Talk:Manifold

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

I don't agree with the new headings

6 Symplectic manifolds

7 Complex manifolds

8 Kähler and Calabi-Yau manifolds

9 Lie groups

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

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

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

First paragraph

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

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

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

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

looks easier to understand than:

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

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

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

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

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

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

Reasons for rewriting the introduction

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

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

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

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

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

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

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

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

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

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

Such a simple space is the simplest example of a manifold.

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

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

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

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

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

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

topological manifolds

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

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

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

examples

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