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That discussion is getting lost in the middle of the previous section, so it may be best to start a new thread. There are several ways of defining manifolds. I take it we are discussing the following two:
Are we in agreement on this, or should other approaches be added? Tkuvho ( talk) 13:47, 31 May 2012 (UTC)
Tkuvho: In a better world than this, the "most general audience" might be able to fall back on their previous experience with "elementary calculus". In the real world, the most general audience does not know calculus. The lead paragraph needs to tell the reader that 1) manifolds are objects mathematicians study and 2) they are spaces that are locally Euclidean. The most general audience does understand what the word "local" means. I'm not so sure about "Euclidean" but without that we're lost. We also need to give a few simple examples. While it is true that, by the embedding theorem, every manifold can be embedded in a Euclidean space, that is not necessarily the best way to understand, say, the projective plane. Under most definitions of a graph, the lemniscate is a graph of . There are, of course, various specialized definitions of the word "graph". Rick Norwood ( talk) 19:02, 31 May 2012 (UTC)
What is a smooth manifold? In a recent American book I read that Poincare was not acquainted with this notion (which he himself introduced) and that the "modern" definition was only given by Veblen in the late 1920s: a manifold is a topological space which satisfies a long series of axioms. For what sins must students try and find their way through all these twists and turns? Actually, in Poincare's Analysis Situs there is an absolutely clear definition of a smooth manifold which is much more useful than the "abstract" one. A smooth k-dimensional submanifold of the Euclidean space RN is a subset which in a neighbourhood of each of its points is the graph of a smooth mapping of Rk into RN−k (where Rk and RN−k are coordinate subspaces). This is a straightforward generalization of the commonest smooth curves on the plane (such as the circle x2 + y2 = 1) and of curves and surfaces in three-dimensional space. Between smooth manifolds are naturally defined smooth mappings. Diffeomorphisms are mappings which, together with their inverses, are smooth. An "abstract" smooth manifold is a smooth submanifold of a Euclidean space considered up to diffeomorphism. There are no "more abstract" finite-dimensional smooth manifolds in the world (Whitney's theorem). Why do we keep on tormenting students with the abstract definition? Would it not be better to prove for them the theorem on the explicit classification of closed two-dimensional manifolds (surfaces)? Tkuvho ( talk) 14:08, 31 May 2012 (UTC)
Considering the current lead, which rightly emphasizes a general (i.e. topological) perspective on manifolds, doesn't the top picture feel out of place and misleading? It's quite appropriate on Spherical geometry, but not on this article, I believe. In particular, it suggests an incorrect interpretation of the word "Euclidean" in the phrase "resembles the Euclidean space". Bikasuishin ( talk) 18:52, 22 March 2009 (UTC)
I agree that the picture is inappropriate, but worse still was the caption to the picture, which emphasised metric properties rather than topological ones, so I have deleted most of it, leaving only the part which seemed reasonably relevant. What is left of the caption would be better illustrated by a plain picture of the earth (without angles marked on it) perhaps accompanied by a plane map. If anyone knows of a suitable free picture it would be an improvement. I am not saying that a picture of the earth is necessarily the best image to have, merely that if we are to have one then one not displaying metric information would be much better. JamesBWatson ( talk) 22:13, 23 March 2009 (UTC)
I like the idea of the Earth and a map, agree that the triangles are misleading. Ideally we would see a globe, with a point on its surface, and a map with the same point highlighted. The point should be some easily recognisable point, such as London or Tokyo. Rick Norwood ( talk) 15:37, 8 June 2012 (UTC)
Sorry, but "subsets of n-dimensional Euclidean space" doesn't work. The letter T is a subset of the Euclidean plane, but not a manifold. I'm going to see what I can do. Rick Norwood ( talk) 22:30, 31 May 2012 (UTC)
I've made changes based on your comments above. Rick Norwood ( talk) 12:21, 1 June 2012 (UTC)
The current version of the lede describes a manifold as follows: "manifold is an object that can be described by identifying parts of it with subsets of n-dimensional Euclidean space". It seems odd to insist on identifying parts of it with subsets of n-dimensional Euclidean space when in fact all of it can be identified with a subset of N-dimensional Euclidean space. This does not serve the interest of comprehension of the most general reader. Tkuvho ( talk) 11:10, 1 June 2012 (UTC)
Tkuvho: No, in general an n-manifold cannot be identified with a subset of N-dimensional space, unless you intend capital N to be a different number from little n. For example, the projective plane is a 2-manifold, but can only be embedded in a Euclidean space of four or more dimensions. Rick Norwood ( talk) 12:09, 1 June 2012 (UTC)
Great work, guys. The article is much improved. Rick Norwood ( talk) 11:46, 4 June 2012 (UTC)
I've already said in the WT:WPM thread that I think we should go back to the old stable version of the lead. The lead has since undergone several complete rewrites, all without much discussion (nor even any clear reason). Further discussions should be based on the last stable version, before the latest rounds of misguided attempts at rewriting. Sławomir Biały ( talk) 13:40, 8 June 2012 (UTC)
I like the old lead, but I like the current lead as well, and it has a technically correct definition of a manifold, which the old lead lacks. If memory serves, the rewrites began when someone complained that the old lead was unreadable. The first rewrites were not very good, but as you note we've been able to repair some of the damage. Rick Norwood ( talk) 11:48, 9 June 2012 (UTC)
The following discussion is copied from Wikipedia talk:WikiProject Mathematics. D.Lazard ( talk) 15:42, 7 June 2012 (UTC)
