Talk:Manifold/Archive 2

Archive 1

Archive 2

Archive 3

Archive 4

Archive 5

Let us cool down

It seems that the atmosphere got a bit too heated up for a fruitful collaboration. So, I propose we all stop editing the article and the talk page for a week, calm down, and come back here on 22:23, 2 August 2005 (UTC).

Old talk page is in talk:manifold/rewrite/freezer

Thank you, Jitse Niesen (talk) 22:24, 26 July 2005 (UTC)

I very much agree with Jitse. Now, from my limited reading of the discussion, it looks to me that the roots for the argument are: for one, KSmrq has been a bit too overprotective about others modifying his edits; and for two, when Marcus did not like something, he used to rewrite the thing first rather than list his objections on the talk page and wait for comments. Probably avoiding both of these in the future will make for a more pleasant experience. :) Oleg Alexandrov 02:49, 27 July 2005 (UTC)

• Lethe wonders what on earth has been going on over here...*

In previous discussions I made a remark about "German pride", with friendly intent (including a happy-face emoticon). It seems that was a cultural blunder, for which I am embarrassed and regretful. As soon as it came to my attention I apologized personally to one party, and promised a public apology as well. So to everyone who saw my remark as offensive, I am truly and deeply sorry; I greatly admire the contributions of Germany to mathematics, and meant nothing more than to honor that.

I believe the world at large would be shocked to discover the passion mathematics can evoke. But, however heated debates of content may become, I hope always — as ever — to treat other participants with respect and good will, so that we may inspire a new generation with our knowledge and enthusiasm. KSmrq 19:21, 2005 August 8 (UTC)

rewrite moved to manifold

Please note that it looks as if manifold/rewrite has been merged into manifold by Deryck Chan (no, don't ask me why and what, I discovered this by accident). Oleg Alexandrov 03:22, 4 August 2005 (UTC)

"Construction" and "More Examples"

I think that these two sections overlap in their content too much and are causing the article to look cluttered. Indeed, most of the examples deal directly with constructions. I prefer to look at them as examples of constructions rather than standalone examples. A new "construction" which just appeared due to Joshua deals with the double line (non-Hausdorff); I'm wondering how technically complicated this construction is to a newcomer? - Gauge 06:58, 5 August 2005 (UTC)

Good points. My first thought is to move the "construction" section further down, under the definition of a topological manifold (which still needs to be written), and have the "construction" section refer to the examples. About Joshua's example: you indeed need to know quite some topology to understand it, but on the other hand I can see the use of including at least one pathological example. Perhaps we should just introduce the example better, saying that the details don't matter that much but that it's included to show that not manifolds are nice. -- Jitse Niesen (talk) 12:08, 5 August 2005 (UTC)

I wrote the section on the double-dotted line primarily because it's listed in the comments right at the beginning of the More examples section as something one might want. I know that it's written at a fairly advanced, technical level. I'm trying to improve my mathematical exposition. After I wrote the example, my father (a computer programmer and amateur mathematician) read the entry and said, "Did you know what you were saying when you wrote that? I have no idea what you said!" Thus I know it's not written to help someone who's not actively studying mathematicians. It does comply with the "For the rest: graduates" target audience listed at the beginning of the article. I've been trying to think how to describe it, in a way that's more accessible but still correct, while making it clear that it's a pathology. JPB 19:06, 5 August 2005 (UTC)

I know. I intentionally used the allready present examples to illustrate the constructions. This article should be more than a list of manifolds. There are a lot of manifolds, but there should allways be a reason why a specific manifold is given as an example. I would prefer if this reason is explained and then illustrated by the example. So I would prefer a general section of Hausdorffness and the line with two origins as an example. Scientific method has an example, which is separated into smaller pieces, each illustrating a single step of the method and linking to the other pieces. Maybe we could use a similar (improved) technique here. Markus Schmaus 20:48, 6 August 2005 (UTC)

I like the idea of explaining a certain construction and then placing the example immediately afterwards, instead of a section for constructions and a separate section for examples. This would make the examples more immediately relevant to the discussion and avoid the "list of manifolds" trap . - Gauge 00:16, 7 August 2005 (UTC)

New picture at circle example

I like the new picture by KSmrq. I thought putting everything in one picture would make it too cluttered, but I now see I was wrong. Two remarks though: I think the lines are a bit too thick, and the domains of the charts in the picture do not match with the description in the text, where all charts cover half of the circle (I understand that one needs to leave a gap between the domains of the blue and green map in the picture, but I wonder whether the gap can be shrunk). -- Jitse Niesen (talk) 12:08, 5 August 2005 (UTC)

Nice picture indeed. And I agree with Jitse's remarks about tweaking it a bit. Oleg Alexandrov 15:31, 5 August 2005 (UTC)

Thanks for the compliments. Now, about the tweaks:

1. I played with the thickness in designing the new illustration, the main purpose of which was to get unambiguous (and consistent) colors. A quirk of human color perception is that we don't discriminate tiny patches of color well, so I finally chose the bloated lines for better "readability".
2. I also explored the gap possibilities, and arrived at this design as the compromise "reading" the clearest. It does show all the essentials of the charts and their overlaps. The text refers to a chart on, say, the top half of the circle; and, actually, the yellow does span about that much of the black.

If it helps, think of this as the children's coloring book version of differential geometry, with thick lines, bright primary colors, and deliberate simplification. I finally had to quit tinkering, declare it "good enough", release it, and move on. (Illustrations can be an astounding timesink.) I might get motivated to come back to it; meanwhile, if anyone else has the time and inclination, be bold!

But maybe it's not so bad. Think back on all the mathematical illustrations you've seen. :-( I had an algebraic topology class with a professor who explained that his drawings weren't great but they had good labels. :-D Almost nobody has the time or talent to compete with George Francis' A Topological Picturebook (ISBN 0387964266) and Ralph Abraham's Dynamics,the Geometry of Behavior, or to create marvelous mathematical sculture like Helaman Ferguson. KSmrq 20:24, 2005 August 5 (UTC)

Ad 1. I understand what you're getting at; in fact, I prodded Oleg to thicken lines for the same reason. However, I still think the lines are probably too thick, especially since the colours are not essential. However, it is a detail, and as you say, creating pictures costs a lot of time, perhaps better spent on other matters.

Ad 2. Fair enough. My comment was mainly in case you hadn't thought about it (I hope you don't mind). Jitse Niesen (talk) 21:17, 5 August 2005 (UTC)

Far from minding, I'm happy to see I'm not the only one concerned with such details.

I have now added a second image to that section, illustrating the two chart atlas I introduced in the text. Despite the time I spent making the figure, it may contribute more clutter than clarity. Anyway, it now exists on Commons for whatever purpose. KSmrq 22:44, 2005 August 5 (UTC)

Time to merge the stuff?

