Talk:Diffeomorphism

message: HELP : can't get the links to Manifold#Differentiable_manifolds to work. When I click (try atlases for instance) I get to the top of the Manifold page. What's wrong ? Pascalromon 14:19, 27 Nov 2003 (UTC)
thread_title:  Resolved question concerning linking 2003 

message: I'm linking into here from guage theory, trying to wrap my head around some of these topics. However, IMHO, all of them are written by math nerds for other math nerds. Can't anyone provide a single example of these concepts for us non-math-nerds? The second sentance of ANY article should not start "the mathematical definition is as follows"!
Maury 22:26, 28 November 2005 (UTC)
replies: text: The definition is very simple. Just read carefully, it says a function which is invertible, and both the function and the inverse are smooth. I assume you know what the words "invertible function" and "smooth" mean. Otherwise, this article will not be helpful to you. Oleg Alexandrov (talk) 22:33, 28 November 2005 (UTC)
replies: text:  And there is an example there, and a good discussion at ==Local description==. Oleg Alexandrov (talk) 22:34, 28 November 2005 (UTC)

text: I don't think it was buried. It was the third sentence in the article (if you count the wording "The mathematical definition is as follows." as a sentence). But I do agree that it looks better now. Oleg Alexandrov (talk) 20:18, 29 November 2005 (UTC)
replies: text:  I've added three nice examples. The example about the reals modulo the integers was too algebraic, and not very informative. Since all manifolds are locally diffeomorphic to real space we can do all practical calculations in coordinate charts. Thus I have given three examples of maps from two-space to two-space. For any calculation I have ever had to make (and I work in applications of singularity theory to differential geometry) I have always used the Jacobian method in my three examples. It's all well and good saying that a map and it's inverse must be differentiable, but how does one compute an explicit inverse for a map? With the Jacobian method it's automatic: it tells you that the map is differentiable, and that there exists a differentiable inverse. Dharma6662000 (talk) 19:24, 16 August 2008 (UTC)
thread_title:  Any chance of a lay example? 

message: The current definition of a diffeomorphism is given in terms of a map between manifolds, but a diffeomorphism can be defined over much more general spaces (namely, normed vector spaces - see e.g. Marsden, Ratiu, & Abraham, "Manifolds, Tensor Analysis, and Applications").  Why is the more specific case used here?  Trevorgoodchild (talk) 05:23, 29 January 2008 (UTC)
replies: text: I think the most general case is even more general than that, you can have diffeomorphisms over infinite-dimensional manifolds (which look locally like a normed vector space). But I guess going that general could make the article hard to understand. The most used cases are in manifolds over R^n anyway. If you want, you can add somewhere at the bottom a section on generalizations, but I'd prefer that at least the first several sections in the article be about the "usual" R^n manifolds case. Oleg Alexandrov (talk) 15:44, 29 January 2008 (UTC)
thread_title:  Manifolds or just normed vector spaces? 

message: The "model example" section says that the diffeomorphism f "happens to be" a homoeomorphism.  From the definition, it seems to me that all diffeomorphisms are homoeomorphisms, since all differentiable functions are continuous.  If that is the case, "happens to be" seems misleading. LachlanA (talk) 22:45, 6 May 2008 (UTC)
replies: text: Yeah you're totally right: every diffeomorphism is a homeomorphism, but the converse is false. I'm sure you knew, but just to make sure: a function and its inverse must be differentiable (resp. continuous) for it to be a diffeomorphism (resp. homeomorphism). I've changed the article, and made your point clear. I replaced the sentence in question with: "f being a diffeomorphism is a stronger condition than f being a homeomorphism. For a diffeomorphism we need f and its inverse to be differentiable. For a homeomorphism we only require that f and its inverse be continuous. Dharma6662000 (talk) 19:28, 16 August 2008 (UTC)
thread_title:  Relationship with homoeomorphisms 

message: I am quite sure that two manifolds are diffeomorphic iff there is a smooth bijective function between them that also has a smooth inverse (not as in the article where only a bijective differentiable function with differentiable inverse is required). For example, think about it this way: two mathematical objects are equivalent (and considered the same) iff there is an isomorphism between them. According to the definition given in the article, two differentiable manifolds may be equivalent without really having the same differentiable structure. 
In fact, C0 functions are just continuous, so according to the article, two manifolds are diffeomorphic if they are homeomorphic. I am therefore changing this. If you think otherwise, please give your reasons.
Topology Expert (talk) 14:03, 6 October 2008 (UTC)
thread_title:  Diffeomorphic manifolds=a smooth bijection with smooth inverse between them 

message: I flagged the statement
:In both cases, the diffeomorphism group is locally homeomorphic to the space of Cr vector fields on the manifold (where r is the order of differentiability considered).
as dubious.  To the best of my knowledge, there are diffeomorphisms in any neighborhood of the identity that are not in the image of the exponential map.  Sławomir Biały (talk) 14:13, 4 December 2009 (UTC)
replies: text: In any case, it seems unlikely that this can be true in both cases, as asserted in the article: the strong topology is not locally homeomorphic to the weak topology on a non-compact manifold.  It seems like a way forward would be to write a short section that describes each topology a little bit more carefully, making sure also to provide references in order to verify the content and also to avoid possible misunderstandings.  Sławomir Biały (talk) 16:44, 4 December 2009 (UTC)
replies: text: I think I have resolved the problem.  The article wasn't claiming that the local homeomorphism came from the exponential mapping of the group. Sławomir Biały (talk) 02:53, 5 December 2009 (UTC)
thread_title:  Dubious 

message: I know very little about this topic, but hope to learn more. As a user of Wikipedia, I would like to know what the following notation means:
  
g_{|U \cap X} = f_{|U \cap X}
I mention this, not so much to get my question answered (I know, this is not a discussion group) but to point out that Wikipedia apparently offers no help with notation. I cannot click on a symbol and get its definition. I have been wondering if there might be some way to close up that gap. Dratman (talk) 20:05, 19 September 2012 (UTC)
thread_title:  Question on notation