Talk:Manifold/Archive 3

message: I quote Rick Norwood: "Also, the sides of a line are different from the ends of a line." Apparently you do agree that a line has ends. For me if the line is ]-∞, ∞[, then any set ]-∞, a[ with a in ]-∞, ∞[ is one end of the line. The other end is any non-overlapping set from the other infinity. The line has these things, but perhaps "end" is not the best name for it. What better name do you suggest?--MarSch 16:06, 28 October 2005 (UTC)
thread_title: about ends of a line

message: One more thing. Rick, you should be less bold. Both here and at mathematics, there was an insane amount of discussion of many people for months and if not longer about how to best write things. As such, please be more conservative with big rewrites. Thanks. Oleg Alexandrov (talk) 02:19, 29 October 2005 (UTC)
replies: text: Yes, I would not like to see things change so hastily without a thorough discussion beforehand. To future editors: please make sure you read the old talk archives before making serious changes to this page. There have been literally months of careful debate over various parts of this article, and we can all be more productive if we discuss major changes here on the talk page before they are made. - Gauge 07:21, 29 October 2005 (UTC)
replies: text: I agree.  Rick Norwood 23:34, 29 October 2005 (UTC)
thread_title:  And my reaction.  

message: "One model of the universe visualizes it as a 3-manifold. This is not easy to picture -- you can't build a universe in a classroom -- but imagine the universe as the interior of two balloons, and then imagine one of the balloons turned inside out and the skin put inside the other balloon. One strange result is that you can start at the center of one balloon, travel through the overlap into the other balloon, cross the other balloon, reenter the overlap, and return where you started. In other words, if this model of the universe is correct it would be theoretically possible to circumnavigate the universe. Many science fiction stories have been based on this idea."
This looks like a broken attempt to describe a klein bottle, but the wording doesn't really make sense.
At this prominent position of the intro and together with the sphere and the circle as examples we get the impression that all manifolds would be  bounded, which is not true as every euclidean space is a manifold, too.
84.160.236.106 17:50, 30 October 2005 (UTC)
thread_title:  This part is unclear and missleading 

message: "A (topological) n-manifold is a separable metric space each of whose points has an open neighborhood homeomorphic to either Rn or to Rn+..."  3-Manifolds, John Hempel.
Clearly the requirement that the neighborhoods be open means that in a connected space, such as the circle written as the union of two open neighborhoods, those neighborhoods must have open, non-empty intersection.
I could cite half a dozen other books that say the same thing. Rick Norwood 15:17, 31 October 2005 (UTC)
replies: text: I'm guessing here that this comment is directed at me. I completely agree. The problem I'm having is with your blindly putting "overlapping" in, without regard for context. When gluing a manifold together the pieces you start with are completely separate. Only after you've glued them together do they overlap anywhere. This also implies that you CANNOT glue them together where they overlap and accomplish anything. This would be a non-action. When you glue you specify where you glue and this can also be done by MAKING the separate copies overlap and then glue them together where they overlap. It is a clumsy way of saying it, but at least it is correct and uses the word "overlapping" which should make you happy. --MarSch 12:01, 1 November 2005 (UTC)
thread_title:  We've been here before. 

message: "Ad 3) If you had read the old discussion pages, specifically, Talk:Manifold/Archive2#Time to merge the stuff?, you would have seen that I there give the reference Abraham, Marsden and Ratiu, Manifold, Tensor Analysis, and Applications, Definition 3.1.7. 
Ad 4) The same book starts with defining manifolds modeled on Banach spaces with possibly infinite dimensions (Definition 3.1.3). 
-- Jitse Niesen (talk) 14:06, 31 October 2005 (UTC)
replies: text: As I'm sure you know, the book you cite is notorious for the number of errors it contains.  However, both of the definitions you cite are correct, and both agree with my points.  Definition 3.1.7 says that a manifold is made up of open subsets of a Banach space.  A Banach space is a normed linear space and a linear space is Rn with additional structure added, so every manifold is made up of pieces homeomorphic to Rn.  Definition 3.1.3 mentions that the dimension may be infinite.  That is not a problem, but you cannot have some of the pieces of finite dimension and some of infinite dimension, nor can you have some pieces of countably infinite dimension and some of uncountably infinite dimension.  (I do not think we need to go into the difference between the topological dimension and the banach space dimension.)  The important thing is that, according to your own reference, every piece has the same dimension and if pieces intersect, they intersect in open sets.  Rick Norwood 18:43, 31 October 2005 (UTC)
replies: text: Somebody had better say 'connected' very quickly. Charles Matthews 18:54, 31 October 2005 (UTC)

