Talk:Manifold/Archive 7

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crutches

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:

• atlases, charts, transition maps, etc.

• subset of Euclidean space.

Are we in agreement on this, or should other approaches be added? Tkuvho (talk) 13:47, 31 May 2012 (UTC)

response to your comment In your beginning paragraph, you do not give an explanation of `local' even by example. The first paragraph of a Wikipedia page should be directed at the most general audience with deference given to the reader who has no prior experience with the subject of the article: Note that your version of the lede also uses the term "local" without explaining it. I fully agree that the lede should be directed at the most general audience. For this reason, it is preferable to start with a definition that builds upon their previous experiences, namely euclidean space and elementary calculus. The definition as a subset of euclidean space clearly beats out the "chart and transition functions" definition, which is an entirely unfamiliar concept to someone with elementary calculus experience. Talk about ocean waves may not necessarily help understand either transition functions or manifolds. Tkuvho (talk) 15:11, 31 May 2012 (UTC)

`Local' is suggested by geographical map or chart. Mathematically, `local' means in a neighborhood of a point. For a manifold, the most natural neighborhoods of a point are the domains of the charts, in other words, the regions of the manifold being represented by charts (or geographical maps)Lost-n-translation (talk) 16:21, 31 May 2012 (UTC)

You have written `you can choose (say, orthonormal) coordinates near every point so that a neighborhood on the sphere is the graph of a suitable function.' So you are also implicitly using the the notion of a chart to define a manifold. One can't avoid charts. They are essential to the notion of a manifold. Why suppress them?Lost-n-translation (talk) 19:59, 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 $(x^{2}+y^{2})^{2}=2a^{2}(x^{2}-y^{2})\,$. There are, of course, various specialized definitions of the word "graph".Rick Norwood (talk) 19:02, 31 May 2012 (UTC)

what arnold wrote

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)

Hi Tkuhvo! I don't understand why this section has been entitled `what Arnold wrote'. Did you mean to provide a specific citation of something that Vladimir Arnol'd wrote?Lost-n-translation (talk) 16:15, 31 May 2012 (UTC)

Misleading top picture?

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)

There is an overall dearth of appropriate images in this article. I agree that the lead image is inappropriate. There are some rather spectacular images (maybe open source?) due to William Thurston of the Poincare dodecahedral space (and other compact three-manifolds). Perhaps one of these should be incorporated. Sławomir Biały (talk) 00:28, 23 March 2009 (UTC)

Where are these images to be found? Algebraist 00:30, 23 March 2009 (UTC)

I'm not quite sure. I've seen Thurston's "hyperbolic space explorer" (demonstrated at a conference 8 years ago). I assume it was open source, but I haven't looked into it since then. Anyway, if push comes to shove, Brakke's surface evolver is able to make nice images, is open source, and is of substantial importance for contemporary research. Sławomir Biały (talk) 00:57, 23 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)

Whereas metric properties should be de-emphasized if possible, the sphere isn't terribly interesting otherwise as a topological manifold. It might be interesting to compare with the lead image of differentiable manifold, which deliberately illustrates a non-differentiable chart on the sphere, and so is appropriate to the subject matter by way of contrast. Anyway, if push comes to shove, certainly OS images of genus g>0 surfaces can be rustled up, if there is demand. Sławomir Biały (talk) 01:20, 24 March 2009 (UTC)

What about a picture with a topological circle (maybe a little wavy) and a genus 2 or 3 compact Riemann surface to illustrate what manifolds are, as well as a figure-8 curve and an immersion of the Klein bottle in 3-space (or maybe just two mutually intersecting tori) to illustrate what manifolds are not? Bikasuishin (talk) 11:05, 24 March 2009 (UTC)

How about a table of compact manifolds? In 1D, we can show a circle (with waves) as a manifold. In 2D, a sphere, a sphere with a handle (if available), a sphere with two handles, and then the pattern is clear. I suggest that we should tread with more caution in treating non-orientable and immersed cases. Whereas the Klein bottle is a manifold, it is not a "submanifold" of R³, but this potentially misses an important opportunity to illustrate the power of the manifold concept to a potentially non-mathematical audience. Sławomir Biały (talk) 23:22, 24 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)

I'm going to give it a try.

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 was trying to rid the first paragraph of the word `local' because of Tkuhvo's objections as well as a *whole host* of objections that appear in the rest of these talk pages.

A torus is not something with which most lay people are familiar. Better to say something like surface of a doughnut or not use it at all.Lost-n-translation (talk) 11:57, 1 June 2012 (UTC)

You said earlier that you didn't want to discuss manifolds with boundary at a first pass, but in fact planet earth (from the point of view suggested by your phrasing) is a manifold with boundary and your maps are describing the boundary.Lost-n-translation (talk) 12:00, 1 June 2012 (UTC)

I've made changes based on your comments above.Rick Norwood (talk) 12:21, 1 June 2012 (UTC)

Hi Rick. This morning, I made the above comments and then started editing the page. When I tried to save, I found that you had beat me to it. (You must be working much faster than I am). So this time I edited the page first, and am only now going to explain my edits. (Please hold on!)Lost-n-translation (talk) 16:16, 1 June 2012 (UTC)

When I saw your second sentence I was tempted to change the first sentence to `...a mathematical object that is *only* locally Euclidean', because otherwise the connection to the second sentence isn't very strong. But then I wondered `do we really want to define a manifold by what it is not?'

I like the use of the globe thing, but perhaps lower doen in the lede. In fact, it is this exact thing that I was trying to hint at in a paragraph that I think that you erased...

The phrase `inside of a circle' is a little too non-mathematical for my tastes and probably most research mathematicians. Also, I thought that using an open submanifold (the disc/surface of the ocean) was too suggestive of a manifold with boundary?

