Talk:Manifold

Talk:Manifold/Archive1

Template:To do

Contents

1 Revert to old version of manifold 2 Recent changes 3 Charts 4 The empty manifold 5 Misc article problems 6 Hausdorff 7 Volume form 8 Algebra of scalars 9 Algebraic variety 10 Algebra of scalars 11 solving things 11.1 Charts 11.2 More headings 11.3 proposed combined and further expanded new definition 12 new recent changes 13 Proposal for rewrite

Revert to old version of manifold

moved from Wikipedia talk:WikiProject Mathematics

Manifold recently underwent a rather thorough rewriting, by MarSch. The diff is here (http://en.wikipedia.org/w/index.php?title=Manifold&diff=14073889&oldid=13927023).

Let me summarize how the article changed (but I would request you read the diffs for yourselves).

• Less emphasis is given on plain words and more on terminology: witness "near" vs "locally", "space" vs. "Euclidean space Rⁿ".
  • This is about the intro. I also added the intuitive view about gluing. "A general manifold can be obtained by gluing together open balls." --MarSch 12:32, 25 May 2005 (UTC)

What you added is mathematically correct. However, a nonmathematician will think upon reading this that a manifold looks like a stack of oranges in a grocery store. But that was a good attempt, and I think I know how to improve on it. Oleg Alexandrov 00:01, 26 May 2005 (UTC)

• More emphasis is given, right from the introduction on applications of manifolds to classical mechanics, mentioned are in the second pararaph pseudo-Riemannian manifolds, and how they work to model spacetime in general relativity.
  • this stuff I moved to the intro, because I think it is important to explain why manifolds are interesting.

I did not say what you did was wrong. I just summarized your edits. Now, to mention just several issues, more immediate ones like relativity was enough. No need for two sentences about configuration spaces and other things. Oleg Alexandrov 00:01, 26 May 2005 (UTC)

• The intuitive discussion about "Intrinsic versus extrinsic" has been shortened.
  • Yes, I removed some lines which I deemed vague and also didn't make this a subheading of history since it isn't.--MarSch 12:32, 25 May 2005 (UTC)

The notions of "intrinsic" and "extrinsic" are very important and very hard to explain nonmathematically. That paragraph did a very good job with that. I am very sad you did not appreciate that and decided to cut it down. Oleg Alexandrov 00:01, 26 May 2005 (UTC)

• Charts and transition maps have been given more prominence, as well as the symbol C^k.
  • Instead of spelling out the definition of chart a couple of times I thought it better to define it once. This allowed me to clean up the definition of topological manifold considerably. I have used Ck where appropriate in formal definitions. --MarSch 12:32, 25 May 2005 (UTC)

When I write a math paper, I do exactly that. However, by making things more formal and putting the heavy definitons upfront, you made the whole thing much harder to understand. To be concise and rigurous is good in a math paper or book, not when you write an essay. Oleg Alexandrov 00:01, 26 May 2005 (UTC)

• The fact that a manifold is Hausdorff has been emphasised.
  • This is simply not true. I have used Hausdorffness in the definition just as the older version did. Actually I moved the discussion about Hausdorffness to a separate heading, maybe that is what you mean.--MarSch 12:32, 25 May 2005 (UTC)

That's what I mean. You replaced two sentences in two places with an entire section. Now, I am aware that the Hausdofness assumption is very important, and most differential geometry will not hold without it. But again, to put more value on this technicality than on the intuitive explanation of intrinsic and extrinsic is not good in an article aimed at the level of say college students. Oleg Alexandrov 00:01, 26 May 2005 (UTC)

• I am not sure I understand the sentence:

A topological n-manifold is a Hausdorff space in which every point p is in the interior or in the boundary.

• • "At each point in the interior there is an Rn-chart and at each point in the boundary there is an R0+×Rn-1-chart." Explains what they are. I agree that the formulation can be improved here.--MarSch 12:32, 25 May 2005 (UTC)

What you wrote before was plainly incorrect. The expanded explanation is correct. But I would not insist so early on manifolds with boundary. The previous version of the article wisely talked about it later. Oleg Alexandrov 00:01, 26 May 2005 (UTC)

Again this is simply not the case. The old article started with a topological manifold with boundary and I have simply preserved it. I agree that it might be a good idea to change this. I wish you would take a better look at what was _really_ in the old version and not simply assume that I changed things for the worse. --MarSch 10:18, 26 May 2005 (UTC)