See Talk:Manifold#How_far_till_this_is_done.3F. Wonder if there is any way to merge the histories too. Oleg Alexandrov 04:53, 9 August 2005 (UTC)

The rewrite page looks half-finished to me, yet the effort seems to have been largely abandoned. The history of both the original page and the rewrite makes me, personally, tend to be shy, not bold. As I said on the manifold page, parallel efforts make no sense. If there is to be no determined effort to complete the rewrite, the best parts should be incorporated into the original page. I would not support replacing the original. But I would prefer that interested parties devote anywhere from a few days to at most a week finishing a satisfactory rewrite. If there is substantial contribution, then we can reconsider the question.

I think the original page is much stronger for specifying Rⁿ rather than "a simple space" in most examples and definitions. I still remain unconvinced that there is any mainstream use of a definition that allows the dimension to vary from component to component. The examples seem disorganized, and I have no sense of a compelling pedagogical narrative. I believe counterexamples are instructive; where are they? And so on.

As a rule, it seems more productive to make incremental changes to a large article. If this rewrite were as good as it aspires to be, would it not have attracted more edits? Are editors "voting with their feet", by staying away? I don't know, but I don't think we should leave the rewrite dangling for long. KSmrq 07:42, 2005 August 9 (UTC)

I'd vote for merge. I disagree with 'Rⁿ instead of simple space, as the simple space could at least also be a halfspace or maybe an infinite dimensional vectorspace for some other type of manifold. But I do agree with the the other issues. But those issues can also be resolved while/after merging. Markus Schmaus 12:45, 10 August 2005 (UTC)

I was too vague. The idea is not to eliminate halfspaces and so on as variations; the idea is to begin with a specific, important, practical, familiar space, which is Euclidean space. The Bourbaki progression from general to specific is great for economy, but lousy for teaching. In the history of mathematics, and in the learning of it, we muddle our way through special cases before we appreciate grand generalizations. It is tempting in a general-purpose manifold article to jump straight to the broadest possible definition, but I believe this project was instigated to fight that temptation. KSmrq 00:50, 2005 August 11 (UTC)

Let us indeed push for a merge. I hope to be able to spend some time on the article tomorrow and Saturday. In what places do you think manifold/rewrite is worse than manifold? The "Construction" and "More examples" sections need to be brought together, but that material is not in manifold at all. "Intrinsic and extrinsic view" is indeed rather out of place, but for the rest it seems ready to go live (not perfect, of course, just better than the current article).

Re "the original page is much stronger for specifying Rⁿ rather than "a simple space" in most examples and definitions": what do you have in mind? The only place in the rewrite where this applies to, as far as I can see, is the third paragraph of "Introduction" ("The demand that it locally looks like a simple space is quite strong."). I think we can safely replace simple by Euclidean there: the statement is still true and it is already mentioned right at the start that there are other possibilities.

Re "I still remain unconvinced that there is any mainstream use of a definition that allows the dimension to vary from component to component": I went to the library and the second book I looked in says that the dimension can vary with the component (Abraham, Marsden and Ratiu, Manifold, Tensor Analysis, and Applications, Definition 3.1.7). The first book I took out (Hirsch, Differential Topology) implies that the dimension must be the same. Admittedly, manifolds with varying dimension do not have much use, but I fear writing that the dimension is the same may confuse students using books like Abraham, Marsden and Ratiu. Perhaps we should write that the dimension is everywhere the same and attach a footnote that this is not entirely true under some definitions? -- Jitse Niesen (talk) 21:10, 10 August 2005 (UTC)

I don't have a point-by-point comparison, though that might be helpful. I'm especially looking at the big picture, the overall organization, flow, coherence, and focus.

You may not have noticed: I shuffled some "Introduction" paragraphs and reworded the newly positioned first sentence to say

"A manifold can be characterised as looking like Euclidean space, or some other relatively simple space, in a close-up view."

I find this small rewording makes a big difference compared to the previous

"A manifold can be characterised as looking like a relatively simple space in a close-up view."

"Euclidean space" is solid ground, "simple space" alone is not. (A reader might expect a technical meaning, like simple group.) I'll scan for more such tweaks as I go. (I'm mildly concerned that "looks like" might be misinterpreted in too strong a sense, precluding bending. I'll try to insert a little language early on to correct for that.)

Thanks for the research! I notice that some authors say "n-manifold", while others just quietly assume fixed dimension. Apparently few authors need the extra generality, nor worry about confusion. By taking a graduated approach we may achieve universal happiness: begin with simple restricted versions for accessibility and progress to extra structure, fancier "simple spaces", and flexible dimension for generality. I would not go with a footnote; I would define an n-manifold, and say that for most authors a manifold is taken to be an n-manifold. This is akin to saying, by analogy, "in this chapter, all our manifolds are smooth". We can then say that other authors (cite AMR) admit broader definitions, where dimension need not be finite, or can vary with component. That way, we always tell the whole truth, yet build from simple to general. KSmrq 01:54, 2005 August 11 (UTC)

Introduction example

The cone is an bad choice of counterexample for the introduction, for two reasons. First, the reader does not know if we mean an object with a circular cross section tapering to a point, or a full quadric "double cone" as is common in geometry, or something else. (And neither do I!) Second, the point may be a problem for a differentiable manifold, but a (half-)cone is a perfectly good topological manifold, homeomorphic to a disc. KSmrq 01:01, 2005 August 10 (UTC)

Yes, I realize this, but I couldn't think of a better example at the moment. Your figure eight does the trick. -- Jitse Niesen (talk) 21:10, 10 August 2005 (UTC)

Further inspection reveals more problems. Consider

"[T]he surface of the Earth looks flat when you are standing on it."

Somebody ought to get outside more! Almost nowhere on Earth is terrain that flat. What's a reader in Switzerland or Alaska going to think? If we're appealing to intuition based on everyday experience, we blew it.

Also, I see mention of different kinds of manifolds, but my remark about manifolds-with-boundary has disappeared. These are far too important to omit. KSmrq 01:20, 2005 August 10 (UTC)

The surface of the Earth looks flat as in flat earth. But maybe "looks like a disk" would be more appropriate.

A manifold with boundary is just one type of manifold. A manifold looks locally like a simple space (or can be covered by an atlas with domains in a simple space) and if this simple space is a halfspace, we are talking about a manifold with boundary. It should be mentioned side by side with topological and diffeological manifolds. Markus Schmaus 12:58, 10 August 2005 (UTC)

Good point on the earth not being flat (unless you're from Holland). Do you think it can be fixed? For instance, "suppose that the earth were a perfect sphere, without hills and mountains." It seems a nice way to introduce manifolds.

Regarding the disappearance of manifolds with boundary: mea culpa, mea culpa, mea culpa maxima. I seem to remember that I had a good reason to remove it, but I wanted to see them mentioned somewhere else and somehow it slipped my mind. -- Jitse Niesen (talk) 21:10, 10 August 2005 (UTC)

I haven't come up with anything that isn't too fussy. "Looks like a disk" is no better than flat, but maybe readers will understand they shouldn't take us too literally. Since we talk about chart and atlas as cartography of the Earth, I very much want to keep the Earth example. Maybe, using Markus' idea, we could say something like "people once thought the Earth was flat", and go from there.

