Talk:Diffeomorphism

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z(x) is a point on the unit circle in the Complex Plane. This is the circle centered on 0 (the origin), and having radius, 1. For example, the x-value 0 maps onto the point z=1; the x-value 0.5 maps onto the point z=i, the x-value 1 (we'll allow it to sneak in) maps onto the point z=-1, and the x-value -0.5 maps onto the point z=-i. So using this sequence of x-values we have traversed the unit circle counterclockwise, E, N, W, S. If we allowed the x-value, x=-1, it would also map to z=-1, since exp( i*pi*1) = exp( -i*pi*1) = -1 (Euler's identity, q.v.). This unit circle is called S1, because a circle is a 1-dimensional "sphere" (embedded in 2-dimensional space), a sphere (S2) is a 2-dimensional object (embedded in 3-dimensional space), etc. The integer part of the cosets (the "Z" part), when inserted into the formula, sends you to z=+1, z=-1, (E and W), again and again, contributing nothing new. So we have mapped a line segment (-1 to +1) on the Real axis, onto the unit circle, of which it is a diameter. If this segment is traversed from -1 to +1, then the unit circle will be traced from -1, counterclockwise, all the way around to -1 again.

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true construction R/Z doesn't suffer this defect. Like I mentioned, 0+Z and 1+Z are the same cosets, and so the same points of R/Z, however, they are not special. It's also true that 0.5+Z and 1.5+Z are the same points, and 0.9+Z and 1.9+Z are the same points, etc. Since I started with an infinite line R, instead of a line segment, there are no special points anywhere, at least not from a topological point of view. When you recall that R and Z are groups, both have a special point, 0, the identity. R/Z then has a special point as well, but this is OK for groups. You could get rid of this special point by viewing R as an

1783:'s book for example. For the circle, ordering of points must be taken into account. A quick search of mathscinet gave the generalisations of Boothy, Hatakeyama and Michor & Vizman to symplectic and contact manifolds. Here it seems that, rather than taking an exotic definition of smooth manifold, compact or not, the assumptions of the original authors in the literature need to be unearthed. What is true is that the old article does not give page numbers in the sources, or any direct citations. This would help avoid situations like this. 1947:, your curve with a cusp has no natural manifold structure. You can make it a manifold in many ways, the way you've chosen (writing it as the graph of a non-smooth function and then mandating that it has the differentiable structure to make the function a diffeomorphism onto its image) allows one to make sense of a smooth structure, but it's in no way natural, because if you make a different choice of bijection with the reals, you can get a different smooth structure. To be precise -- in your terminology, there is no 84: 74: 53: 2542:

The addition of a fermion is poorly explained, and does not show how one gets a "group" out of the resulting Lie superalgebra. The second section seems to be about diffeomorphism groups in a particular context (relativity theory maybe), and so again seems poorly placed and poorly explained. I'm leaving this here for discussion, since it's possible some of this content should be worked into the article in some other way.

1323:(the lede). In Knowledge (in general at least), unless it is stated that manifolds must be Hausdorff in a given article, one can assume otherwise (I actually would prefer to assume that a manifold is second countable and Hausdorff but Knowledge has separate articles for seperate assumptions of what a manifold is). If you assume the Hausdorff condition, just add it to the lede and I can add back that example. 1876:? It is definitely an imbedded submanifold if not an immersed one. Furthermore, according to Rybu, the projection map from the curve with a the cusp to the x-axis (along wih the obvious atlas) is not natural? Why not (it would be 'natural' if we embedded the 'curve with a cusp' in itself (with the previous mentioned differential structure))? To determine whether two manifolds 22: 3154: 1045:) 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. 2981: 3493:

This article ambiguously switches between the categories of differentiable manifolds and smooth manifolds at various points. I looked at past revisions and there seems to have been a bit of a mini edit war concerning the adjectives "differentiable", "continuously differentiable", and "smooth" in the

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The following content was recently added to the text. I feel that it is out of place there because the first section does not clearly say what it is trying to do (namely that the Lie algebra of the diffeomorphism group is the space of vector fields, which can be mentioned in a better way elsewhere).

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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

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but it's not a natural smooth structure. In this sense the smooth structure you're putting on the above curve would be considered unnatural. The natural smooth structure on a subset of a manifold is the one coming from the ambient manifold. This is a common notion for sets as well as submanifolds.

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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

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R/Z : Divide any real number by any integer; throw away the integer part of the answer; what remains is a number, x, between -1 and +1: -1 < x < +1; this set of numbers is R/Z. Now use the formula, z(x) = exp( i*pi*x ); "exp" = "e to the"; "*" = multiply ); for any value of x in our set, R/Z,

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I believe the best course of action is to clarify that there are two notions of diffeomorphism, a diffeomorphism of smooth manifolds and a diffeomorphism of differentiable manifolds, and that the respective notion of diffeomorphism is the isomorphism in each category. We could also mention $ C^r$

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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

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Charles's description has a special point because he started with a line segment. A line segment has two special points: its endpoints. When you join the two endpoints with glue, they become a single point, and a real circle doesn't have any special points, from a topological standpoint. But the

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I think you are just fighting the terminology used in the field of manifold theory. There are a lot of words that are near synonyms. Often they are synonyms, depending on the context. We aren't writing a textbook here, just giving the basic ideas and pointing people to references. If one makes

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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

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on a Cantor set, or any continuum? If you have no standards for what the words you use mean, it's not clear they belong in a mathematics article. I suppose undefined terms have a place, but here we're talking about something technical, not heuristic. Ah well. Somebody else can clean this up.

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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

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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

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I am not disagreeing with you but I usually assume differentiable manifolds to take any appropriate differentiable structure. I think that this is the point of view taken by most mathematicians. Of course, in the context of submanifolds, this may be different. However, prior to me deleting the

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I have added a Notes section in the article (for some strange reason the Hirsh and Smale references had been left orphaned), together with a direct link to Banyaga for the point on transitivity. I added the comments on Choquet and Rado, having sourced them in the literature. These days I would

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The article has many issues, even after my recent edits. The main issue was that, in the first line of the lead it defined diffeomorphism of non-differentiable manifolds. In my edit, I get rid of the ambiguous word "smooth", which can be understood either as "differentiable" or "indefinitely

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Rybu, I don't mean to confuse anyone by being overly pedantic, but I believe that a diffeomorphism of smooth manifolds is not a synonym for diffeomorphism of C^1 manifolds. I'm simply worried that people unfamiliar with the material may get tripped up on the unstated fact there are multiple

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I mention this, not so much to get my question answered (I know, this is not a discussion group) but to point out that Knowledge 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.