I added Poincare's original definition of a differentiable manifold at Manifold#Poincar.C3.A9.27s_original_definition. Poincare defined a manifold as a subset of euclidean space which is locally a graph (see details there). This definition is arguably more accessible to a general reader than the more abstract definition involving atlases, charts, and transition functions. The lede could profit from focusing on the subset-of-R^n definition instead of the abstract definition. However, another editor feels that the reader does not need the crutch of Euclidean space to understand the concept of a manifold, and my changes to the lede were repeatedly reverted. Which definition should the lede be based on? Tkuvho ( talk) 11:37, 4 June 2012 (UTC)
11:44, 7 June 2012 (UTC)
I do not like the definition through graphs of functions, because it is less intuitive (at least for me) and it uses implicitly the implicit function theorem, which is far of being trivial (it is needed to show that a circle, defined as usual by its implicit equation, is a manifold). On the other hand, I do not like either the use of "scale" in the first sentence of the graph, because it appears in neither formal definition. Thus, I propose for the first sentence: "In mathematics (specifically in geometry and topology), a manifold is a mathematical object that, near each point of it, looks like Euclidean space". This has the advantage to be very close to the definition by charts (except that nothing is said on the transition maps, which are needed only for technical reasons). In fact the definition by charts and atlas is simply a formalization of this informal definition. D.Lazard ( talk) 16:12, 4 June 2012 (UTC)
I don't really think the lead is perfect at present. In fact, it seems to be worse than the version from three years ago. I'd like to discuss possibly bringing back this earlier revision of the lead. In any event, I don't think it is a good idea to emphasize Poincare's original definition of manifold. Not many sources do this, and at least the motivational examples section of the article would need to be rewritten from this point of view. Sławomir Biały ( talk) 16:37, 4 June 2012 (UTC)
May I point out that this whole discussion should be taking place at talk:manifold, not here. T R 12:28, 7 June 2012 (UTC)
End of copied discussion. D.Lazard ( talk) 15:42, 7 June 2012 (UTC)
Good work, Lost-n-translation. Rick Norwood ( talk) 12:11, 22 June 2012 (UTC)
Footnotes are used primarily for references. Wikipedia does allow explanatary footnotes, but the two new footnotes in the lead are not helpful, so I am removing them. Riemannian manifolds are discussed below. "Anamorphosis" is not a more common word than "homeomorphism". And, as for "small", the word is meaningless, so better to remove it rather than explain that "small" can mean "large". Rick Norwood ( talk) 12:04, 8 June 2012 (UTC)
Rick Norwood wrote in an edit summary: "I am not at all sure that we want a technical definition in the first paragraph". IMO an accurate definition is absolutely needed in the lead. In fact there are more than thousand pages linked here, most of them devoted to advanced notions. It follows that a large proportion of readers should be people with good mathematical knowledge, who only need to be reminded the exact definition of a manifold. Thus an accurate definition has to be easily accessed. Without restructuring completely the article, this may be done only in the lead. Fortunately, contrarily to many mathematical notions, this may be done in a single sentence without specific technicalities. D.Lazard ( talk) 18:00, 8 June 2012 (UTC)
The article currently suggests that "curve" is synonymous with 1-manifold, and "surface" is synonymous with 2-manifold. Since this is not true, I suggest that these terms should be removed from the lead, or a better alternative should be proposed. Sławomir Biały ( talk) 17:06, 23 June 2012 (UTC)
I'm not sure why you say "this is not true". Here is what Munkres says in Topology, Second Edition, page 225, in the section that introduces the term "manifold". "A 1-manifold is often called a curve, and a 2-manifold is called a surface." You disagree? Rick Norwood ( talk) 17:54, 23 June 2012 (UTC)
I think that's straining at a gnat. If you want to go into how many different ways mathematical words are used, consider "normal". We have a good, standard text that calls 1-manifolds curves and 2-manifolds surfaces. That seems a substantial reference, even if some writers use the words differently. Rick Norwood ( talk) 13:29, 24 June 2012 (UTC)
Surely this is false generally? "Topological object" would be better, or "mathematical" in case "topological" is too intimidating or is also incorrect for some less-common objects that are also named "manifold". ChalkboardCowboy ( talk) 16:46, 24 June 2012 (UTC)
"Geometrical object" has been recently edited into "object". I object to that. Indeed, for a non mathematician, it is not clear what a mathematician means by "object". In fact, in current English an object is a tangible entity, what is not a mathematical object. " Mathematical object" would be correct, but I am afraid that it will appear as jargon for many reader: who, except professional mathematicians, think of a field (mathematics) a an object? However, the mathematical objects occurring in geometry are very close to the usual notion of object: Nobody would oppose to call "object" a triangle or a sphere. Manifolds are very similar, as spheres and triangles are manifolds (with boundary in the case of the triangles). The use of "geometrical object" has another advantage: it suggest that the theory of manifolds is a part of modern geometry (even if some think that is is broader). D.Lazard ( talk) 13:00, 25 June 2012 (UTC)
I also agree with Rick Norwood's point that if a single adjective is in dispute and it is not necessary to convey meaning, then it's better to drop it altogether. Lost-n-translation ( talk) 17:41, 25 June 2012 (UTC)
First, let me suggest when the topic is a single word, "revert" is not the best option. Just edit, to include the word or omit it. Second, going back and forth like this is not helpful. Can we find a compromise? A "manifold" is a special case of what general class? "Set of points"? "Topological space"? I have no strong opinion one way or the other, and find "geometrical object" entirely clear. But if we can't agree on that, let's find something we can agree on. Rick Norwood ( talk) 14:47, 25 June 2012 (UTC)
How about this? Sławomir Biały ( talk) 15:11, 25 June 2012 (UTC)
So something like this:
In mathematics, a manifold is a topological space that near each point resembles Euclidean space of a fixed dimension, called the dimension of the manifold. More precisely, each point of a manifold of dimension n has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.