The figure 'X' thing is ok. I think it's great to tell the reader what a manifold is not, but maybe it raises too many questions: `What letter should it look like?' 'The letter `I' of course.' I'm not sure why you insist on using the torus as an example. But maybe you're thinking `line' is to `plane' as `circle' is to torus. But you wrote `sphere'....I will try to fix.Lost-n-translation (talk) 16:28, 1 June 2012 (UTC)

Oh, also I have tried to retain the `small enough scale' bit as a tribute to our forebearers. Apparently, many versions of this page included the `scale' thing. Lost-n-translation (talk) 16:31, 1 June 2012 (UTC)

parts with subsets

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)

Well, of course, I would love to say `open' set, but I shied away from it fearing criticism. That is, I would like it to read something `open subsets of the manifold are identified with open subsets of (little) n-space via a homeomorphism...Lost-n-translation (talk) 11:48, 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)

Rick, yes, he does intend N to be different than n. This is why I said open set. He wants all manifolds to be (only) thought of as some subset of a larger dimensional Euclidean space. This is true (see, e.g. Munkres, Intro to Topology) but misleading.Lost-n-translation (talk) 12:43, 1 June 2012 (UTC)

new lead

Great work, guys. The article is much improved. Rick Norwood (talk) 11:46, 4 June 2012 (UTC)

(Comment already posted to Wikipedia talk:WikiProject Mathematics‎‎) 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) 16:17, 4 June 2012 (UTC)

I have edited and expanded the lead in the spirit of my preceding comment. I hope that this will be considered as an improvement. D.Lazard (talk) 09:09, 6 June 2012 (UTC)

I'm not so happy with the sentence "For example the whole surface of the planet Earth is not flat nor Euclidean, but in a small region, one may consider it as a plane" (nor the picture, for that matter). The point is that small enough regions can be smoothly represented by a plane. The fact that the angles in the plane might be close to angles on the sphere is not part of the definition, not part of the concept, but the figure seems to suggest that at a small enough scale, triangles have angles adding to (close to?) 180 degrees.This is not part of the definition, can't be, because there is no a priori concept of distance on a manifold, and no a priori concept of straight line. (by the way, we are talking about smooth manifolds with a given degree of smoothness, C-1, right? The word "smooth" is important. Richard Gill (talk) 14:40, 7 June 2012 (UTC)

I agree that the figure is misleading here, as angles are meaningful only for Riemannian manifolds, while the lead is only on (topological) manifolds (differentiable manifolds, smooth manifolds, analytic manifolds, Riemannian manifolds, ...) are all manifolds with additional structure). But I am not competent to modifying it. "Small region" is also misleading, as the region may be the whole Earth but a point, or, as in most world maps, the whole Earth but a meridian. I'll add a note to clarify that. D.Lazard (talk) 15:44, 7 June 2012 (UTC)

I've been meaning to comment that the use of the term Euclidean space in the first sentence of the lead also implies structure that is not part of the definition of a manifold without added structure: it implies a positive-definite metric. Surely a vector space is what a manifold resembles locally. Further, is it a vector space over specifically reals, or perhaps other permitted fields (complex etc.?) — Quondum^☏ 18:10, 7 June 2012 (UTC)

"Euclidean space", "vector space over the reals" and Rⁿ are equivalent here and they have all additional structures. "Euclidean space", as the space of elementary geometry may be more intuitive for many users. But you are right, it is tempting to think "resemblance" as a metric one. I do not see how to modify the first sentence for avoiding this ambiguity, while keeping it easy to read. Thus I'll think of a footnote to clarify this point. D.Lazard (talk) 07:08, 8 June 2012 (UTC)

The current approach to the lede is thoroughly misguided as it assumes that the reader knows what a topological space is. The talk about homeomorhisms is meaningless otherwise. What can be reasonably explained to a wide audience which is calculus-literate is the notion of the graph of a differentiable function and its role in the definition of a differentiable manifold. Otherwise one is displaying one's erudition instead of informing the reader. Tkuvho (talk) 08:36, 8 June 2012 (UTC)

Unfortunately, the concept being defined is inherently a topological one. One cannot explain a concept rooted in an abstract one without building on the abstract concept. To explain the concept in terms of more familiar ones that carry with them excess structure is too inexact for encyclopedic purposes, which trump pedagogical value. The explanantion in terms of a graph suffers from other deficiencies, as may be seen from my earlier reaction; specifically, a graph or function implies that one set of variables/coordinates is dependent on another (and the correspondence of these to what is being explained must be made explicit), charts, it implies embedding in a higher-dimensional space, and it risks compounding the problem of "Euclidean space" (i.e. that a metric may be assumed). Putting all this into context in the lead becomes too cumbersome to use the idea of a graph or function to define a manifold. As a non-mathematician, I find having to sort out the wheat from the chaff too confusing. You may think these aspects need no explanation, but I disagree that your objectives of reaching the desired audience will be met with the concept of a graph. — Quondum^☏ 12:58, 8 June 2012 (UTC)

There is a confusion here between the notion of smooth manifolds - which usually loosely means C_1 - and manifolds in general, no differentiability required, only continuity ie C_0. Richard Gill (talk) 16:41, 8 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)

A lot of work has gone into improving the lead. I think you should propose a specific lead, and explain why you think it better than the current lead. Rick Norwood (talk) 14:08, 8 June 2012 (UTC)

The current lead is not better than this one from three years ago. Regardless of how much work went into bringing the lead into its present form, as far as I can tell, this "work" consisted of entirely rewriting the lead at least three times. At least two of these revisions were a clear erosion of quality. Granted some work seems to have been done to repair the damage, but this seems like a wasted effort. Sławomir Biały (talk) 16:55, 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)

Poincare's definition of manifold

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)

Having the historical definition in a section on history makes sense, but for example that definition makes it quite hard to see that the graph of the absolute value function, as a subset of $\mathbb {R} ^{2}$ is a manifold (not differentiable at 0), or the unit circle as a subset of $\mathbb {R} ^{2}$ (not locally a graph).

A similar thing happens with the concept of function; the historical definitions were simultaneously more limited in some ways and more broad in other ways than the modern definition, so we can't start the article with them. — Carl (CBM · talk) 12:09, 4 June 2012 (UTC)

I for one have serious difficulty understanding what is meant by the wording. Use of the term "graph" in place of "function" confuses. Also the implication that every manifold is globally embeddable in a Euclidean space should not be implicit in the (modern) definition, even if this is (nontrivially) provable. So, no, not Poincaré's definition in the lead. — Quondum^☏ 12:35, 4 June 2012 (UTC)

The lede as it currently stands (i.e. using a map on the surface of the earth as an example) is utterly perfect. My vote is: leave it like it currently is - non-mathematicians will be able to access it admirably from there. --Matt Westwood 13:38, 4 June 2012 (UTC)

@Carl: the graph of the absolute value function is not really relevant as it is not a smooth manifold (actually as an abstract Riemannian manifold it is perfectly differentiable at 0 also). The circle is indeed locally a graph, either over the x-axis or over the y-axis. As Whitney proved, the two definitions are exactly equivalent. This means that the atlas definition is only different from Poicare's definition in that it is harder to follow. It is neither more limited nor more broad.