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Hi there. Manifolds in this article (as standard on Knowledge unless otherwise noted) are considered to be 2nd countable Hausdorff. Although the diffeomorphism groups of non-Hausdorff manifolds are occasionally considered in research literature, more often than not they're considered

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respectively (the space is Hausdorff apart from (0,a) and (0,b)). Continuity of the homemorphism means that there are disjoint neighbourhoods about (0,a) and (0,b); a contradiction. Since every diffeomorphism is a homeomorphism, this is a counterexample to the claim in the article.

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I might be confused, but I am quite sure that it is still possible to give the curve a differentiable structure since the projection map yields a local homeomorphism with the curve and the reals (and the transition maps are smooth since there is only one (the identity map of

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would have to be multi-valued.. contradiction. Did you erase that example? If so it'd be best to change it back as it's an important example. You've been making so many modifications it's hard for me to keep track. I'll see if I can find it in the revision history.

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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.

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is defined at 0 ( f '(0) = 0 ). It is the derivative of the inverse function that cannot be defined at zero, because the zero value of the derivative of the original function f '(), is in the denominator of the of derivative of the the inverse function. From

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This should ideally be done soon, given the mishmash state of the article. As an example, as of 2/15/24 the first two sentences contradict each other, with the first saying "smooth manifolds" and the second stating the definition for differentiable manifolds.

1717:, this is a contradiction if you assume that the curve is a smooth manifold which it is with the differentiable structure that I mentioned). But first could you please tell me why that curve is not a smooth manifold with that differentiable structure? 1307:

beside-the-point. Generally speaking the people who work on diffeomorphism groups are living in the Hausdorff world, and if they do consider non-Hausdorff manifolds they make a point of it (rather than the other way around that you're considering).

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We need some experts on diffeomorphism groups. Currently, the content has no meaning and is incorrect. For instance, the article claims the title of this section. I wanted to believe it but it seems somewhat dodgy. Here is a counterexample:

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Rybu, Thanks for catching that problem on my side, my browser did not render the blackboard bold R. The technical part of the statement that I question, is the statement: "derivative vanishes at 0". I believe that the derivative of

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example (the one with the curve having a cusp), the article had not mentioned anywhere that the reason that this curve is not diffeomorphic to a straight line is because one is not a smooth manifold. Furthermore, they did not state

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it can't be a smooth manifold (which is not true from what I know (I could be wrong though)). Also, the article contradicts itself since it says that for dimensions 1, 2 and 3, homeomorphic smooth manifolds must be diffeomorphic

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is not a diffeomorphism from its image to itself because its derivative equals 0 at x = 0 (and hence its inverse is not differentiable at x = 0). This is an example of a homeomorphism that is not a diffeomorphism. See the

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Taking a standard definition of manifolds, where 1-manifolds are either intervals or the circle (cf Milnor's "Topology from the Differentiable Viewpoint", Guillemin & Pollack, etc), the statement is that in dimension :

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is not diffeomorphic to the real line, because this curve is not a submanifold. The proof follows from the implicit function theorem -- if it was a submanifold, you'd be able to write it as the graph of a smooth function

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I am not an expert on the subject and I don't know of many of the important results to do with diffeomorphism groups. If someone knows, could he/she please include them in the article (provided there is a reference)?

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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:

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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.

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is not a diffeomorphism from to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0). This is an example of a homeomorphism that is not a diffeomorphism.

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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.

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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.

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and bend it round into a circle, identifying the end points. Of course that gives you one special point, and there should be no special points at all. Still, it is the correct topological picture.

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diffeomorphic to the straightline and according to your reason, this does not make any sense (since you can't speak of whether two curves are diffeomorphic unless they are both smooth manifolds).

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There's a bit of a problem with conventions going on here. As I mentioned above, smoothness of a map is something defined for arbitrary subsets of manifolds. I mention the curve is not a

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are diffeomorphic, we should not have to know where they are imbedded in. I don't think that the reason given in the article is appropriate; the curve with the cusp is a submanifold of

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I agree, the curve is trivially not an immersed submanifold (at (0,0), the differential is not defined). Nor can it be an embedded submanifold. But the article wrote that this curve is

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and there are representations for tensors and mixed indices. The diffeomorphism group can be extended by supersymmetry by adding a new fermion parameter in which case the rule is:

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the curve with the cusp is not a smooth manifold by referring to submanifolds. Now what you have written has the appropriate hypothesis so I am not going to remove it again.

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The text is fine, you're misquoting it though. Apparently the blackboard bold R is not showing up on your end -- some kind of rendering problem in your browser, maybe?

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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 ?

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difeomorphic to a straight line. Wrong, since the projection map yields a diffeomorphism (if you consider the natural differential structures on the two manifolds)

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they mean the derivative exists but is zero. This implies (via the inverse function theorem) that f is not a diffeomorphism. So I'm happy with the text as-is.

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The derivative a diffeomorphism has to be invertible at every point. Vanishing in this elementary case precludes that possibility. As for the TeX question:

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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.

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I fixed both these things and I will remove the statement given as the title of this section. If there are any objections, please post your reasons here.

1758:. Being a submanifold and being possible to turn a set into a manifold, are two completely different things. The 1st is relevant, the 2nd is not. The 1139:

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

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1. The assumptions on the manifolds are given in the first chapter of the book. An analogous result for symplectomorphisms appears on page 109.

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I think I have resolved the problem. The article wasn't claiming that the local homeomorphism came from the exponential mapping of the group.

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are the same would refer to that "special point" that Charles was mentioning where the two ends of the line segment forming the circle meet?