I had hoped that there was a way to get rid of the second sentence so that all concerned are happy. As it stands, I think it may be a little too technical. Sławomir Biały ( talk) 00:16, 26 June 2012 (UTC)
Shouldn't ' charts' have their own math article?! All Clues Key ( talk) 01:40, 7 September 2012 (UTC)
The recent on Manifold#Manifold with boundary introduces an interesting question: does the smoothness of the boundary make a difference to whether something might be considered a differentiable manifold? IMO it seems evident that a manifold with boundary could qualify as differentiable irrespective of the shape of its boundary; differentiability does not generally extend to the boundary (though one could choose to use the one-sided limit). The differentiability of the boundary as a manifold does not seem to be relevant to the definition; it would not necessarily itself be a differentiable manifold. My guess is that the edit comment "rounded corners mean you don't need to worry about the distinction between topological and differentiable manifolds" does not follow. — Quondum 10:34, 17 September 2012 (UTC)
For this article, I think "with rounded corners" is unnecessarily confusing. For smooth manifolds, there are different smoothness classes one can impose at the boundary and interior. But mostly I have seen smoothness assumed to extend up to the boundary. Sławomir Biały ( talk) 12:54, 17 September 2012 (UTC)
"Manifolds need not be connected (all in "one piece"); an example is a pair of separate circles. In this example we see that a manifold need not have any well-defined notion of distance, for there is no way to define the distance between points that don't lie in the same piece."
why not? what about.. the euclidean distance?!!!?!!!-- 93.66.195.134 ( talk) 21:00, 11 October 2012 (UTC)
The statement "we see that a manifold need not have any well-defined notion of distance" is incorrect (at least for what the article calls a 'topological manifold'). Taking a manifold as a Hausdorff, second countable space locally homeomorphic to R^n for some n, this shows that manifolds are locally compact; that means that they are regular, so by the Urysohn metrization theorem they are metrizable. Thus there is always a metric function for manifolds. — Preceding unsigned comment added by Forgetful functor187 ( talk • contribs) 04:01, 9 November 2012 (UTC)
In the lede is the phrase "Examples include the plane, the sphere, and the torus, [...]". I would suggest that it should be written "Examples include the plane, the surface of a sphere, and the surface of a torus, [...]" as a sphere and torus each could be viewed as either a surface or as a solid, but only the surface view is what is intended here. — al-Shimoni ( talk) 06:40, 11 December 2012 (UTC)
I never comment on Wikipedia, but I read just the first half and was already very impressed. I am a graduate student and took differential topology, and boy do I wish I looked at this article earlier. It's simplicity and clarity in explaining what can normally be very complicated concepts is a model for Wikipedia pages. I admittedly do not know the technical details about why this page was removed from being a featured article candidate, but it is by far one of the best written articles I have ever come across on wikipedia. Thus, I would like to thank the writers and contributors; you guys deserve kudos. — Preceding unsigned comment added by 146.201.205.212 ( talk) 21:08, 23 January 2013 (UTC)
A recent edit has changed, in the lead, "Euclidean space" into "coordinate space". I have reverted it for the following reasons: In higher mathematics, "Euclidean space" roughly means "metric affine space". But, for most people, it simply means the usual space of geometry over the reals, and is much more intuitive than "coordinate space". Thus the modification makes the lead unnecessary WP:TECHNICAL. Moreover, the edit suggests that one may consider manifolds over arbitrary fields, which is wrong. Apparently, the motivation of the edit was that that the Euclidean metric is not used in the definition of a manifold. But the coordinate space over the reals has also a natural Euclidean metric (the dot product) and has a further structure (of a vector space equipped with a basis) which is not used in manifold theory. Thus the version that I have reverted is not only too technical, but also less correct than the previous one. D.Lazard ( talk) 09:55, 19 April 2013 (UTC)
I think we can improve the Zeroth Law of Thermodynamics in a way that satisfies both Chjoaygame and Prokaryotes. For quite some time, I had been proposing a deletion of the entire reference to a one dimensional manifold, but suddenly I grasped what people were trying to say. Now that I understand it, we need to make two decisions:
I actually took the trouble to look up Serrin's article, and wasn't much impressed. He didn't explain it well either. Apparently (and I am just guessing here) some authors claim that the zeroth law establishes the existence of temperature. What I think Serrin was trying to say is that one needs to establish a few more concepts before jumping from the zeroth law to the idea that a temperature scale can be defined. For example, what is the meaning of the cryptic phrase "For suitable systems...?". What I think he meant was that the state variable must change when the temperature changes.