Well, not quite. In the article, there was first the section "Technical definition" which had a lot of wording without mentioning boundary. After that, you are right, the topological manifold did talk about boundary. But then, later, in the section ==differential manifold== one again does not deal with the boundary. And that is a very long and well written section. When you rewrote the article however, you merged together a lot of the things in ==topologica manifold== and in ==differential manifold== and now the boundary was present everywhere. Oleg Alexandrov 15:15, 26 May 2005 (UTC)

Can I say the new version is certainly bad? Probably no. Is it more rigurous and stresses more on heavy use of manifolds in math and physics? Yes. Is it more understandable for non-specialists? No. Is this article still a fine encyclopedic essay? Maybe, but worse than before. I would like to discuss if we maybe could move to the older version.

And I couple of words about MarSch. MarSch is a very smart guy, knowing lots of differential geometry, and probably many other things. MarSch is a very bold guy, having rewritten such a prominent article as mathematics (I did not see how it went), and MarSch desired to edit the main Wikipedia page, but lacked administrative powers to do that. MarSch has very good technical skills, and likes the technical parts to be more prominent in articles (there was a discussion right here about [[Laplace operator]). My own view is that MarSch needs to still absorb a bit more the existing culture on Wikipedia before making drastic changes. Oleg Alexandrov 19:16, 22 May 2005 (UTC)

Sigh. Agree with Oleg. Maybe be not so terribly bold would be a better byword. I guess WP has no formal process for "Vote For Reversion"? linas 03:36, 23 May 2005 (UTC)

That would be going to the talk page and bringing it up there ;) Dysprosia 10:26, 23 May 2005 (UTC)

I am surprised, again, that someone dislikes my edits so much that they want to revert them completely. I have tried to improve the logic and structure of the article, thereby hoping to make it more accessible to all and I like to think I succeeded. I have responded to the few objections Oleg mentioned above, but please take a look at the changes for yourself. --MarSch 12:32, 25 May 2005 (UTC)

MarSch, I will try my best not to be confrontational. But you see, in a sence you are not giving me a choice. You did a thorugh rewriting of a very prominent math article, much rewritten and much discussed before. And you did so without any consultation on the article's talk page.

I did not say your contributions are overall bad. I just don't know how to harmonize what you wrote with what was before. I will not do it now, but I plan to revert to the old version of manifold, then we can work together to improve on it with the changes being done gradually and discussed in advance on the talk page. Oleg Alexandrov 00:01, 26 May 2005 (UTC)

I think the biggest problem though is the way you and me view things. You seem to be writing for the fellow mathematician who knows at least as much as you about the given subject. I try to write for the general public, for the level of college students. I have six years of teaching experience as teaching assistant and lecturer, and I think I can safely say that no college student will understand much of manifold the way it is now written. Oleg Alexandrov 00:06, 26 May 2005 (UTC)

I think this discussion should really continue on Talk:Manifold. If the biggest problem is the discussion of 'atlases' - my guess - then we can apply the usual idea of having a less formal discussion on the Manifold page, and full technical details on atlas (topology), with a See main article ... link to it. Charles Matthews 15:28, 26 May 2005 (UTC)

moved my comment down --MarSch 17:14, 4 Jun 2005 (UTC)

Recent changes

Lost in the debate was my primary objections to MarSch's edits. First was cutting the "intrinsic vs extrinsic part"; that still to be discussed. The second was the "cleanup" taken by MarSch as result of which charts were defined first and the treatment of topological manifolds and differential manifolds were rather combined. Now I reverted that to the way it was before. More work is needed, but this is a better starting point. Oleg Alexandrov 21:27, 26 May 2005 (UTC)

I stared at the article a bit more and I do like the old version better. It flows. Whereas, in MarSch's edits the stuff about charts seemd suoperfluous (its already wikilinked to the right article), and the topology discussion got interrupted by statements about differentiability before moving back to topology. However, the article does have problems, I'll list below. (Sorry to take so long to respond. I've gotten hopelessly tangled in stupid VfD issues, see Harmonics theory. I normally run away from controversy. Sigh.) linas 05:10, 2 Jun 2005 (UTC)