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I know very little about this topic, but hope to learn more. As a user of Knowledge, I would like to know what the following notation means:

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are actually the same coset, so that this quotient space eventually gets back to where it started, just like a circle. Speaking of circles,

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torturous revisions to adhere to terminology that is more specific than researchers use in practice, I don't think we'd be helping anyone.

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that includes noncommutativity. I think this has little place in an article about the diffeomorphism group of an (ordinary) manifold.

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I have got to go know (I have more to say but I might not be able to respond very soon; if I do I will let you know on your talk page).

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Thank you so much ... your added explanation in the article, and your patient explanation on this page helped. ... So where the cosets

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carefully, making sure also to provide references in order to verify the content and also to avoid possible misunderstandings.

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needs some cleaning up, but according to Knowledge I'm using the word submanifold in the intrinsic embedded submanifold sense.

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under the main definition in the article. If you'd like a reference, see for example the Guillemin and Pollack textbook

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are not diffeomorphic because one of the spaces is not even a manifold is useless and unecessary. If one of them is

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so shouldn't the projection map's inverse be smooth? You seem to assume that unless the curve is a submanifold of

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admits a local smooth extension to an open neighbourhood of the point in the manifold. I elaborated on this in

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Alephtwo, there's no problem in the text. Perhaps it's a terminology issue. When someone says the derivative

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1 the group of diffeomorphisms of compact support acts n-transitively on the manifold. This is discussed in

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a simple representation of the diffeomorphism group is given by the (infinitesimal) generators is found:

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Update: I have incorporated some of the content about the Lie algebra into an earlier paragraph of the

1525:)). So with this (natural) differentiable structure, the curve is diffeomorphic to the straight line. 83: 3540:

that there are multiple notions of diffeomorphism depending on the notion of manifolds one considers?

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By the way, I noticed that you added that example back. But with manifolds, the definition of a map

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Another issue is the factual accuracy of the article. The article claimed (or said in effect that):

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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Newcomer to the subject here, but isn't the diffeomorphism in the example above rather given by

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If in about a week or so I see no disambiguation either here or in the article, then I plan to

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set), the 'curve with the cusp' is an immersed submanifold. Surely this is natural (and it

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A fibre bundle is a space (which is wrong); fibre bundles are ordered quadruples (if you

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to two-space. For any calculation I have ever had to make (and I work in applications of

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I think I followed the definition at the beginning through and including this statement:

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it says a function which is invertible, and both the function and the inverse are smooth

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question as two whether they are diffeomorphic. Therefore, I will remove that example.

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For connected manifolds, the diffeomorphism group acts transitively on the manifold???

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There was quite a big discussion on what manifolds should be according to Knowledge

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In both cases, the diffeomorphism group is locally homeomorphic to the space of

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and its inverse to be differentiable. For a homeomorphism we only require that

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And there is an example there, and a good discussion at ==Local description==.

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Maybe I 'clouded up' my main point with other discussion. My main point was:

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Diffeomorphic manifolds=a smooth bijection with smooth inverse between them

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are charts over a particular open set). The curve in question is a smooth

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actually add the exact references for both statements, time permitting.

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A differentiable bijection is not necessarily a diffeomorphism, e.g.

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By making a small change to the coordinate x at each point in space:

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A differentiable bijection is not necessarily a diffeomorphism, e.g.

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the definition, it should be the second sentance of the article. The

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I am quite sure that two manifolds are diffeomorphic iff there is a

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Let me take some guesses at what this means in layperson's terms:

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As a subspace of itelf (with the projection map as a chart over

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Accoding to Rybu, the curve with a cusp is not a submanifold of

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Any continuum has a smooth structure making it diffeomorphic to

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A curve with a cusp, in particular the curve, x^2 - y^3 = 0, is

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Unresolved question concerning unexplained mathematical symbols

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and update the article with one of the above two definitions.

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For functions of a single variable, the theorem states that,

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denote the image of (0,b) under this homeomorphism. Clearly,

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Please be gentle on me, my WikiStress is higher than usual.

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This is an issue of standards. From the point of view of

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The "model example" section says that the diffeomorphism

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1, then your diffeomorphism can be given by the formula

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should be blocked off into its own para. Like this...

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inverse (not as in the article where only a bijective

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differentiable. I hope that this solves the problem.

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being smooth is when f o d o f^(-1) is smooth (where

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What do you suggest would be an appropriate edit to

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being a homeomorphism. For a diffeomorphism we need

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being a diffeomorphism is a stronger condition than

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of groups, so that it is actually an isomorphism of