And it is important to provide either one or two examples of a state variable that changes. The dispute we are having is whether we need one example (pressure), or two examples (pressure and volume). We cannot resolve this until we have resolved the two aforementioned questions. In other words, why argue about a sentence when a couple of paragraphs need work?-- guyvan52 ( talk) 16:53, 21 March 2014 (UTC)
The text refers to semicircles (verbally and mathematically) while the diagram illustrates shorter arcs. Not too serious, but it detracts. — Preceding unsigned comment added by Pierreva ( talk • contribs) 03:08, 21 October 2013 (UTC)
I was going to suggest that simple change, then I noticed the reference to the interval (-1,1), and the following two paragraphs both only make sense in the context of semicircles. I'm afraid it is the graphic that is the smallest change target. — Preceding unsigned comment added by Pierreva ( talk • contribs) 18:11, 22 October 2013 (UTC)
I agree. for the given circle (x^2) + (y^2) = 1, y is positive for the entire top half of the circle. You can easily see this by plugging in y = 0 and noting x = +1 or x = -1 are solutions. However, the diagram shows a yellow arc that only covers 1/4 of the top of the circle. — Preceding unsigned comment added by ArchetypeRyan ( talk • contribs) 04:05, 19 September 2014 (UTC)
Not strictly true: the south pole (for example) can be depicted as a circle. — Preceding unsigned comment added by 92.2.211.93 ( talk • contribs) 2014-10-16T17:39:48
I've removed the animated GIF of "boy's surface" because of the distraction it causes. This action is consistent with MOS:ACCESS but counter to the wishes of User:Slawekb
https://en.wikipedia.org/?title=Manifold&diff=581403804&oldid=581391247 : (The image and its caption accompany the text of the lead. If you don't like this particular image of Boy's surface, then find another one.)
Any opinions on this apart from the two of us?
-- Catskul ( talk) 23:09, 14 November 2013 (UTC)
I think the idea that there are 2-manifolds that are not realizable as surfaces in Euclidean space is actually quite important for understanding why there is a mathematical notion of "manifold" at all. Certainly, one can study manifolds without worrying about their embedding properties (although there are people who build their careers entirely on the latter), but to someone with no idea what a manifold even is, I think it is very important to realize that they do represent a significant generalization of the elementary notion of a surface. Sławomir Biały ( talk) 13:31, 17 November 2013 (UTC)
The topological characteristics of a manifold are captured in its Betti numbers and torsion coefficients. Only the first of these is mentioned in this article. I have started a discussion at Talk:Homology (mathematics)#Betti numbers and torsion coefficients and would be grateful for any contributions. — Cheers, Steelpillow ( Talk) 15:37, 12 January 2015 (UTC)
The lead says that the Klein bottle and real projective plane cannot be realized in three dimensions. Where does this term "realized" come from? Sure they cannot be embedded/bijected, but they sure can be injected. In the theory of abstract polytopes the idea of "realization" describes all such injections into real space, however degenerate. Does topology differ, or should the lead be amended accordingly? — Cheers, Steelpillow ( Talk) 15:23, 12 January 2015 (UTC)
I'm sure I've misunderstand the following statement from the introduction:
> Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane which cannot.
It seems to suggest Klein bottles cannot be represented in three dimensional space but then links to the article on Klein bottles with several real world models of them. Is:
Cheers, -- TFJamMan ( talk) 07:56, 5 April 2016 (UTC)
The subsection "Transition Maps" and its neighbors (especially "Atlases") need a bit of work, but for now I'm just curious: does a transition map send a region of R^n to another region of R^n, or does a transition map send a part of the manifold to another part of the manifold? In this article, the first convention is used. But in the Wikipedia article Atlas (topology) the other convention is used - with a picture!
