# What is Diffeomorphisms: Definition and 30 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 Do diffeomorphisms have to be one-to-one functions?

The definition of a diffeomorphism involves the differentiable inverse of a function, so must the original function be one-to-one to make its inverse a function, or can the inverse be a relation and not a function? Sorry if it's a silly question, I am just a second semester calc student who...
2. ### I Active Diffeomorphisms of Schwarzschild Metric

I am trying to understand active diffeomorphism by looking at Schwarzschild metric as an example. Consider the Schwarzschild metric given by the metric $$g(r,t) = (1-\frac{r_s}{r}) dt^2 - \frac{1}{(1-\frac{r_s}{r})} dr^2 - r^2 d\Omega^2$$ We identify the metric new metric at r with the old...
3. ### 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...
4. ### 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...
5. ### A Diffeomorphisms & the Lie derivative

I've been studying a bit of differential geometry in order to try and gain a deeper understanding of the mathematics of general relativity (GR). As you may guess from this, I am approaching this subject from a physicist's perspective so I apologise in advance for any lack of rigour. As I...
6. ### 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...
7. ### Diffeomorphisms and active transformations

I've been reading Sean Carroll's notes on General Relativity, http://arxiv.org/pdf/gr-qc/9712019.pdf . I've got to chapter 5 (page 133) and am reading the section on diffeomorphisms in which Sean relates diffeomorphisms to active transformations. When he says this does he mean that one defines a...
8. ### Active/Passive Diffeomorphisms – clarification on Rovelli’s

I am new here but tried to go through some of the posts on subject matter: I apologize if I am overlooking your input as I am sure you must have clarified already my naive doubts ! I just completed a first reading of Carlo Rovelli's Quantum Gravity book (hardcover edition, 2004). I find the...
9. ### Symmetry (killing vector) preserving diffeomorphisms

Suppose that on a Riemannian manifold (M,g) there is a killing vector such that ##\mathcal{L}_{\xi} g = 0.## How would one then characterize the group of diffeomorphisms ##f: M \to M## such that $$\mathcal{L}_{f^* \xi} (f^*g) = 0?$$ How would one describe them? Do they have a name...
10. ### Breaking of time diffeomorphisms in inflation

Hello here, I am currently working on the topic of inflation. It seems that at the stage of inflation, the universe can be described as a de Sitter space. In such a space, all spacetime diffeomorphisms are preserved. (That is something I don't really understand but I keep reading that so I...
11. ### Diffeomorphisms beginner

I'm a complete rookie here, and i'd like some help. For starters , can a diffeomorphic mapping be represented via a matrix , like say a transformation? If so, how would it be parameterised?
12. ### Diffeomorphisms, Differential Structure, ETC.

I'm trying to understand the difference between diffeomorphisms, diffeomorphism invariance, reparameterization invariance, and differential structures and how each of these terms relate to physics. Perhaps there's a book out there that explains the differences between these constructs and the...
13. ### Diffeomorphisms to poincare

Hi, I'm sorry to have to ask this, but I can't seem to reason this one out by myself at the moment. Given the metric is the Minkowski spacetime, is the group of diffeomorphisms the poincare group, or are there diffeomorphisms for flat-space that are not metric preserving? I would really...
14. ### Prove that diffeomorphisms are between manifolds with the same dimension

My definition of diffeomorphism is a one-to-one mapping f:U->V, such that f and f^{-1} are both continuously differentiable. Now, how to prove that if f is a diffeomorphism between euclidean sets U and V, then U and V must be in spaces with equal dimension (using the implicit function theorem)?
15. ### Proving g is a One-Param Group of Diffeomorphisms on a Manifold

Homework Statement Let M be a differentiable manifold and g: \Re \times M \rightarrow M, (t,x) \rightleftharpoons g^{t}x be a map such that the following conditions are satisfied. i) g is a differentiable map. ii) The map \Re \rightarrow Aut(M), t \rightleftharpoons g^{t} is a...
16. ### Which of the functions are diffeomorphisms

Homework Statement Decide which ones of the following maps f: are diffeomorphisms. f(x) = 2x, x^2, x3, e^x, e^x + x. Homework Equations The Attempt at a Solution I think 2x, x^3 are diffeomorphisms. They are bijective and their inverses are differentiable x^2 and e^x are not...
17. ### Do Homeomorphisms , Diffeomorphisms Preserve Intersections?