101:, a collaborative effort to improve the coverage of 1982:imbedded in itelf as it is in the natural manifold 3463: 3391: 3311:Subgroups of the diffeomorphism group include the 3295: 3148: 2975: 2819: 2726: 2649: 2411: 2364: 2309: 2287: 2267: 2226: 2161: 2116: 2067: 2001: 1903: 1868: 1750: 1649: 1629: 1609: 1583: 1550: 1488: 1468: 1448: 1413: 1377: 1116: 913: 843: 815: 756: 698: 669: 642: 614: 586: 558: 509: 478:{\displaystyle \mathbb {R} \setminus \mathbb {Z} } 477: 409: 382: 346: 319: 285: 222: 202: 2830:There is also a `vector' matrix representation: 2563:section. There should be a separate article on 1184:bijective function between them that also has a 2820:{\displaystyle \left=L_{h.\nabla f-f.\nabla h}} 2475:is the order of differentiability considered). 2275:is a continuously differentiable function and 764:. Then you just have to check that this is a 485:. As Charles points out, what we have here is 947:"the mathematical definition is as follows"! 8: 3489:Smooth manifolds vs differentiable manifolds 1617:subsets of manifolds) if for every point in 1253:and there are disjoint neighbourhoods about 710:. A 1-dimensional sphere is also called a 2319:the inverse is continuously differentiable 47: 3464:{\displaystyle g_{|U\cap X}=f_{|U\cap X}} 3445: 3444: 3421: 3420: 3414: 3383: 3370: 3360: 3355: 3333: 3327: 3221: 3197: 3178: 3167: 3137: 3129: 3116: 3092: 3082: 3070: 3057: 3042: 3029: 3013: 3003: 2994: 2964: 2940: 2930: 2918: 2905: 2890: 2880: 2875: 2862: 2857: 2847: 2838: 2787: 2769: 2756: 2745: 2715: 2702: 2687: 2674: 2668: 2632: 2616: 2603: 2597: 2403: 2397: 2346: 2302: 2280: 2260: 2218: 2197: 2155: 2154: 2152: 2108: 2087: 2059: 2038: 1993: 1987: 1895: 1889: 1860: 1854: 1744: 1743: 1741: 1642: 1622: 1596: 1564: 1542: 1538: 1537: 1534: 1481: 1461: 1426: 1391: 1363: 1350: 1344: 1286:consider structure groups acting on them) 1109: 896: 884: 837: 836: 828: 809: 808: 800: 739: 727: 690: 686: 685: 682: 661: 655: 636: 635: 627: 608: 607: 599: 580: 579: 571: 552: 551: 549: 510:{\displaystyle \mathbb {R} /\mathbb {Z} } 503: 502: 497: 493: 492: 490: 471: 470: 463: 462: 460: 401: 395: 374: 368: 338: 332: 311: 305: 274: 263: 262: 257: 253: 252: 250: 215: 195: 158:Resolved question concerning linking 2003 1659:diffeomorphisms of subsets of manifolds 1456:near the origin, but in the first case 1063:Manifolds or just normed vector spaces? 467: 439:. You can think of it this way: take a 49: 19: 2986:and a `spinor' matrix representation: 2317:is invertible in a neighborhood of a, 3542:2601:644:400:8430:1937:78A9:EDC5:C955 2471:vector fields on the manifold (where 540:. This quotient group is set of all 7: 3533:different notions of diffeomorphism. 3504:2601:644:400:8430:C5F:BA8C:DFC9:46DF 914:{\displaystyle x\mapsto e^{2\pi ix}} 757:{\displaystyle x\mapsto e^{2\pi ix}} 451:set minus is usually written with a 354:means)" ... or is it something like: 95:This article is within the scope of 1476:is not smooth, and in the 2nd case 38:It is of interest to the following 3587:High-priority mathematics articles 3285: 3270: 3255: 3243: 3228: 3109: 3085: 3063: 3059: 2957: 2933: 2911: 2907: 2809: 2794: 2708: 2704: 14: 1100:Relationship with homoeomorphisms 115:Knowledge:WikiProject Mathematics 1922:a manifold, there should not be 1551:{\displaystyle \mathbb {R} ^{n}} 867:, and R/Z the set of Z-orbits. - 699:{\displaystyle \mathbb {S} ^{n}} 587:{\displaystyle 0.5+\mathbb {Z} } 210:) if there is a diffeomorphism 118:Template:WikiProject Mathematics 82: 72: 51: 20: 3498:manifolds as a generalization. 1160:and its inverse be continuous. 135:This article has been rated as 3484:20:05, 19 September 2012 (UTC) 3446: 3422: 3345: 3339: 3104: 3098: 3054: 3048: 3010: 2996: 2952: 2946: 2902: 2896: 2854: 2840: 2699: 2693: 2644: 2638: 2609: 2359: 2356: 2208: 2202: 2098: 2092: 2049: 2043: 1575: 1443: 1437: 1408: 1402: 889: 844:{\displaystyle 1+\mathbb {Z} } 816:{\displaystyle 0+\mathbb {Z} } 732: 643:{\displaystyle 1+\mathbb {Z} } 615:{\displaystyle 0+\mathbb {Z} } 1: 3567:12:00, 23 February 2024 (UTC) 3550:04:03, 23 February 2024 (UTC) 3528:01:36, 18 February 2024 (UTC) 3512:09:31, 15 February 2024 (UTC) 2419:.I don't know if this helps. 