I don't know what to believe anymore. - Norbornene ( talk) 15:38, 4 January 2017 (UTC)
The article states: Two atlases are said to be equivalent if their union is also an atlas. I cannot imagine what it means. In the first place is according to the definition of an atlas any extension of an atlas with a chart again an atlas. And couldn't it be the case that compatibility is meant? So, something is missing here. Madyno ( talk) 21:24, 25 September 2017 (UTC)
As far as I can see, nothing in the definitions puts a condition on the transition maps. Madyno ( talk) 12:55, 26 September 2017 (UTC)
Quite strange, you don't get my point. I know what you're saying. The point is, it isn't mentioned in the definitions. Madyno ( talk) 13:27, 27 September 2017 (UTC)
"The surface of the Earth requires (at least) two charts to include every point"
Not true, see, for example "Mercator_projection". There's a bit more precision (or hand waving) required in this explanation. — Preceding unsigned comment added by 125.239.100.117 ( talk) 08:50, 2 November 2017 (UTC)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | ← | Archive 5 | Archive 6 | Archive 7 | Archive 8 |
That discussion is getting lost in the middle of the previous section, so it may be best to start a new thread. There are several ways of defining manifolds. I take it we are discussing the following two:
Are we in agreement on this, or should other approaches be added? Tkuvho ( talk) 13:47, 31 May 2012 (UTC)
Tkuvho: In a better world than this, the "most general audience" might be able to fall back on their previous experience with "elementary calculus". In the real world, the most general audience does not know calculus. The lead paragraph needs to tell the reader that 1) manifolds are objects mathematicians study and 2) they are spaces that are locally Euclidean. The most general audience does understand what the word "local" means. I'm not so sure about "Euclidean" but without that we're lost. We also need to give a few simple examples. While it is true that, by the embedding theorem, every manifold can be embedded in a Euclidean space, that is not necessarily the best way to understand, say, the projective plane. Under most definitions of a graph, the lemniscate is a graph of . There are, of course, various specialized definitions of the word "graph". Rick Norwood ( talk) 19:02, 31 May 2012 (UTC)
What is a smooth manifold? In a recent American book I read that Poincare was not acquainted with this notion (which he himself introduced) and that the "modern" definition was only given by Veblen in the late 1920s: a manifold is a topological space which satisfies a long series of axioms. For what sins must students try and find their way through all these twists and turns? Actually, in Poincare's Analysis Situs there is an absolutely clear definition of a smooth manifold which is much more useful than the "abstract" one. A smooth k-dimensional submanifold of the Euclidean space RN is a subset which in a neighbourhood of each of its points is the graph of a smooth mapping of Rk into RN−k (where Rk and RN−k are coordinate subspaces). This is a straightforward generalization of the commonest smooth curves on the plane (such as the circle x2 + y2 = 1) and of curves and surfaces in three-dimensional space. Between smooth manifolds are naturally defined smooth mappings. Diffeomorphisms are mappings which, together with their inverses, are smooth. An "abstract" smooth manifold is a smooth submanifold of a Euclidean space considered up to diffeomorphism. There are no "more abstract" finite-dimensional smooth manifolds in the world (Whitney's theorem). Why do we keep on tormenting students with the abstract definition? Would it not be better to prove for them the theorem on the explicit classification of closed two-dimensional manifolds (surfaces)? Tkuvho ( talk) 14:08, 31 May 2012 (UTC)
Considering the current lead, which rightly emphasizes a general (i.e. topological) perspective on manifolds, doesn't the top picture feel out of place and misleading? It's quite appropriate on Spherical geometry, but not on this article, I believe. In particular, it suggests an incorrect interpretation of the word "Euclidean" in the phrase "resembles the Euclidean space". Bikasuishin ( talk) 18:52, 22 March 2009 (UTC)
I agree that the picture is inappropriate, but worse still was the caption to the picture, which emphasised metric properties rather than topological ones, so I have deleted most of it, leaving only the part which seemed reasonably relevant. What is left of the caption would be better illustrated by a plain picture of the earth (without angles marked on it) perhaps accompanied by a plane map. If anyone knows of a suitable free picture it would be an improvement. I am not saying that a picture of the earth is necessarily the best image to have, merely that if we are to have one then one not displaying metric information would be much better. JamesBWatson ( talk) 22:13, 23 March 2009 (UTC)
I like the idea of the Earth and a map, agree that the triangles are misleading. Ideally we would see a globe, with a point on its surface, and a map with the same point highlighted. The point should be some easily recognisable point, such as London or Tokyo. Rick Norwood ( talk) 15:37, 8 June 2012 (UTC)
Sorry, but "subsets of n-dimensional Euclidean space" doesn't work. The letter T is a subset of the Euclidean plane, but not a manifold. I'm going to see what I can do. Rick Norwood ( talk) 22:30, 31 May 2012 (UTC)
I've made changes based on your comments above. Rick Norwood ( talk) 12:21, 1 June 2012 (UTC)
The current version of the lede describes a manifold as follows: "manifold is an object that can be described by identifying parts of it with subsets of n-dimensional Euclidean space". It seems odd to insist on identifying parts of it with subsets of n-dimensional Euclidean space when in fact all of it can be identified with a subset of N-dimensional Euclidean space. This does not serve the interest of comprehension of the most general reader. Tkuvho ( talk) 11:10, 1 June 2012 (UTC)
Tkuvho: No, in general an n-manifold cannot be identified with a subset of N-dimensional space, unless you intend capital N to be a different number from little n. For example, the projective plane is a 2-manifold, but can only be embedded in a Euclidean space of four or more dimensions. Rick Norwood ( talk) 12:09, 1 June 2012 (UTC)
Great work, guys. The article is much improved. Rick Norwood ( talk) 11:46, 4 June 2012 (UTC)
I've already said in the WT:WPM thread that I think we should go back to the old stable version of the lead. The lead has since undergone several complete rewrites, all without much discussion (nor even any clear reason). Further discussions should be based on the last stable version, before the latest rounds of misguided attempts at rewriting. Sławomir Biały ( talk) 13:40, 8 June 2012 (UTC)
I like the old lead, but I like the current lead as well, and it has a technically correct definition of a manifold, which the old lead lacks. If memory serves, the rewrites began when someone complained that the old lead was unreadable. The first rewrites were not very good, but as you note we've been able to repair some of the damage. Rick Norwood ( talk) 11:48, 9 June 2012 (UTC)