Charts

I would like to see Atlas (topology) expanded; the layman's intro is good, but a good mathematical definition would be a lot better. e.g. a discussion of the transition functions, and a discussion of the Jacobi identity-like thing (the Bianchi identity-like thing?) or whatever its called where you have three open sets intersecting, and the charts have to be three-way consistent. Maybe MarSch could apply himself there? linas 05:35, 2 Jun 2005 (UTC)

I concur. The detailed material on atlases is better placed in the separate atlas article where they can be given a thorough treatment without interrupting the flow of this article. I suggest perhaps using the section called "Additional structures and generalizations" to (briefly!) give a description of differentiable manifolds, and then a link to a separate article: differentiable manifolds. My view is that we should relegate all additional structures not present in the basic definition of topological manifolds to separate articles. - Gauge 04:11, 5 Jun 2005 (UTC)

I propose to wait moving everything to a separate article until the sections here get "too big", and not because of lack of headings. Since we need to summarize these articles here anyway I see little point in splitting of things which are too small to summarize. --MarSch 11:49, 5 Jun 2005 (UTC)

The empty manifold

Should we allow (topological) manifolds to be empty? The current definition does allow this, but is contradicted by the following paragraph. The classification of 1-manifolds further down the page assumes they are nonempty, and there are other articles that also assume manifolds are nonempty. The only statement I have seen that relies on empty manifolds is the one about the boundary of an n-manifold being an an (n-1)-manifold. --Zundark 16:25, 1 Jun 2005 (UTC)

Not sure, but I think we should allow it. Don't know what it's dimension would be, though. --MarSch 11:55, 5 Jun 2005 (UTC)

It's a manifold of every dimension, since every point has an open neighbourhood homeomorphic to Rⁿ for every non-negative integer n. But its covering dimension is -1. I think we should disallow it. --Zundark, 5 Jun 2005

Okay, maybe we should disallow it. Unfortunately this means that a manifold without boundary is not a special case of a manifold with boundary. --MarSch 16:50, 9 Jun 2005 (UTC)

No, a manifold without boundary would still be a manifold with boundary, since we would disallow the empty manifold in both cases. Perhaps what you meant to say is that the boundary of a manifold may not be a manifold (because it may be empty) - this is what I was referring to above in my 1 June comment. --Zundark 08:09, 10 Jun 2005 (UTC)

You are entirely correct. --MarSch 09:39, 10 Jun 2005 (UTC)

Misc article problems

Article states: (Readers should see the topology glossary for definitions of topological terms used in this article.) Can we cut this sentence out? linas 05:23, 2 Jun 2005 (UTC)

This was added before we had articles on the individual terms. It's no longer needed, so I've removed it. --Zundark 09:39, 2 Jun 2005 (UTC)

Hausdorff

Article states: ...it is tempting to think that being locally homeomorphic to a Euclidean space implies being a Hausdorff space. A counterexample is created by deleting zero from the real line and replacing it with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This construction, called the real line with two origins is not Hausdorff, because the two origins cannot be separated.

These sentences seem to imply that the "real line with two origins" is homeomorphic to the real line. It is not clear that it is, or that it isn't. Wording should be fixed. linas 05:23, 2 Jun 2005 (UTC)

Volume form

Just an idea, but maybe the article should provide a few sentances on differential geometry and calculus on manifolds, maybe mentioning that a manifold has concepts of area and volume? linas 06:06, 2 Jun 2005 (UTC)

Well, it doesn't really. Charles Matthews 10:07, 2 Jun 2005 (UTC)

To be pedantic, what a smooth manifold has is a class of Lebesgue measures, such that every diffeomorphism leaves the class invariant. So volume forms are defined up to smooth invertible functions. But that's not the same as having a preferred volume form. Charles Matthews 12:49, 2 Jun 2005 (UTC)

Didn't mean to suggest there is a natural volume form; what I'm trying to suggest is that half the stuff in the section on differentiable manifolds should be moved to the article on charts, and that instead a paragraph or two should be added on calculus on manifolds. Surely a mention of calculus on manifolds is more important and interesting that algebra of scalars which is neat and all that but of somewhat lower priority. Since we appear not have a calculus on manifolds article, I figured that at a bare minimum, the article on volume form is at least not embarassingly thin and thus linkable. Also, adding your pedantic defn. above to the article seems maybe worthwhile. I dunno. linas 06:25, 4 Jun 2005 (UTC)

I think you mean to mention things like Differential forms and tensor fields and I agree. --MarSch 12:01, 5 Jun 2005 (UTC)

