Talk:Differentiable manifold

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[edit] Smooth and Analytic manifolds

I think we need to mention smooth manifold (all derivatives exist) and analytic manifold (charts are analytic functions). --Salix alba (talk) 21:39, 29 January 2006 (UTC)

[edit] Differentiable manifolds and differentiable functions

I have added to these sections.

  • I have tried to simplify the definition using the notion of global and local.
  • I have defined the transition maps explicitly

(anyone interested can proof read this since there are subscripts etc)

  • I have defined differentiablity of maps by using the directional derivatives, local coordinates and the

differential map.

  • I made a comment on the definition of differentiablity using transition functions,

(there is a short section that seems to indicate this approach)

but this ought to be done explicity or dropped in my opinion. I think it is a mistake for the unitiated and unnecessary for the expert.

The real difficulty is how detailed should this page be?

for instance tangent bundle is given a cursory definition, but tangent vectors are not defined as far as I can tell.

I will continue working on this page if there are no objections. Geomprof 19:24, 15 March 2006 (UTC)

  • Edits look good so far, very nice to actually see some well informed good writing on wikipedia, quite a rare thing. A slight concern about the introduction, as it starts at quite a high level. We tend to try to keep first paragraph as simple as posible. Also it would be good if a link to manifold could be included somewhere, as this is probably the most accessable article. --Salix alba (talk) 21:25, 16 March 2006 (UTC)
  • In the definition one might mention that by necessity the inverse images of transition maps are differentiable, so that all transition maps are diffeomorphisms. MotherFunctor 05:45, 17 May 2006 (UTC)


Geomprof, you're doing great. Very informative. Thanks. Mct mht 09:01, 8 April 2006 (UTC)

[edit] Regarding symplectic structures

The claim is made that every surface has a symplectic structure and then the volume form is used to prove this -but non-orientable surfaces do not have well defined volume forms. I think this statement should be changed. Any comments? Geomprof 21:21, 16 March 2006 (UTC)

the volume form is a pseudo form and IIRC it exists also for nonorientable manifolds. --MarSch 10:29, 24 April 2006 (UTC)
I don't understand. I thought a volume form was an orientation form that has unit total integral. (?) MotherFunctor 05:43, 17 May 2006 (UTC)
Just to clarify, for a surface, a two-form IS a top-dimensional form, i.e. a volume form. A symplectic form must be (1) closed, and (2) nondegenerate. Any two-form on a surface must be closed, because it is top-dimensional and hence there are no non-zero three-forms. A two-form on a surface being nondegenerate is exactly equivalent to it being a non-vanishing volume form for the surface. (A volume form is any top-dimensional form, i.e. something which allows you to integrate on the manifold.) The existence of a non-vanishing volume form is one way of defining what it means for a manifold to be orientable. So, every symplectic surface must be orientable, but in fact this is true for any symplectic manifold. --Yggdrasil014 15:07, 27 July 2006 (UTC)


I just happened across this page (actually, I saw the link to pages requiring expert attention). Perhaps I am misreading what was written here, but it is simply not the case that every manifold has a symplectic structure. For one thing, any such manifold must be even dimensional. However, what is true (and quite important) is that the cotangent bundle of any manifold can be given a symplectic structure in a natural way. Greg Woodhouse 23:01, 22 March 2007 (UTC)

[edit] Composition Link

Can someone clarify what exactly is meant by the composition link? There are three separate definitions for mathematics-related composition on the disambig page, and so the link as it is not very helpful. All we need to know is what definition the author was using for the word. SingCal 06:59, 27 July 2006 (UTC)

[edit] Comment on definitions

Definitely lots of fine and extensive work has been done on this page. It seems like it could be simplified by referring readers to the article on Manifolds for the definitions of charts and transition maps, and then focusing the definition here on the fact that a differentiable structure is essentially an atlas (or a maximal atlas) in which all transition maps are of a specified differentiability. It seems like several different articles related to manifolds go through the details of charts and transition maps, and maybe this isn't necessary. Just a suggestion. But like I said, very nice work has been done in each of these articles. --Yggdrasil014 18:32, 29 July 2006 (UTC)