@Quondum: y=f(x) is a function; the set of points (x,y) satisfying y=f(x) is its graph in the plane. I think most calculus students are more comfortable with the notion of a graph of a function than with transition functions between charts.

@WestwoodMatt: The current lede does not really tell you what a manifold is. Note that the abstract definition ends up using differentiable functions in the end, as well: the transition functions have to be differentiable functions. The only difference is the abolition of intuition in the abstract definition, according to Arnold. Tkuvho (talk) 13:45, 4 June 2012 (UTC)

A circle is not locally a graph, there's no neighborhood of the 3 o'clock point around which the curve passes the vertical line test. It could be that you mean that the circle is the image of the real line under a suitable embedding, but that is not what "is the graph" means, because the circle is not the graph of that embedding (the graph is at best a noncircular subset of $\mathbb {R} ^{3}$). Whitney's theorem is about embeddings of manifolds, but the embeddings are not generally graphs of functions. — Carl (CBM · talk) 19:08, 4 June 2012 (UTC)

In this setting, a graph means that there exists locally an affine coordinate system in which the manifold is a graph. Nevertheless, under the naive meaning of "graph" as it is used elsewhere in mathematics, it is clearly problematic to say this. Sławomir Biały (talk) 19:36, 4 June 2012 (UTC)

@Carl: you are correct that Whitney's theorem is about embeddings of manifolds. Indeed embeddings are locally graphs of functions by the implicit function theorem (that's the content of the implicit function theorem). Tkuvho (talk) 14:27, 5 June 2012 (UTC)

Sławomir Biały already mentioned what you seem to be ignoring, which is that you are not talking about things that are locally graphs of functions in the usual sense of the term. The "original definition" of a manifold is not going to be more enlightening if it requires readers to apply unusual or field-specific definitions to the terms it uses. As it is usually considered, the implicit function theorem doesn't apply to the side points of the unit circle, because a certain matrix isn't invertible at those points. In fact they use this as an example in implicit function theorem. — Carl (CBM · talk) 02:21, 6 June 2012 (UTC)

Carl, what you seem to be ignoring that our page implicit function theorem is only a special case of a more general implicit function theorem applicable to any smooth submanifold or regular parametrisation thereof. Thus, whenever the gradient of the defining expression is nonzero, the implicit function theorem applies. I explained this in terms of your example, namely the circle, at Manifold#Poincar.C3.A9.27s_original_definition. I usually defer to your judgments when it comes to issues of mathematical logic. Have some common sense to acknowledge that this is not a field you are an expert in, and that your original opposition was based on a misconception. No "unusual or field-specific definitions" here. Tkuvho (talk) 11:27, 7 June 2012 (UTC)

I am quite happy to believe that the way you're using the terminology is common in the area. I'm simply saying that it is not as clear to people outside the area as one might think. — Carl (CBM · talk)

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)

@D.Lazard: Thanks for your input. I respect your sentiment in "not liking" the implicit function theorem. However, this theorem is standard for an advanced calculus course. The lede shouldn't be an occasion for pleasing the personal tastes of this or that editor, but rather dictated by the goal of greatest possible accessibility. Certainly the chart definition is an indispensible technical tool, but again the goal of the lede is not necessarily to provide technical tools. Rather, it is to give the reader an idea of the subject matter of the page. Tkuvho (talk) 16:22, 4 June 2012 (UTC)

@Tkuvho: I agree with you that the lead should "dictated by the goal of greatest possible accessibility". But it should, in a non technical formulation, be as close as possible as the technical definition. I "like" the implicit function theorem, what I do not like is to use it implicitly where it is not really relevant. IMO, the "greatest possible accessibility" implies to use only mathematical notions which are unavoidable for given an idea of the subject. Here "near every point" is unavoidable because neighborhoods appear in every definition. On the other hand, "scale" is not needed. The definition through graphs involves a (at least partial) choice of coordinates, which is also not needed. D.Lazard (talk) 16:57, 4 June 2012 (UTC)

I didn't put the "scale" in. Feel free to delete it. As far as choice of coordinates is concerned, it is unnecessary. One can use a coordinate plane in the ambient R^n without the need to choose new coordinates. Tkuvho (talk) 14:30, 5 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)

The current version of the lede expects the reader to know what a homeomorphism is, what a topological space is, and what a neighborhood is. Is this more accessible than the graph of a function? Tkuvho (talk) 11:30, 6 June 2012 (UTC)

I think you are arguing that "graph of a function" would be easier to understand for people outside the area. I do know what a manifold is, but I don't find the "graph" explanation clearer even for one-dimensional manifolds, and it's much harder for me to visualize a 3-dimensional manifold as a graph of a function than as something locally homeomorphic to $\mathbb {R} ^{3}$. (And either way we have to know what a neighborhood is, because it's "locally a graph of a function".) — Carl (CBM · talk) 11:44, 7 June 2012 (UTC)

May I point out that this whole discussion should be taking place at talk:manifold, not here.TR 12:28, 7 June 2012 (UTC)

End of copied discussion. D.Lazard (talk) 15:42, 7 June 2012 (UTC)

I went back to the original source, Analysis Situs, and found that the description offered in this article differed significantly from what Poincare actually wrote. (This is not a surprise because it seems that the original version of this section was based solely on some oblique references made to it over 100 years later in a critique of modern mathematical education!) First of all, Poincare gives two definitions of manifold. Second he never explicitly says that a manifold is a subset of Euclidean space. Instead he says that a general manifold is obtained by forming chains of manifolds. It is clear from reading Analysis Situs that this notion of chaining is what Poincare thought was important and new about this definition (as opposed to the other that he gives).Lost-n-translation (talk) 02:56, 22 June 2012 (UTC)

Good work, Lost-n-translation. Rick Norwood (talk) 12:11, 22 June 2012 (UTC)

footnotes

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)

Technical definition in the lead

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)

Hi D. Lazard! Did you read all of this talk page *before* you started messing around with editing the page. This page has apparently been somewhat unstable because it is serving two types of people: lay persons who are very interested in understanding this difficult topic and serious students of mathematics. It is very hard to simultaneously serve both of these populations. There has been a tug of war and it's not helpful to tug until you have carefully studied the issue.Lost-n-translation (talk) 20:57, 11 June 2012 (UTC)

Lazard. Can you tell me what you find wrong with the following:http://en.wikipedia.org/w/index.php?title=Manifold&oldid=495918196? In particular, this version of the page clearly directs the reader to a formal definition. — Preceding unsigned comment added by Lost-n-translation (talk • contribs) 21:01, 11 June 2012 (UTC)