Hi, Everyone: Say f:M-->N is a non-identity homeomorphism. Does f preserve intersections, both number and sign-wise? Maybe a more precise statement (for a 2n-manifold), given the intersection form Q on H_n . Is it the case that Q(a,b)=Q(f(a),f(b))? Thanks.
18. ### Help with Diffeomorphisms

So I'm trying to prove eqn (223) in the notes attached in this thread: https://www.physicsforums.com/showthread.php?t=457123 I took the equation ( \phi^* ( \eta) ) ( X) = \eta ( \phi_* (X)) and expanded in a coordinate basis as follows ( \phi^* ( \eta) )_\mu dx^\mu X = \eta_\alpha...
19. ### Diffeomorphisms and regular values

Homework Statement 1) If f:M -> N, where M is a compact and boundaryless manifold, N is a connected manifold has regular values y and z. And h:N -> N is a diffeomorphism which is smoothly isoptopic to the identity and carries y to z, then why is z a regular value of the composition h o f...
20. ### Simple question relating diffeomorphisms and homeomorphisms.

Consider a Euclidean space or a manifold or whatever. Furthermore, consider two regions on this space. If one can construct a diffeomorphism between the points from one region to the other, does this imply that the two regions are homeomorphic? My gut feeling is "yes," but I would like a...
21. ### Large diffeomorphisms in general relativity

"Large" diffeomorphisms in general relativity We had a discussion regarding "large diffeomorphisms" in a different thread but it think we should ask this question here. For a 2-torus there are the so-called "Dehn twists"; a Dehn twist is generated via cutting the 2-torus, rotating one of the...
22. ### Diffeomorphisms homework problem

Homework Statement Let A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} . Write A as a composition of primitive linear diffeomorphisms. Homework Equations The Attempt at a Solution I want to make sure that I understand this question properly. Are they saying to find a...
23. ### Diffeomorphisms and embedded submanifolds

The tex-code in my post was behaving really weirdly - some parts of the code seemed to have "exchanged places", so I chose to delete what I had written. I apologize for the inconvenience...
24. ### What are valid coordinate transforms (diffeomorphisms)?

The thread bcrowell had on time reversal in GR got me thinking about this. Some limitations are obvious: mapping two events onto one, discontinuity,... I will use x*, t*, etc. to refere to tansformed coordinates (primes always confuse me with derivatives). Similarly, the transform x* = t, t*...
25. ### What is the CFT equivalent of active diffeomorphisms in AdS?

Is it renormalization running? Or SU(N) Gauge transformation?
26. ### Time Parameterizations and Diffeomorphisms

Hello! It's my first post here, as I am currently reading some material, but have not been able to really grasp it. Sorry, if this is a rather dumb question. I have a dynamical system (Newtonian) that is defined on some manifold M times R (time-dependent system). Say that time is labeled...
27. ### Differentiable structures and diffeomorphisms

The definition of having multiple differentiable structures is that given two atlases, {(U_i ,\phi_i)} and {(V_j,\psi_j)} (where the open sets are the first entry and the homeomorphisms to an open subset of Rn are the second entry), that the union {(U_i,V_j;\phi_i,\psi_j)} is not necessarily...
28. ### Integral curves and one-parameter groups of diffeomorphisms

I think I understand why a vector field must have a unique set of integral curves, but I don't see why they must define a one-parameter group of diffeomorphisms. Let X be a vector field on a manifold M, and p a point in M. A smooth curve C through p is said to be an integral curve of X if...
29. ### Inclusions of submflds. and diffeomorphisms.

Hi, everyone: I am a little confused about the issue of the inclusion map on submanifolds. AFAIK, if S is a submanifold of M , then if we give S the weak (or initial) topology of the inclusion, then the inclusion map is a homeomorphism.(his is the way I understand, of making...
30. ### Geodesics and Diffeomorphisms

Hi, everyone: I was just going over some work on Hyperbolic geometry, and noticed that the geodesics in the disk model are the same as the geodesics in the upper- half plane, i.e, half-circles or line segments, both perpendicular to the boundary. Now, I know the two...