Algebra of scalars

Wording needs to be fixed and clarified. For a Ck manifold M, you can form the Ck functions to the real or complex numbers. What does "..to the.." mean? Which Ck functions are these? Are these the charts? Are these some other functions defned on the manifold? Its known that the set of scalar-valued functions on a manifold form an algebra, is that what this paragraph is talking about? linas 06:06, 2 Jun 2005 (UTC)

The functions with k derivatives, basically: see smooth function. The kth derivative is assumed continuous. Charles Matthews 10:06, 2 Jun 2005 (UTC)

The sentence above is indeed mathematically correct, but I would say not well worded. It is not whether "you can form" the functions; they exist whether you want it or not. And indeed, saying "to the reals" sounds clumsy, better say "real-valued". I modified this in the article, wonder how it looks now. Oleg Alexandrov 14:55, 2 Jun 2005 (UTC)

Thanks Oleg. Charles, due to the recent edit controversy, I just thought I'd jot down a list of to-do cleanup items for this article; the questions were rhetorical. linas 06:17, 4 Jun 2005 (UTC)

Algebraic variety

Should this article mention (probably in the "see also" section) algebraic variety ? Or at least algebraic geometry? linas 06:06, 2 Jun 2005 (UTC)

At least in the see also section algebraic variety. Don't know about alg. geom. --MarSch 11:51, 4 Jun 2005 (UTC)

Algebra of scalars

I find this section in the article not very helpful. I mean, all the other sections talk about really important concepts about manifolds (tangent space, classification); this one I think is a minor thing, and I belive we could be better off without it. Comments? Oleg Alexandrov 16:52, 4 Jun 2005 (UTC)

Agreed. It seems out of place here as a separate section. - Gauge 04:16, 5 Jun 2005 (UTC)

I put this in here, because of the possibility to recover from the algebra of scalars the manifold itself. Unfortunately I don't know very much about this yet, specifically under what conditions this is possible. Also another def of the tangent bundle is as derivations on the algebra of scalars. In that way it is more basic than the tangent bundle, so I think that this is also "really important".

As an aside: It is the Wikipedia way to try to preserve all information. Even if you think something of minor importance, you should try to preserve it. Please try to be a bit more carefull in your suggestions about "things we can do without". --MarSch 11:36, 5 Jun 2005 (UTC)

You can preserve that information, but in a different place. As you said, you don't know how to expand that section, or how to explain its significance. So for now, we are better off without it. Oleg Alexandrov 15:17, 5 Jun 2005 (UTC)

I _have_ just explained its significance. You can stick a sect-stub on it if you want. --MarSch 16:47, 9 Jun 2005 (UTC)

Actually this section is also a prerequisite for the exterior calculus that linas wants to add, since these are the 0-forms. --MarSch 16:54, 9 Jun 2005 (UTC)

solving things

I would like to alert some editors to this template, which expresses the beliefs of a lot of wikipediansa about technical details: Template:Technical That said, I would like to start trying to sort out our differences. Therefore I will start simple.

Charts

• I think that a good article on manifolds should not ignore charts.MarSch

I thought the section ==Differentiable manifolds== mentions charts in the right place. Where else would you want them? Oleg Alexandrov 16:47, 4 Jun 2005 (UTC)

topological manifolds also have charts. Transition maps are automatically homeomorphism. There is stuff we can say about charts and we should make a section for that. When it grows too large we can split off some to the article about charts, but we are not there yet. Charts are also "really important". --MarSch 11:40, 5 Jun 2005 (UTC)

More headings

• For ease of future editing and also searching, I like to use headings for important subsubjects. This helps give an article structure. Please comment on this. --MarSch 12:01, 4 Jun 2005 (UTC)

Perfectly agree. So far, this article uses 11 headings. Would you be specific where more heading would be needed, and how they should be called? I can see that the section ==Differentiable manifolds== is kind of long; but not very long, one screen and a bit on my computer, and things there flow nicely without headings. So, just please be more specific. Oleg Alexandrov 16:52, 4 Jun 2005 (UTC)

I dislike all the loose facts about manifolds which are thrown haphazardly either into the topological or the differentiable heading. I would like those to be collected under new headings and keep the def in its own heading. Also the bit of explanation about Hausdorffness needs to be isolated so that it can be improved. It is so much easier to edit small sections than large ones, since you have less to worry about. If we want to link to chart and atlas as subarticles, I think we should also give each of those a heading. If they deserve an article than they definately deserve a heading... --MarSch 17:34, 4 Jun 2005 (UTC)

OK, let us figure this out piece by piece.

