Talk:Manifold/rewrite/freezer

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Since there are lots of different kinds of manifolds would it be a good idea to have an exmaple for each kind and then the definition in a section on that kind of manifold? That way the examples would be spread throughout different sections. What do you think about this? --MarSch 15:05, 26 Jun 2005 (UTC)

I don't agree with the new headings

6 Symplectic manifolds

7 Complex manifolds

8 Kähler and Calabi-Yau manifolds

9 Lie groups

Having sections made up of just one or two sentences looks kind of tasteless. Wonder what others think. Oleg Alexandrov 20:22, 26 Jun 2005 (UTC)

They are supposed to be filled in more. --MarSch 28 June 2005 11:31 (UTC)

I have moved (pseudo)Riemannian symplectic and Lie group to diff. manifold. Kaehler and Calabi-Yau should probably be put into complex manifold. --MarSch 29 June 2005 16:25 (UTC)

Contents 1 First paragraph 2 What are the key ideas to express in this article? 3 Reasons for rewriting the introduction 4 topological manifolds 5 examples 6 Ruthless cuts 7 Merging atlas (topology)? 8 Jitse's edits of 14 July 2005 9 (the?) Euclidean space 10 colors of circle charts 11 Construction 12 Large table of contents 13 Intro again 14 Varieties and Schemes 15 KSmrq's introduction 16 Counterexamples 17 Etymology 18 "Markus' edits"

[edit] First paragraph

I find the very first paragraph in the article very hard to comprehend. Besides, from what I know, any manifold, whether differentiable or simply topological, looks locally like the Euclidean space. Comments? Oleg Alexandrov 30 June 2005 16:14 (UTC)

Well, the idea was to make a separate entry for topological manifold, so then in this article manifold is not synonimous to topological manifold. Another kind of manifold is for example Banach manifold. This would be a generalization to allow infinite dimensions. Also algebraic varieties and schemes come to mind. What exactly do you find hard to comprehend. Perhaps you can specify a sentence that bugs you. --MarSch 30 June 2005 17:11 (UTC)

I thought the introductory paragraphs to an article should be as simple as possible, illustrating the most important features of the objects in question in an accessible manner. I think you focused on being most general at the expence of simplicity. Don't you think that saying

A manifold is a space which looks locally like the Euclidean space. The simplest example of the manifold is the Euliclidean space itself

looks easier to understand than:

a manifold is a space that locally looks like a specific space, which is deemed simple. For example a topological manifold locally looks like Euclidean space. The simplest example of a manifold is the space it locally looks like itself.

All those fine details about some manifolds being topological, some differentiable, and that the concept itself can be generalized to an infinite number of dimensions, can be talked about much later after the reader understands the main concept. Oleg Alexandrov 1 July 2005 02:00 (UTC)

The intro should be as simple as possible (but not simpler), but your suggestion is to pretend that only topological manifolds exist and that is simpler. A manifold is not a space that locally looks like Euclidean space. You left out the last sentence in your example:

"In mathematics, a manifold is a space that locally looks like a specific space, called here a simple space. For example, a topological manifold locally looks like Euclidean space, so the corresponding simple spaces are all Euclidean. The simplest example of a manifold is the corresponding simple space itself. Thus the simplest example of a topological manifold is Euclidean space."

I don't see how this is more difficult, since everything about topological manifolds is explained just as in your suggestion, so that information is not more difficult. The other information is that this is not the only example. Don't you think this is an interesting "teaser"? --MarSch 1 July 2005 10:03 (UTC)

I think the current introduction strikes the correct balance between generality and simplicity. It is definitely not true that all manifolds locally look like Euclidean space, and therefore it is misleading to assert that they are as Oleg suggests. I believe that the notion of a "simple space" is a good way to simplify the exposition without losing generality. As for having examples of each different type of manifold, this may be obstructed by the fact that some examples may be technically difficult to construct and explain to beginners. - Gauge 1 July 2005 20:03 (UTC)

[edit] What are the key ideas to express in this article?

Hi all, I think you want to focus on explaining, first, that the notion of a manifold is the simplest known way to solve an unavoidable technical problem which arises when we try specify respectively location, rates of change, or distances in more or less abstract spaces. You can say that the technical problem might initially seem like a mere annoyance, but the way we overcome it leads to a fundamental insight into a basic feature of many other important topics in mathematics, topics which, like manifolds, turn out to have many practical applications.

I think the circle example is adequate for getting across the idea of overlapping neighborhoods on Mⁿ, each homeomorphic to a ball in Rⁿ, but to explain why specifying locations is tricky in general "spaces", the S² example is probably best. High school students may even have noticed that all plane maps of the globe leave out some points.

You could start with a three dimensional picture showing equilevels of some smooth bounded function on a round sphere (even better would be a Java applet allowing the reader to rotate the globe), and propose the problem of mathematically stating what the temperature on the surface of the Earth. You could show a figure mapping the temperature in the Northern hemisphere to the plane by orthogonal projection (because that's easy for most people to understand), and point out that that this assigns coordinates to points in the Northern hemisphere, but in a somewhat arbitrary manner. You can say the observation that such descriptions must leave out at least one point (you could say that stereographic projection leaves out exactly one point) motivates the following ideas:

• a "cartographic map" of the globe is a suitable one-one "function" (map) from a topological disk neighborhood on the sphere to R²,
• this associates (in a continuous way) a unique pair of reals with each point in our disk neighborhood, so we can tell points apart,
• we need to stitch together these local descriptions to a global description,
• we can do that using transition functions on overlaps of our disks to provide a "concordance" between competing local descriptions.

You can point out that we neccessarily have multiple representations of location (at some place in our space), because each point in the manifold belongs to the (disk) domain of infinitely many local coordinate charts, hence the name "manifold", expressing "many-foldness". You can say that key insights here are that

• we must allow multiple representations,
• this is acceptable as long as we know how to convert between any two representations which happen to apply to a given (overlap) region.

You can say that this is a basic feature of much of mathematics. For high school students, you could point out that angles exhibit multiplicity, and that generations of students have agonized over branches of trigonometric functions because this complication is unavoidable. For undergraduate students, you could point out another example: a fundamental feature of linear algebra is that we need to allow for many bases of a given subspace, and many matrix representations of a given operator, but we can once again deal with this complication since we know how to convert between the various representations. (Linas, you could jump in here and ensure that this article links nicely to one explaining monodromy, branch cuts and so forth in a similar style.)

Of course, you need to express this in simple nontechnical language (as far as possible), hopefully with some pictures, which is a challenge! And you need to explain why this procedure is not circular.

If you can do that, you have conveyed some idea of the motivation, definition, and utility of the notion of a topological manifold. Next, you can say, suppose we have some manifold (maybe picture a three holed torus) on which we have defined some function, call it "temperature". We might want to study how temperatures are changing from place to place. To do this, we need additional structure. Which we can do by...

After explaining both topological and smooth manifolds, you can point out that this illustrates another fundamental aspect of much of mathematics, not just manifold theory: levels of structure. That is, to get more precise information, e.g. passing from "the temperature at (0,1,-3) is 5" to "the temperature there is changing most rapidly in the (0,1,-1) direction at a rate...", we need to add more structure to our definition. Turning things around, we can say that the notion of a topological manifold underlies the notion of a smooth manifold.

By the way, after topological and smooth manifolds, it would be natural to introduce a third level of structure, Riemannian manifold, where of course the added structure allows us to discuss how far apart two different locations are. Here, we have the new "technical problem" that the "pedestrian" notion of distance on the globe is neccessarily path dependent, which we can overcome by...

Another observation which could only be aimed at graduate students is that intuitively, most people would want to see minimal atlases, e.g. you can point out how many topological disks are need to cover S^{n. More generally, most people would want to minimize the multiplicity of representation. But to make definitions sufficient to get a theory off the ground, it is often best to do just the opposite, throw in all possible representations, for example in defining a maximal atlas.}

^{As a matter of fact, it might be best to begin with a higher level of structure, and then to systematically remove structure. One strategy would be to (re)-write detailed articles on topological manifold, smooth manifold, and Riemannian manifold incorporating some of the ideas above, and use this article to sketch the key features and lessons learned as above, and then to point out tell the reader that he can find more examples and discussion in the specialized articles, and that he might want to start with Riemannian manifold, even though this has a more complicated definition, because without further experience he might not be able to easily separate geometrical intuition about "continous change" versus "rates of change" versus "distance". ---CH (talk) 4 July 2005 20:02 (UTC)}

^{I like very much the recent changes by the anon to this article. Though I did not read in very much detail what CH suggests, I like that too. It is a breath of fresh air in this article which was getting very abstract recently. Oleg Alexandrov 5 July 2005 02:22 (UTC)}

^{The main idea I get from CH's exposé is that we need to talk more about the motivation. I wholeheartedly agree. -- Jitse Niesen (talk) 5 July 2005 15:34 (UTC)}

^{Oleg and Jitse: thanks! Everyone: is it OK with you if I try to provide a new introduction along the lines above? ---CH (talk) 00:01, 11 July 2005 (UTC)}