2024:12:45, 15 November 2008 (UTC) 1966:10:02, 15 November 2008 (UTC) 1938:08:58, 15 November 2008 (UTC) 1378:{\displaystyle x^{2}-y^{3}=0} 1024:20:18, 29 November 2005 (UTC) 1009:13:06, 29 November 2005 (UTC) 984:22:34, 28 November 2005 (UTC) 970:22:33, 28 November 2005 (UTC) 954:22:26, 28 November 2005 (UTC) 871:21:23, 29 November 2005 (UTC) 109:and see a list of open tasks. 3582:C-Class mathematics articles 2529:02:53, 5 December 2009 (UTC) 2509:16:44, 4 December 2009 (UTC) 2491:14:13, 4 December 2009 (UTC) 2162:{\displaystyle \mathbb {R} } 1845:11:02, 10 October 2008 (UTC) 1817:07:39, 10 October 2008 (UTC) 1801:07:30, 10 October 2008 (UTC) 1786:Banyaga's book is available 1772:04:58, 10 October 2008 (UTC) 1751:{\displaystyle \mathbb {R} } 1727:02:35, 10 October 2008 (UTC) 1689:02:23, 10 October 2008 (UTC) 1675:04:58, 10 October 2008 (UTC) 1335:02:23, 10 October 2008 (UTC) 1094:15:44, 29 January 2008 (UTC) 1078:05:23, 29 January 2008 (UTC) 942:Any chance of a lay example? 559:{\displaystyle \mathbb {Z} } 2412:{\displaystyle f^{\prime }} 2295:has nonzero derivative at a 1507:19:46, 9 October 2008 (UTC) 1302:03:16, 9 October 2008 (UTC) 1216:14:03, 6 October 2008 (UTC) 1197:differentiable structure. 1170:19:28, 16 August 2008 (UTC) 1055:19:24, 16 August 2008 (UTC) 3603: 3315:and (in 4 dimensions) the 2737:which form a group since: 2579:12:00, 22 April 2012 (UTC) 2554:11:52, 22 April 2012 (UTC) 2227:{\displaystyle f(x)=x^{3}} 2117:{\displaystyle f(x)=x^{3}} 2068:{\displaystyle f(x)=x^{3}} 2012:Could you please comment? 594:. Notice that the cosets 361:word salad (mental health) 2565:superdiffeomorphism group 2079:Possibly this should be: 996:But it's buried. If this 166:14:19, 27 Nov 2003 (UTC) 134: 67: 46: 2463:I flagged the statement 2250:Statement of the theorem 2237:inverse function theorem 2127:inverse function theorem 1584:{\displaystyle f:X\to Y} 936:12:25, 31 May 2006 (UTC) 926:11:03, 31 May 2006 (UTC) 855:15:50, 6 Jun 2005 (UTC) 788:18:36, Jun 1, 2005 (UTC) 447:16:40, 1 Jun 2005 (UTC) 432:16:12, 1 Jun 2005 (UTC) 141:project's priority scale 2585:=== Representations === 2453:16:39, 8 May 2009 (UTC) 2429:15:21, 8 May 2009 (UTC) 2388:15:05, 8 May 2009 (UTC) 2183:20:03, 7 May 2009 (UTC) 2142:17:52, 7 May 2009 (UTC) 1134:22:45, 6 May 2008 (UTC) 203:{\displaystyle \simeq } 98:WikiProject Mathematics 3465: 3393: 3297: 3150: 2977: 2821: 2728: 2651: 2413: 2366: 2311: 2289: 2269: 2228: 2163: 2118: 2069: 2029:Possible error in text 2003: 1905: 1870: 1752: 1651: 1631: 1611: 1585: 1552: 1490: 1470: 1450: 1449:{\displaystyle x=f(y)} 1415: 1414:{\displaystyle y=f(x)} 1379: 1118: 915: 845: 817: 758: 700: 671: 644: 616: 588: 560: 511: 479: 411: 384: 363:) is diffeomorphic to 348: 321: 287: 224: 204: 190:(symbol being usually 28:This article is rated 3466: 3394: 3298: 3151: 2978: 2822: 2729: 2652: 2414: 2367: 2312: 2290: 2270: 2229: 2164: 2119: 2070: 2004: 2002:{\displaystyle R^{2}} 1945:Differential Topology 1906: 1904:{\displaystyle R^{2}} 1871: 1869:{\displaystyle R^{2}} 1753: 1663:Differential topology 1652: 1632: 1612: 1586: 1553: 1491: 1471: 1451: 1416: 1380: 1231:line with two origins 1119: 1043:differential geometry 916: 846: 818: 780:. Hope that helps. - 759: 706:is the n-dimensional 701: 672: 670:{\displaystyle S^{n}} 645: 617: 589: 561: 512: 480: 412: 410:{\displaystyle S^{1}} 385: 383:{\displaystyle S^{1}} 349: 347:{\displaystyle S^{1}} 322: 320:{\displaystyle S^{1}} 288: 225: 205: 3413: 3403:Question on notation 3326: 3166: 2993: 2837: 2744: 2667: 2596: 2561:Diffeomorphism group 2396: 2365:{\displaystyle f'()} 2345: 2301: 2279: 2259: 2196: 2151: 2086: 2037: 1986: 1888: 1853: 1740: 1641: 1621: 1595: 1563: 1533: 1480: 1460: 1425: 1390: 1343: 1108: 883: 827: 799: 726: 681: 654: 626: 598: 570: 548: 489: 459: 394: 367: 331: 304: 249: 214: 194: 121:mathematics articles 3365: 3145: 2885: 2867: 2173:) is the TeX code. 1610:{\displaystyle X,Y} 3461: 3389: 3351: 3293: 3146: 3125: 2973: 2871: 2853: 2817: 2724: 2647: 2409: 2362: 2307: 2285: 2265: 2224: 2159: 2114: 2065: 1999: 1901: 1866: 1748: 1647: 1627: 1607: 1581: 1548: 1486: 1466: 1446: 1411: 1375: 1114: 1039:singularity theory 911: 841: 813: 754: 696: 667: 640: 612: 584: 556: 507: 475: 407: 380: 344: 317: 283: 220: 200: 90:Mathematics portal 34:content assessment 3307:=== Subgroups === 3123: 3077: 2971: 2925: 2722: 2310:{\displaystyle f} 2288:{\displaystyle f} 2268:{\displaystyle f} 1650:{\displaystyle f} 1630:{\displaystyle X} 1489:{\displaystyle f} 1469:{\displaystyle f} 1117:{\displaystyle f} 