I wonder, which loose facts you mean. So far, this article has two important sections, the section about topological manifolds, and then the section on differential manifolds, which are like topological manifolds, and in addition have smooth charts and transition maps. This structure makes perfect sence to me.

You can see what I want by looking at the article before you did a partial revert[1] (http://en.wikipedia.org/w/index.php?title=Manifold&oldid=14282278#Why_require_Hausdorffness). The loose facts are under "Homogenous, second-countable and paracompact". A lot of the section "diff manif" is about charts. Actually it is more than half of it. --MarSch 12:17, 5 Jun 2005 (UTC)

I don't see what to improve about the Hausdorfness. It has its own paragraph, and is rather well explained. Are you implying you need a heading called ==Hausdorfness== only for one paragraph? I think that would make things too fragmented. And that paragraph is in proper place, where the topological issues are discussed, before moving on to the differential structure.

Yes, I am implying that we need ====Hausdoffness==== or some such. We can say a whole lot more about it. Also linas doesn't think the example is so clearly explained. I don't see how it fragments things. This could grow into a new article on non-Hausdorff manifolds.--MarSch 12:17, 5 Jun 2005 (UTC)

Again, at least to me, the two sections "Topological manifolds" and "Differential manifolds" make perfect sence, as the latter builds on the former. Could you be more specific what exactly you don't like about the current arrangement? Oleg Alexandrov 23:39, 4 Jun 2005 (UTC)

PS If I did not address some other concerns you mentioned, I will. Let us take things piece by piece. Oleg Alexandrov 23:39, 4 Jun 2005 (UTC)

proposed combined and further expanded new definition

I have tried to integrate the chart-def with the homeomorphic neighborhood-def and also tried to explain the defs more. See below:

Topological manifold without boundary

The prototypical example of a topological manifold without boundary is Euclidean space. A general manifold without boundary looks locally, as a topological space, like Euclidean space. This is formalized by requiring that a manifold without boundary is a topological space in which every point has an open neighbourhood homeomorphic to (an open subset of) Rn. Another way of saying this, using charts, is that a manifold without boundary is a topological space in which at every point there is an Rn-chart.

Topological manifold with boundary

More generally it is possible to allow a topological manifold to have a boundary. The prototypical example of a topological manifold with boundary is Euclidean half-space. Most points in Euclidean half-space, those not on the boundary, have a neighbourhood homeomorphic to Euclidean space in addition to having a neighbourhood homeomorphic to Euclidean half-space, but the points on the boundary only have neighbourhoods homeomorphic to Euclidean half-space and not to Euclidean space. Thus we need to allow for two kinds of points in our topological manifold with boundary: points in the interior and points in the boundary. Points in the interior will, as before, have neighbourhoods homeomorphic to Euclidean space, but may also have neighbourhoods homeomorphic to Euclidean half-space. Points in the boundary will have neighbourhoods homeomorphic to Euclidean half-space. Thus a topological manifold with is a topological space in which at each point there is an Rn-chart or an R0+×Rn-1-chart. The set of points at which there are only R0+×Rn-1-charts is called the boundary and its complement is called the interior.

• What do you think of this? --MarSch 17:13, 4 Jun 2005 (UTC)

Basically, this extends the section "Topological manifolds". Overall does not look too bad. But let us clarify first the overall article structure. I still think the split into "Topological manifolds" and "differential manifolds" is the right one. After we discuss that (see above) we can come to this.

By the way, the section about "topological manifold with boundary" is rather long. In my view, we better have the section about "topological manifolds" without mentioning boundary at all, then the section "differential manifolds" again without mentioning the boundary, and only later a new heading in which the boundary is discussed. Oleg Alexandrov 23:46, 4 Jun 2005 (UTC)

If we were to go with my suggestion of splitting off nearly all of the content on differentiable manifolds to a separate article, these additions would be a great way to elaborate upon the definition of topological manifolds. I think that the distinction between manifolds with/without boundary is very important, and as such they should be treated separately without favoring one over the other. In particular, this means that I prefer the headings "Topological manifolds without boundary" and "Topological manifolds with boundary" instead of the former being the "default" definition. - Gauge 04:25, 5 Jun 2005 (UTC)

new recent changes

I think the thing Oleg and linas didn't like about my previous changes was that I removed the section titled technical description. I've removed the sentence about what follows below, because that is not usefull at all. I think this section is very bad. It is not technical which it claims. It includes some history and some other random facts. Can we relocate all info and then delete this then empty section?