Lost-n-translation: I can not accept the wording "you started messing". I'll respond to your post only when you will respect Wikipedia policies WP:No personal attack and WP:Civility. D.Lazard (talk) 01:18, 12 June 2012 (UTC)

Hi Lazard. Thanks for the link to WP:No personal attack. I read through the list of what is thought to be a personal attack and I didn't see anything that applied. Could you please explain why you linked to this page? Now if you don't think that it was civil to say that you were messing around with the page, then I apologize for any offense. We had spent many hours recently reading the talk page and editing the page in just the previous week. Lost-n-translation (talk) 16:40, 12 June 2012 (UTC)

If you look at the talk pages and revision history, you'll see that periodically an editor puts a technical definition in the lede and then someone who has the intuition but no serious training removes the technical definition and sometimes this editor will put nonsense in the lede. I would rather have a stable nontechnical lede that makes some sense instead of an unstable and sometimes false lede. Don't you agree?Lost-n-translation (talk) 17:06, 12 June 2012 (UTC)

I acknowledge that you apologize and you have removed the personal attack.

About my edits: I came to this page after Tkuvho asked some arbitration on Wikipedia talk:WikiProject Mathematics‎ (see the thread #Poincare's definition of manifold). From my reading of the talk page, I convincing myself that your preferred lead was better than that proposed by Tkuvho, but that some issues needed to be solved:

• The first paragraph was misleading: the use of "small" and "scale" (twice) suggested that the metric structure (distance and angles) of the Euclidean space was involved in the "resemblance". This was enforced by the figure which displays angles. In fact this paragraph would more convenient to introduce the tangent space of a differential manifold than for the general notion of manifold. This was the motivation of my first edit. IMO, the new version of the first sentence is as easy for the layman and closer to the true mathematical definition.
• Even rewritten, the first sentence remained misleading: The word "resemble" is ambiguous. For the layman, it suggest wrongly that the "distance" does not change too much. For people with some math. knowledge, it means nothing. I do not see any other way to solve this issue than using mathematics. The way, which I have proposed, of giving a translation of the first sentence in mathematical language, has multiple advantages. For the layman, it is clearly a translation ("in mathematical terms ...") and thus he knows that he may skip the translation in a first reading. For people with a little more mathematical knowledge, it indicates which are the minimal mathematical notions (the "context" in MOS:LEAD), that he has to learn to really understand the notion. For a more advanced mathematician, this gives him immediately a workable definition.
• In the previous lead there was no mention of the fundamental notions of "charts" and "transition map". As they may easily be introduced informally on the example of planet Earth, it would be a pity of not doing that.

As far as I understand the posts in #new lead, there is some consensus that these edits are improvement, even if some prefer something different D.Lazard (talk) 10:18, 13 June 2012 (UTC)

As a mathematician, I completely agree with your point of view and I like the way that the page looks now. But there are others who do not agree and my fear is that they will come and ruin the work that you have done. I hope that you will continue to monitor the page. I am done with it.Lost-n-translation (talk) 14:49, 13 June 2012 (UTC)

Curve and surface

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 suggest: "Lines, circles and more generally smooth curves, but not figure eights, are one-dimensional manifolds or 1-manifolds. Planes, spheres, torus and more generally smooth surfaces are examples of 2-manifolds."

"Smooth surfaces and curves" are usually supposed to be indefinitely differentiable; thus they are examples, as there are manifolds that are not smooth for this meaning of "smooth". D.Lazard (talk) 17:29, 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)

Many mathematicians would consider a figure 8 a curve, though it is not a topological manifold. Many would also consider algebraic curves to be curves, although typically these have topological dimension 2. The term "surface" suffers from a similar ambiguity. Sławomir Biały (talk) 20:30, 23 June 2012 (UTC)

This implies that my suggestion above should be rewritten:

"Lines, circles and more generally smooth real curves, but not figure eights, are one-dimensional manifolds or 1-manifolds. Planes, spheres, torus and more generally smooth real surfaces are examples of 2-manifolds."

D.Lazard (talk) 20:50, 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)

I beg to differ. Referring to a curve as a 1-manifold seems to be extremely idiosyncratic. Sławomir Biały (talk) 13:43, 24 June 2012 (UTC)

I think the article's intent is the other way around: it's saying 1-manifolds are curves, not that curves are 1-manifolds. But I agree that this is not especially clear and I don't think it's helpful to mention curves at all. Of course Munkres is authoritative, but it's also true that 1-manifolds are not often called anything, because there's really not much to say about them. The use of "surface" for "2-manifold" is widespread, and it gives the layperson something pretty close to the right idea. "Curve" is loaded with all kinds of extra meaning for the layperson, not to mention being the name of a primary object of study in another mathematical field. I suggest removing everything after the colon in "Lines and circles, but not figure eights, are one-dimensional manifolds: 1-manifolds, also called curves." ChalkboardCowboy (talk) 17:07, 24 June 2012 (UTC)

That seems like the right balance. If no one objects, I will implement this. Sławomir Biały (talk) 18:42, 24 June 2012 (UTC)

I've gone ahead and revised the lead based on this discussion. Sławomir Biały (talk) 15:22, 25 June 2012 (UTC)

Lead: "...a manifold is a geometrical object..."

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)

This seems like a good point. It seems better to state simply:

In mathematics (specifically in geometry and topology), a manifold is an object that, ...

All our bases are covered with this, because the correct adjective (geometric/topological) applies implicitly from the parentheses. I'll leave someone more involved to decide and implement, though. — Quondum^☏ 17:13, 24 June 2012 (UTC)

Sounds good to me. Sławomir Biały (talk) 18:41, 24 June 2012 (UTC)

Objects

"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 don't ever remember hearing the phrase "geometrical object". It sounds somewhat funny. Lazard, I don't see why your argument against the term "mathematical object" does not apply to "geometrical object" as well. In American English the term "geometrical object" sounds as much like jargon as does "mathematical object". But, in fact, a manifold is more of a "topological object" at its core.

In sum, I agree for Slawomir Bialy (for the first time). It is better to leave off the word "geometrical".Lost-n-translation (talk) 14:20, 25 June 2012 (UTC)

I do not understand your objection against "geometrical object". It appears in more than 20 Wikipedia articles, without counting the occurences of "geometric object". If it sounds somewhat funny, why is it so widely used by editors? D.Lazard (talk) 14:35, 25 June 2012 (UTC)

It sounds funny to my ears. I am a practicing mathematician and most of my colleagues refer to me as a "geometer". I travel all of over North America and Europe attending academic conferences. I never hear the term "geometrical object". But not all editors are like me, and so some things don't sound funny to them.