^{Are you saing the introduction needs to be rewritten again? I sort of like the way things are now. But OK, you guys decide.}

^{What really needs work is the part starting with "Topological manifold" and going down. With due respect to MarSch, but splitting that part into minisections and reffering to "subarticles" for more details turns me off. I'd say those sections be all merged into one big section and referring to other articles (not subarticles!) wherever appropriate. Oleg Alexandrov 01:23, 11 July 2005 (UTC)}

^{[edit] Reasons for rewriting the introduction}

• ^{I think instead of saying "manifold is made from pieces from a simpler space" it is better to give a concete example of such a simple space first, (the Euclidean space), and then say that spaces can be more general than that.}

• ^{I took out the 3-sphere example since the examples are discussed in the section right below. Oleg Alexandrov 2 July 2005 05:23 (UTC)}

^{By the way, I saw Gauge's comments above after I changed the introduction. I did not mean to say that all manifolds are differentiable, only that the wording was so abstract that it was hard to follow. Oleg Alexandrov 2 July 2005 05:25 (UTC)}

^{I've rewritten the intro from Gauge's last version, starting with gluing simple spaces. I think I will expand some about this. We need people to think about pieces of paper and glue.--MarSch 2 July 2005 13:40 (UTC)}

^{Oleg removed the 2nd and "4"th sentences [1]}

^{In mathematics, a manifold is a space that is glued together from specific simple spaces. Such a simple space is the simplest example of a manifold. For example a topological manifold is glued together from Euclidean spaces and Euclidean spaces are the simplest examples of topological manifolds.}

^{saying: "removed excessive repetition. The words "simple" and "space" are ambiguous enough, at least don't use them as often."}

^{I don't see how this removing of the simplest examples improves the article. --MarSch 2 July 2005 16:04 (UTC)}

^{Very good point, and thanks for not reverting right away. I think it is good to have that example, but not right in the definition. The problem is the following:}

^{Readers are rather unfamiliar with the words "simple" and "space" at this point. I mean, these concepts are rather ambiguous. Then, it is not good to build upon them. You wrote:}

^{Such a simple space is the simplest example of a manifold.}

^{But the notion of simple space was not defined previously. Then, this becomes a bit like a tautology (the simple space is the simplest space). I think it is good what you do below, where concrete examples are given. This particular example above, stated as it is, is not I think very helpful, even if I agree that the example is indeed important. Oleg Alexandrov 2 July 2005 16:57 (UTC)}

^{Comments? Oleg Alexandrov 2 July 2005 16:57 (UTC)}

^{It took me a while, but I think I understand what you are getting at. Are you saying that it looks too much like manifold and simple space are defined at the same time, first manifold as a function of simple space and then in the next sentence vice versa?}

^{Perhaps we need something to introduce simple space first. Something like "Imagine something simple, now take lots and glue them together in weird ways ;)" or better "Manifolds are a way to get complicated spaces from simple spaces" --MarSch 2 July 2005 17:23 (UTC)}

^{Yes, you got my point right. But I don't find your suggestion very satisfactory. You suggest making things more and more abstract. That's why in the previous version of the introduction I wrote, I started with the topological manifold, obtained by patching together the well known Euclidean space. Once the reader understands that, you can say that instead of the Euclidean space you use a different space, then you get a different manifold. Oleg Alexandrov 3 July 2005 01:14 (UTC)}

^{The idea to get across is that manifolds provide a way to generalize something. Constructing complicated things from simple things. Not completely concrete, but hardly abstract. Constructing complicated topological manifolds from simple Euclidean spaces. Very concrete, but has lost all meaning. Perhaps you can give a better formulation. I remember that earlier you argued against the use of "topological space" in favour of "space" in the intro. Now you argue the opposite with the same argumentation. I'm arguing the opposite in both cases so there must be a difference, but think about it. What I think is the difference is that the earlier debate was about a sentence that sounded too much like a definition and I think that you shouldn't be vague in definitions. Many mathematicians use space as a synonym for topological space. Now we are arguing about a non-technical explanation. In non-technical explanations space refers to what is intuitively understood by a space. At least this is what I would want. The word gluing should make clear that this is an intuitive explanation, not a technical definition. I dislike instantiating the word space (which is intuitively clear) with something technical such as topological manifold or even Euclidean space (which have technical meanings). Perhaps you can write up some suggested formulations. --MarSch 3 July 2005 11:50 (UTC)}

^{What I forgot to say: The meaning of simple soace shouldn't be defined. Instead it should be intuitively clear what is meant. Something simple, anything, it doesn't really matter what it is. The word simple is what is important. Not the word space.--MarSch 3 July 2005 11:55 (UTC)}

^{OK. All I really mind is the tautology I deleted. By the way, the part about the 3-d sphere, Greenland, and Antarctica is unnecessary. That is explained right below, in the circle manifold example with the picture. I like that circle explanation more than the one with Greenland.}

^{And one more thing. Please don't use the exclamation sign in article. The ! should only be used for factorial notation. It is not as if you are giving a clown presentation in front of awe-strikken middle school kids. Oleg Alexandrov}

^{Obviously I do mind the deletion. I consider it essential information. If you don't have any suggestions for a better wording then it should be restored. The example you are referring to is perhaps out of place in the intro, or perhaps it needs to be integrated better. However the big difference with the full example of the circle is that it is much shorter and more intuitive. Once you understand this you will more easily understand the circle example. Also it is very concrete. Obviously the circle exmaple cannot be put into the intro--MarSch 3 July 2005 16:03 (UTC)}

^{I did some changes to the article at the points in question. The introduction is still too long I think. Oleg Alexandrov 3 July 2005 16:53 (UTC)}

^{"The simplest example of a manifold is the one made up of just one space." Which space, is the immediate question. How does this convey that you are talking about a simple space? Every manifold is one space. Your edit to my unencyclopedic tone is fine. Oh, I think the length of the intro is fine right now, but i don't really care about that. It lacks cohesion at the moment, with my gluing example in the middle. Let's await some more input. --MarSch 3 July 2005 17:02 (UTC)}

^{The current introduction is incorrect. The sphere is a manifold, no matter whether or not it is equipped with an atlas, just as a vector space is a vector space even if it is not equipped with a basis. A space does not have to be defined as being glued together in order to be a manifold. Furthermore not every space glued together is a manifold. If I clue a string to a plane, the result is not a manifold.}

^{A space is a manifold, if and only if it locally looks like a specific simple space. I agree that this is not understandable for highschool students and we should work to make it more understandable, but it must not become incorrect. Correctness is more important than simplicity. A difficult definition is hard to understand, a wrong definition is useless. Markus Schmaus 4 July 2005 15:32 (UTC)}

^{I do not think Markus is correct. A group is a set with a product, a vector space is an abelian group, a field, with an action of one on the other, and the manifold is a topological space with an atlas. Take one away, and you've just used the forgetful functor to move to a different category. You're no longer talking about a manifold, now you've just got a topological space. All textbooks give the definition of a manifold this way: space + maximal atlas. -July 5, 2005 16:07 (UTC)}

^{Well, it is an introduction, not a definition. I do not quite understand "The sphere is a manifold, no matter whether or not it is equipped with an atlas." If you define manifold by "it looks locally like a simple space," then you are also using charts, aren't you?}

^{However, perhaps we should indeed use the "looks locally like" definition instead of gluing. It is not that hard to explain that the sphere looks locally like a flat plane, with a reference to the surface of the Earth. By the way, I think we should replace paper by rubber in the gluing example (try wrapping a spherical gift in paper, even high-quality paper). -- Jitse Niesen (talk) 4 July 2005 16:29 (UTC)}

^{Yeah Schmaus is incorrect. There are topological manifolds that have different differentiable structures. S7 has 28 such. A topological space is a set with a topology. A sphere can be a point set, a top. space, a top. manifold, a diff. manifold and many more things. What I do agree to, is that for a given topological space, it either is or isn't a topological manifold. However if you just take a top. space as top. manifold, then how do you define what a diff. manifold is? --MarSch 6 July 2005 09:16 (UTC)}

^{It is possible to do differential geometry without refering to coordinates.}

^{The morphisms of the cathegory of topological manifolds are the same as those of topological spaces. In order define this cathegory, we do not need to define an aditional structure, but we need only to select those topological spaces, which we also consider as manifolds.}

^{The differential structure of differential manifolds is most often defined by pulling back the differential structure using an atlas (if possible). But it could also be done by embedding the manifold into some larger space, defining the tagent spaces in the natural way (if possible) and using those to define the differential structure on the manifold. The zeros of x2 + y2 + z2 = r2 form a differential manifold, the differential structure is clear and we don't need an atlas.}

In the end, when talking about a specific manifold, we do not refer to a pair of a topological space with an atlas, but to an isomorphism class of such pairs. The isomorphism class does not have a distinguished atlas other than the maximal atlas, but the maximal atlas isn't very usefull and I don't view a torus as being glued together from a maximal atlas.