I've found two subsections which Oleg deleted by doing his partial revert. I've put them back. This does not mean that I want these as they are now, but we shouldn't be throwing the info out. The first is the section on Hausdorffness. Since Hausdorffness is no longer in the core def we might want to mention that most mathematicians do include Hausdorffness in the def. And second-countability also maybe? Not sure. The other section is random facts on (diff.) manifolds, ah I see there something about 2nd countability.

I've put in the expanded def. on topological manifolds. We should probably have something similar for diff. manifolds. Both a def with substitutions of subdefs and a short def. We should collect all random facts into either the section about topo or about diff manifolds. It will probably turn out that diff manifold could be split off. Maybe we should also split topo manifold off. We could then expand on some more general manifolds like the Banach and Frechet manifolds. Maybe a section for Lie groups. And of course a section about complex manifolds. Also needed is the canonical picture for clarifying what a transition map is. Tell me what you think --MarSch 14:45, 11 Jun 2005 (UTC)

The common assumptions between the various possible definitions are as follows: we always require a manifold to be a kind of topological space with second-countability, but not always Hausdorff (although I'd never heard of a non-Hausdorff manifold before seeing the foliation example). All of the definitions also require that a manifold is locally (complex or real Euclidean, or Banach space, Hilbert space, etc), and we also require the notion of transition maps. It seems to me that a very general definition could use bundle-theoretic methods. In particular, perhaps the fibres could be viewed as torsors for the various linear groups? This, however, would be quite abstract to newcomers. Perhaps we should direct newcomers to pages treating more specific types of manifolds before trying to understand this (necessarily very abstract and general) page? Or perhaps make a separate page giving the more general definition, whatever it may be? - Gauge 21:56, 17 Jun 2005 (UTC)

The concept of manifold is indeed a complex one. And recently I started understanding the motivation behind MarSch's changes. There is a tradeoff between keeping an article acessible, and making it more concise/general/rigurous/complete. MarSch seems to be leaning against the latter, and above, Gauge seems to be leaning in the same direction.

However, since this is an encyclopedia, I think one should lean towards acessibility at the expence of the other things. There are lots of good books in differential geometry out there, but there are not many accessible essays not directed towards the fellow mathematician who knows at least as much as the author abut a given subject.

And by the way, since people in general have diverse opinions about how a given article should look like, I suggest that any big changes to a math article be discussed in advance on the talk page. In this way we will avoid the recent debacle with this article. I acknowlege that some of it was also due to my impatience, but a preliminary discussion would have saved lots of energy spent arguing. Oleg Alexandrov

I am in favor of making this page the "more accessible" one. It seems to me that a casual user would just want to know about the most commonly-used definition, namely real or complex finite-dimensional manifolds that have the nice topological properties such as the Hausdorff property. However, I also think that we should have another article (or articles) treating more general cases and giving more technical details. This scaling-up of technical difficulty is necessary due to the widely-varying backgrounds of our audience. We should be able to come up with something that will make everyone (newcomers and experts alike) happy. - Gauge 04:16, 18 Jun 2005 (UTC)

I would not mind having another more technical article in addition to manifold. I think Charles suggested above that some of the more technical issues could be discussed at atlas (topology). Or somewhere else. Any suggestions? Oleg Alexandrov 04:57, 18 Jun 2005 (UTC)

I would like this article to be technically complete and also start with a description (which should read like a popular science book) for non-mathematicians This would also be highly interesting to mathematicians and physicists. It should explain how to visualize manifolds, as in the bit about exterior and interior viewpoints, elaborate on the gluing which is mentioned in the intro and generally use our geometrical intuition to clarify the why of manifolds. But the absence of that is no reason to start culling the rigorous mathematics. For the more mathematically knowledgeable the rigorous definition should be explained as clearly as possible and as simply as possible, but without excluding things merely because they are too technical that are very relevant to this article. I see these two things as complementary and mutually reinforcing. I hope we can work to improve both, but I find "hiding things away" in separate articles a simplistic "solution". --MarSch 12:35, 19 Jun 2005 (UTC)