I agree a manifold with boundary is, in one sense, just another variation. In other senses, it's special. Some authors describe the interior of the space with Rⁿ, and the boundary region with a halfspace. This may avoid technical problems with using an open subset of the halfspace. Also, most kinds of manifolds admit a version with boundary, so it cuts across the other variations. Maybe Jitse will agree to insert appropriate language in an appropriate place as penance. :-)

Among the variations, we have a choice of ordering. We could start with topological and add structure to get smooth, or start with smooth and remove structure (add flexibility) to get topological. The choice may be influence by the availability of good examples. KSmrq 02:19, 2005 August 11 (UTC)

OK, I've leveraged Markus' flat Earth as proposed. Meanwhile, I spotted this non sequitur:

"Note that T is simply a function between open intervals, so we know what it means for T to be differentiable."

Right; as if high school students know all about open intervals and differentiability! I'm thinking we should remove this from the introduction and introduce it later. Either that or flag it as a comment for advanced readers. KSmrq 03:10, 2005 August 11 (UTC)

Running edit commentary

I seem to be attempting a series of small edits, and I thought it best to say a word about them as I go.

I've added a couple of sketchy paragraphs to the circle example that probably don't belong exactly there, but since I don't presently have a better home, there they are. The flow of the text is roughly thus:

1. Circle as manifold, showing the idea of charts and overlap.
2. Free from single atlas, free from geometric projection.
3. Need for multiple charts.
4. Free from connected.
5. Free from closed (but avoid technical term, "compact", just yet).
6. Free from finite.
7. Free from real.

As noted, this overloads the circle example; but I think the progression may be helpful.

Whenever I give a talk or write a paper, I concentrate on clearly conveying one idea. I expect to see that reflected in the table of contents. The organization of this page doesn't yet have that focus and clarity. A standard rhetorical device is to organize in threes, as in the famous beginning, middle, end dictum. For example, common advice for talks is: (1) tell them what you're going to tell them, (2) tell them, (3) tell them what you told them. Also popular are balanced pairs, as in "this and that".

Just so you know where I'm headed. KSmrq 03:48, 2005 August 10 (UTC)

Do you think it would be a good idea to merge the article into manifold and edit in place? I mean, the merged article does not need to be perfect, just not horrible.

That's just a question. Since you do the work, you decide. :) Oleg Alexandrov 04:28, 10 August 2005 (UTC)

See my earlier answer. At the moment it looks like I'm working alone, but I have no way of knowing if that will last; there seemed to be half-a-dozen folks interested before, either in the original article or its rewrite. I hope to have contributed a few good nuggets of text, a figure or two, and some structural help before the effort is abandoned. KSmrq 04:43, 2005 August 10 (UTC)

Circle slope chart

[Copied from User talk:KSmrq]

By the way, it took me pretty long to grasp the above example [the slope chart picture]. Maybe something like

<?xml version="1.0" encoding="utf-8"?>
<svg width="1014" height="592">
<ellipse cx="505.12020519966717" cy="296.47827308529054" rx="95.72940535020797" ry="95.72940535020797" style="stroke:rgb(0,0,0);fill:none;stroke-width:3"/>
<line x1="409.3907998494592" y1="296.4782730852906" x2="600.8496105498751" y2="105.0194623848746" style="stroke:rgb(180,0,0);stroke-width:1"/>
<line x1="600.8496105498751" y1="105.0194623848746" x2="594.0" y2="118.0" style="stroke:rgb(180,0,0);stroke-width:1"/>
<line x1="600.8496105498751" y1="105.0194623848746" x2="587.0" y2="112.0" style="stroke:rgb(180,0,0);stroke-width:1"/>
<line x1="409.3907998494592" y1="296.4782730852906" x2="600.8496105498751" y2="392.2076784354985" style="stroke:rgb(0,0,178);stroke-width:1"/>
<line x1="600.8496105498751" y1="392.2076784354985" x2="586.0" y2="390.0" style="stroke:rgb(0,0,178);stroke-width:1"/>
<line x1="600.8496105498751" y1="392.2076784354985" x2="590.0" y2="382.0" style="stroke:rgb(0,0,178);stroke-width:1"/>
<path d="M 406.0 293.0 H 411.0 V 298.0 H 406.0 Z" style="fill:rgb(0,124,0);stroke:rgb(0,124,0);stroke-width:1"/>
<line x1="601.0" y1="1309.0" x2="601.0" y2="-718.0" style="stroke:rgb(0,0,0);stroke-width:1"/>
<path d="M 598.0 102.0 H 603.0 V 107.0 H 598.0 Z" style="fill:rgb(180,0,0);stroke:rgb(180,0,0);stroke-width:1"/>
<text x="606.8496105498751" y="123.0194623848746" style="font-size:0;fill:rgb(180,0,0);font-weight:normal">1</text><path d="M 598.0 389.0 H 603.0 V 394.0 H 598.0 Z" style="fill:rgb(0,0,178);stroke:rgb(0,0,178);stroke-width:1"/>
<text x="606.8496105498751" y="410.2076784354985" style="font-size:0;fill:rgb(0,0,178);font-weight:normal">1/2</text><path d="M 502.0 198.0 H 507.0 V 203.0 H 502.0 Z" style="fill:rgb(180,0,0);stroke:rgb(180,0,0);stroke-width:1"/>
<path d="M 560.0 370.0 H 565.0 V 375.0 H 560.0 Z" style="fill:rgb(0,0,178);stroke:rgb(0,0,178);stroke-width:1"/>
</svg>