An atlas of a manifold is usefull, just as a basis of a vector space is usefull, but not allways needed. Markus Schmaus 6 July 2005 11:34 (UTC)

The fact that real smooth manifolds embed is a theorem. This does not work for complex manifolds. Please tell me what a differentiable function from your embedded 2-sphere to itself is.--MarSch 6 July 2005 17:03 (UTC)

[edit] topological manifolds

for what it's worth, I think that topological manifolds are of minor significance, and can by missed in the introductory paragraph. when I say "manifold", I always mean "differential manifold". perhaps I'm being too closed-minded though. are there any interesting results about topological manifolds that I don't know about? Someone tell me something interesting about topological manifolds.

Anyway, even if topological manifolds can be interesting, they still have a distinct name: "topological manifold". other manifolds probably ought to be assumed to be differential manifolds. --Lethe | Talk July 2, 2005 08:19 (UTC)

differentiable manifolds are also topological manifolds. Just with differentiable transition maps. I think we should always try to be as clear as possible. While in common usage manifold may most often refer to diff. manifold, it is better I think to write it out. --MarSch 2 July 2005 13:15 (UTC)

[edit] examples

I've begun some work on the next few examples. Basically I've laid out some of the stuff to define it rigorously or alternatively prove that it is a top. manifold. Of course that is a bit formula heavy. Since this is supposed to be a non-technical article, this is probably not the way to go. However the technical stuff can be moved to top. manifold or to diff. manifold. Maybe someone else can give a less technical description a try? --MarSch 2 July 2005 17:19 (UTC)

[edit] Ruthless cuts

I agree with the "ruthless cuts". I had mentioned to MarSch again and again that the sphere discussion is unnecessary as it is explained in the circle example right below. Oleg Alexandrov 5 July 2005 16:07 (UTC)

And I didn't like much the "lots of glue" approach either. Oleg Alexandrov 5 July 2005 16:40 (UTC)

The new intro is pretty good. I think we should also mention gluing. Perhaps it can be fitted right in, where the maps overlap. Finding an atlas for a sphere is kind of the inverse to gluing. How do you get new manifolds? By gluing. How do you prove that something is a manifold? By finding what gluing was used, that is finding an atlas.

Oleg, don't you think you are exagerating(sp) as if hyperbole is nothing with the again and again? You said you didn't like the paper gluing and that is was redundant. That doesn't mean I agree or that I am deaf just because I don't jump up and down when you say so. --MarSch 6 July 2005 09:19 (UTC)

I like the changes by Jitse, too.

Cutting and glueing is an important concept for manifolds, not just glueing together charts, but also glueing handles to spheres or moebius strips, with border, into holes. I think it would be best to write a seperate section about cutting and glueing in the main part.

P.S.: 129.187.163.33 was me. Markus Schmaus 6 July 2005 12:12 (UTC)

I'm a big fan of conciseness and I'd like to keep the intro short. Of course, gluing is important and it probably should get a section, but I think it is too much to talk about both gluing and looks-locally-like in the introduction. But if you think that you can condense it to one sentence, put it in and we can discuss it. I find it rather hard to consider it in abstracto. -- Jitse Niesen (talk) 6 July 2005 20:10 (UTC)

[edit] Merging atlas (topology)?

Should we also merge atlas (topology) into this article? Markus Schmaus 6 July 2005 12:13 (UTC)

I would think that would make the article more complicated. I guess one can mention about atlasses, but refer to atlas (topology) for more details. Either way, importing from atlas (topology) wording of the form:

By definition, a smooth differentiable structure (or differential structure) on a manifold M is such a maximal atlas of charts, all related by smooth coordinate changes on the overlaps.

makes me a bit nervous. :) Oleg Alexandrov 6 July 2005 15:10 (UTC)

Actually I wasn't thinking about moving content, but redirecting atlas (topology) to manifold and describing and defining "atlas" and "manifold" in the same article. Is there anything about an atlas which should not be in an article on manifolds? Markus Schmaus 6 July 2005 16:50 (UTC)

I don't see anything in atlas which is not about manifolds. Please also note that I have added a section on atlases to the rewrite of differentiable manifold. I suggest you write first and then if atlas turns out to be superfluous we can redirect.--MarSch 6 July 2005 17:09 (UTC)

I am fine with merging contents, as long as things don't get way too technical. It was sort of agreed that manifold should be an entry article for differential geometry, so the really heavy math should go elsewhere.

So, again, you are welcome to merge the atlas here, as long as the contents is still somewhat readable by nonprofessional differential geometers. Oleg Alexandrov 7 July 2005 02:18 (UTC)

While we are on the topic of atlases, I have a question: what is the precise relationship between a sheaf taking values in topological spaces and an atlas? If we require the sheaf maps to be homeomorphisms on the open sets, do we obtain an atlas? Note that abstract varieties and schemes use sheaves, not atlases, for their "gluing" constructions. I could imagine defining a category of topological spaces in such a way as to induce the correct restriction maps. - Gauge 22:19, 10 July 2005 (UTC)

I really like Markus' writing about charts and transition maps. I was thinking that the latter sections on different types of manifolds could just be compressed into one section "Types of manifolds". They already have as much detail as I would want them to have in this article (elsewhere we can put all the sophisticated technical details). - Gauge 20:33, 11 July 2005 (UTC)

Agree. I mentioned the same thing above. Oleg Alexandrov 21:13, 11 July 2005 (UTC)

Great work on the atlas etc., Markus. About the sheaf. A sheaf is a functor from the category that is the topology with an additional property (gluing axiom). A scheme is a top. space with a sheaf of rings. A functor to the cat. of rings. So I am not sure what you mean by a sheaf taking values in topological spaces. Do you mean functions to a specific or arbitrary topo. space? Anyway from topo. mani.

"The continuous real-valued functions form a sheaf. An alternative definition of a topological manifold is a topological space with a sheaf of continuous functions locally isomorphic to Euclidean space with its sheaf of continuous functions."

All charts are in the sheaf of continuous functions and every homeomorphism is a chart (possibly require the range to be Rn). I'm not sure this is a sheaf, although it probably is. What happens to homeomorphism that coincide on some common domain? They couldn't be glued, cause the gluing would not be injective if the ranges are required to be Rn, but does this situation exist. Imagine an atlas for the cylinder consisting of two charts glued at both ends where they coincide such that they can be glued to form a homeomorphism of the entire cylinder to an annulus. The first map maps to the square ]-1, 1[^2. Lets stretch it in vertical direction with (x, y) |-> (x, tan(y (1 - |x|/2) \pi/2) ) with inverse (x, y) |-> (x, arctan(y)/(1 - |x|/2) \pi/2). Then these charts cannot be glued to an annulus. They must intersect somewhere and are thus non-injective and thus cannot be homeo's. --MarSch 17:32, 16 July 2005 (UTC)

Thanks for the response. I was wondering if there was a way to define topological manifolds just using sheaves instead of charts. It is not entirely clear to me what is meant by "a sheaf of continuous functions locally isomorphic to Euclidean space with its sheaf of continuous functions." Are we comparing the sheaves or the spaces here? - Gauge 18:13, 19 July 2005 (UTC)

[edit] Jitse's edits of 14 July 2005

I made some changes, but I'm not so sure whether it's a good idea, so I'm solliciting opinions:

• I moved the circle example to above the excellent section on atlases etc. The idea is that that section is rather abstract and it might help the reader if he has a concrete example in mind (however, we are writing an encyclopedia).
• I added some formulas to the example. I think the formulas are easy enough to be understood, though the one for the transition map needs a bit more explanation. On the other hand, it is often said that after every formula costs you half the readers in a popular science text.
• I made the picture that I actually had in mind when writing the example (though it can probably use some annotations). However, Oleg has already uploaded a picture some days ago, which is now included in the article, so I did not upload mine. You can see both pictures at [2]. I honestly don't know which to prefer: Oleg's is much simpler, but mine has some more information.

Awaiting your comments, Jitse Niesen (talk) 22:04, 14 July 2005 (UTC)

I think your picture explains better than mine what charts are all about. So I have no problem if you replace my picture with yours. Oleg Alexandrov 22:09, 14 July 2005 (UTC)

Actually I think the pictures compliment eachother. I would like both in the article for maximum clarity. Of course their colors need to be coordinated.--MarSch 17:35, 16 July 2005 (UTC)

The text of the circle example is badly out of sync with itself and some formulas. Will fix when final colors of the charts are established.--MarSch 17:46, 16 July 2005 (UTC)

[edit] (the?) Euclidean space

I think we should say for example "homeomorphic to Euclidean space" and not "homeomorphic to the Euclidean space". There is a Euclidean space for each natural number. Similarly "the" should be removed from other sentences.--MarSch 17:40, 16 July 2005 (UTC)

[edit] colors of circle charts

Currently the text doesn't speak about yellow but instead about brown. I would call the left chart yellow. If it is actually some light version of brown, then it should be changed to yellow. In the bottom picture with all charts at once, the top and bottom charts (red and green, or brown?) are difficult to distinguish, because of their thinness. It would help if their thickness was increased slightly to match that of the top picture. Personally I can distinguish the top(red) and bottom(green) charts in the top picture. I don't know about other colorblind people, but an alternative would be to have blue, yellow, black and a fourth chart which is red, green, brown or any desirable mixture of these.--MarSch 08:46, 18 July 2005 (UTC)

I have to say that I did not think hard about the choice of colours and I didn't take colourblindness into account. I assume we shouldn't use both red and green? The problem with black is that the circle itself is already black. So, we can use blue and yellow, and then we need two other colours. What about red and grey?