text: Good.  We agree.  (It was to take care of the possibility of a manifold that was not connected that I included the qualifier "if pieces intersect".)  The introduction should certainly include the possibility that the "n" in Rn could be ω.  The introduction should mention the infinite dimensional case.  With all this in mind, I am going to attempt another rewrite.  I'll try to change as little as possible and keep it short.  Another thing that needs fixing -- the word "chart" is only used in the differentiable case, in the topological case, use "open n-ball".  But I'll leave that for later.  One thing at a time. Rick Norwood 22:24, 31 October 2005 (UTC)

text: (Had an edit conflict. Sorry about the ordering of comments.) The correct title uses the plural: "Manifolds, Tensor Analysis, and Applications", . I can't speak to the errors from direct experience, not having a copy at hand; however, the Amazon reviews complain of typos. One mathematics professor describes it to his students as "A very well respected, application oriented text that immediately works in infinite dimensions." There is a SIAM professional review, but I can't read my downloaded copy for some reason. Marsden has a web page saying a third edition is in preparation; perhaps it will have fewer typos. My personal experience of Ralph Abraham is positive, and Marsden is exceptional. It is hardly likely they would make major mistakes on definitions at the core of the book, especially aspects that are essential to the applications they discuss. Our prior discussion proposed that we distinguish n-manifolds, which are mainstream, while acknowledging these more general manifolds with mixed and infinite dimension, which are outliers but valid and notable. --KSmrqT 22:53, 31 October 2005 (UTC)
thread_title:  Abraham, Marsden, and Ratiu 

message: I have noticed that one problem in Wikipedia, not just here but in many other articles as well, is that it is easy to spend countless hours on the opening paragraph and give not nearly enough attention to the rest of the article.  I am going to move on, and fix the last two paragraphs of the introduction.  Rick Norwood 01:12, 1 November 2005 (UTC)
replies: text: I prefer the term "map" for a function. "Transition function" sounds awkward, which means that I probably haven't heard people use it often/ever. If there are places where "map" should be replaced with "chart", please do so, keeping in mind that charts are maps. - Gauge 04:45, 7 November 2005 (UTC)

text: In most of my experiences with topology, functions between topological spaces (differentiable or not) have been called maps. Whether a "map" conjures up geographical notions is in the eye of the beholder, I suppose, but personally it never occurred to me to make this association. We can link to the word map easily enough. If the term "function" really is preferred to "map" in topological manifolds, then we can make a special note about that case, but in general usage I almost always have seen "map" used instead. By the way, could you give some references to this usage of "function" in topological manifold theory?  - Gauge 04:12, 9 November 2005 (UTC)
thread_title:  Moving right along 

message: This is called a quotient space, and it sometimes, but not always, results in a manifold.  The potential for misleading the reader is so great that I don't think this paragraph sould be in an introductory article.  If we want it in the article, then we need to present necessary and sufficient conditions for when a quotient space is a manifold, and some examples, such as the torus formed as a quotient space of a square.  Rick Norwood 19:36, 18 November 2005 (UTC)
replies: text: I changed the paragraph to stress this point. This topic is so large that it really needs its own article. I suppose quotient space would be the appropriate starting point. - Gauge 02:08, 17 December 2005 (UTC)
replies: text: Rick: I think it may be misleading to call a manifold resulting from a quotient construction a "quotient space". Quotient spaces in general need not be manifolds, and I think we should keep this distinction clear. Secondly, the point I was trying to make in the remark about "pathological spaces" is that quotients are not even remotely nice in general; here I think of the bad foliations arising in noncommutative geometry, for example. There are also examples of nonmeasurable sets arising naturally as quotient spaces. - Gauge 02:37, 23 December 2005 (UTC)