or

<?xml version="1.0" encoding="utf-8"?>
<svg width="1014" height="592">
<ellipse cx="570.0" cy="296.0" rx="3.0" ry="3.0" style="stroke:rgb(0,124,124);stroke-width:1;fill:none"/>
<ellipse cx="507.0" cy="296.0" rx="63.375" ry="63.375" style="stroke:rgb(0,0,0);fill:none;stroke-width:1"/>
<path d="M 478.0 235.0 H 483.0 V 240.0 H 478.0 Z" style="fill:rgb(0,0,0);stroke:rgb(0,0,0);stroke-width:1"/>
<line x1="507.0" y1="1309.0" x2="507.0" y2="-718.0" style="stroke:rgb(0,0,0);stroke-width:1"/>
<ellipse cx="507.0" cy="255.0" rx="3.0" ry="3.0" style="fill:rgb(180,0,0)"/>
<ellipse cx="507.0" cy="255.0" rx="3.0" ry="3.0" style="stroke:rgb(180,0,0);stroke-width:1;fill:none"/>
<ellipse cx="444.0" cy="296.0" rx="3.0" ry="3.0" style="stroke:rgb(0,124,124);stroke-width:1;fill:none"/>
<ellipse cx="507.0" cy="199.0" rx="3.0" ry="3.0" style="fill:rgb(0,0,178)"/>
<ellipse cx="507.0" cy="199.0" rx="3.0" ry="3.0" style="stroke:rgb(0,0,178);stroke-width:1;fill:none"/>
<line x1="443.625" y1="296.0" x2="481.41930190822734" y2="238.0170843253154" style="stroke:rgb(153,153,224);stroke-width:1"/>
<line x1="443.625" y1="296.0" x2="507.0" y2="198.7719234024745" style="stroke:rgb(0,0,178);stroke-width:1"/>
<line x1="507.0" y1="198.7719234024745" x2="503.0" y2="213.0" style="stroke:rgb(0,0,178);stroke-width:1"/>
<line x1="507.0" y1="198.7719234024745" x2="495.0" y2="208.0" style="stroke:rgb(0,0,178);stroke-width:1"/>
<line x1="570.375" y1="296.0" x2="481.41930190822734" y2="238.0170843253154" style="stroke:rgb(225,153,153);stroke-width:1"/>
<line x1="570.375" y1="296.0" x2="507.0" y2="254.69104104953345" style="stroke:rgb(180,0,0);stroke-width:1"/>
<line x1="507.0" y1="254.69104104953345" x2="521.0" y2="259.0" style="stroke:rgb(180,0,0);stroke-width:1"/>
<line x1="507.0" y1="254.69104104953345" x2="516.0" y2="266.0" style="stroke:rgb(180,0,0);stroke-width:1"/>
</svg>

might be easier to understand.

Sorry for cluttering your talk page. The SVGs don't use all the things you mentioned above, but just illustrate what I'm thinking about. Markus Schmaus 21:55, 7 August 2005 (UTC)

If we're going to talk about the specifics of this illustration much, we should move the conversation to the project. Briefly, yes, this mapping is equivalent to (scaled) stereographic projection, and that's simpler to illustrate. (The factor of 1/2 might be awkward.) But to do so would fight the point I make in the text, that charts need not be based on geometric projection. We don't want readers thinking of homeomorphisms or diffeomorphisms or manifold charts in general as embedded geometry; this example is (among other things) a small step in that direction. It's not such a bad thing if a little mental effort is required to understand the slope chart, because we soon introduce much more abstract examples where imagination is essential. I'm hoping high school students have already learned about the slope-intercept description of a line. (True in my time, but maybe no longer. Sigh.) Also, this particular circle parameterization is heavily used in applications; it is important in older algebraic geometry, and is one of the motivating examples for NURBS.

An alternative would be to drop the four-chart description and go straight to stereographic projection. KSmrq 23:51, 2005 August 7 (UTC)

---

I understand your point and agree with your reasons. But is the slope chart an ideal example for a chart which is not a geometric projection, as it is equivalent to the stereographic projection? What about the angle chart? It cannot be understood as a geometric projection. Or if this is too easy what about the signed distance between a point and (1, 0) in R²? Markus Schmaus 13:16, 10 August 2005 (UTC)

It is really easy to define the mapping θ→(cos θ,sin θ), but not so easy to define the inverse mapping, which is what we need for the chart and for the transition. I expect high school students will have seen trigonometry, but I'm not sure how much. The slope map is the simplest thing I know that comes close to an angle map. I'd have to see what you mean by signed distance, the formulae you'd want to use, to make a judgment. Is the chart a diffeomorphism? If so, is it easy to see why; if not, is that an advantage for distinguishing topological manifolds from differential manifolds? KSmrq 19:40, 2005 August 10 (UTC)

I am enthusiastic about the slope chart example. It is intuitive to high school students who have probably already learned the notion of "slope" and can be understood even without the accompanying parametrization. - Gauge 21:02, 20 August 2005 (UTC)

Clarification needed

Under Intrinsic vs. Extrinsic Views:

The circle can be defined intrinsically by gluing together two copies of the line. We do this by identifying non-zero points in the first copy by their multiplicative inverse in the second copy. This circle is not embedded in anything.

This makes no sense to me. What "line" are we talking about here?

On a different note, we really should try to clarify the relationship between sheaves and atlases if we are going to include varieties and schemes as examples of manifolds according to the definition given here. A scheme is locally built up of affine schemes, which are topological spaces with the Zariski topology, and the atlas would consist of maps that are (open?) immersions of affines. What are the transition maps? I'm guessing that they should be regular morphisms, but I don't know enough geometry yet to know for sure. Btw, the notion of exclusively using something like "Euclidean space" goes out the window in these situations because affine schemes almost never look like Euclidean spaces (not Hausdorff, for instance).

Exclusive use of Euclidean spaces is a straw man opponent, never my position. But you raise the question of if we're going to include varieties and schemes as manifolds. In my view, these objects are closely related, and their similarity to manifolds is worth remarking, but I don't think they are manifolds. Another closely related object is the fiber bundle, which is locally a product space; but, although these may be manifolds, it's not the local construction per se that makes them so. (I notice "Construction" omits them, despite the obvious importance of, say, tangent bundles.) If pressed, I'd waffle on orbifolds as well; there's a reason they have a name of their own.

Currently omitted, and scandalously so, are Riemannian manifolds (requiring a metric) and pseudo-Riemannian manifolds (lest we be mobbed by irate physicists). And we probably ought to mention the specializations to homogeneous spaces and symmetric spaces. These four missing object categories are all important uses of manifolds. KSmrq 07:53, 2005 August 11 (UTC)

Exclusive use of Euclidean spaces is a straw man opponent, never my position.

Fair enough, but I never said that it was your position.

If pressed, I'd waffle on orbifolds as well; there's a reason they have a name of their own.

Yeah, I reneged my earlier position on orbifolds as manifolds. I agree that orbifolds are different beasts. As for Riemannian manifolds: they deserve to be mentioned, but trying to give a technical description of them goes beyond what I perceive to be the scope of this article. Symmetric spaces are also important and deserve mention, and it should not be too difficult to give a rough description and a pointer to another article providing more details. - Gauge 17:58, 11 August 2005 (UTC)

Finally, according to the simple space definition, orbifolds are not generalizations of manifolds... they are just manifolds, where we admit quotients of Euclidean space by finite group actions as "simple" enough. - Gauge 03:35, 11 August 2005 (UTC)

Actually, regarding orbifolds this is not quite good enough; firstly, the finite groups at the various singularities may differ; secondly, the group actions must be compatible in the sense that a restriction of open sets induces a group monomorphism. There are more conditions but I won't list them here. See Thurston Geometry and Topology of 3-Manifolds pg. 300. However, there is a theorem (also in the reference pg. 302) that if M is a manifold and Γ is a group acting properly discontinuously on M, then M/Γ is an orbifold. - Gauge 04:24, 11 August 2005 (UTC)

