vs homeomorphism

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...
30. ### CPT and Diffeomorphism Invariance

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...
31. ### Diffeomorphism From Tangent Bundle to Product

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...
32. ### Adjoint operators and diffeomorphism

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...
33. ### How a local diffeomorphism preserves orientation

Hi, I'm currently reading Manfredo do Carmo's "Differential Geometry of curves and surfaces" and I'm stocked with problem 2 of section 2-6 which is "Let S_2 be an orientable regular surface and \varphi:S_1 \rightarrow S_2 be a differentiable map which is a local diffeomorphism at every p \in...
34. ### Is a change of coordinates a diffeomorphism?

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?
35. ### Isometric diffeomorphism on HPn/S^4n+3

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...
36. ### Sufficient condition(s) for a non-local diffeomorphism?

I hope I'm posting this question in the right forum; anyway, here goes: What would be a sufficient condition for a diffeomorphism to be non-local--specifically, for it to be valid over a given domain? In the particular case I'm examining, the mapping I'd like to be a non-local diffeo is given...
37. ### Topology of the diffeomorphism group

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...
38. ### Why not diffeomorphism group representation theory?

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...
39. ### Why doesn't Diffeomorphism Invariance lead to Scale Invariance?

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...
40. ### Is nonholonomic Xformation = Diffeomorphism

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...
41. ### F is diffeomorphism implys df is injective?

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?
42. ### Diffeomorphism of a disk and a square?

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...
43. ### Diffeomorphism Invariance in GR

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.
44. ### Diffeomorphism and constant curvature

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

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...
46. ### Gravity as a diffeomorphism invariant gauge theory (new Krasnov paper)

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...
47. ### Help for proving a mapping is a diffeomorphism

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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49. ### A Question in a proof on Lebesgue integral under diffeomorphism

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...
50. ### Is a diffeomorphism if the inverse function

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 ->...