text: A ring is an additive category with one object, of course ;-) Yes, I agree that the above things you mention are quotient spaces. My gripe is that quotient spaces need not be manifolds, and we should make this clear. This is the sentence that I think may be misleading:  "There is, however, no reason to expect the result of this operation to be a manifold. If it is, it is called a quotient space."  I would rather say that you get a quotient space regardless, but it may not always be a manifold. - Gauge 00:12, 25 December 2005 (UTC)

text: Okay, I took a stab at it. Let me know what you think. - Gauge 01:46, 26 December 2005 (UTC)
thread_title:  gluing along boundaries 

message: I am not a native speaker of English but wouldn't it be rather an Euclidean space? 84.160.245.39 20:13, 29 December 2005 (UTC)
replies: text: Yes, it should be "an". Rick Norwood 22:54, 29 December 2005 (UTC)
replies: text: Like 'an Yuletide greeting from an user'? No, it is pronounced 'yewclidean', so it is a Euclidean space. Charles Matthews 22:57, 29 December 2005 (UTC)
replies: text: And an 'appy 'olliday to you. Rick Norwood 23:11, 29 December 2005 (UTC)

text:  I think this was also discussed in the archives. My ear agrees with Charles Matthews; the correct article is "a", not "an". Also, omitting the article would be undesirable; it would shift the meaning. --KSmrqT 05:14, 30 December 2005 (UTC)
thread_title:  a Euclidean space 

message: Ithink we need to have a brief section on what a manifold with boundary is. --Salix alba (talk) 10:30, 28 January 2006 (UTC)
thread_title:  Manifold with boundary 

message: I saw this article on WP:FAC. It doesn't look like it's going to pass, but I have two comments so it can be improved:
# The first sentence, while technically accurate, does little to capture the spirit of what a manifold is for people without a strong mathematical background. People don't think of manifolds as glued together collections of charts, or as topological spaces with a complete atlas of charts, they think of manifolds as spaces locally homeomorphic to n-dimensional Euclidean space. So why not, something like "A manifold is a mathematical space that in a small neighborhood of any point on the manifold looks like n-dimensional Euclidean space, while the global structure looks quite different. An example is the torus..." That was pretty cumbersome, but you get the idea. Why mention charts? They seem more fundamental to the definition than to the idea.
# Why not discuss the classification of 2-manifolds in terms of Euler characteristic (or genus for orientable manifolds)? I realize that this is a page about all kinds of manifolds, but it is a very easily described, famous result that non-mathematicians might find it enlightening to hear about. –Joke 15:03, 31 January 2006 (UTC)
replies: text: Yes sounds good to me. How about a manifold is a mathematical space which looks locally like a n-dimensional Euclidean space. Probably worth avoiding neigbourhood in first line. --Salix alba (talk) 15:17, 31 January 2006 (UTC)
replies: text:  Thanks for stopping by to play. An uncountable number of people have spent an uncountable number of months considering an uncountable number of variations of the first line and the rest of the intro. Well, that's a slight exaggeration; but it has been beaten to death, as a review of the talk page and its archives will show. The best way to improve the article at this point is to concentrate attention on the substance, the organization and content of the body (which has been beaten on, but far less). --KSmrqT 15:37, 31 January 2006 (UTC)

text: The article as it stands is a compromise between people who came to the idea of a manifold via point set topology and people who came to the idea of a manifold via calculus on manifolds.  The opening paragraph was a compromise that I don't think either side was entirely satisfied with, but at least it was something both sides could accept, after a long and heated debate.  Maybe it is time to revisit the issue, especially in the light of the negative reaction from non-mathematician reviewers (as contrasted with the view of people with a little math (even if they are in 10th grade) which is that this article is about as readable as an article on manifolds gets).  Spivak, anyone?  Rick Norwood 20:26, 31 January 2006 (UTC)
thread_title:  Comments 

message: I don't claim to be an expert on grammer, but there seem to be a lot of Consider .... statements about, is this good style? --Salix alba (talk) 16:22, 31 January 2006 (UTC)
replies: text: It is bad style. That's chatty mathematicsese. –Joke 02:22, 2 February 2006 (UTC)
thread_title:  Consider 