In any case, I still don't understand why "geometrical object" sounds less like jargon than "mathematical object". And as has been mentioned above and below, a manifold is really---at its core---a "topological object". By saying that it's a "geometrical object", one gives the false impression that it belongs mainly to geometry.

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)

Maybe I'm just muddying the waters, but is it not true that there exist (topological) manifolds that are not geometric? The phrase "geometrical object" excludes such a possibility, and as such would be unfortunate in the defining sentence. — Quondum^☏ 15:00, 25 June 2012 (UTC)

How about this? Sławomir Biały (talk) 15:11, 25 June 2012 (UTC)

I like it, except that the leap from "mathematical object" to "point" is too big. The inference would be that it is an object somehow related to points, but as the connection is not stated, it causes a double-take. From "topological object" the leap would not be too large. — Quondum^☏ 15:27, 25 June 2012 (UTC)

I like it too, with the same qualification. Could we replace mathematical object with topological space? Not only is it correct, which I think is important, but the word "space" gives an appropriate intuitive idea, and gives context for the discussion of "points" which comes next. Unless the word "topological" is deemed too scary, or unless there are things commonly called manifolds which are not topological spaces (I've never heard of this, but math is a big place), I think this works well. ChalkboardCowboy (talk) 17:55, 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)

I wouldn't object to removing the second sentence. It's not particularly helpful to the layperson, and mathematically sophisticated readers are by now used to scanning for the "mathematical definition" heading anyway. My opinion is that (in the lead) we don't have to say everything, but we should take care not to say anything that's wrong. — ChalkboardCowboy^[T] 01:15, 26 June 2012 (UTC)

I am too. What is lost is the clear distinction between a metric and a non-metric Euclidean space implied by the word "homeomorphic", but I'm happy that this can be left until the article body. I really like the first sentence – it's a perfect nutshell by my reckoning. — Quondum^☏ 12:55, 26 June 2012 (UTC)

I agree with Sławomir text. But I object to remove its second sentence: "resemble" has no mathematical meaning, even informally. As almost all possible readers probably have at least a minimal mathematical background, the first sentence alone has an empty meaning for them. Let me recall that more than 1000 articles link here, and almost all are mathematical articles, often of high math. level. Thus, without its second sentence, Sławomir text would break MOS:LEAD, where one may read, among other relevant assertions: "The lead should be able to stand alone as a concise overview".

On the other hand, I appreciate two things in Sławomir text, compared to the present version of the lead: firstly, it avoids the use of "object", which is, in any case, somewhat ambiguous. Secondly the two technical conditions in the definition are removed from the lead. This is not a problem, because the spaces that do not satisfy them are somewhat pathological and are never encountered in practice by people interested by manifolds.

In summary, I find Sławomir text almost perfect, and, if there is a consensus for this text, I suggest that any edit aimed to change its math level to either less math or more math should be considered as vandalism. This would allow to avoid to reiterate a time consuming discussion that has been done several times.

D.Lazard (talk) 14:19, 26 June 2012 (UTC)

I'll be happy with or without the second sentence. The parts that were bothering me are fixed either way. — ChalkboardCowboy^[T] 00:58, 27 June 2012 (UTC)

Slawomir's text seems ok to me. If one looks above on the talk page, one sees that the use of the word `resemble' has been controversial for a long time. But other choices seem clumsy. I think that the second sentence should suffice for readers who come to this page from high level math pages. There are at least two audiences for this page and not all are on a high level.Lost-n-translation (talk) 01:39, 27 June 2012 (UTC)

Charts

Shouldn't 'charts' have their own math article?! All Clues Key (talk) 01:40, 7 September 2012 (UTC)

They do: see Chart (topology). I've edited Chart (disambiguation) to try and make this a bit clearer. Jowa fan (talk) 02:17, 7 September 2012 (UTC)

Thanks for clarifying. I've added numerous internal links to the Atlas page, and the Charts section. All Clues Key (talk) 18:11, 7 September 2012 (UTC)

"with rounded corners"

The recent edit 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)

I was thinking that "(differentiable) manifold with boundary" is sometimes taken to mean that the boundary is a (differentiable) manifold too. But I don't have a reference for this, and I'm far from an expert on this particular topic. Jowa fan (talk) 10:43, 17 September 2012 (UTC)

Since nearly everything I know on the topic comes from WP, I label myself similarly. The section defines the concept without this requirement, but I've no idea what the range of standard definitions encompasses. — Quondum 11:08, 17 September 2012 (UTC)

I am also not a specialist of the subject, but, requiring that the boundary should be a differential manifold is certainly too strong for most applications, such as fluid mechanics. For example requiring that the boundary should be a manifold would prevent to study the mechanic of a cubic pool. But some conditions are certainly required. From some work that I have recently done about cylindrical algebraic decomposition, I guess that 1/ the charts should be homeomorphisms from open subsets sets of the interior of the manifold onto open sets of Rⁿ that can be prolongated into an homeomorphism of the closure onto a subset of Rⁿ and 2/ the transition maps may be prolongated differentially on the boundary. But, again, one has to look on textbooks which define varieties with boundary and use them effectively. D.Lazard (talk) 12:00, 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 $C^{\infty }$ smoothness assumed to extend up to the boundary. Sławomir Biały (talk) 12:54, 17 September 2012 (UTC)

I have read the second paragraph of the section (I had skipped it in a first reading). For the definition of this paragraph, the paper sheet with hard corners is a smooth (and even analytic) manifold with boundary. Moreover the boundary is also, per se, a smooth manifold (take 4 charts, each containing exactly one corner; the transitions maps are smooth and even analytic). The only things that are not smooth are the embedding of the boundary in the Euclidean space and the homeomorphism of the closed square onto the closed disc. This is clear enough to me for removing the tag and the mention "with rounded corners" (I agree with Sławomir that it is confusing) D.Lazard (talk) 14:29, 17 September 2012 (UTC)

obviously false sentence!

