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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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?
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...
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...
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)?
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...
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...
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.
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...
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...
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...
"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...
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...
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...
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*...
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...
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...
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...
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...
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...