1068:case used here? 223:{\displaystyle f} 155: 154: 151: 150: 147: 146: 3594: 3470: 3468: 3467: 3462: 3460: 3459: 3449: 3436: 3435: 3425: 3398: 3396: 3395: 3390: 3388: 3387: 3375: 3374: 3364: 3359: 3338: 3337: 3302: 3300: 3299: 3294: 3292: 3291: 3213: 3209: 3208: 3207: 3189: 3188: 3155: 3153: 3152: 3147: 3144: 3136: 3124: 3122: 3121: 3120: 3107: 3097: 3096: 3083: 3078: 3076: 3075: 3074: 3058: 3047: 3046: 3037: 3036: 3021: 3020: 3008: 3007: 2982: 2980: 2979: 2974: 2972: 2970: 2969: 2968: 2955: 2945: 2944: 2931: 2926: 2924: 2923: 2922: 2906: 2895: 2894: 2884: 2879: 2866: 2861: 2852: 2851: 2826: 2824: 2823: 2818: 2816: 2815: 2779: 2775: 2774: 2773: 2761: 2760: 2733: 2731: 2730: 2725: 2723: 2721: 2720: 2719: 2703: 2692: 2691: 2679: 2678: 2656: 2654: 2653: 2648: 2637: 2636: 2621: 2620: 2608: 2607: 2571: 2546: 2521: 2501: 2483: 2418: 2416: 2415: 2410: 2408: 2407: 2371: 2369: 2368: 2363: 2355: 2316: 2314: 2313: 2308: 2294: 2292: 2291: 2286: 2274: 2272: 2271: 2266: 2233: 2231: 2230: 2225: 2223: 2222: 2168: 2166: 2165: 2160: 2158: 2123: 2121: 2120: 2115: 2113: 2112: 2074: 2072: 2071: 2066: 2064: 2063: 2008: 2006: 2005: 2000: 1998: 1997: 1914:Saying that two 1910: 1908: 1907: 1902: 1900: 1899: 1875: 1873: 1872: 1867: 1865: 1864: 1781:Augustin Banyaga 1757: 1755: 1754: 1749: 1747: 1656: 1654: 1653: 1648: 1636: 1634: 1633: 1628: 1616: 1614: 1613: 1608: 1590: 1588: 1587: 1582: 1559:ie: A function 1557: 1555: 1554: 1549: 1547: 1546: 1541: 1495: 1493: 1492: 1487: 1475: 1473: 1472: 1467: 1455: 1453: 1452: 1447: 1420: 1418: 1417: 1412: 1384: 1382: 1381: 1376: 1368: 1367: 1355: 1354: 1123: 1121: 1120: 1115: 920: 918: 917: 912: 910: 909: 850: 848: 847: 842: 840: 822: 820: 819: 814: 812: 763: 761: 760: 755: 753: 752: 705: 703: 702: 697: 695: 694: 689: 676: 674: 673: 668: 666: 665: 649: 647: 646: 641: 639: 621: 619: 618: 613: 611: 593: 591: 590: 585: 583: 565: 563: 562: 557: 555: 521:of the additive 516: 514: 513: 508: 506: 501: 496: 484: 482: 481: 476: 474: 466: 445:Charles Matthews 416: 414: 413: 408: 406: 405: 389: 387: 386: 381: 379: 378: 353: 351: 350: 345: 343: 342: 326: 324: 323: 318: 316: 315: 292: 290: 289: 284: 279: 278: 266: 261: 256: 229: 227: 226: 221: 209: 207: 206: 201: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 3602: 3601: 3597: 3596: 3595: 3593: 3592: 3591: 3572: 3571: 3491: 3440: 3416: 3411: 3410: 3405: 3379: 3366: 3329: 3324: 3323: 3313:conformal group 3217: 3193: 3174: 3173: 3169: 3164: 3163: 3112: 3108: 3088: 3084: 3066: 3062: 3038: 3025: 3009: 2999: 2991: 2990: 2960: 2956: 2936: 2932: 2914: 2910: 2886: 2843: 2835: 2834: 2783: 2765: 2752: 2751: 2747: 2742: 2741: 2711: 2707: 2683: 2670: 2665: 2664: 2628: 2612: 2599: 2594: 2593: 2569: 2544: 2539: 2537:Recent addition 2519: 2499: 2481: 2461: 2399: 2394: 2393: 2348: 2343: 2342: 2299: 2298: 2277: 2276: 2257: 2256: 2214: 2194: 2193: 2149: 2148: 2104: 2084: 2083: 2055: 2035: 2034: 2031: 2016:Topology Expert 1989: 1984: 1983: 1951:. What is the 1930:Topology Expert 1891: 1886: 1885: 1856: 1851: 1850: 1837:Topology Expert 1738: 1737: 1719:Topology Expert 1681:Topology Expert 1639: 1638: 1619: 1618: 1593: 1592: 1591:is smooth (for 1561: 1560: 1536: 1531: 1530: 1478: 1477: 1458: 1457: 1423: 1422: 1388: 1387: 1359: 1346: 1341: 1340: 1339:FYI, the curve 1327:Topology Expert 1294:Topology Expert 1223: 1208:Topology Expert 1178: 1106: 1105: 1102: 1086:Oleg Alexandrov 1070:Trevorgoodchild 1065: 1016:Oleg Alexandrov 976:Oleg Alexandrov 962:Oleg Alexandrov 944: 923:155.198.196.187 892: 881: 880: 825: 824: 797: 796: 735: 724: 723: 716:complex numbers 684: 679: 678: 657: 652: 651: 624: 623: 596: 595: 568: 567: 546: 545: 534:normal subgroup 487: 486: 457: 456: 397: 392: 391: 370: 365: 364: 334: 329: 328: 307: 302: 301: 270: 247: 246: 212: 211: 192: 191: 172: 160: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 3600: 3598: 3590: 3589: 3584: 3574: 3573: 3570: 3569: 3554: 3553: 3552: 3534: 3490: 3487: 3458: 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2695: 2690: 2686: 2682: 2677: 2673: 2658: 2657: 2646: 2643: 2640: 2635: 2631: 2627: 2624: 2619: 2615: 2611: 2606: 2602: 2587: 2586: 2582: 2581: 2570:Sławomir Biały 2545:Sławomir Biały 2538: 2535: 2534: 2533: 2532: 2531: 2520:Sławomir Biały 2512: 2511: 2500:Sławomir Biały 2482:Sławomir Biały 2477: 2476: 2460: 2457: 2456: 2455: 2436: 2435: 2434: 2433: 2432: 2431: 2406: 2402: 