message: I would like to suggest that we work on a new intro here, and then take it to the article, rather than doing our rewrite in the article.  Here is what I would like to see.  Comments?
:In mathematics, a manifold is a geometric object which is simple locally but which may be complicated when viewed as a whole, and which may have additional structure.  A sphere -- an idealized version of the surface of the Earth -- is a manifold.  Locally, the Earth seems to be flat, but viewed as a whole, it is round.  And the surface of the Earth can be given additional structure: lines of latitude and longitude, for example.
:Every manifold has a dimension.  One-dimensional manifolds are called 1-manifolds.  A straight line is a 1-manifold.  So is a circle.  Two-dimensional manifolds are called 2-manifolds.  A sphere is a 2-manifold.  So is the surface of a donut, a torus.  A solid ball is a 3-manifold.  So is the universe we live in (except for black holes).  Einstein suggested that space-time may be a four-dimensional manifold. 
:Examples of manifolds with additional structure include differentiable manifolds, symplectic manifolds which serve as the phase space in classical mechanics, and four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity.
:A technical mathematical definition of a manifold is given below.
Rick Norwood 22:05, 31 January 2006 (UTC)
thread_title:  New intro 

message: * JA: Perhaps it would be better to write a couple of entirely independent and separate articles.  A couple of title suggestions may be found in the head.  Jon Awbrey 22:26, 31 January 2006 (UTC)
thread_title:  Mani-fold, mani-spindle, mani-mutilate 

message: I really don't like the opening sentence about "patchwork."  A N-dimensional manifold is a second countable Hausdorf space with the property that every point has a neighborhood homeomorphic to an open ball in N-dimensional Euclidian space.  The most important property that distinguishes a manifold is that it locally looks like some specific space at every point.  As opposed to, for instance, an analytic variety, which is the solution space of a system of M polynomials in N variables, and may have cusps and other interesting things on it.  I am not going to leap in and edit a "featured article" which has had peer review, but I feel strongly that the opening sentence should be changed to something meaning "locally like some other space" and the "patchwork" metaphor eliminated, which could mean lots of things, many of them non-manifold-like. Hermitian 03:22, 2 February 2006 (UTC)
replies: text: The article is not featured. You're absolutely right. There seems to be some misconception in the article and here on talk that a topological manifold requires a notion of "charts" and an "atlas." This is false: all you need to know is that every point has a neighborhood homeomorphic to an open set in Rn. –Joke 03:24, 2 February 2006 (UTC)
replies: text: Yes, I read too quickly.  It was a "former featured article candidate."  The basic notion of manifold is that of a topological manifold, which is locally n-dimensional.  We can add a differentiable structure (not necessarily uniquely up to diffeomorphism) and get a differentiable manifold.  We can also add other things, like a connection, a metric, a continuous group operation, or whatever, and get various other refinements.  Take a look at the PlanetMath  article on manifolds.  It is concise, includes everything necessary, and nothing in it is wrong or misleading.  We should strive for something similar, with additional English text explanation of the various concepts suitable for the non-technical reader, and the incorporation of additional topics of interest.  A paragraph on "Exotic Spheres" would be nice.Hermitian 20:05, 2 February 2006 (UTC)

text: There is something to be said for how transition functions describe what happens at the overlaps, otherwise it is hard to say how these Euclidean spaces are fitting together. I am inclined to agree that a "patchwork" is probably more difficult to understand than a space locally homeomorphic to Rn or otherwise, but I would be conservative and propose your changes here before making them in the article, and then give adequate time (1-2 days at least) for a reply from interested parties. - Gauge 03:43, 2 February 2006 (UTC)
thread_title: Small Quibble

message: Using Prerequisites is okay in a technical article, as is written in Make Technical Articles Accessible. I'm not saying don't even try to read this, simply stating the obvious fact that the ordinary lay person will not get head or tail of the even easier sections of the article without a strong familiarity with High School math. Just look at the concepts referred to in Motivational Example. Feel free to clear up the prerequisites section, but please don't remove it. Loom91 07:37, 2 February 2006 (UTC)
replies: text: I don't agree. Putting a "there be dragons here" sign at the top of an article about one of the most important topics in mathematics (and mathematical physics) over the past two centuries is not acceptable. In fact, I don't think I've ever seen a technical article with a prerequisites tag, not, say Einstein field equations which needs it if anything does.–Joke
Agree with the above (unsigned) post. I've never seen a math article on wikipedia with prerequisites explicitly mentioned, and I am not convinced that we should start now, especially when much more sophisticated articles don't list them.  - Gauge 18:04, 2 February 2006 (UTC)
replies: text: Agree w/Joke, Gauge (if my opinion is worth anything here). There are certainly ways of making things clear and understandable without making them dumb an stupid. (which is what has happened to some of the more pop math articles). One should cater to the smart but ignorant, rather than the stupid and ignorant. linas 00:28, 3 February 2006 (UTC)