Oleg, it would be useful if you could include the Matlab source of the pictures you make, so that they can be edited mercilessly beyond recognition, see for instance Image:Sphere with charts.png.

I do not like all the formulas that MarSch put in, especially the pr_1|S_red bit. As I wrote above, I was reluctant to include any formulas at all, as they tend to scare quite some people away. -- Jitse Niesen (talk) 12:11, 18 July 2005 (UTC)

I usually post the code, but that one was a dirty hack. I hope to clean the code up today, and modify the picture according to the suggestions.

I vote to cut out MarSch's formulas. That's the first example after all, no need to start with the heavy stuff. Oleg Alexandrov 15:32, 18 July 2005 (UTC)

I uploaded the picture with thicker lines, and the source code. How do you like the yellow? Oleg Alexandrov 02:16, 19 July 2005 (UTC)

The formulas mentioned above do seem excessive, especially for an introductory example. When I was new to this stuff I found such formulas intimidating at first. - Gauge 18:17, 19 July 2005 (UTC)

The formulas are now gone. Oleg, I'm not sure it's worth the effort to document the Matlab source; the source that I upload is always without comments. -- Jitse Niesen (talk) 19:01, 19 July 2005 (UTC)

I must say I hate the yellow. I vote to go back to brown to be consistent with Jitse's picture right above mine. Oleg Alexandrov 21:09, 19 July 2005 (UTC)

Aye, it's a wee bit on the bright side. I think the brown in my picture is not a pretty colour either, but it is a bit clearer (at least to me) than the yellow. I have the same problem when writing overhead slides: after I've used red, black, blue and green, I seem to run out of colours... -- Jitse Niesen (talk) 22:15, 19 July 2005 (UTC)

In illustrations for other articles I've used the opponent colors shown to the right. I give the sRGB values on the description page, and also in another section here, but thought it might be helpful to see the appearance. These values don't have good separation for common types of color-blindness, so it's best to use them in a supplementary role. For example, use left, right, up, down instead of naming colors in the text. The linguistically "universal" colors in addition to these are brown, orange, purple, pink, and gray. There are so many conflicting considerations to juggle I have never settled on an ideal set. If you want to check an image for color-blind safety, one option is Vischeck. Another quick and easy check is to convert an image to grayscale. KSmrq 19:50, 2005 July 22 (UTC)

[edit] Construction

I've started a section on the construction of manifolds. So far I presented the constructions "zeros of a function" and "gluing charts", but there are at least three more constructions I'd like to include, "gluing manifolds with bounderies", such as gluing handles or moebius strips to spheres, "quotient manifolds", such as R²/Z², the torus, and the "carthesian product" of manifolds. Markus Schmaus 19:47, 19 July 2005 (UTC)

Excellent! I was just thinking about writing on constructing the circle with gluing, so I can do something else now. However, I am foreseeing one problem with the article: all the examples are in fact differentiable manifolds, so we have to be careful that it doesn't end up being an article about differentiable manifolds (or perhaps this is not such a bad thing). Are these constructions also valid for more general manifolds? Unfortunately, the little I do know about manifolds is all differentiable manifolds ... -- Jitse Niesen (talk) 20:31, 19 July 2005 (UTC)

[edit] Large table of contents

I removed subsection status on some of the headings just to make the table of contents a bit smaller. In particular, not every example requires its own subsection. I also implemented the "types of manifolds" idea to reduce clutter, and indented a couple of the examples to make them stand out from the rest of that section. - Gauge 04:32, 20 July 2005 (UTC)

I don't like using only boldface to indicate a subheading, as this removes structure. This article is much from finished and I don't think we should care about the length of the table of contents now. Markus Schmaus 16:16, 23 July 2005 (UTC)

[edit] Intro again

[Moved to new section by Markus Schmaus 19:38, 20 July 2005 (UTC)]

Wikipedia philosphy says be bold, so I have written a completely new introduction. The opening sentence was miserable, if really intended for students with little experience. The word "space" is used in a technical sense familiar to mathematicians, but to no one else. The phrase "specific simple space" is impossible to read aloud without sounding like a snake, and is so vague and general as to be useless. Far from enticing me into the article, it repelled me — and I like this stuff! Try to imagine yourself a bright, curious mind of 16 years, exploring Wikipedia to learn more mathematics than your pathetic school provides. Which style of intro would you find more compelling?

In the same vein, the motivational example of the circle throws up unnecessary obstacles, though I have not yet tried to rewrite it. We're talking about a circle, folks; do we really need to use notation for sets, ℝ², and "such that"? For motivation? I think not. Can we also trim some of the notation from the compatibility maps? I think so. As a minor note, better colors for the picture might be sRGB #bc1e47 (red), #fec200 (yellow), #009246 (green), #0081cd (blue). These are taken from cross-cultural studies of the psychophysics of human vision, the theory of opponent colors. The term "yellow" would then almost surely find agreement. Better still, use "top", "bottom", "left", "right" in the text.

I have left the previous material in the article rather than delete it, but it needs to be modified if retained. Happy editing. :-D KSmrq 01:01, 2005 July 20 (UTC)

Thanks for your edits. I think the new intro is much better. - Gauge 04:36, 20 July 2005 (UTC)

Much appreciated. On a fine point, I wrote "rubber band" which you changed to "closed loop of string". I deliberately chose the former over the latter as I wrote, because a rubber band is seamless, but we need a knot to close the string. And, yes, I know I later dangled a string from a sphere — but without a knot. So I'd prefer to go back to "rubber band", if that's internationally acceptable; or perhaps we can find a better universally understood real-world example. KSmrq 05:08, 2005 July 20 (UTC)

The reason I don't like rubber band as an example here is that it is not very 1-dimensional. A rubber band can look more like a cylindrical strip which is not locally 1-dimensional everywhere. I chose "closed loop" of string to indicate that it is already closed - no knot is needed. I suppose the only problem is that string isn't manufactured in closed loops  :-) - Gauge 03:08, 21 July 2005 (UTC)

We have skinny rubber bands and thick string. Either way, we're lying. :-) KSmrq 08:45, 2005 July 22 (UTC)

If the opening sentence was misserable it had to be changed. "Generalizes a surface" is good, but "is like a patchwork quilt" is wrong. Though it is possible to construct a manifold as a patchwork quilt, that's only one possibility among others. Even though each manifold admits an atlas, a topological space is a manifold regardless whether an atlas is specified or not. "Looks simple from a closeup view", on the other hand, is true for any construction. I like the second paragraph. But the introduction is too long. Dimension should get a section on its own and only clutters the introduction. That's also the case with applications. There's a subtle error in the fith paragraph, only conected manifolds have to be the same dimension everywhere. That a string connected to a plane is not a manifold is not a special case, but simple due to the fact that the point where the string meets the plane, the space isn't simple.

Carl Friedrich Gauss and Bernhard Riemann, who are mentioned in the history section as the the founding fathers of manifolds, were German, if one of them coined "Mannigfaltigkeit", I doubt it "traces back to Old English". So could you please tell me who first introduced the term manifold/Mannigfaltigkeit.

The two important facts about manifolds are:

• A manifold is a space. It does not have to be embedded into a larger space, but it is a space of its own right. The very own space we're living in is a manfold (at least according to general relativity).
• When only a small part of the manifold is regarded, it looks simple.

Markus Schmaus 20:04, 20 July 2005 (UTC)

Quoting Markus: Even though each manifold admits an atlas, a topological space is a manifold regardless whether an atlas is specified or not

I thought that MarSch already pointed out that this is incorrect. The same topological space can give rise to different manifolds, depending on the atlas one chooses. See exotic sphere for more information. - Gauge 03:14, 21 July 2005 (UTC)

I thought that I made it clear that MarSch missed the point. A topological space is a topological manifold and there's only one topological 7-sphere. Its properties do not depend on an atlas.