"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)

No notion of distance is incorrect

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)

The fact that one may define a distance on any variety does not imply that it is unique. Well-defined does not means correctly defined but defined in an unambiguously way. --D.Lazard (talk) 07:25, 9 November 2012 (UTC)

I've deleted the relevant sentence. I think it gives a misleading impression (readers could be left wondering if disconnected metric spaces exist at all), the point it's making isn't valuable, and it's an unsourced statement. Jowa fan (talk) 23:38, 9 November 2012 (UTC)

Lede statement on sphere and torus

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)

It was rather insightful to mark proposed wording with color:#600 (like {{!xt}}). I am not completely sure about “torus”, but in mathematics, “sphere” could be viewed only as a 2-manifold. A 3-dimensional subset of an Euclidean space is called “ball (mathematics)”. They can be sometimes confused by geographers, geologists, engineers and so, but not by mathematicians. “The surface of a sphere” is a nonsensical gibberish or, at best, a pleonasm. Incnis Mrsi (talk) 09:47, 20 April 2013 (UTC)

Great Article!

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)

"Euclidean space" or "coordinate space"?

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)

The coordinate space over the reals has a natural Euclidean metric, which is not used in manifold theory, and, unlike an abstract Euclidean space, is has the coordinate structure, which is used in manifold theory. Why should the lead link the article about the structure which is not used, but avoid linking article about coordinate structure which is frequently used? Also, complex manifolds, algebraic manifolds, p-adic manifolds, and probably others do exists, so the main D.Lazard’s argument against my version is plainly wrong. If there will be no further objections, I’ll reinstate my version of the lead. Incnis Mrsi (talk) 10:23, 19 April 2013 (UTC)

I don't think of the new revision as an improvement. I don't think the distinction between a Euclidean space and real coordinate space is especially mathematically significant, and for most readers Euclidean space is likely to be clearer. The proposed revision leaves vague the main case of real coordinate space until after examples have been given (which I think defeats the purpose of those examples). For almost all mathematicians, the word "manifold" will mean "manifold over the reals" (probably evenly split between whether it has a differentiable structure or not), not a possibly p-adic manifold or complex manifold. If a mathematician means one of those things, then he will say "p-adic manifold" or "complex manifold". We shouldn't emphasize unusual cases in the lead of an article: things there should appear in proportion to their prominence. Sławomir Biały (talk) 12:18, 19 April 2013 (UTC)

Per WP:LEAD, the lead has two main functions: introduce the topic in an accessible way and summarize the content of the article. Euclidean space is a more familiar concept than coordinate space and is the better term to use in an accessible introduction. Euclidean space is a concept used throughout the article, while coordinate space is not. Even definitions of topological manifolds and scheme-theoretic analogs make reference to Euclidean space. So it is appropriate that Euclidean space be used in the lead. From a mathematical point of view, coordinate space is merely a representation of the geometric object called Euclidean space. It seems wrong to define a geometric concept, such as a manifold, in terms of a particular representation of another geometrical object, albeit a common one. --Mark viking (talk) 17:30, 19 April 2013 (UTC)

The real n-space has some structures which an Euclidean space has not. These are: the origin, n coordinates (or, dually, the standard basis), an orientation (or, the same, an order on coordinates or basis elements). You can assert that it is merely a representation of the geometric object called Euclidean space, but it means that you just do not understand that ℝⁿ is an object on its own standing, of several structures which are not Euclidean. Could you provide citations for definitions of topological manifolds and scheme-theoretic analogs making reference to Euclidean space? Incnis Mrsi (talk) 17:46, 19 April 2013 (UTC)

The structure desired is primarily the local geometry, not the local coordinate system. Having a locally $\mathbb {R} ^{n}$ coordinate system is a particular sense in which you've accomplished creating such a local geometry. This is a useful perspective to take when you seek to generalize the concept: You could make sheaves of all kinds of magmas. But you wouldn't if you were trying to produce the algebraic analogue of manifold semantics, because you're looking for the coordinate system you use to have laws corresponding to certain geometric properties. Effective featurelessness, cohesive behavior, and what else you like about Euclidean space are why they're chosen as building blocks of things, not coordinatizability, which you want for entirely its own reasons. Recent Advances in the Foundations of Euclidean Plane Geometry (Bruck 1953) breaks down the study of planar ternary rings in terms of generalizing the semantics of Euclidean space, because while coordinatizability is a given, without those semantics non-desarguessian planes can be taken for a formal aberration, not spaces in their own right. Geometric Algebra (Artin 1957) also takes the approach of deriving fields from affine spaces, not the reverse, because it's motivation for fields themselves, and it's the motive, which should not be hidden, for wanting things to be patched together from them. ᛭ LokiClock (talk) 18:28, 21 April 2013 (UTC)

Do you really contribute to this discussion, not to some new one? Euclidean space is a particular structure over the real numbers. Nobody wants to derive real numbers from affine spaces or such, but somebody can be interested in non-real-based manifolds. When one chooses “Euclidean space” instead of an (abstract) “coordinate space”, one loses all manifolds over non-Archimedean fields, as well as part of algebraic manifolds (those which are over finite fields, algebraic numbers, and possibly something else). Incnis Mrsi (talk) 18:48, 21 April 2013 (UTC)

Insisting on the lead of the article referring to abstract coordinate spaces in order to accommodate rather exotic objects like manifolds over non-Archimedean fields makes about as much sense in this article as it would to insist that the lead of the article integer should accommodate notions like p-adic integer. It's just not an appropriate focus for the lead of the article. Sławomir Biały (talk) 20:56, 21 April 2013 (UTC)

Partially answered at user talk: Sławomir Biały #Talk: Manifold because an off-topic starts here. The only correct point, although missed by the poster himself, is existence of two articles integer about the ring Z = O(Q) and ring of integers about O(whatever). We have differentiable manifold and topological manifold articles where the manifold structure is more specific. What should we have in the “Manifold” article? IMHO a WP:CONCEPTDAB-like article. Incnis Mrsi (talk) 06:26, 22 April 2013 (UTC)

I agree with Slawomir: Talking on "manifolds over non-Archimedan field" in the lead is confusing for most readers (including myself, see my post introducing this thread). Moreover, one may be interested by such extensions only if one has well understood the classical definition. IMO, the place of "manifolds over non-Archimedan field" is as a subsection of the section "Generalization", possibly with a {{main}} template, if a separate article is written. Emphasizing in the lead on such a particular generalization of the primary topic would break the WP:DUE policy. D.Lazard (talk) 10:20, 22 April 2013 (UTC)

Incnis - Euclidean space is not a structure over the real numbers. Euclidean space is a natural geometry for the real numbers, and the real numbers are a natural algebra for Euclidean space. Given the "what links here" articles are like Dimension (mathematics and physics), Dynamical system, or General relativity, it's unfair to suppose the reader takes this view of the real numbers, and implicitly make the transposition. Euclidean space might be their only reason for finding the real numbers so special. Besides, one isn't losing p-adic manifolds in the long term, one gains them in the first place by generalization of what you consider to be Euclidean space or the topology of the real numbers. What you lose is the analogy between generalizations and the original, because you've lost the original geometric notion. ᛭ LokiClock (talk) 13:47, 22 April 2013 (UTC)