Orbifolds are not manifolds, varieties are not manifolds, schemes are not manifolds, bundles are manifolds. I think any definition of manifold that allows otherwise is a nonstandard definition. It's nice to give preference to the "intuitive idea" of a manifold, in which case you might allow those others, but not at the cost of a correct definition. My 2 cents. -Lethe | Talk 20:17, August 11, 2005 (UTC)

This article is about the concept of gluing simple spaces together, not for discussing topological manifolds or differentiable manifolds. Manifold is not supposed to be used as a synonym for either in this article. Manifold is thus not a technical term and has no rigorous definition, only its instances do. In this way schemes are manifolds. Orbifolds may or may not be manifolds, they sure are closely related and deserve mention. --MarSch 13:31, 14 August 2005 (UTC)

Oh, almost forgot. "The line" is R, just as "the circle" is S and "the plane" is R^2. Now does it make sense to you(please sign your edits)? --MarSch 15:35, 14 August 2005 (UTC)

That was me; I think the signature was pushed down the page by other edits. I am happy now that "the line" refers back to the circle example. There is still a fundamental issue that I don't think we have yet resolved:

Question: Do we require manifolds to be topological spaces a priori, or do we require only a set with appropriate chart maps whose images can be used to define a topology?

The current wording of the article now suggests the latter, and I am in favor of this as it is a more general definition. I prefer to start with the simplest possible structure like this and build up to the smooth and/or Riemannian cases from there, as these latter things are the most complex (from a foundational point of view). It would be nice to settle on a definition of "a manifold", if only to be able to distinguish manifolds from non-manifolds. My feeling right now is that schemes should not be labeled as manifolds unless we can demonstrate that

• a) An atlas (in our sense) can be recovered from the structure sheaf of any scheme, complete with a description of what the transition maps should be
• b) The structure sheaf and the atlas can be computed from one another for any scheme. - Gauge 21:35, 20 August 2005 (UTC)

TOC and construction section

It seems now would be the time to shorten the table of contents to a reasonable length. In the construction section, I have difficulty distinguishing immediately between the "Ideas" and the "Examples of the Ideas" because, despite the use of indentation in the examples themselves, the headers are still on the left since they are sections. Ideally, we could indent the headers as well. Does anyone know how to do this? - Gauge 03:48, 11 August 2005 (UTC)

Before we resort to technical wizardry, let's try some old-fashioned editing. Isn't it strange that "Construction" and "More examples" list the same items? If we fix that properly, the TOC will be half its present length. KSmrq 06:48, 2005 August 11 (UTC)

"Hausdorff" section

This section could be expanded to "Local topological criteria" or some similar title and refer not only to Hausdorff-ness, but also paracompactness (useful for partitions of unity), local metrizability, and so on. In other words, brief explanations of any local "niceness" conditions could go here. I'm not going to be around for the next week, so feel free to overwrite the old article while I am away. - Gauge 00:55, 13 August 2005 (UTC)

hi guys

I'm glad to see all the hard work that has gone into improving this article. Are we ready to replace the original yet? Either that or very close I think. Anything worthwhile that is not yet incorporated should be added to this version. But it probably already has a home here at or one of the other two articles. Talking about those, I see no activity there. How come? I also think the time has come to do a request for comments or whatever it's called. --MarSch 15:41, 14 August 2005 (UTC)

I've been delaying working on the other articles until the dust settles on this one. It will then be clearer to me what should be included and excluded from the other articles. - Gauge 21:40, 20 August 2005 (UTC)

Charts construction

I can't see any independent "charts" construction, in fact the sphere in the example is constructed as zeros of x² + y² + z² - 1. Markus Schmaus 12:08, 15 August 2005 (UTC)

Well, I wanted to reorganize stuff but I ran out of time yesterday so I didn't explain my edit. My goal was to get rid of the "More examples" section (and to clean up the "extrinsic and intrinsic" section in the process). One way to do that is simply deleting it, but I decided against it because the sphere is one of the few examples fully explained (though it might also have to do with the fact that I wrote it myself). I admit that I hadn't thought about putting it in the "zeros of a function" section. However, it does seem to me that there is an independent "charts" construction (the name sucks though): for instance, if you define the torus by the parametrization

x(u, v) = (R + r cos v) cos u

y(u, v) = (R + r cos v) sin u

z(u, v) = r sin v.

How would you call this? -- Jitse Niesen (talk) 13:53, 15 August 2005 (UTC)

Hey Jitse, you need to be more precise. What is the domain of this map? It might be R^2, then we have an immersion I think of R^2 into a subset of R^3 diffeomorphic to a torus, or the domain might be the torus T^2, in which case this would be an embedding of the 2-torus into R^3. In the first case, you might imagine cutting R^2 into small pieces which are all mapped injectively into R^3. Those restrictions would form an atlas of this torus.

--MarSch 12:43, 16 August 2005 (UTC)

I meant the first possibility: domain is R^2, chop it up suitably and you get an atlas, thus proving that the torus is a manifold. With torus, I mean the image of R^2 under the map, not S^1 x S^1. -- Jitse Niesen (talk) 16:37, 16 August 2005 (UTC)

In other words a covering map. Markus Schmaus 18:18, 17 August 2005 (UTC)

Zeros of a function construction

[moved by Markus Schmaus]

Markus, I disagree with the notion of a "zeros of a function" construction". The implicit function theorem is used to prove the existence of charts. It always comes down to charts.

--MarSch 12:43, 16 August 2005 (UTC)

What do you disagree with? That a topological manifold is a locally Euclidean topological space? Markus Schmaus 18:18, 17 August 2005 (UTC)

See Talk:Manifold/rewrite/freezer#Intro_Again for previous relevant discussion. Some quotes from there:

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

Gauge: 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.

Gauge: 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.

I agree that this is still an issue that we need to resolve. I am in favor of requiring the presence of an atlas (or an equivalent) in order to call a topological space a manifold, even if the space is homeomorphic to a subset of some ambient space consisting of the zeros of some function. Markus has shown that topological manifolds which are homeomorphic as topological spaces are isomorphic as topological manifolds because the charts transfer. However, if an atlas carries additional conditions on the chart maps (such as differentiability), a similar argument will not hold, and in fact there are differentiable manifolds that are homeomorphic as topological spaces (exotic spheres, exotic R⁴ and so on). The difference is that the atlases are not uniquely determined by the topology as in the case of topological manifolds. Thus, the atlas is intrinsic to the definition of most other types of manifolds. Therefore, I am arguing for the sake of consistency that every manifold must be assigned some atlas (or some equivalent thereof). I am not convinced that such a topological embedding yields a canonical choice of atlas. - Gauge 22:36, 20 August 2005 (UTC)

intro criticism

Current intro reads:

In mathematics, a manifold generalizes the idea of a surface. Technically, it can be constructed using multiple overlapping pieces to form a whole and is, in this sense, like a patchwork. On a small scale manifolds are always simple; on a large scale, they have rich flexibility.