Even though there are 28 differential manifolds, which are as topological spaces homeomorphic to the topological 7-sphere, there is only one topological 7-sphere. Markus Schmaus 12:32, 21 July 2005 (UTC)

I believe that this is incorrect. Where is the proof that two homeomorphic topological spaces with distinct atlases are necessarily isomorphic as topological manifolds? From what I understand, morphisms in the manifold category depend not only on the underlying topological spaces but also on the atlases. For example, for f: X → Y to be a morphism between differentiable manifolds, we require that at each point x ∈ X there is an X_i ⊂ X and a Y_j ⊂ Y such that x ∈ X_i and f(x) ∈ Y_j with chart morphisms (homeomorphisms) φ_i: U_i → X_i and ψ_j: V_j → Y_j such that φ_i⁻¹(X_i ∩ f⁻¹(Y_j)) is open and the composite ψ_j⁻¹fφ_i: φ_i⁻¹(X_i ∩ f⁻¹(Y_j)) ⊂ Rⁿ → R^m is differentiable at φ_i⁻¹(x). For topological manifolds, the latter maps have to be continuous instead of differentiable. In either case, the morphisms depend on the chosen atlas. Therefore a topological manifold is only determined from a topological space once you choose an atlas, unless you can prove that any two topological manifolds with the same underlying topological space are isomorphic in the manifold category using morphisms like the above. Comments anyone? - Gauge 18:32, 21 July 2005 (UTC)

Let M be a manifold with atlas (φ_i: U_i → X_i)_i∈I and N be a manifold with atlas (ψ_j: V_j → Y_j)_j∈J, such that M is homeomorphic to N as topological space and let f be that homeomorphism.

For all i in I and j in J

φ_i and ψ_j are homeomorphisms,

U_i, X_i, V_j, and Y_j are open subsets,

and since f is a homeomorphism

f⁻¹(Y_j) is open.

and as ψ_j, φ_i, and f are homeomorphisms, the composition

ψ_j⁻¹ f φ_i

is also a homeomorphism.

Hence M and N are isomorphic as a manifold. Markus Schmaus 19:31, 21 July 2005 (UTC)

Thank you for the proof. This is very interesting. Perhaps this fine point could be mentioned in the article on topological manifolds? Best, Gauge 20:13, 21 July 2005 (UTC)

Markus: Thank you very much for your careful consideration. I'm glad you liked at least parts of it! I hope you can also appreciate the point I'm trying to make with the whole opening. We can nitpick specific words, phrases, or examples, or what should be included and what omitted; but if we claim to be talking to a lay audience, not specialists, then we must radically depart from "mathematics-speak". So, let's see how we can make it better still.

I believe "like a patchwork quilt" is acceptable. I don't care, and don't discuss, how the manifold is constructed. Maybe you found one in your backyard, and are curious if it is a manifold. We have pieces, we have overlap, we have consistency; that's "like a patchwork quilt". It is a tangible, common, real-world analogy, an overview to build intuition for the detailed mathematics to follow. Is there a better one? Let's hear it!

For anyone who has seen differential calculus, "looks simple from a closeup view" will be a familiar idea, and it is a valuable perspective on manifolds. But we're not interested in closeups alone; in many applications it is vital to connect the individual closeup views. Another pitfall is that "simple" for you may be anything but simple for a young student. Mathematics is replete with cautionary tales of "simple", "obvious", and "trivial"; I sometimes use these words, but they make me nervous.

I think dimension and applications are important to include. John Q. Public has never seen a manifold before, has no idea what we might be talking about. We say "generalizes a surface". Fine; only how? What is there about a surface that can be generalized, or that needs to be generalized? I have seen a miserable tradition in mathematics that presents a long string of definitions and proofs with the confidence that the student will be content to assume there's a point to it all. It does not work well even for graduate students, and is hopeless for the general public. An introduction must give a preview of what's to come, and why. Hence dimensions and applications.

Is the intro too long? I'm all for brevity, though I find it hard work to accomplish. But consider that for many readers the intro may be all they read. It cannot make them a technical expert, but should at least prepare them for cocktail parties. :-)

I'm troubled by your assertion about dimension. All the manifolds I've worked with were locally homeomorphic to ℝ^d, for a fixed dimension d. For many applications it is vital to have d fixed. I'll quote two definitions chosen from many similar examples:

1. A topological (C⁰) manifold is a separable Hausdorff space such that there is a d-dimensional chart at every point. The dimension of the manifold is the same as the dimension of the charts. Thus there is a collection of charts {μ_α : U_α → ℝ^d | α ∊ I} such that {U_α | α ∊ I} is a covering of the space. — Bishop, Goldberg. Tensor Analysis on Manifolds.
2. A topological n-manifold (without boundary) is a paracompact Hausdorff space in which each point has an open neighborhood homeomorphic to ℝⁿ (called a coordinate neighborhood in the manifold). — Spanier. Algebraic Topology.

About origins: Point taken. Let me be more precise. The etymology of the English word "manifold" traces back to Old English. Old English is a Germanic language, and Indo-European. I don't know first-hand the actual history of who in mathematics first chose the term, nor do I know why. Jeff Miller's site says:

• MANIFOLD was introduced as Mannigfaltigkeit by Bernhard Riemann (1826–1866) in Grundlagen für eine Allgemeine Theorie der Functionen, published (posthumously) in 1867, Werke p. 3 [Mark Dunn].

However, I believe the German and English are roughly equivalent, and may even share a common etymology. Since I am writing in English, I gave an English derivation. I find that abstract mathematical terms are more meaningful when linked to non-mathematical ideas. This is my attempt to make such a connection. My apologies if I offended Riemann!

About important facts: I give general relativity as an example, and nowhere introduce embedding. In fact, SO(3) is used as an example of an abstract space. Do you feel we should add or change something? If so, what? As for locally "looks simple", the challenge is to say something that will be gutsy and meaningful for a lay reader. The two mathematical definitions I quoted above do a fine job for the cognoscenti, and support your view. We clearly [sic] cannot use them in the intro! I'm also thinking of the bigger picture, such as sheaves and schemes, where treatment of overlap is a recurring theme.

Let me be clear. I replaced the old intro to make a vivid point, to demonstrate the kind of prose needed for a general reader. If that sensibility is retained, I have no objection to my edit being completely discarded in favor of something stronger. All I ask is that we remember who we're addressing: not fellow mathematicians, but curious web surfers and bright kids — maybe future mathematicians. We want to engage, inspire, inform, not terrorize, not anesthetize. KSmrq 05:15, 2005 July 21 (UTC)

Easy issues first. I'm a native German speaker. "Mannigfaltig" in German doesn't mean "many times", but rather "multifaceted". It certainly doesn't derive from a single point represented in many charts, but rather from the many facets of a manifold.

If I understand you correctly, the surface article you linked to not what you had in mind. It states a surface is a two dimensional manifold, which is correct in some sense, but a surface is most often defined to be the twodimensional equivalent of a curve, a map from R² into space.

Manifolds generalize surfaces in several ways. First, manifolds allow for higher dimensions. Second, a surface has to be embedded into some space, a manifold doesn't have to. Third, we can also use other spaces but Rⁿ as the base space, for example a half space, which gives rise to manifolds with boundaries.

The reason I object to the quilt model of a manifold is becaus it gives rise to statements like the this:

The consistency of manifolds is a strong demand. For example, we cannot dangle a string (a 1-manifold) from a sphere (a 2-manifold) and call the whole a manifold.

Why is a string connected to a sphere not a manifold? According to statement, that's due to ominous consistency conditions. They are not specified and are probably quite complicated.

But in fact the answer is much easier. At the point where the string meets the sphere the space isn't flat. Markus Schmaus 14:02, 21 July 2005 (UTC)

It can be helpful that German is your native tongue. I will curb my frustration when my English is misconstrued. The word attributed to Riemann is "Mannigfaltigkeit", the noun form. Either way, the sense seems to be a little ambiguous, but very much consistent with the English, as in "many", "diverse", "varied". My exact words are "referring to the many pieces used." I did not say "represented in many charts," nor did I mean to say that. Unfortunately, your edit weakens the intro. I previously said that it is important to tell readers why we use the word manifold. Let me elaborate. English-speakers most frequently encounter the word as part of their automobile, which is nonsense here. Nonsense is hard to remember. I would be delighted if you could find original words by Riemann explaining why he chose the term!

It is unfortunate that the current Wikipedia surface article defines a surface as a 2-manifold. I'm not happy with the current state of that page, but it is still the correct link. Did you mean to imply that every surface is a parametric surface? It is well-known in algebraic geometry that most implicit surfaces have no parametric form. (Surely Gauss accepted a sphere as a surface, and did not need to call it a manifold.)

I'm not sure what point you are making with your discussion of how manifolds generalize surfaces. I believe I have already agreed with each fact you state, and each already plays a role in the intro.

What is "ominous" and "not specified" and "quite complicated" about having the same dimension? (And, please, don't characterize my statements; that tends to replace reason with emotion, surely not what we need here.) Do you disagree with all the reputable mathematicians who expect consistency of dimension? You do not rebut my quotations, and I would very much like to know of a more relaxed definition if one exists. It is true that the point of attachment is a problem in itself, but that's an unnecessary diversion. The string and sphere could be separate components and the argument I explicitly give would still be valid: different dimensions. Although a manifold need not be connected, I feel it would be bad exposition to introduce that complication at this point in the intro. Thus I dangle the string from the sphere.

It is tempting to discuss (in lay terms) the idea of every point having a neighborhood homeomorphic to ℝⁿ, but I resist the temptation because it would make the intro longer, and because I don't see how to do it without getting technical. We get to it soon enough once we reach the formal part of the article.