The following discussion is copied from Wikipedia talk:WikiProject Mathematics. D.Lazard ( talk) 15:42, 7 June 2012 (UTC)
I added Poincare's original definition of a differentiable manifold at Manifold#Poincar.C3.A9.27s_original_definition. Poincare defined a manifold as a subset of euclidean space which is locally a graph (see details there). This definition is arguably more accessible to a general reader than the more abstract definition involving atlases, charts, and transition functions. The lede could profit from focusing on the subset-of-R^n definition instead of the abstract definition. However, another editor feels that the reader does not need the crutch of Euclidean space to understand the concept of a manifold, and my changes to the lede were repeatedly reverted. Which definition should the lede be based on? Tkuvho ( talk) 11:37, 4 June 2012 (UTC)
11:44, 7 June 2012 (UTC)
I do not like the definition through graphs of functions, because it is less intuitive (at least for me) and it uses implicitly the implicit function theorem, which is far of being trivial (it is needed to show that a circle, defined as usual by its implicit equation, is a manifold). On the other hand, I do not like either the use of "scale" in the first sentence of the graph, because it appears in neither formal definition. Thus, I propose for the first sentence: "In mathematics (specifically in geometry and topology), a manifold is a mathematical object that, near each point of it, looks like Euclidean space". This has the advantage to be very close to the definition by charts (except that nothing is said on the transition maps, which are needed only for technical reasons). In fact the definition by charts and atlas is simply a formalization of this informal definition. D.Lazard ( talk) 16:12, 4 June 2012 (UTC)
I don't really think the lead is perfect at present. In fact, it seems to be worse than the version from three years ago. I'd like to discuss possibly bringing back this earlier revision of the lead. In any event, I don't think it is a good idea to emphasize Poincare's original definition of manifold. Not many sources do this, and at least the motivational examples section of the article would need to be rewritten from this point of view. Sławomir Biały ( talk) 16:37, 4 June 2012 (UTC)
May I point out that this whole discussion should be taking place at talk:manifold, not here. T R 12:28, 7 June 2012 (UTC)
End of copied discussion. D.Lazard ( talk) 15:42, 7 June 2012 (UTC)
Good work, Lost-n-translation. Rick Norwood ( talk) 12:11, 22 June 2012 (UTC)
Footnotes are used primarily for references. Wikipedia does allow explanatary footnotes, but the two new footnotes in the lead are not helpful, so I am removing them. Riemannian manifolds are discussed below. "Anamorphosis" is not a more common word than "homeomorphism". And, as for "small", the word is meaningless, so better to remove it rather than explain that "small" can mean "large". Rick Norwood ( talk) 12:04, 8 June 2012 (UTC)
Rick Norwood wrote in an edit summary: "I am not at all sure that we want a technical definition in the first paragraph". IMO an accurate definition is absolutely needed in the lead. In fact there are more than thousand pages linked here, most of them devoted to advanced notions. It follows that a large proportion of readers should be people with good mathematical knowledge, who only need to be reminded the exact definition of a manifold. Thus an accurate definition has to be easily accessed. Without restructuring completely the article, this may be done only in the lead. Fortunately, contrarily to many mathematical notions, this may be done in a single sentence without specific technicalities. D.Lazard ( talk) 18:00, 8 June 2012 (UTC)
The article currently suggests that "curve" is synonymous with 1-manifold, and "surface" is synonymous with 2-manifold. Since this is not true, I suggest that these terms should be removed from the lead, or a better alternative should be proposed. Sławomir Biały ( talk) 17:06, 23 June 2012 (UTC)
I'm not sure why you say "this is not true". Here is what Munkres says in Topology, Second Edition, page 225, in the section that introduces the term "manifold". "A 1-manifold is often called a curve, and a 2-manifold is called a surface." You disagree? Rick Norwood ( talk) 17:54, 23 June 2012 (UTC)
I think that's straining at a gnat. If you want to go into how many different ways mathematical words are used, consider "normal". We have a good, standard text that calls 1-manifolds curves and 2-manifolds surfaces. That seems a substantial reference, even if some writers use the words differently. Rick Norwood ( talk) 13:29, 24 June 2012 (UTC)
Surely this is false generally? "Topological object" would be better, or "mathematical" in case "topological" is too intimidating or is also incorrect for some less-common objects that are also named "manifold". ChalkboardCowboy ( talk) 16:46, 24 June 2012 (UTC)
"Geometrical object" has been recently edited into "object". I object to that. Indeed, for a non mathematician, it is not clear what a mathematician means by "object". In fact, in current English an object is a tangible entity, what is not a mathematical object. " Mathematical object" would be correct, but I am afraid that it will appear as jargon for many reader: who, except professional mathematicians, think of a field (mathematics) a an object? However, the mathematical objects occurring in geometry are very close to the usual notion of object: Nobody would oppose to call "object" a triangle or a sphere. Manifolds are very similar, as spheres and triangles are manifolds (with boundary in the case of the triangles). The use of "geometrical object" has another advantage: it suggest that the theory of manifolds is a part of modern geometry (even if some think that is is broader). D.Lazard ( talk) 13:00, 25 June 2012 (UTC)
I also agree with Rick Norwood's point that if a single adjective is in dispute and it is not necessary to convey meaning, then it's better to drop it altogether. Lost-n-translation ( talk) 17:41, 25 June 2012 (UTC)
First, let me suggest when the topic is a single word, "revert" is not the best option. Just edit, to include the word or omit it. Second, going back and forth like this is not helpful. Can we find a compromise? A "manifold" is a special case of what general class? "Set of points"? "Topological space"? I have no strong opinion one way or the other, and find "geometrical object" entirely clear. But if we can't agree on that, let's find something we can agree on. Rick Norwood ( talk) 14:47, 25 June 2012 (UTC)
How about this? Sławomir Biały ( talk) 15:11, 25 June 2012 (UTC)
So something like this:
In mathematics, a manifold is a topological space that near each point resembles Euclidean space of a fixed dimension, called the dimension of the manifold. More precisely, each point of a manifold of dimension n has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.