• First off, I dislike the "manifold generalizes surface" idea.
  • I guess pedantically speaking going from dimension 2 to arbitrary (finite) dimension is a generalization, but it is trivial. It doesn't help to talk about it in this way. What could be said is "2-dimensional manifolds are surfaces", but I still don't find it very helpful.
• Second I dislike the word "technically".
  • Does this add insight or information? Or does it merely look expensive?
• Going on to gluing (overlapping pieces).
  • pieces of what? What does overlapping mean in this context? Constructed how? Gluing should be mentioned here! Or sewing or whatever your metaphor of choice is. We should also be specifying what those pieces are made of. Pieces of paper or cloth or whatever. Unfortunately my high-grade paper example is gone.
• Patchwork
  • I like this metaphor. I like multiple metaphors.
• rich flexibility
  • this is way too vague for my taste. What is meant is that different manifolds can differ globally, but that they are all equal locally. The way it stands I would read as each single manifold having rich flexibility, which makes little sense.

I also dislike the "In the remainder of this article...". Completely unnecessary. I would like to integrate the parts about terminology coming from cartography, which is another valuable metaphor, into the part currently headed "introduction" and remove that heading. I think we should fully exploit the cartography metaphor, by covering the Earth with some charts (as I did in a previous version). Please comment. --MarSch 13:08, 16 August 2005 (UTC)

The history of the lead paragraphs and introduction section consists of extensive edits and debates, with the current version something of a truce — whether uneasy or happy I do not know. I strongly recommend that we let sleeping dogs lie. After the rest of the article is satisfactory perhaps we will want to revisit this question.

For example, early on I introduced the word "technically" to which you object, and myself have since come to feel dissatisfied with it. Were this my personal writing, I would revise it; as it is not, I feel it best to let it stand. Likewise, I introduced the simile (not metaphor) of a "patchwork quilt". In the course of discussion that changed to fiberglass and yarn and then back to "patchwork", with other materials, arrangements, and attachments considered along the way. It got pretty exciting. I disagree with your reading of "rich flexibility" as making little sense; the meaning is what you expect, and I rather doubt readers imagine a manifold flapping in the breeze. But again, this wording was arrived at through honing of language that raised even more objections.

Here's a metaphor you may appreciate: The beginning of this article is a Bermuda triangle, into which much editing effort has mysteriously disappeared! Yet the end of the article, despite my exhortations (but without my edits), still does not mention Riemannian manifolds, which seems much more serious a problem. Maybe we should reverse the order of the sections until we are done editing, and see if that helps. :-D --KSmrq 14:53, 2005 August 16 (UTC)

Hi KSmrq, what _is_ the difference between metaphor and simile? I don't think the fact that any particular other part of this article is still lacking is any reason to not discuss the intro and try to improve it. Riemannian manifolds should be discussed in the diff. mani. article anyway, which should be summarized here. Unfortunately a lot of the headings I created for that purpose have reverted back into the dotted list very similar to the original article. Thus it looks finished but isn't. I don't understand how you can say that rich flexibility is clear. Even when I interpret it as it was no doubt intended I find it a particularly bad way of expression.--MarSch 09:48, 18 August 2005 (UTC)

To say "A is like B" is to use a simile; to say "A is B" is a metaphor. Thus when the lead says "… is, in this sense, like a patchwork", that is a simile — a noting of similarity between two things; whereas when I commented "… is a Bermuda triangle", that is metaphor — one thing standing in the place of another. As a mathematician is careful about the difference between isomorphism and equality, so a writer is careful about the difference between a simile and a metaphor. To be even more precise, we hope the patchwork is more than a simile; we intend it as an analogy.

The shortcomings of the main content are a very good reason to neither discuss nor edit the intro at this time. Again, I strongly urge you to look through the history of the article and the frozen talk page.

My response continues below. --KSmrq 13:34, 2005 August 18 (UTC)

I have already read everything in there before my recent edits after my break. Therefore instead of editing the intro I'm here stating my views and trying to discuss. You seem only interested in delaying this. Unfortunately no one else seems to have anything to say.--MarSch 12:59, 20 August 2005 (UTC)

I would agree with KSmrq about letting the introduction stand for now. There is a lot of work to be done below. Oleg Alexandrov 21:12, 20 August 2005 (UTC)

Dito. I usually write introductions last and the current one is adequate. -- Jitse Niesen (talk) 22:36, 20 August 2005 (UTC)

Goals

Now on to more productive topics. When you say "it looks finished but isn't", I agree, and have said so previously on this page. The question is, what to do about it? This is one area where wiki collaboration is awkward, because reorganization is not local edits, but an all-or-nothing commitment which is either accepted or reverted. Nevertheless, I'd be delighted if we can work together and bring this rewrite to a satisfactory conclusion. To that end, can you provide a link to the version of manifold that you consider your best effort, so we have a common point of reference? I'll read that and compare it to the state of the rewrite. Also, it might help if you could (briefly) state which topics or aspects of presentation are most important in your view. Here's my concept:

1. The lead should work for a total non-mathematician who knows nothing about topology or calculus or set theory, to give them a general sense of what we mean by "manifold".
2. The intro should work for a bright teenager with nothing more than basic algebra (quadratic equations) and elementary geometry (perhaps Euclid's Elements), and should deepen the orientation. It should introduce terminology, but informally.
3. I like the idea of an example or two to show why we need this patchwork thing. Some important points:
   • We can't just bend a line segment and make a circle; the topology won't work.
   • We want to be able to do geometry and calculus, and need a way to carry over "flat" tools.
   • We need to be able to work "self-contained" (intrinsically), without an ambient space.
4. By now I'm impatient, and want to see a formal definition of a basic version of a manifold. I hope we can still make some sense to the youngster, but I'm ready to talk to the undergraduates. One idea of a basic version is a differentiable manifold, since calculus on manifolds is so important. (A topological manifold is a generalization by removing structure.) An alternative basic version would be a topological n-manifold, including the Hausdorff condition, something along these lines:
   • 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.
   • 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.
5. Now I'd like one or two more brief technical examples, and also counterexamples (to clarify the conditions).
6. This much alone should be the core of the article. Everything else is relatively brief.
7. We now enumerate different kinds of extra structure, or alternative definitions, and name (and link) the flavors of manifold so produced. Here, or in the intro, we should draw connections to other parts of mathematics. This section can talk to graduate students and beyond.
8. Wrap up with "History", "References", "See also", and "External links".

I want an article that's just long enough to clearly say what it has to say. It should flow naturally from start to finish. It should be engaging, but not sloppy. It should be well illustrated. (Also, I'd like to complete my part in this quite soon.)