I greatly appreciate your interest in working toward an excellent intro, and I hope it will continue. However, there is still an enormous amount of work to do in the body of the article. Once that is complete, the role of the intro will be more clear, and we can be sure that everything that must be said has been said. I apologize if my rewrite of the intro has diverted attention from that bigger task; perhaps it was premature. KSmrq 08:45, 2005 July 22 (UTC)

Thanks to KSmrq for his efforts. The new intro is generally a nice read, but I am not easily satisfied. Here are my reactions on the new intro and some of the points raised above:

1. I like the first three sentences very much, including the patchwork quilt "definition", though I wonder whether the patches in a patchwork really overlap. That it does not include the intrinsic view, which is a bit more advanced, is acceptable to me.
2. The "competing" description, of a manifold looking like a simple space from closeby, is not so easy to understand. The word space is indeed liable to be misunderstood by non-mathematicians and the word simple needs to be qualified in some sense (perhaps "relatively simple"?).
3. I am not so sure what the fourth sentence ("Benefits include local convenience and global consistency, balancing flexibility with familiarity") tries to say.
4. In some books, the definition for manifold does not require the dimension to be the same. I have no books at hand to be precise (I'll go to the library if KSmrq wants), but this is the case in the lecture notes for my undergrad courses. which state (my translation, note also that this definition does not a priori assume that a manifold is a topological space)

   X is a nonempty set. A chart of X is a triple (U,φ,E) with U a subset of X which is mapped bijectively by φ on the real finite dimensional linear space E. Two charts are compatible if they map the overlap to open regions in the linear space, and the transition maps are continuous. An atlas is a set of compatible charts covering X, and a manifold is a set admitting an atlas.

5. I agree that the Old English etymology is misleading. It seems to me that the mathematical meaning of manifold simply derives from the nonmathematical meaning, as used for instance in the Bible: "O LORD, how manifold are thy works! in wisdom hast thou made them all: the earth is full of thy riches." (Psalm 104:24, King James Version). By the way, a completely different word is used in Dutch (variëteit) and French (variété).
6. The introduction is too long to serve as a lead section (the bit above the first heading), which is typically one or two paragraphs. So it seems logical to make a first section, called Introduction, put the material in there, and retain only a part in the lead section. Unfortunately, I could not find a clean way to separate one or two paragraphs from the introduction.
7. I think we should cater for the people looking for a more technical definition for a topological or differentiable manifold by placing a comment right at the top. Now, they have to go way down before they find anything exact.
8. KSmrq, do you really think that the formulas in the circle example are an obstacle? I did have my misgivings about it when I added them (see discussion above), but I tried to imagine myself "a bright, curious mind of 16 years," and I concluded that I wouldn't have troubles understanding them at that age (at least the formula for a circle as a set, the one for the transition map should be explained better if it is to be retained).

That's all for now. Happy editing! -- Jitse Niesen (talk) 11:20, 22 July 2005 (UTC)

Further on the etymology, following KSmrq's edit, I looked in the Oxford English Dictionary. It includes dozens of spellings of manifold in Old and Middle English and confirms that it derives from Riemann in the mathematical meaning. The principal meanings are listed as "Varied or diverse in appearance, form, or character; having various forms, features, component parts, relations, applications, etc.;" and "Numerous and varied; of many kinds or varieties." (both tagged obsolete). -- Jitse Niesen (talk) 12:26, 22 July 2005 (UTC)

Regarding dimension, I think that most mathematicians would agree that they expect manifolds to have the same dimension everywhere. I quote from dimension#Manifolds: "A connected topological manifold is locally homeomorphic to Euclidean n-space, and the number n is called the manifold's dimension. One can show that this yields a uniquely defined dimension for every connected topological manifold." In the case of more than one connected component, I think it is reasonable to say that allowing for the dimension to vary between the components is not what is expected and that it unnecessarily introduces extra technical complications.

I like the patchwork quilt analogy because it gives a newcomer a visual starting point. The inaccuracies can be clarified later. This is a standard pedagogical approach. I do think that we need to split the lead section somehow so that the table of contents is not pushed down so far as it is now. I would prefer to have at most four sentences before the toc, and the rest of the introduction below. - Gauge 18:24, 22 July 2005 (UTC)

---

Thanks for the kind words, Jitse and Gauge. It seems that Markus disagrees, so passionately that he has essentially reverted, only worse. It is not a good use of my time to discuss or edit the intro further, at least not now, and certainly not in a revert war. But no hard feelings; I like to hope my efforts to date will eventually help make a better article.

About the circle example: Yes, I do think S := { (x,y) ∊ ℝ² | x² + y² = 1 } is an obstacle. A circle itself is not mysterious, but the notation assumes, for example, that the reader knows what ℝ² means. An alternative is:

Consider a circle defined as the set of points in the plane at a distance of one unit from the center. If x and y are the coordinates of such a point, then we have x² + y² = 1.

I'd like to rely less on coordinates, but this version would present a lower barrier. And it gives the essential equation needed to derive the transition maps. I suppose I was spoiled by the old Scientific American magazine, where equations were forbidden. I came to imagine that mathematics could be discussed without them. :-)

Incidentally, we could use a rational circle, x=(1−t²)/(1+t²), y=2t/(1+t²). Negations and permutations give the other charts. It makes the transitions rational, but I doubt it's worth the unfamiliarity. Still, if we wanted to show a two-chart circle, this would be the way to go. (The other chart would negate x, with transition t ↦ 1/t.)

Were I to come back to this, I'd first look to see if the big picture was complete. Then I'd see how it "felt", if it was pleasant to read, or better yet, enticing. The last check would be for accuracy. Keep in touch, and let me know if you'd like me to take another look. Meanwhile, I've got some gaping holes to fill elsewhere, and a new specialized wiki to help sort out. KSmrq 22:42, 2005 July 22 (UTC)

I must say that I disapprove of Markus' latest edit that essentially removed KSmrq's introduction in favor of an entirely new one. You probably can't convince me that "The global structure of a manifold can be quite complicated, it may bend and even wrap. But on a small scale it has to be plain and flat" is better than the patchwork quilt analogy. I am disappointed that algebraic varieties and schemes are included as manifolds when the point I have explained below has not yet been addressed. The separate etymology section should appear under "History" somewhere... not in the introduction. Also I do not agree with the comment "We have a section on atlas and charts surpassing atlas. When this article is finished, atlas should simply redirect". This is not what was the consensus as I understood it: instead, the technical details of the atlas were supposed to belong to the atlas article, and this article should avoid technical details where possible. I advocate reverting the introduction to the previous one, while preserving any new information Markus may have added. Is everyone with me on this? - Gauge 02:25, 23 July 2005 (UTC)

And you probably can't convince me that a "patchwork quilt" is good description for a sphere.

I don't think my version is very good, but after our short discussion on whether a topological manifold needs an atlas, I was more convinced than ever, that saying a manifold is a patchwork is a bad idea.

But how about "a manifold can be consistently covered by multiple overlapping pieces". I don't think that's perfect but something along that lines could be a compromice.

Technically, you are saying a manifold is a space together with an atlas, while I insist, that an atlas is not essential, which I proofed above. But I do agree that any manifold admits an atlas, and I'd be happy with saying a manifold is a space, which admits an atlas. Markus Schmaus 16:10, 23 July 2005 (UTC)

I don't understand why you don't agree with "patchwork quilt", but willing to accept "covered by multiple overlapping pieces". Furthermore, what you proved is that for a topological manifold, it does not matter what atlas it has. This does not hold for differentiable manifolds, and even for topological manifolds having an atlas is essential because otherwise it is not a manifold (or perhaps you have a different characterization of topological manifolds in mind which does not rely on atlases).

I agree with Gauge that atlas should not direct here. In fact, I would remove the part on maximal atlases. The sentence "The global structure of a manifold can be quite complicated, it may bend and even wrap" is fine with me, but I'm not a fan of the next sentence, "But on a small scale it has to be plain and flat". In fact, I don't quite know what you want to say with "plain" (first I thought you meant "plane", but then I got so confused that I removed the word in the end) and "flat" (which is rather misleading methinks). -- Jitse Niesen (talk) 16:39, 23 July 2005 (UTC)

I'm not willing to accept "is covered" but "can be covered".

The characterization of a topological manifold as locally homeomorphic to Rⁿ does not depend on picking an atlas, even though this entails that a topological manifold admits an atlas. Just as being a vector space does not depend on picking a basis, even though any vector space has a basis.

Defining the sphere as the zeros of x² + y² + z² makes the sphere a differential manifold without relying on an atlas.