text: I don't agree that an article about manifolds is unavoidably technical. Sure, some if it is, but I think any reader can grasp the basic idea of what a manifold is, and most readers can grasp the examples with a little thinking and high school algebra, which isn't exactly technical. –Joke 14:09, 3 February 2006 (UTC)
replies: text: You will notice that I specifically stated High School ALgebra as the prerquisite. You also agree that the page requires AT LEAST high school algebra. A fifth grade child is highly unlikely to understand even the simplest possible explanation of Manifolds. There's a reason people get education. 
And as you yourself say, some (in my opinion a large) part of the article HAS to be highly technical. The prerequisites I mentioned are required to understand even motivational example section. Unless the article is written in a way that DOES NOT require Precalculus to be understood, the prerequisites header is at par with Make technical articles accessible. I just don't get your problem with the prerequisite header! Please stop pointlessly reverting. Loom91 11:53, 4 February 2006 (UTC)
thread_title:  Prerequisites 

message: In case you don't want to read through the above, this is a summary of my reservations about the article. I have to say I'm surprised they're so controversial.
*The history section is poor, and is at the end. This is an encyclopedia article about one of the most important topics in mathematics. Articles like particle physics, evolution, big bang all have prominent history sections. It is sort of expected of a major article, and this should be no different, particularly given how central the subject is to modern mathematics. I realize that nobody here may know much about the history – I certainly don't – but that's no excuse for marginalizing it.
*The article is at an inconsistent technical level. For example, it is mentioned that topological manifolds are generally required to be Hausdorff and second countable, but it is never mentioned what the compatibility condition for transition maps is.
*The idea of an atlas of charts is too prominent. In fact, it is used in the introduction when it is only relevant to a differentiable manifold. A topological manifold does not require an atlas of charts. Why not just use some kind of "locally" hand waving in the introduction?
*The organization needs to be improved. The examples are good, but it is often not clear what the motivation for the examples is, they're just thrown out there in the "construction" section. Often, the first thing you want to know about a mathematical object is not how to construct it, but what it actually is in some deep sense. Maybe it would be best to talk about submanifolds the way Riemann originally did, as surfaces in some higher dimensional Euclidean space, probably defined as zeros of functions, and then point out with examples that a lot of the associated concepts (the metric, the extrinsic geometry) are no longer part of what we think of as a manifold, but rather extra structure.
What do you think? –Joke 14:50, 2 February 2006 (UTC)
replies: text: While we're on the subject of reservations, let me add my own. I've always thought this article tries to be too general. The presentation makes it look like topological manifolds are a special type of manifold, and that not all manifolds need to be topological. In my mind this is fundamentally flawed; all manifolds are topological (in that they are locally Euclidean, Hausdorff topological spaces). Specializations result by adding additional structure (like an atlas of smooth or analytic functions). I've never liked having topological manifold as a separate article. -- Fropuff 17:09, 2 February 2006 (UTC)
replies: text: Very good point. --Salix alba (talk) 18:05, 2 February 2006 (UTC)

text: Well, "all" manifolds are topological only if you allow them to be locally homeomorphic to a finite or infinite-dimensional Banach space, if you ask me. I think of this article as more about explaining the "manifold concept" in mathematics rather than the rigorous technical properties of topological manifolds. The latter article was created so that technical details that only serve to confuse the idea to newcomers can be placed there. Obviously we will have to be careful about how much detail we put on this page to reach FAC status again. - Gauge 00:04, 3 February 2006 (UTC)
replies: text:  Just curious, are there any good examples of ones which are not homeomorphic to a Banach space? --Salix alba (talk) 00:47, 3 February 2006 (UTC)
replies: text:  Well the obvious ones would be manifolds with boundary, although for pedagogical reasons it is probably better to introduce these later rather than sooner. Under the same considerations, I can agree to write as if all manifolds are locally homeomorphic to finite-dimensional Euclidean space (or half-space) so long as this article isn't inundated with technical detail that belongs at topological manifold. - Gauge 05:06, 3 February 2006 (UTC)