2376: 2373: 2361: 2358: 2354: 2351: 2336: 2335: 2334: 2333: 2327: 2326: 2325: 2324: 2323: 2322: 2306: 2284: 2264: 2251: 2243: 2242: 2241: 2240: 2221: 2217: 2213: 2210: 2207: 2204: 2201: 2186: 2185: 2157: 2111: 2107: 2103: 2100: 2097: 2094: 2091: 2078: 2062: 2058: 2054: 2051: 2048: 2045: 2042: 2030: 2027: 1996: 1992: 1969: 1968: 1898: 1894: 1863: 1859: 1820: 1819: 1775: 1774: 1746: 1678: 1677: 1646: 1626: 1606: 1603: 1600: 1580: 1577: 1574: 1571: 1568: 1545: 1540: 1518: 1517: 1485: 1465: 1445: 1442: 1439: 1436: 1433: 1430: 1410: 1407: 1404: 1401: 1398: 1395: 1374: 1371: 1366: 1362: 1358: 1353: 1349: 1313: 1312: 1288: 1287: 1279: 1278: 1222: 1219: 1194:differentiable 1192:function with 1190:differentiable 1177: 1174: 1173: 1172: 1113: 1101: 1098: 1097: 1096: 1064: 1061: 1060: 1059: 1058: 1057: 1027: 1026: 989: 988: 987: 986: 943: 940: 939: 938: 908: 905: 902: 899: 895: 891: 888: 877: 876: 875: 874: 873: 872: 839: 835: 832: 811: 807: 804: 790: 789: 751: 748: 745: 742: 738: 734: 731: 693: 688: 664: 660: 638: 634: 631: 610: 606: 603: 582: 578: 575: 554: 519:quotient group 505: 500: 495: 473: 469: 465: 437:quotient group 419: 418: 404: 400: 377: 373: 356: 355: 341: 337: 314: 310: 294: 293: 282: 277: 273: 269: 265: 260: 255: 219: 199: 178:Two manifolds 171: 168: 159: 156: 153: 152: 149: 148: 145: 144: 133: 127: 126: 124: 107:the discussion 94: 93: 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 3599: 3588: 3585: 3583: 3580: 3579: 3577: 3568: 3564: 3560: 3555: 3551: 3547: 3543: 3539: 3535: 3531: 3530: 3529: 3525: 3521: 3516: 3515: 3514: 3513: 3509: 3505: 3499: 3495: 3488: 3486: 3485: 3481: 3477: 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1730: 1729: 1728: 1724: 1720: 1716: 1712: 1708: 1704: 1700: 1696: 1691: 1690: 1686: 1682: 1676: 1672: 1668: 1664: 1660: 1644: 1624: 1604: 1601: 1598: 1578: 1572: 1569: 1566: 1543: 1528: 1527: 1526: 1524: 1515: 1511: 1510: 1509: 1508: 1504: 1500: 1483: 1463: 1440: 1434: 1431: 1428: 1405: 1399: 1396: 1393: 1372: 1369: 1364: 1360: 1356: 1351: 1347: 1337: 1336: 1332: 1328: 1324: 1322: 1318: 1310: 1309: 1308: 1304: 1303: 1299: 1295: 1291: 1285: 1281: 1280: 1276: 1272: 1271: 1270: 1267: 1263: 1260: 1256: 1252: 1248: 1244: 1239: 1236: 1235:bug-eyed line 1232: 1227: 1220: 1218: 1217: 1213: 1209: 1205: 1203: 1198: 1195: 1191: 1187: 1183: 1175: 1171: 1167: 1163: 1162:Dharma6662000 1159: 1155: 1151: 1147: 1142: 1138: 1137: 1136: 1135: 1131: 1127: 1111: 1099: 1095: 1091: 1087: 1082: 1081: 1080: 1079: 1075: 1071: 1062: 1056: 1052: 1048: 1047:Dharma6662000 1044: 1040: 1035: 1034: 1033: 1032: 1031: 1025: 1021: 1017: 1013: 1012: 1011: 1010: 1007: 1003: 999: 994: 993: 985: 981: 977: 973: 972: 971: 967: 963: 958: 957: 956: 955: 952: 948: 941: 937: 934: 930: 929: 928: 927: 924: 906: 903: 900: 897: 893: 886: 870: 866: 862: 857: 856: 854: 833: 830: 805: 802: 794: 793: 792: 791: 787: 783: 779: 775: 771: 767: 766:homeomorphism 749: 746: 743: 740: 736: 729: 721: 717: 713: 709: 691: 677:or sometimes 662: 658: 632: 629: 604: 601: 576: 573: 543: 539: 535: 531: 528: 524: 520: 498: 454: 450: 449: 448: 446: 442: 438: 433: 431: 426: 424: 402: 398: 375: 371: 362: 358: 357: 339: 335: 312: 308: 299: 298: 297: 280: 275: 271: 267: 258: 245: 244: 243: 240: 239: 238:. For example 235: 231: 217: 197: 189: 188:diffeomorphic 183: 179: 175: 169: 167: 165: 157: 142: 138: 137:High-priority 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 62:High‑priority 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 3537: 3500: 3496: 3494:definition. 3492: 3472: 3409: 3406: 3310: 3158: 2985: 2829: 2736: 2659: 2588: 2560: 2540: 2478: 2472: 2468: 2462: 2440: 2318: 2254: 2170: 2131: 2081: 2077: 2032: 2014: 2011: 1979: 1975: 1973: 1970: 1952: 1948: 1944: 1928: 1923: 1919: 1915: 1913: 1881: 1877: 1848: 1835: 1832: 1827: 1824: 1821: 1785: 1776: 1733: 1710: 1706: 1702: 1698: 1694: 1692: 1679: 1662: 1658: 1522: 1519: 1513: 1338: 1325: 1314: 1305: 1292: 1289: 1283: 1274: 1268: 1264: 1258: 1254: 1250: 1246: 1242: 1234: 1230: 1228: 1224: 1206: 1201: 1199: 1193: 1189: 1185: 1181: 1179: 1157: 1153: 1149: 1145: 1140: 1103: 1066: 1028: 1001: 997: 995: 991: 990: 949: 945: 878: 861:affine space 527:real numbers 441:line segment 434: 427: 420: 295: 241: 237: 233: 187: 185: 181: 177: 176: 173: 161: 136: 96: 40:WikiProjects 1822:Dear Rybu, 1760:submanifold 1734:submanifold 1319:. Also see 774:isomorphism 