I dug a little bit around and discovered diffeology. I have never heard of it before, but it is another way to define a differential manifold not using an atlas. Markus Schmaus 17:29, 23 July 2005 (UTC)

From the diffeology article: "Differentiable manifolds also generalize smoothness. They are normally defined as topological manifolds with an atlas, whose transition maps are smooth, which is used to pull back the differential structure" (emphasis mine). The point is not whether differentiable manifolds can be defined without an atlas (I already pointed out that they can also be defined with sheaves), but that the classical definition that most mathematicians are still using is the atlas definition. Markus is correct that the choice of an atlas for a topological manifold doesn't matter, but to be a manifold I will contend that it must have some atlas. Defining the sphere as a set of zeroes is implicitly embedding the sphere into another space. It has already been pointed out several times that a topological sphere does not have a unique differentiable manifold structure. How is the differentiable manifold structure induced by the embedding, and (more to the point) how can one understand the corresponding differentiable manifold structure on a topological sphere without mentioning this embedding? Topological spaces are not equipped with embeddings a priori. That a sphere with a differentiable manifold structure has a topological embedding into euclidean space does not at all imply that the differentiable structure was induced by this embedding. - Gauge 19:35, 23 July 2005 (UTC)

If this were an article on differential manifolds I would agree, but this is an article about manifolds in general. Being a manifold is property of a topological space, just as being abelian is a property of a group, but neither requires additional structure. So you don't have to pick some atlas for a topological manifold.

Yes, defining the sphere as a set of zeroes is implicitly embedding the sphere into another space and a topological sphere does not have a unique differentiable manifold structure. But this embedding induces a unique differential structure. A tangent vector at a point x is an equivalence class of curves through that point which entirely lie in the manifold and have the same differential in Rⁿ. Differentiating a function along this vector is differentiating the composition of the function with a representative of this vector, if possible. If a function on the other hand thus defines a linear map on the tangent space at the point x, it is differentiable and the linear map is its differential. A function is differentiable if it is differentiable at each point.

For the 7-sphere this differential structure is the same as the differential structure You get when using the atlas, which is mentioned in the n-sphere section, for pulling back the differential structure of R⁷. And this is the differential manifold I'd consider the differential 7-sphere, even though there are other differential manifolds homeomorphic to it.

It is possible to understand the corresponding differentiable manifold structure on a topological sphere by using an atlas without mentioning this embedding and it is possible to understand this structure using the embedding without mentioning this atlas. And maybe it is possible to understand this structure using diffeology mentioning neither. Markus Schmaus 20:45, 23 July 2005 (UTC)

I cannot agree with your statement "Being a manifold is property of a topological space, just as being abelian is a property of a group, but neither requires additional structure" as I have never seen a definition of topological manifolds that does not include the atlas condition. If you could put out an authoritative reference that says this I would be interested. I understand your point that manifold structure doesn't have to be defined by an atlas. I still prefer the atlas definition as the primary definition because it is the one that appears in most (if not all) the references I've seen, but perhaps as a compromise we can mention somewhere in the article that manifolds can also be defined by embeddings, sheaves, diffeology, or whatever else. I don't think we should attempt to explain what these are (except maybe the embedding) in this article, but we can at least mention them. - Gauge 23:12, 23 July 2005 (UTC)

Besides, you don't have to start from a topological space. You can also defined a manifold as any set admitting an atlas, and then the atlas will define the topological structure. -- Jitse Niesen (talk) 23:28, 23 July 2005 (UTC)

[edit] Varieties and Schemes

I am concerned that the article lists varieties and schemes as examples of manifolds. Although the definition of a manifold is not explicitly given in the article, most definitions that I've seen make use of an atlas. What is the corresponding atlas that defines a scheme or variety as a manifold? I noticed that Planetmath has a sheaf-theoretic definition of a manifold here. - Gauge 20:28, 21 July 2005 (UTC)

[edit] KSmrq's introduction

I really liked KSmrq's introduction. I am surprised it is gone now. This article is again slowly becoming the same abstract mess which necessitated a rewrite to start with. Oleg Alexandrov 17:47, 23 July 2005 (UTC)

I agree. KSmrq's introduction had some quirks to work out, but I think it was far better than what we have now. I thank Markus for his input (especially on etymology) and his attempt at a fresh introduction, but I think it would be better if we could go back to KSmrq's introduction as something that we are all willing to work with. With your permission, Markus, could we please go back to the old introduction? I will try to work with you on the various problems you had with the previous introduction. - Gauge 19:48, 23 July 2005 (UTC)

When I first read KSmrq's introduction I thought about instantly reverting it. There were several issues I disagreed with, from the etymology to the patchwork. Mostly subtle points, but those are the worst as they are hard to spot and hard to overcome. I left it in the hopes those issues could be solved. But your statement "From what I understand, morphisms in the manifold category depend not only on the underlying topological spaces but also on the atlases." convinced me that KSmrq introduction gave the wrong impression of manifolds.

As I stated before I find correctness more important than simplicity, what good is an simple explanation if it explains the wrong concept. I'd rather start with a correct statement and try to express it more simple while retaining correctness than start with a simple statement and try to reintroduce correctness. Markus Schmaus 21:14, 23 July 2005 (UTC)

Done! I hope. There was an edit conflict and then the server didn't respond and it looks like a triple update. But nevermind.

Details: I found a little time again, so I read Riemann's lecture and also searched for better similes. Since Markus is really attached to the local simplicity idea, and I somewhat agree, I tried to work that in. For lack of a better idea, I've shortened the lead by moving all but the first two paragraphs to the retained Introduction section, which now needs work.

I did an informal probe of the first person I met, and sure enough, the immediate reaction to manifold from a non-mathematician was a part of an automobile; even when prodded, he only vaguely recalled the "many" sense. So I have used a sentence which I am now fairly sure agrees with Riemann (having just read him), but which helps the term stick to the ribs.

The etymology section had some nice observations, though the English language was slightly bruised in the process. (I'm not complaining; my German would be far worse!) I have cleaned it up and appended it to the history section, which seems more appropriate.

I am reminded once again how nice it can be to read original sources. To see Riemann himself trying to find ways to explain this new way of looking at space(s) to an audience of mathematicians, well it convinces me we are right not to use "space" in an opening line for a lay audience.

To my eye, this lead is stronger than any of our previous attempts. I hope we agree enough to work from here rather than reverting. Of course, there remains good material in the section now labelled "Introduction" that needs a cleanup.

Unfortunately, my little window of time is closing again. I'll try to respond to any comments in my next window. Happy editing. KSmrq 21:38, 2005 July 23 (UTC)

This article is becoming too complicated. Markus, no offence, but you remind me of a fellow who has the same initials as you, and that is MarSch. Why is the cartezian product of manifolds written before even the simplest example of the sphere is described? Why is the torus mentioned as a product manifolds before even it was explained that this is a manifold (see below in the example section, still to be written). Oleg Alexandrov 21:27, 23 July 2005 (UTC)

I agree that the order at the moment is not ideal, but we can think about the correct order later, and perhaps discuss whether some parts need to go to different articles. -- Jitse Niesen (talk) 22:22, 23 July 2005 (UTC)

KSmrq: those are some great edits. I am enthusiastic about the new introduction. The only nitpick that I have so far is maybe we could use shingles instead of fibreglass or yarn? I don't want the charts to be too thin  ;-) Btw, sorry about the recent strange edit burst from me. The server was timing out on me. - Gauge 23:22, 23 July 2005 (UTC)

Well, the sphere is described twice before the torus and a torus is a manifold because it is the product of two manifolds. We don't have to look at every aspect of a sphere or a torus, this can be done on sphere and torus.

I agree that the whole section on construction is very difficult and cartesian product is very techincal, but we should mention this somewhere. Markus Schmaus 18:11, 24 July 2005 (UTC)

I see your point. But still probably it is better to introduce first the torus as itself, and only later, after discussing the product of manifolds, give the (now described torus) as an example of a product manifold. By the way, my tone was not the most constructive, sorry. And about the product manifold again, I will not mind if at some point it gets its own article, with this article just mentioning briefly what is going on and referring to that one. Let us see how it goes. Oleg Alexandrov 20:31, 24 July 2005 (UTC)

No problem, lately, I wasn't allways constructive myself.

One reason I used the formulas was because cartesian product did not contain the cartesian product of functions. I added this section today, now I can simply say that the cartesian product of charts is a chart for the product manifold. Markus Schmaus 23:15, 24 July 2005 (UTC)

[edit] Counterexamples

"Similarly, a cone is not a manifold because the tip of the cone looks different". This sentence seems to end too early but I couldn't find a way to fix it. The cone may not be the best example because of the ambiguity between bounded cones (with a base) and unbounded cones (like a light-cone in relativity). I would like to say something like "it looks different from the rest of the space", but with bounded cones you have this corner at the base to deal with. - Gauge 05:12, 25 July 2005 (UTC)

I agree. I had to finish quickly. The other problem is that the (unbounded) cone is a topological manifold (I think), but not a differentiable manifold. My reasoning was twofold (but not manifold): we should also give some counterexamples, otherwise it seems that everything is a manifold, and I did not like the counterexample of the cylinder precisely because of the ambiguity between bounded cylinders (which are only a manifold with boundary) and unbounded cylinders (which are bona fide manifolds). -- Jitse Niesen (talk) 09:53, 25 July 2005 (UTC)

[edit] Etymology

KSmrq, I like your merge and in general I like your changes, but what makes you think he was refering to repetition of points and repetitions of something like a curve or a surface? Markus Schmaus 21:41, 23 July 2005 (UTC)

Thanks. If that's so, I find it hard to understand why you then destroy what I wrote, removing the language I feel is important. This is not a one-man project, reflecting a single vision. I have tried very hard to accomodate your views, and those expressed by others; I see little respect from you for my views. Or perhaps you simply do not understand why your edits are so disruptive.