text: Well, I consider Banach manifolds to be generalizations rather than specializations of manifolds, which I assume to be finite-dimensional Euclidean spaces. Like I said I think the article tries to be too general. IMHO, of course. -- Fropuff 02:46, 3 February 2006 (UTC)
thread_title: Summary of my reservations

message: * JA: Please discuss article-related issues at the article talk page, not my user talk page.  I copy your remarks and reply here.
Please be careful in reverting
Hi, you reverted my edits to the intro on manifold, which is fine. But when you do it, don't revert other edits, such as , which is in accordance with Wikipedia style guidelines for linking years, minor edits like  and , and this edit  which has nothing to do with the intro. If you disagree with these edits, you should comment on talk and explain which you are reverting and why. Don't undo other people's work simply because it is more convenient just to revert to the most obvious version. –Joke 14:06, 2 February 2006 (UTC)
thread_title: On being careful

message: My suggestion that we take this slowly has not been accepted.  I think the most important step, in light of the strong statement by other wikipedians that the lead is unreadable, is a new lead.  I've written one.
Next, again with an eye toward an accessable learning curve, I plan to move the history section into second place, followed by a mathematical definition. Rick Norwood 20:03, 2 February 2006 (UTC)
replies: text: Why does moving history there make it more accessible? I find it quite a hard section to understand, because it uses quite a lot of technical terms. -- Jitse Niesen (talk) 23:22, 2 February 2006 (UTC)
replies: text: Agree with Jitse, for reasons I stated earlier. Looking at what is actually written in the "History" section makes it clear to me that any potential readers will stop reading the article at that point due to a deluge of technical details and references. Do we really expect people to know things like curvature, ambient space, surface, analytic continuation, abelian varieties, Hamiltonian and Lagrangian mechanics, differential geometry and Lie group theory? This is terrible as motivation and only serves to educate a posteriori once one has been in mathematics and has seen these things. It serves no purpose at the beginning of the article. - Gauge 00:14, 3 February 2006 (UTC)

text: I don't think it necessarily has to use so many technical terms, at least without a quick hand-waving explanation. The key point is that the history section should have some sense of narrative, not simply be a collection of trivia. I think characters like Felix Klein, Arthur Cayley, Hermann Grassmann, Sophus Lie, Gaspard Monge and Leonhard Euler could merit a mention. Although perhaps they didn't advance the definition of manifolds, they certainly advanced the way in which we think of problems involving manifolds. Although maybe it would be best to write a history of differential geometry rather than a history section here? I will try and work on it later in the day – no time now. –Joke 14:20, 3 February 2006 (UTC)
thread_title:  Since going slow is no longer an option. 

message: * JA: Rick, et aliases, "We" now have a function (mathematics) article that I would not give a sixth grader to cut their dog's teeth on.  Has it ever occurred to you, just maybe, to ask why people liked the way this article was written, what their mathematical and pedia-gogical reasons might have been for the design decisions that they made, that you are now trashing without a thought?  Who knows, those choice may be indefensible, but you could at least lend an ear.  Who knows, you might actually learn something in the process.  I am sure that some of the same folks, or reasonable proxies thereof, are still around, if you give them half a chance.  If this is really such a cornerstone of mathematics, then what's the rush to shift it?  I'm all for pushing the envelope, do it every day, but this particular package is looking more and more like some amateur hack job every time I revisit it.  Jon Awbrey 20:38, 2 February 2006 (UTC)
replies: text: Jon, you do not like the way I write.  You have said so on many occasions, with what you evidently consider disarming frankness.  But the facts are not on your side when you accuse me of rushing and of not taking other people's opinions into account. Rick Norwood 21:44, 2 February 2006 (UTC)

text: Clearly, Jon, we will have to agree to disagre on the subject of which of us edits unilaterally and which of us always tries to work with others and find a consensus. Rick Norwood 23:03, 2 February 2006 (UTC)
replies: text: What are your specific criticisms of Rick's work, Jon? If they are about function (mathematics), please list them over there. - Gauge 00:20, 3 February 2006 (UTC)
thread_title: Solera Process at Wiki Whinery