164:Pascalromon 112:Mathematics 103:mathematics 59:Mathematics 3576:Categories 1707:1-manifold 1002:definition 921:? Thanks, 778:Lie groups 390:(whatever 327:(whatever 2171:\mathbb R 1976:the whole 1200:In fact, 453:backslash 435:It's the 3559:D.Lazard 2441:vanishes 2380:Alephtwo 2134:Alephtwo 1321:manifold 1311:Hi Rybu, 1126:LachlanA 865:Z action 538:integers 3538:clarify 3476:Dratman 2459:Dubious 2421:Mathsci 2297:, then 1809:Mathsci 1793:Mathsci 1249:is not 1233:or the 863:with a 720:modulus 423:be bold 417:means)" 139:on the 30:C-class 2375:thanks 1916:spaces 1703:f^(-1) 1237:; see 1186:smooth 1182:smooth 931:Yes. - 853:Vonkje 770:smooth 712:circle 708:sphere 542:cosets 530:modulo 517:, the 430:Vonkje 36:scale. 2321:, ... 1284:don't 1006:Maury 951:Maury 933:lethe 869:lethe 782:Lethe 768:, is 523:group 230:from 3563:talk 3546:talk 3524:talk 3520:Rybu 3508:talk 3480:talk 2575:talk 2550:talk 2525:talk 2505:talk 2487:talk 2449:talk 2445:Rybu 2425:talk 2384:talk 2179:talk 2175:Rybu 2138:talk 2020:talk 1962:talk 1958:Rybu 1934:talk 1880:and 1841:talk 1813:talk 1797:talk 1788:here 1768:talk 1764:Rybu 1723:talk 1701:and 1685:talk 1671:talk 1667:Rybu 1503:talk 1499:Rybu 1331:talk 1317:here 1298:talk 1257:and 1229:The 1212:talk 1166:talk 1130:talk 1090:talk 1074:talk 1051:talk 1020:talk 980:talk 966:talk 823:and 786:Talk 622:and 532:the 186:are 182:and 131:High 2255:if 2009:)? 1924:any 1920:not 1828:why 1778:--> 1514:not 1421:or 1275:not 1141:and 1041:to 718:of 574:0.5 544:of 536:of 525:of 234:to 3578:: 3565:) 3548:) 3526:) 3510:) 3482:) 3454:∩ 3430:∩ 3385:μ 3372:ν 3362:μ 3357:ν 3335:μ 3289:ψ 3286:∇ 3277:− 3274:η 3271:∇ 3259:η 3256:Γ 3253:ψ 3244:∇ 3235:− 3229:∇ 3205:η 3186:ψ 3142:β 3139:α 3134:ν 3131:μ 3127:σ 3118:ν 3110:∂ 3094:μ 3086:∂ 3080:− 3072:τ 3064:∂ 3060:∂ 3044:τ 3034:β 3031:α 3027:δ 3018:β 3015:α 2966:ν 2958:∂ 2942:μ 2934:∂ 2928:− 2920:τ 2912:∂ 2908:∂ 2892:τ 2882:μ 2877:ν 2873:δ 2864:μ 2859:ν 2810:∇ 2801:− 2795:∇ 2717:μ 2709:∂ 2705:∂ 2689:μ 2634:μ 2626:ε 2618:μ 2610:→ 2605:μ 2577:) 2552:) 2527:) 2507:) 2489:) 2451:) 2427:) 2405:′ 2386:) 2378:-- 2181:) 2140:) 2132:-- 2129:. 2022:) 1980:is 1964:) 1936:) 1843:) 1815:) 1799:) 1770:) 1725:) 1687:) 1673:) 1665:. 1637:, 1576:→ 1505:) 1357:− 1333:) 1300:) 1214:) 1168:) 1132:) 1092:) 1076:) 1053:) 1022:) 998:is 982:) 968:) 901:π 890:↦ 784:| 744:π 733:↦ 468:∖ 268:≃ 198:≃ 3561:( 3544:( 3522:( 3506:( 3478:( 3457:X 3451:U 3447:| 3442:f 3438:= 3433:X 3427:U 3423:| 3418:g 3381:P 3377:+ 3368:x 3353:M 3349:= 3346:) 3343:x 3340:( 3331:h 3283:. 3280:f 3268:. 3265:h 3262:, 3250:+ 3247:h 3241:. 3238:f 3232:f 3226:. 3223:h 3219:L 3215:= 3211:] 3202:, 3199:f 3195:L 3191:, 3183:, 3180:h 3176:L 3171:[ 3114:x 3105:) 3102:x 3099:( 3090:h 3068:x 3055:) 3052:x 3049:( 3040:h 3023:= 3011:) 3005:h 3001:L 2997:( 2962:x 2953:) 2950:x 2947:( 2938:h 2916:x 2903:) 2900:x 2897:( 2888:h 2869:= 2855:) 2849:h 2845:L 2841:( 2813:h 2807:. 2804:f 2798:f 2792:. 2789:h 2785:L 2781:= 2777:] 2771:f 2767:L 2763:, 2758:h 2754:L 2749:[ 2713:x 2700:) 2697:x 2694:( 2685:h 2681:= 2676:h 2672:L 2645:) 2642:x 2639:( 2630:h 2623:+ 2614:x 2601:x 2573:( 2548:( 2523:( 2503:( 2485:( 2473:r 2469:C 2447:( 2423:( 2401:f 2382:( 2372:? 2360:) 2357:( 2353:′ 2350:f 2305:f 2283:f 2263:f 2239:: 2220:3 2216:x 2212:= 2209:) 2206:x 2203:( 2200:f 2177:( 2169:( 2156:R 2136:( 2110:3 2106:x 2102:= 2099:) 2096:x 2093:( 2090:f 2061:3 2057:x 2053:= 2050:) 2047:x 2044:( 2041:f 2018:( 1995:2 1991:R 1960:( 1932:( 1897:2 1893:R 1882:N 1878:M 1862:2 1858:R 1839:( 1811:( 1795:( 1766:( 1745:R 1721:( 1714:( 1711:R 1699:f 1695:d 1683:( 1669:( 1645:f 1625:X 1605:Y 1602:, 1599:X 1579:Y 1573:X 1570:: 1567:f 1544:n 1539:R 1523:R 1501:( 1484:f 1464:f 1444:) 1441:y 1438:( 1435:f 1432:= 1429:x 1409:) 1406:x 1403:( 1400:f 1397:= 1394:y 1373:0 1370:= 1365:3 1361:y 1352:2 1348:x 1329:( 1296:( 1259:y 1255:x 1251:y 1247:x 1243:x 1210:( 1202:C 1164:( 1158:f 1154:f 1150:f 1146:f 1144:" 1128:( 1112:f 1088:( 1072:( 1049:( 1018:( 978:( 964:( 907:x 904:i 898:2 894:e 887:x 838:Z 834:+ 831:1 810:Z 806:+ 803:0 750:x 747:i 741:2 737:e 730:x 692:n 687:S 663:n 659:S 637:Z 633:+ 630:1 609:Z 605:+ 602:0 581:Z 577:+ 553:Z 504:Z 499:/ 494:R 472:Z 464:R 403:1 399:S 376:1 372:S 340:1 336:S 313:1 309:S 281:. 276:1 272:S 264:Z 259:/ 254:R 236:N 232:M 218:f 184:N 180:M 143:. 42::

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