I had hoped that there was a way to get rid of the second sentence so that all concerned are happy. As it stands, I think it may be a little too technical. Sławomir Biały ( talk) 00:16, 26 June 2012 (UTC)
Shouldn't ' charts' have their own math article?! All Clues Key ( talk) 01:40, 7 September 2012 (UTC)
The recent on Manifold#Manifold with boundary introduces an interesting question: does the smoothness of the boundary make a difference to whether something might be considered a differentiable manifold? IMO it seems evident that a manifold with boundary could qualify as differentiable irrespective of the shape of its boundary; differentiability does not generally extend to the boundary (though one could choose to use the one-sided limit). The differentiability of the boundary as a manifold does not seem to be relevant to the definition; it would not necessarily itself be a differentiable manifold. My guess is that the edit comment "rounded corners mean you don't need to worry about the distinction between topological and differentiable manifolds" does not follow. — Quondum 10:34, 17 September 2012 (UTC)
For this article, I think "with rounded corners" is unnecessarily confusing. For smooth manifolds, there are different smoothness classes one can impose at the boundary and interior. But mostly I have seen smoothness assumed to extend up to the boundary. Sławomir Biały ( talk) 12:54, 17 September 2012 (UTC)
"Manifolds need not be connected (all in "one piece"); an example is a pair of separate circles. In this example we see that a manifold need not have any well-defined notion of distance, for there is no way to define the distance between points that don't lie in the same piece."
why not? what about.. the euclidean distance?!!!?!!!-- 93.66.195.134 ( talk) 21:00, 11 October 2012 (UTC)
The statement "we see that a manifold need not have any well-defined notion of distance" is incorrect (at least for what the article calls a 'topological manifold'). Taking a manifold as a Hausdorff, second countable space locally homeomorphic to R^n for some n, this shows that manifolds are locally compact; that means that they are regular, so by the Urysohn metrization theorem they are metrizable. Thus there is always a metric function for manifolds. — Preceding unsigned comment added by Forgetful functor187 ( talk • contribs) 04:01, 9 November 2012 (UTC)
In the lede is the phrase "Examples include the plane, the sphere, and the torus, [...]". I would suggest that it should be written "Examples include the plane, the surface of a sphere, and the surface of a torus, [...]" as a sphere and torus each could be viewed as either a surface or as a solid, but only the surface view is what is intended here. — al-Shimoni ( talk) 06:40, 11 December 2012 (UTC)
I never comment on Wikipedia, but I read just the first half and was already very impressed. I am a graduate student and took differential topology, and boy do I wish I looked at this article earlier. It's simplicity and clarity in explaining what can normally be very complicated concepts is a model for Wikipedia pages. I admittedly do not know the technical details about why this page was removed from being a featured article candidate, but it is by far one of the best written articles I have ever come across on wikipedia. Thus, I would like to thank the writers and contributors; you guys deserve kudos. — Preceding unsigned comment added by 146.201.205.212 ( talk) 21:08, 23 January 2013 (UTC)
A recent edit has changed, in the lead, "Euclidean space" into "coordinate space". I have reverted it for the following reasons: In higher mathematics, "Euclidean space" roughly means "metric affine space". But, for most people, it simply means the usual space of geometry over the reals, and is much more intuitive than "coordinate space". Thus the modification makes the lead unnecessary WP:TECHNICAL. Moreover, the edit suggests that one may consider manifolds over arbitrary fields, which is wrong. Apparently, the motivation of the edit was that that the Euclidean metric is not used in the definition of a manifold. But the coordinate space over the reals has also a natural Euclidean metric (the dot product) and has a further structure (of a vector space equipped with a basis) which is not used in manifold theory. Thus the version that I have reverted is not only too technical, but also less correct than the previous one. D.Lazard ( talk) 09:55, 19 April 2013 (UTC)
I think we can improve the Zeroth Law of Thermodynamics in a way that satisfies both Chjoaygame and Prokaryotes. For quite some time, I had been proposing a deletion of the entire reference to a one dimensional manifold, but suddenly I grasped what people were trying to say. Now that I understand it, we need to make two decisions:
I actually took the trouble to look up Serrin's article, and wasn't much impressed. He didn't explain it well either. Apparently (and I am just guessing here) some authors claim that the zeroth law establishes the existence of temperature. What I think Serrin was trying to say is that one needs to establish a few more concepts before jumping from the zeroth law to the idea that a temperature scale can be defined. For example, what is the meaning of the cryptic phrase "For suitable systems...?". What I think he meant was that the state variable must change when the temperature changes.