So, maybe that's a lot to ask, and maybe is not exactly what others have in mind. For example, currently the rewrite includes a long section on construction, with many examples; I'm inclined to spin that off into a separate article. Everyone, do we agree on goals? --KSmrq 13:34, 2005 August 18 (UTC)

Yes, that is a good outline in my opinion. Some clarifications, to make sure I don't misunderstand: with "We can't just bend a line segment and make a circle" you mean that one needs two charts to cover the circle. If we talk about calculus on manifolds, we should include the definition of differentiable manifolds (which can be done in one sentence: a topological manifold with differentiable transition maps). -- Jitse Niesen (talk) 18:45, 19 August 2005 (UTC)

Thanks for responding; I was feeling a bit lonely here! :-)

Correct, I mean two charts; R¹ is not homeomorphic to S¹. Differentiable transition maps do not imply the charts are differentiable, which we also need; still, it should be easy enough. One other important point I forgot to mention: We sometimes need boundaries. This is not just another flavor because it cuts across all the flavors. --KSmrq 19:26, 2005 August 19 (UTC)

The older versions of this rewrite of which I was thinking are very bare when I look again. There is no version to which I would like to point. I only think that the section "Other types and generalizations of manifolds" is too compactly written. I would like to see some more headings. Certainly for complex manifolds and schemes/varieties. These sections should summarize the articles on those subjects. Those articles are far from complete themselves, though.

I don't think there should be a lead and an intro as there is now. There should be just one intro. It should use simile/analogy/metaphor to tap into physical intuition. I dislike stating exactly who should be able to read something. Everything should be made as simple as possible. I would like this whole article to be easy for five-year-olds, but that is probably pushing it. I would like to point out that the original purpose of the rewrite was to separate technical stuff from the intuitive notion of a manifold. For that purpose we have (the rewrites of) [topological manifold] and [differentiable manifold] (I have the impression that you still haven't looked there). I'm wondering if we shouldn't move a lot of the "construction" section there. Also those articles are for holding rigorous definitions. This article on the other hand is for pointing out the common theme connecting all different kinds of manifold. It is interesting to think about boundaries in that light. All examples of manifolds with boundary have simple spaces that are vector spaces. Is there such a thing as a scheme with boundary? An affine scheme with boundary?--MarSch 13:47, 20 August 2005 (UTC)

I don't understand "Differentiable transition maps do not imply the charts are differentiable, which we also need." I also don't see the problem with having a lead and an introduction. The lead has to be short, but we have more to say, so we have an introduction as well. I think the definitions are not overly technical, once you have explained what charts and transition maps are; the only technical bits are Hausdorff, paracompact, etc. I think some of the constructions can indeed be moved to other articles. -- Jitse Niesen (talk) 22:36, 20 August 2005 (UTC)

I agree that the constructions should be moved to other articles, and recommend moving them to articles about the respective manifolds (torus, cylinder, klein bottle, etc.) Their specific constructions only serve to clutter the article. The circle/sphere can remain as our main illustrative example. - Gauge 23:40, 20 August 2005 (UTC)

Great! I glimpse light at the end of the tunnel. :-D

• Jitse: Two charts can be differentiable where they overlap, with differentiable transition maps, yet have singularities elsewhere; our calculus tools would not survive.
• MarSch:
  1. We separated lead and intro because the TOC was pushed too low before.
  2. In spirit, I agree with writing for a broad audience throughout; realistically, we can't hope to do that for, say, Kähler manifolds. My gradations reflect Jitse's original partition goals, stated at the top of the rewrite.
  3. Articles Manifold/rewrite/topological manifold and Manifold/rewrite/differentiable manifold are clearly important, but not on my present agenda.
  4. We had decided, though not yet rewritten, to say that schemes are related to manifolds. Still, I probably should say "cuts across many of the other flavors", not all.

Everybody, I'd prefer someone besides me (Gauge?) to move constructions. --KSmrq 10:59, 2005 August 21 (UTC)

Grammar questions

I am not a native speaker of English, and this is why I have these maybe naive questions.

The text currenty says:

• A manifold can be characterised as looking like Euclidean space
• rotations in three-dimensional space.

Should these be respectively

• A manifold can be characterised as looking like an Euclidean space
• rotations in the three-dimensional space?

That is, should there be the articles "an" and "the" in places? Thanks. Oleg Alexandrov 17:30, 17 August 2005 (UTC)

The English language baffles even native speakers. The US spelling is "characterize", for example. As to your questions, both changes are definitely wrong.

• The wrong article is used in "an Euclidean space", because "Euclidean" starts with a sound like "yell", which calls for "a", not "an". The present "looking like Euclidean space" may suggest 3D space, which is misleading; so "looking like a Euclidean space" may be better.

  an ape, a bag, a cat, a dog, an egg, a fig, a gas,

  a hat (but an honor — because the "h" is not pronounced),

  an ion, a jig, a keg, a lid, a man, a nod,

  an orb (but a one — because it is pronounced like "won"),

  a pen, a quip, a rag, a sun, a tip,

  an urn (but a union — because it is pronounced with a "you" sound),

  a van, a win, a yes, a zoo.

Of course there must be no "an". My mistake. So you agree that there should be a "a" article. Oleg Alexandrov 18:31, 18 August 2005 (UTC)

Yes. --KSmrq 06:15, 2005 August 19 (UTC)

• Saying "in the three-dimensional space" is a more subtle mistake. It inadvertently raise the question of what 3D space (we haven't mentioned one), whereas omitting the article "the" allows the proper assumption, our ordinary Euclidean space. The article "a" would be quite wrong, because it implies we have a choice of space, much as if we had said "some". --KSmrq 18:14, 2005 August 18 (UTC)

Differentiable manifold requirements

I agree with Jitse that differentiable transition maps is all that is required (and all that makes sense/has meaning). The charts are declared to be C^k exactly when the transitions maps are C^k. Perhaps an example to enlighten us of what you mean?--MarSch 11:33, 21 August 2005 (UTC)

(MarSch: I've moved this to a new section because it's not really about goals.) The question turns on a subtle point: Do we require a transition map from a chart to itself? If not, a chart inherits C^k from the transition maps only where it overlaps another chart; C^k in one part of a chart does not imply C^k everywhere. And there is no requirement that every point of a manifold be covered by two charts. In fact, we may have only one chart for the whole manifold. So we must stipulate both: charts (and their inverses) and transition maps are C^k. Alternatively, we make it clear that a C^k "transition map" must also exist from a chart to itself.

Here's the way Bishop and Goldberg describe it:

• Two charts μ: U→R^d and τ:V→R^e on a topological space X are C^∞-related if d=e and either U∩V=∅; (the empty set) or μ∘τ⁻¹ and τ∘μ⁻¹ are C^∞ maps.
• A C^∞ atlas is one for which every pair of charts is C^∞-related. A chart is admissible to a C^∞ atlas if it is C^∞-related to every chart in the atlas. In particular the members of a C^∞ atlas are themselves admissible. (Emphasis added.)