What means “real numbers [are] so special”? Yes, they are intimately related with metric geometry, particularly with aforementioned Archimedean property and the concept of a complete metric space, but metric geometry is not a very important thing in the theory of manifolds. Topology is crucially important, differential calculus is important, linear algebra is important, and hence fields are important. A metric structure is important only in Riemannian geometry, a rather special part of the theory of manifolds. Incnis Mrsi (talk) 17:38, 22 April 2013 (UTC)

By setting coordinate spaces as your basic notion for manifolds, you're implicitly emphasizing the real numbers, and the properties that arise from them. Saying fields, linear algebra, and differential calculus are important begs the question - what makes fields give rise to linear and differential algebras, and why are linear algebras good? If you look at a sphere up close, you're thinking it's a manifold because of the plane geometry, of the behavior of the lines and points in the neighborhood. So maybe there are other geometries that behave closely to Euclidean space. Wondering if the real numbers aren't so special can lead you to finite fields as coordinate systems for geometries that are Euclidean for all intents and purposes, while also justifying the real numbers because it's the most complete option (pun intended). ᛭ LokiClock (talk) 19:07, 22 April 2013 (UTC)

Also, I want to clarify the original statement I made. Your concern about people being interested in manifolds over other fields is exactly what I was addressing - the reason you might call something with local field structure a manifold is precisely because they are the algebras for what can be thought of as generalized Euclidean spaces. The algebraic properties required to coordinatize a space are derived from the axioms of the geometry in the sources I cited. The difference between the axiomatic affine spaces studied in incidence geometry and the affine spaces which are vector spaces whose origins have been forgotten (by introducing new automorphisms) is that fields coordinatize spaces that satisfy more axioms of Euclidean geometry, special cases of the axiomatic affine spaces. And given that the first irrational numbers were derived geometrically, to show they must be included in a good number system, one would want to derive the reals from affine space. ᛭ LokiClock (talk) 16:19, 22 April 2013 (UTC)

Indeed, the first irrational was derived from Euclidean plane. You are correct in the point that real numbers are its natural algebra, and these are namely real numbers over which the structure of a quadratic/bilinear-symmetric form yields a useful geometry. Incnis Mrsi (talk) 17:38, 22 April 2013 (UTC)

Note that, in his book "Geometric algebra", Emil Artin showed (as far as I remember) that the axioms of Euclidean geometry allow to construct naturally a field, which is isomorphic to the reals, and, conversely, that the geometry which is constructed from the affine space over the reals, equipped with the dot product, is isomorphic to Euclidean geometry. To Incnis Mrsi: Nobody contest that manifolds may and have been generalized to other fields. The problem is the importance/notability of this generalization, which does not allow to mention it reasonably in the lead. D.Lazard (talk) 17:51, 22 April 2013 (UTC)

Let's get the zeroth law section right

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:

1. Do we want to delete all mention of whether the zeroth law defines "temperature as a numerical scale for a concept of hotness which exists on a one-dimensional manifold with a sense of greater hotness"?
2. If the answer to the previous question is "yes", how do we provide examples that allow people to understand the concept?

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)

I think this was posted in the wrong talk page--this article is about mathematical manifolds. --Mark viking (talk) 17:24, 21 March 2014 (UTC)

Figure 1 does not illustrate text

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)

That's true. I've replaced semicircles with arcs in the prose. Thanks, --Mark viking (talk) 03:57, 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)

The surface of the Earth requires (at least) two charts to include every point

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

Read the rest of the sentence. That circle is an example of "duplication of coverage". — Cheers, Steelpillow (Talk) 15:27, 12 January 2015 (UTC)

Animated GIF

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/w/index.php?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)

Presumably the onus is on you to make a more suitable image and convert the gif to video, per the guideline. The manual of style should not be used to dictate what kinds of informative content to have in articles. This image has informative value. Sławomir Biały (talk) 00:35, 15 November 2013 (UTC)

The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.

As I understand it there is no onus on editors to replace offending content. If, for example, a statement is unsourced, an editor is not obligated to find a source which negates the content before removing. While the content I am attempting to remove has value, it is neither critical to the article nor mentioned anywhere in the text.

Despite my belief that replacement is not a requirement, consensus is needed. So in an attempt to achieve consensus, I suggest replacement of the original with the following:

-- Catskul (talk) 16:30, 15 November 2013 (UTC)

Aesthetically, that image is not really an improvement over what is there now. I will work on a better replacement when I get the time. Sławomir Biały (talk) 00:08, 16 November 2013 (UTC)

I'm going to add in an opinion here. The point that some 2d manifolds cannot be embedded in 3d space without self-intersection is not central to the concept of a manifold; it is an essentially topological result that applies only for an arbitrarily constrained choice of embedding space for a some manifolds. I'd say that a far more significant (or more generally applicable) points topologically are that manifolds can be closed or can be non-orientable, which are not even mentioned in the lead. I'd think that it would be sufficient to mention one or two illustrative cases such a the Klein bottle without mentioning properties of selected embeddings. —Quondum 13:01, 16 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 aspects of manifolds are important, I agree, but is only one of what they are useful for. The same point about embedding can be illustrated within a Klein bottle without challenging the reader nearly as much. Emphasizing the topological aspects at the expense of the geometric aspects is also not ideal. For example, the real projective plane also represents elliptic geometry, which does not come through at all. All these issues are complex enough that more than a mention in the lead can hint to the reader that this article is hard work to understand. —Quondum 21:32, 17 November 2013 (UTC)

Meanwhile I would second the removal of any animated gif until a suitable alternative can be found. These things do my head in really quite seriously, and I am sure I am not alone. Sometimes I can rely on my browser settings, but far from always. — Cheers, Steelpillow (Talk) 15:32, 12 January 2015 (UTC)

Betti numbers and torsion coefficients

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)

Realization?