And it is important to provide either one or two examples of a state variable that changes. The dispute we are having is whether we need one example (pressure), or two examples (pressure and volume). We cannot resolve this until we have resolved the two aforementioned questions. In other words, why argue about a sentence when a couple of paragraphs need work?-- guyvan52 ( talk) 16:53, 21 March 2014 (UTC)
The text refers to semicircles (verbally and mathematically) while the diagram illustrates shorter arcs. Not too serious, but it detracts. — Preceding unsigned comment added by Pierreva ( talk • contribs) 03:08, 21 October 2013 (UTC)
I was going to suggest that simple change, then I noticed the reference to the interval (-1,1), and the following two paragraphs both only make sense in the context of semicircles. I'm afraid it is the graphic that is the smallest change target. — Preceding unsigned comment added by Pierreva ( talk • contribs) 18:11, 22 October 2013 (UTC)
I agree. for the given circle (x^2) + (y^2) = 1, y is positive for the entire top half of the circle. You can easily see this by plugging in y = 0 and noting x = +1 or x = -1 are solutions. However, the diagram shows a yellow arc that only covers 1/4 of the top of the circle. — Preceding unsigned comment added by ArchetypeRyan ( talk • contribs) 04:05, 19 September 2014 (UTC)
Not strictly true: the south pole (for example) can be depicted as a circle. — Preceding unsigned comment added by 92.2.211.93 ( talk • contribs) 2014-10-16T17:39:48
I've removed the animated GIF of "boy's surface" because of the distraction it causes. This action is consistent with MOS:ACCESS but counter to the wishes of User:Slawekb
https://en.wikipedia.org/?title=Manifold&diff=581403804&oldid=581391247 : (The image and its caption accompany the text of the lead. If you don't like this particular image of Boy's surface, then find another one.)
Any opinions on this apart from the two of us?
-- Catskul ( talk) 23:09, 14 November 2013 (UTC)
I think the idea that there are 2-manifolds that are not realizable as surfaces in Euclidean space is actually quite important for understanding why there is a mathematical notion of "manifold" at all. Certainly, one can study manifolds without worrying about their embedding properties (although there are people who build their careers entirely on the latter), but to someone with no idea what a manifold even is, I think it is very important to realize that they do represent a significant generalization of the elementary notion of a surface. Sławomir Biały ( talk) 13:31, 17 November 2013 (UTC)
The topological characteristics of a manifold are captured in its Betti numbers and torsion coefficients. Only the first of these is mentioned in this article. I have started a discussion at Talk:Homology (mathematics)#Betti numbers and torsion coefficients and would be grateful for any contributions. — Cheers, Steelpillow ( Talk) 15:37, 12 January 2015 (UTC)
The lead says that the Klein bottle and real projective plane cannot be realized in three dimensions. Where does this term "realized" come from? Sure they cannot be embedded/bijected, but they sure can be injected. In the theory of abstract polytopes the idea of "realization" describes all such injections into real space, however degenerate. Does topology differ, or should the lead be amended accordingly? — Cheers, Steelpillow ( Talk) 15:23, 12 January 2015 (UTC)
I'm sure I've misunderstand the following statement from the introduction:
> Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane which cannot.
It seems to suggest Klein bottles cannot be represented in three dimensional space but then links to the article on Klein bottles with several real world models of them. Is:
Cheers, -- TFJamMan ( talk) 07:56, 5 April 2016 (UTC)
The subsection "Transition Maps" and its neighbors (especially "Atlases") need a bit of work, but for now I'm just curious: does a transition map send a region of R^n to another region of R^n, or does a transition map send a part of the manifold to another part of the manifold? In this article, the first convention is used. But in the Wikipedia article Atlas (topology) the other convention is used - with a picture!
I don't know what to believe anymore. - Norbornene ( talk) 15:38, 4 January 2017 (UTC)
The article states: Two atlases are said to be equivalent if their union is also an atlas. I cannot imagine what it means. In the first place is according to the definition of an atlas any extension of an atlas with a chart again an atlas. And couldn't it be the case that compatibility is meant? So, something is missing here. Madyno ( talk) 21:24, 25 September 2017 (UTC)
As far as I can see, nothing in the definitions puts a condition on the transition maps. Madyno ( talk) 12:55, 26 September 2017 (UTC)
Quite strange, you don't get my point. I know what you're saying. The point is, it isn't mentioned in the definitions. Madyno ( talk) 13:27, 27 September 2017 (UTC)
"The surface of the Earth requires (at least) two charts to include every point"
Not true, see, for example "Mercator_projection". There's a bit more precision (or hand waving) required in this explanation. — Preceding unsigned comment added by 125.239.100.117 ( talk) 08:50, 2 November 2017 (UTC)