I disagree with making this article more technically complete. Oleg Alexandrov 15:32, 19 Jun 2005 (UTC)

I completely agree with Oleg. Manifold is a basic article which will be read by a lot of people with different backgrounds. It should be as accessible as possible. By making the article technically more complete we make it less accessible. "Hiding" technical details and complex issues in different articles is exactly the right way (in my opinion) to improve the article. MathMartin 16:10, 19 Jun 2005 (UTC)

So what you're saying is that if you add a technical paragraph at the end of an article, without changing anything else, you have just made the article less accessible? You can't be serious, these things are independent/orthogonal. --MarSch 13:44, 20 Jun 2005 (UTC)

I was away during the recent flurry of edits, so I don't know all the background. My opinion is as folows: I'd like this article to talk about nice manifolds, and include some examples and pictures. I think that that will make the article long enough. At the end, we can include a section on generalizations: relaxing conditions like Hausdorff and second countable, infinite dimensions, complex manifolds, etc. If possible, we can also include a definition there which covers all generalizations, but the section on generalizations should not be more than say one screenful. Splitting the article in topological manifold, differential manifold, etc. and "hiding" things away seems a good approach here. -- Jitse Niesen 16:35, 19 Jun 2005 (UTC)

I agree with Oleg, MathMartin, and Jitse. In particular I like the idea of separate articles for "topological manifold" and "differentiable manifold" that include all of the more technical details. I suggest "Abstract manifold" as an article name for the most general definitions. What are manifolds called when they have atlases of Banach spaces? - Gauge 19:08, 19 Jun 2005 (UTC)

Yes I also agree with putting the technical details in such separate articles, but what about the technical non-details? --MarSch 13:48, 20 Jun 2005 (UTC)

It is okay to mention technical non-details, so long as they are not elaborated upon in this article. We don't want to get into why we make all of these technical restrictions. - Gauge 21:04, 21 Jun 2005 (UTC)

Proposal for rewrite

Well, that depends on what you call a non-detail. We need to be more specific. Here is a proposal of how I'd like to see the page:

• Lead section, History, Intrinsic and extrinsic view. Essentially as it is now.
• Examples: New section, mentioning circle, sphere and perhaps torus, possibly with application (manifolds as configuration space or phase space).
• Technical description. Essentially as it is now.
• Charts. Delete as a separate section from this article.
• Topological manifolds, Differentiable manifolds. Summarize to say two paragraphs each, devolving the rest to separate articles. Perhaps add a section on Riemannian manifolds.
• Tangent space, Algebra of scalars, Classification of manifolds. Move to new article on differentiable manifolds.
• Additional structures and generalizations. Essentially as it is now.

Comments? -- Jitse Niesen 21:07, 22 Jun 2005 (UTC)

I like very much wat Jitse suggests. Pictures needed! I can make pictures like the one in level set method, but I don't know if that's the best style. Anyway, what kind of pictures to put? Oleg Alexandrov 02:51, 23 Jun 2005 (UTC)

What exactly do you mean by this point "* Topological manifolds, Differentiable manifolds. Summarize to say two paragraphs each."? What kind of info do you want to keep?

Since every article on a specific kind of manifold needs to talk about charts and transition maps I see no possibility to remove this section. This is not a technical detail, but an essential part of manifold theory. For a topological manifold you don't need to make any restrictions on the transition maps, so could theoretically ignore them, but I think it is worth mentioning that they are automatically homeomorphisms, even though this is trivial, For a differentiable manifold you cannot ignore them, so I wonder what you want to keep in that section. Not the definition apparently. I think the examples section is a great idea and I think level set-like picture would be nice. --MarSch 11:15, 23 Jun 2005 (UTC)

Charts are introduced in Technical description, and we can discuss them more fully in chart (topology) or perhaps topological manifold. I agree that charts are essential in manifold theory, but I do not want much manifold theory in the article. I am undecided as of yet whether we want any definitions of topological and differentiable manifolds in the article, or whether this would complicate things too much. I will hopefully find time to flesh it out a bit more later, but for the first paragraph of differentiable manifolds, I am thinking along the lines of the current first paragraph, which goes "It is easy to define the notion of a topological manifold, but it is very hard to work with this object. The smooth manifold defined below works better for most applications, in particular it makes possible to apply "calculus" on the manifold." -- Jitse Niesen 12:26, 23 Jun 2005 (UTC)