One problem is clearly a language barrier. I am a native speaker of the English language, and an experienced writer. I choose my words with great care and precision, and attend closely to grammar and spelling. Unfortunately, you continue to misunderstand me, and also seem not to realize when you are writing bad English.

For example, I said nothing about Riemann repeating a curve or surface; that's your misunderstanding. He distinguished between discrete cases, where we can count, and continuous cases, where we must measure. The way you revised my description of Riemann is both bad English writing (including mistakes in grammar and spelling), and a less effective and accurate description of his lecture.

In your view, you are concerned with getting the mathematics "right". That has two serious problems, both of which are crippling progress on this article. The first is that your "right" is not the only one. You have no exclusive claim to the use and definition of manifolds. I quoted two reputable mathematics texts, and you'll notice that they gave slightly different definitions. Every contributor to this article may also have a different view, a different emphasis or different technical requirements. To collaborate here you simply must accomodate other views.

The second serious problem is that we have other concerns besides satisfying your sense of right. Several of us have tried to tell you that, but you show no sign of understanding. The text, especially the early text, must be lively. It must appeal to a broad audience. It must be clear to those with little mathematical background. Either you do not understand this, or you do not realize that your edits disrupt our efforts to achieve this, or you don't care. Therefore, regardless of your views, you must defer to others in these matters.

Alternatively, you can create a page called Manifolds according to Markus Schmaus, which will satisfy you. Sadly, it will not satisfy Wikipedia, and will never qualify for featured page. In fact, it will probably instigate a vote for deletion, citing NPOV.

I speak bluntly because I feel I must. Subtlety and accomodation has not worked. I'd like to see this article achieve featured status. You're in the way of that. Please understand: I do not say that you have nothing to contribute. Rather, I say that unless you back off and let others contribute without disruption, this project is dead. If you cannot work that way, Wikipedia is probably not the right place for your writing. Wikipedia requires a style of collaboration which you have so far failed to exhibit. There is no shame in that; it can be a difficult thing to do, and is not for everyone. To someone who feels strongly "right", it can be painfully uncomfortable. Perhaps you should "own" a page at PlanetMath, which works more the way you like.

It is hard for me to say these things, which may well give offense. I do so in hopes a greater good will emerge: a closer collaboration, a livelier article, a more effective Wikipedia. KSmrq 08:56, 2005 July 26 (UTC)

While I agree that Markus' stress on exactness sometimes borders on pedantry, threatening to endanger the clarity of the text, I am confused by the timing of KSmrq's outburst. It seems to refer to either this edit, in which Markus removes the line "The term manifold itself refers to the manyfold pieces joined together" from the introduction, or this edit to the History section. In neither case, it is clear to me whether Markus' edit makes the article worse or better. Riemann's text is very hard to understand, and I suppose we should try to find a biography of Riemann or some other place where somebody really analyzed the text if we want to explain his text. I want to add to Markus' defense that he agreed to the patchwork metaphor. By the way, I find it odd that Habilitationsschrift is equated to inaugural lecture. Were they the same in Riemann's time? -- Jitse Niesen (talk) 12:17, 26 July 2005 (UTC)

First, Jitse, let me say thanks for your earlier critique. I hope you feel I did justice to your concerns in my rewrite.

For me to say "pedantry" would be to characterize Markus' efforts, which, again, tends to be unhelpful. Likewise "outburst" dismisses the content of what I say. Why now? My dismay at seeing the ruins; maybe not "Ground Zero" in New York, but bad, if you'll forgive an inflammatory metaphor.

Where to begin? Yes, I strongly object to removal of the "manyfold" line. I repeatedly explained why such a line is important. And it is accurate. I previously put a link in the wiki source (edited out by Markus) to a page with Riemann's work. Later I found an English translation by Clifford, which I consider a very strong interpretation given Clifford's skills. On the basis of Riemann's own words, I believe my sentence is accurate. (See more below.) And I explicitly said what my larger concerns are, beyond "right". Removing that line (repeatedly) shows no sensitivity to those concerns. That's a problem. Markus keeps putting words in my mouth about what I have said and what I mean, but he is consistently wrong.

Which brings us to the history/etymology edits. The text is now butchered, and I use the word precisely. What was once a whole lovely creature has now been hacked to bloody pieces and scrambled. For example I put "discrete" first, following Riemann, not Markus. Markus, feeling wiser than Riemann perhaps, moved it back to second in the German. Yet in the parenthetical English he did not swap the words, but made them both read the same! This is someone who "stresses exactness"! The rest is equally bloody. The clarity of the text is not threatened, it is exterminated.

Reading old texts and describing the contents in modern language, we have a few well-known perils. These include translation, shifts in language, shifts in understanding, shifts in definitions, and so on. One reason I rely on Clifford is that he was of the time, an excellent geometer, and able in both German and English. I don't need Markus to interpret Riemann for me, and I certainly don't need him to (mis-)interpret my own words. Despite Markus' misgivings because he does not understand my English, I feel I understand Riemann well enough; the challenge is to distill his meaning for a modern audience, without writing a lecture. In my view, Markus does this poorly.

I, too, was uncomfortable with the translation of "Habilitationsschrift" as "inaugural lecture"; but no comparable English term occurs to me, and this was the one I found on the link given. Thus if someone follows the link they will know they have come to the right place.

Agreeing to "patchwork" is ironically late, considering we had already criticised that term and moved on to better similes, now removed. I suppose a small, late step is better than none, but the total effect is regress, not progress.

For Markus to emphasize his concern with correctness is insulting. Is it really likely that *anyone* with the mathematical sophistication to edit an article on manifolds has no such concern? Notice I have repeatedly brought in authoritative sources, including Riemann and Spanier. That was me, not Markus. In the former case I linked to the complete original text; in the latter (that's with two ells, not "later"), I quoted verbatim. No, Markus elevates his "right" above others, which is not good collaboration.

I have repeatedly tried to model better behavior. My original edit did not remove the previous introduction, it displaced it. I stated the concerns that lead me to do so. As questions arose, I sought out and introduced reliable sources. In response to each issue raised by others, I rewrote. I invited others to work with me and encouraged feedback. And I have acknowledged and complimented the contributions of others, including Markus. I have, until now, tried to be patient.

Markus has behaved otherwise.

My central complaint is that Markus' "right" ignores others', and tramples the life out of the writing. I think he needs to hear that and amend his behavior.

What is your opinion? KSmrq 19:26, 2005 July 26 (UTC)

I never intentionally deleted any link to wikisource. I thought I might have done it unintentionally, so I looked up every edit by KSmrq, but I could not find such a link. Either KSmrq will show me when I deleted such a link, in this case I will apologize for doing so unintentionally, or KSmrq will apologize for unintentionally accusing me of something I never did. Markus Schmaus 21:54, 26 July 2005 (UTC)

I take this as the opion of a single person. I don't think that I'm destroying the project. My contributions on charts and altases got very positive feedback. I know that my stress on exactness borders on pedantry, that's why I'm so good in Math. Take a look on Talk:Linear independence, my pedantry helped to discover a serious error in the definition of linear independence. We could write an article which sounds good and which would for a lay person look correct, such an article might even become featured, but what good is it to feature an incorrect article?

I've tried hard to work towards NPOV, I first stated the problems I have with your new intro, in the hopes you would work towards resolving them. When it became clear, that this was not the case and when it became even clearer that Gauge, missed an important point about topological manifolds due to the patchwork point of view, I decided to do something. Maybe next time I will simply post a NPOV message. I think The "can be constructed" "in this sense" and "can be characterised" can form at least the base for a solution of the NPOV issue. Maybe you should start an article patchwork manifold.

I was happy about the merge with the History section, but I couldn't see how "the language [you] feel is important" is supported by the text. See User:Markus Schmaus/Riemann for how the text supports the changes I made. "Mannigfaltigkeit" and "manifold" are false friends, just as "seriös" and "serious" their meaning differs substantially. In German "Mannigfaltigkeit" does not refer to "repetion", though manifold can refer to repetion in English, it would be wrong to say a manifold is called that way because of any reptetion, it was called this way because Riemann used "Mannigfaltikeit" and Clifford translated it as "manifoldness". Markus Schmaus 14:17, 26 July 2005 (UTC)

I had another close look at On the Hypotheses which lie at the Bases of Geometry. It's not easy to understand, but I'm pretty sure now, that Riemann viewed a n-dimensional manifold as a stack of (n-1) dimensional manifolds. Markus Schmaus 18:08, 24 July 2005 (UTC)

[edit] "Markus' edits"

Just for clarification, those edits, Gauge refered to, weren't mine. They were done by Jitse, but I liked them a lot. Markus Schmaus 12:22, 25 July 2005 (UTC)

Whoops. Thanks for clearing that up. - Gauge 23:38, 25 July 2005 (UTC)