Is a change of coordinates a diffeomorphism?

[mnb96]

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?

 

[chiro]

Hey mnb96.

I'm going to assume yes since a bijection should preserve the dimensionality and provide both a normal map and its inverse to be differentiable over the entire domain and codomain of the bijection.

In fact because of the inverse function theorem, the invertibility requires that the Jacobian never be zero since a bijection must be defined everywhere.

If a change of co-ordinates is simply referring to going from one co-ordinate system with intrinsic dimension N to another with the same intrinsic dimension N with a bijection for every transformation, then it sounds right that it is a diffeomorphism.

 

[mnb96]

Hi chiro,
thanks for your reply.
The way I formulated my question, and the answer you have given seem to suggest that: a change of coordinates is a diffeomorphism of manifolds.

Now I am wondering: is the converse true? Are all diffeomorphisms expressible as change of coordinates?

 

[chiro]

I think the answer is true and my reasoning is that bijections preserve the dimension which means that it will be a valid co-ordinate change (since this has to happen).

The other reason is that a bijection implies an inverse transformation in both directions for all elements of the respective domains in both spaces.

The other thing is that you have continuity requirements for both spaces as well.

Summing this up, these are my reasons why I am inclined to say yes to your answer.

I guess the thing is define what a change of co-ordinates really refers to. To me it just refers to being able to go from co-ordinate system A to B for all points in A and B and that the continuity is preserved in both systems (so that if I go on a continuous trajectory from a1 to a2 in A then the same continuity properties apply in B: i.e. that both paths are continuous and that if the path in A is differentiable, then it is differentiable in B).

So to clarify: if you have a path that is continuous or differentiable in A then that same path when transformed to system B is continuous or differentiable in B respectively.

I can't really think of another way to formalize it unless there is some standard definition that already exists.

 

[micromass]

What are you meaning with "change of coordinates" in the first place. Do you mean that if $\varphi:U\rightarrow \mathbb{R}^n$ and $\psi:V\rightarrow \mathbb{R}^n$ are smooth charts, then $\psi\circ \varphi^{-1}$ is your change of coordinates?? Yes, that is a diffeomorphism indeed since it is smooth and has smooth inverse.

But these are only diffeomorphisms between open subsets of $\mathbb{R}^n$. So do you want to restrict yourself to diffeomorphisms of open subsets of $\mathbb{R}^n$?? I guess I don't get the question very well...

 

[mnb96]

What are you meaning with "change of coordinates" in the first place. Do you mean that if $\varphi:U\rightarrow \mathbb{R}^n$ and $\psi:V\rightarrow \mathbb{R}^n$ are smooth charts, then $\psi\circ \varphi^{-1}$ is your change of coordinates??

Yes, I think this is pretty much what I had in mind.

...But these are only diffeomorphisms between open subsets of $\mathbb{R}^n$. So do you want to restrict yourself to diffeomorphisms of open subsets of $\mathbb{R}^n$??

For the application I had in mind, yes. I need to restrict myself to diffeomorphisms between open subsets of Rⁿ.
...And maybe I also see your point here. You were basically telling me that the very notion of diffeomorphism does not necessarily have to be defined between (sub)manifold of Rⁿ, but instead between manifolds of some more "exotic" space than Rⁿ; and these diffeomorphisms won't be "captured" by change of coordinates.

 

[micromass]

For the application I had in mind, yes. I need to restrict myself to diffeomorphisms between open subsets of Rⁿ.
...And maybe I also see your point here. You were basically telling me that the very notion of diffeomorphism does not necessarily have to be defined between (sub)manifold of Rⁿ, but instead between manifolds of some more "exotic" space than Rⁿ; and these diffeomorphisms won't be "captured" by change of coordinates.

Well, in fact, every manifold is a submanifold of $\mathbb{R}^n$ by Whitney's embedding theorem. But change of coordinates seem to be only between open submanifolds, and not every manifold can be seen as an open submanifold of $\mathbb{R}^n$, of course.

So yes, there are diffeomorphisms out there which are no simple change of coordinates. However, we can restrict our question and ask: are all diffeomorphisms between open submanifolds of $\mathbb{R}^n$ some kind of change of coordinates. The answer is of course yes. Let $U,V\subseteq \mathbb{R}^n$ open and let $F:U\rightarrow V$ be a diffeomorphism. Then F can be seen as a smooth chart of U. But the identity map can also be seen as a smooth chart of U. The change of coordinates between these charts is exactly F.

 

[mnb96]

Hi,
thank micromass. I think I have problems in understanding the definition of open/closed manifold.
I am also confused between open/closed subset and open/closed manifolds.

I was trying to think of the example of polar coordinates. In polar coordinates we have a 1-to-1 mapping between ℝ²-{0,0} and ℝ⁺×[0,2π).

Now, is ℝ² a (open) manifold at all? If not, we have found also a coordinate change that is not a diffeomorphism, as diffeomorphisms seem to be defined as mappings defined between manifolds.