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 agree with Steelpillow. Unless "realized" has some technical meaning that I don't know of, it is a vague statement without any clear meaning. Also, what does "in three dimensions" mean? Presumably it means in a three-dimensional real space, but it doesn't say so, and both the Klein bottle and the real projective plane can be embedded in other three-dimensional real manifolds (even if we restrict "three dimensions" to refer only to three dimensions over the real field). Consequently, I have re-worded the passage in the article. The editor who uses the pseudonym "JamesBWatson" (talk) 14:12, 29 May 2015 (UTC)

To clarify for anyone who reads this, the passage that Steelpillow mentions in the article abstract polytope says "a realization of a regular abstract polytope is a collection of points in space ... together with the face structure induced on it by the polytope, which is at least as symmetrical as the original abstract polytope..." (My emphasis.) As far as I can see, that must include injections which are not one-to-one and introduce more symmetry than in the abstract polytope. The editor who uses the pseudonym "JamesBWatson" (talk) 14:21, 29 May 2015 (UTC)

I am not sure if that quoted passage is quite correct. A few years ago "realization" was shaping up to mean any injection into some parent real space, while preservation of symmetry, number of dimensions and suchlike required the realization to be "faithful". I may have a source somewhere, I'll try to find time to check. — Cheers, Steelpillow (Talk) 18:37, 29 May 2015 (UTC)

There is a notion of a geometric realization as a functor that maps simplicial sets and incidence relations to a topological space formed from suitably glued-together geometric simplices. This is discussed in nLab at geometric realization and mentioned in the article section Simplicial set#Geometric realization. The realization in abstract polytopes that Steelpillow mentions seems an example of this more general concept, although that specialization includes constraints on symmetry. I don't have a source for this, but regarding a Klein bottle as a simplicial set, the article is stating that one cannot map the set to geometric simplices in three dimensional real space that satisfy the incidence relations. This is a fun article on Klein bottle realizations. --Mark viking (talk) 00:44, 30 May 2015 (UTC)

My favourite realization of the Klein bottle is in four-space. We make a Möbius band by taking a two-dimensional strip, twisting one end over in a third dimension and butt-joining the ends to form a twisted cylinder. Take a piece of rod instead of a flat strip. This is already 3D so we twist it over in a fourth dimension before joining the ends to form a twisted hoop. Just as a Möbius band has a definite interior surface between its edges, so too does the hoop. When either is squashed down a dimension it must self-intersect, creating a singularity where not only the boundary self-intersects but also the interior crosses over itself. The interior of the Möbius band is usually obvious to us. Once we understand that a Klein bottle is just the boundary of a solid hoop, its interior becomes equally clear. It is only the intuitive reluctance to introduce a crossover ring singularity in 3-space, when there is no such singularity in the surface, which confuses us into the mistaken idea that it is a "bottle". — Cheers, Steelpillow (Talk) 08:11, 30 May 2015 (UTC)

Klein bottles

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:

1. the Klein bottle model(s) inaccurate?
2. the statement is incorrect?
3. the statement is correct and I've misinterpreted it?
4. other / combination of above?

Cheers, --TFJamMan (talk) 07:56, 5 April 2016 (UTC)

A three-dimensional model (or representation) of the Klein bottle is the image of a differentiable map from the (abstract) Klein bottle into R³, which satisfies some regularity condition. Typically this condition is that every point of the Klein bottle has a neighborhood such that the restriction of the map to this neighborhood is a diffeomorphism of the neighborhood onto its image. It can be proved that such a three-dimensional model cannot be a manifold, that is the model is a surface, which has crossing points, where the surface is not locally homeomorphic to R². Probably, you have been confused by the fact that there are several kinds of surfaces, and those that have crossing points or other singularities are not manifolds. D.Lazard (talk) 09:12, 5 April 2016 (UTC)

In other words, although the Klein bottle can be immersed in ordinary space, such an immersion will always self-intersect - it cannot be embedded without self-intersection. The statement you quote does seem to be a little ambiguous - your scenario 3 seems to be the case, so I'll try to clarify it. — Cheers, Steelpillow (Talk) 09:20, 5 April 2016 (UTC)

A possible source of confusion is that, in Surface, it was asserted that a surface is necessarily non-singular. I have fixed this by rewriting the lead, which was also tagged as too technical. D.Lazard (talk) 10:26, 5 April 2016 (UTC)

Convention for domain and codomain of transition maps?

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)

Please, read Atlas (topology) more carefully: in both articles transition maps map open subsets of R^n to open subset of R^n. D.Lazard (talk) 16:33, 4 January 2017 (UTC)

As I understand it, a transition map describes how to convert from the coordinate system used by one chart to the coordinate system used by an overlapping chart. All are expressed in terms of R^n. A simple example would be converting (transitioning) between a map of Britain using the Greenwich meridian and a map of France using the Paris meridian, when visiting the Channel Islands. — Cheers, Steelpillow (Talk) 17:24, 4 January 2017 (UTC)

Equivalence

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)

In the smooth category, the union of two atlases need not be an atlas, since nothing forces the composite $\phi \circ \psi ^{-1}$ to be smooth if $\phi$ and $\psi$ are just homeomorphisms that define different smooth structures. However, in the topological category, any two atlases are equivalent. Sławomir Biały (talk) 23:18, 25 September 2017 (UTC)

As I understand it, it means that if two atlases both describe the same topological manifold then their union - the bound volume of both sets of charts - will also be an atlas for that manifold. Perhaps I am being too simple-minded? — Cheers, Steelpillow (Talk) 08:23, 26 September 2017 (UTC)

Correct. But also the union of two topological atlases will always be a topological atlas, because the composite of homeomorphisms is a homeomorphism. So the definition of equivalence is fairly vacuous in the topological case. Sławomir Biały (talk) 10:56, 26 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)

For a smooth chart, the transition maps are assumed to be smooth. (For a topological chart, the transition maps are automatically continuous, so this is not needed as an extra hypothesis.) Sławomir Biały (talk) 14:38, 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)

What words would you propose adding to the article, then? Perhaps that will help us to see what you mean. — Cheers, Steelpillow (Talk) 14:04, 27 September 2017 (UTC)

A chart is assumed to "preserve the structure". Usually though, one takes a chart as what defines the structure. For example, a differentiable manifold is a topological space covered by an atlas where the transition functions are differentiable. You can give a different atlas which is differentiable in the sense that its transition maps with itself are differentiable, but for which the transition maps with the other atlas are not. So these are not equivalent atlases. They define two different differentiable structures. Sławomir Biały (talk) 14:50, 27 September 2017 (UTC)

Chart

"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)

The Mercator projection leaves out the poles, and also fails to be continuous on one of the meridians. Sławomir Biały (talk) 10:37, 2 November 2017 (UTC)

Any chart on a sphere must leave out at least one point. Any single wrapping which covers it completely must introduce other discontinuities and so is not a chart. — Cheers, Steelpillow (Talk) 11:26, 2 November 2017 (UTC)