Thus even for a single chart, μ, we require μ∘μ⁻¹ to be a C^∞ map. (Hmm, this seems weaker than requiring both μ and its inverse to be C^∞; is it?) Hope that helps.

A valuable point B&G mention is that a difficult theorem of Whitney guarantees that any C¹ manifold can be made real-analytic (and hence C^∞), in many ways; but (harder still) a C⁰ manifold may fail to become C¹. --KSmrq 13:38, 2005 August 21 (UTC)

But μ∘μ⁻¹ is the identity, so it's always smooth, isn't it? Perhaps it helps to have an example in mind. Take the (single, infinite) cone

$C=\{(x,y,z)\in \mathbf {R} ^{3}:x^{2}+y^{2}=z^{2},z\geq 0\}.$

Take projection on the xy-plane as the one and only chart. Is this a differentiable manifold? Apparently, yes, even though it runs counter to the definition. -- Jitse Niesen (talk) 15:15, 21 August 2005 (UTC)

I don't think this can be diffeomorphic to the same cone with the structure inherited from $\mathbf {R} ^{3}$. Septentrionalis 14:51, 26 September 2005 (UTC)

PS: I meant "intuition" instead of "definition" -- Jitse Niesen (talk) 14:33, 24 August 2005 (UTC)

Hey nice example :) KSmrq, you cannot require that a chart is C^k because it has no meaning separately from transition maps. We say that a chart is C^k iff all transition maps involving it are C^k. This is exactly what I read in the def you quote and your emphasis baffles me. Also as Jitse said the identity (from some part of R^n to itself) is always smooth. Maybe you're confused because you insist on defining manifolds as subsets of R^n? Then you could disagree with Jitse's example.--MarSch 12:22, 23 August 2005 (UTC)

Actually, the more I look into it, the more I'm convinced that, to my surprise, there are no constraints on the charts. I am forced to conclude that any continuous height field, z=f(x,y), is not just a C⁰ (topological) manifold, but in fact C^∞. When did I ever insist on subsets of Rⁿ? Never. But I'm wondering if my thinking is influenced by Riemannian manifolds. Anyway, give me a moment while I rearrange my mental furniture. In the immortal words of the late, great Gilda Radner, "Nevermind." --KSmrq^T 05:03, 2005 August 24 (UTC)

To be honest, I was also certain that a cone is not a differentiable manifold. And I even passed an exam on differentiable geometry! I guess I'm still learning, which means I'm still alive :) -- Jitse Niesen (talk) 14:33, 24 August 2005 (UTC)

It all depends on what structure you insist on inheriting from R^n. (KSmrq, I know you didn't say this, I was just thinking of what might be happening in your head.) Continuing on that line an everywhere non-continuous height function can be used. The only requirement on the charts is that they be invertible, thus bijections. Every structure that you want (including topology) can then be transported. The cone as constructed by Jitse is a topological submanifold of the topological manifold R^3, but it is not a differentiable submanifold, even though it is a differentiable manifold. --MarSch 15:29, 28 August 2005 (UTC)

My head hurts. j/k :-) - Gauge 17:05, 28 August 2005 (UTC)

second sphere example

Do we really need the second sphere example? the one with two charts. It's just stereographic projection, but somehow I find it much less enlightening than the first construction. If there is no strong reason to keep it here I propose to move it to circle.--MarSch 12:30, 23 August 2005 (UTC)

Extra construction sections reduced to examples

I just made some edits to the page, including changing the extra constructions into short examples and leaving out the technical details, which were already present on the respective article pages (torus, real projective space, cylinder, Klein bottle), with the exception of the latter two (I will fill these in as I find time; they are on my talk page if you'd like to do it yourself). Besides those things, the only other things that I see lacking with the article currently are:

1. There should be mention of Riemannian manifolds in the other types and generalizations section, and more importantly...
2. There needs to be a good description of what a "boundary" of a manifold is, since these are necessary to understand some of the methods of construction.

Happy editing, Gauge 05:27, 24 August 2005 (UTC)

Merge!

It's time to move Manifold/rewrite to Manifold. Markus Schmaus 19:55, 7 September 2005 (UTC)

Merge!

It's time to move Manifold/rewrite to Manifold. Markus Schmaus 19:56, 7 September 2005 (UTC)

Merge!

It's time to move Manifold/rewrite to Manifold. Oleg Alexandrov 21:38, 7 September 2005 (UTC)

Merge!

I extend my thanks to everyone who has contributed to make this article what it is today. Merge! - Gauge 03:41, 8 September 2005 (UTC)

Merge!

I concur. We can do two things to preserve history of the old page. Move it to manifold/old, or merge histories from the moment of the split (we loose history on old version after split). move --MarSch 15:13, 8 September 2005 (UTC)

Also could someone move top. and diff. manifold. I can't get it to work --MarSch 15:24, 8 September 2005 (UTC)

Hey people, are we merging this or not? By the way, it is possible to merge the articles while merging the edit histories, see Wikipedia:How to fix cut and paste moves. In this case, merging manifold and manifold/rewrite would produce a bit of a mess since in the last two months both artilcles were edited, but the mess would not be really huge since manifold was not edited a whole lot. So, which way to go, and most importantly, when? Oleg Alexandrov 02:41, 26 September 2005 (UTC)

Actually, I am not sure any longer. I asked at Wikipedia talk:WikiProject Mathematics, and requested people to opine in here. Oleg Alexandrov 04:33, 26 September 2005 (UTC)

I'm not sure the merged edit histories will be harder to follow than an edit war; but then I'm not an admin, and don't have to do it. Another possibility would be to userify the old article [or move it to Mannigfaltig and make it a redirect], move this one, merge in the old one, and leave a note on this talk page. Septentrionalis 14:54, 26 September 2005 (UTC)

OK, if I don't here other opinions anytime soon, I will merge the page histories. I also think that it is the best thing to do. Oleg Alexandrov 23:22, 27 September 2005 (UTC)

I prefer the "merge histories" approach, but it really doesn't matter to me since I'm not the one doing it. I think the recent edits to the rewrite were pretty good, btw, in terms of clarifying what we were trying to say :-) - Gauge 04:21, 28 September 2005 (UTC)

I never tried to merge the histories, but I think it would give a bit of a mess for the last two months. It seems better to me to move manifold to manifold/old and manifold/rewrite to manifold, adding a note at talk:manifold about what happened? Of course, somebody needs to check that everything from the current article has in fact found a place in the new articles. -- Jitse Niesen (talk) 23:29, 28 September 2005 (UTC)

Sounds like a good plan to me. - Gauge 23:17, 2 October 2005 (UTC)

Move of manifold/rewrite to manifold is complete

I implemented Jitse's suggestion. The old article is manifold/old, and its talk is talk:manifold/old. Also see talk:manifold/rewrite/freezer. If people feel like merging the histories later, it can still be done. Oleg Alexandrov (talk) 13:14, 18 October 2005 (UTC)