[edit] Topological manifolds and smooth manifolds

I'm a differential geometer, drawn by the tag that this page needs some expert input. I have a few initial comments. Geometry guy 21:03, 8 February 2007 (UTC)

First, the term differentiable manifold is rather dated. Manifolds which are only once or k-times differentiable are rather a minority interest these days, and to most differential topologists and geometers, the category in which to work is the category of smooth (infinitely differentiable) manifolds. These are the manifolds which arise most often in practice, and they are the manifolds on which one can really do calculus, without worrying about running out of differentiability. For this reason, I suggest that the meat of this article be moved to a Smooth manifold article, and that this article should be a brief summary of definitions (using both atlases and sheaves of functions) of various classes of differentiable manifold.

Secondly, this article does not exist in isolation (okay, an obvious point). It seems to be generally agreed that the manifolds article should be a non-technical article, so it cannot be used as a reference for technical definitions. Instead, an improved version of the topological manifolds article should be used as a reference for basic notions (such as atlases), but much work is needed before it becomes an adequate reference.

Thirdly, I think that a smooth manifolds article is the right place for the most comprehensive list of examples: the examples in the topological manifold article should be limited to those with the greatest topological significance, such as flag manifolds (e.g. projective spaces and grassmannians) and Stieffel manifolds. A good choice of emphasis might encourage differential and algebraic topologists to contribute to that page, which would be very welcome. In particular, the classification problem for topological, piecewise linear and even smooth manifolds probably belongs in the topological manifolds article.

Fourthly, this article needs to be coordinated with articles on Differential geometry (and topology) and Calculus on manifolds: at present these are not properly differentiated (forgive the pun), but even so, there is a bit too much repetition at the moment.

Geometry guy 21:31, 8 February 2007 (UTC)

It all sounds very reasonable to me. Lots of work to be done, though... Turgidson 00:19, 9 February 2007 (UTC)
I like the idea of separating differentiable manifold and smooth manifold as you suggest. Smooth manifolds are by far the most important. As far as technical definitions go, I think both articles should be relatively self contained (but with links to topological manifold as appropriate). Regarding examples: I don't think it hurts to have some overlap between the various articles. Another solution would be to collect all the common examples on a page of its own (e.g. Examples of manifolds) and refer to it from each of the articles. Of course, this has some overlap with our List of manifolds page. -- Fropuff 19:41, 9 February 2007 (UTC)

I'm glad you like this plan, and agree with your comments. As for examples, I agree that an overlap between articles does not hurt. I would prefer to have articles which give the most relevant examples (with overlap not a problem) and then a link to List of manifolds, rather than a new examples of manifolds page. We can then ensure that the list of manifolds includes all the examples mentioned here. Geometry guy 01:37, 10 February 2007 (UTC)

By the way--in a somewhat related vein, I had a little discussion about a month ago here about how to list certain classes of 3-manifolds (and related concepts). Another editor and I couldn't quite agree on how to proceed, and things just petered out. Maybe you'd have some input on that? Turgidson 02:18, 10 February 2007 (UTC)

I looked at it a couple of times, but my only feeling was that even in such a specialized area, you should try as much as possible to be less specialized: the theorems and examples most worth mentioning are those that can most easily be explained.

Meanwhile, I'm coming bit closer to feeling confident to edit the differentiable manifolds page soon... Geometry guy 01:30, 3 March 2007 (UTC)

[edit] Sheaf-theoretic approach

I don't think the text in the (stub) section on the use of sheaves is correct. Or, at any rate, it is confusing. first of all, the use of the sheaf theoretic approach in the context of differentiable manifolds is unusual, though in the case of (complex) analytic manifold it is very natural. The stalks of the sheaf really have nothing to do with the coordinate charts that appear in the more traditional definition, but instead, represent the direct limit of the algebras of functions holomorphic in a neighborhood of a given point. Sheaves are generally used in alebraic geometry where they arise naturally as the ring of functions defined on an open set (in the Zariski topology, of course). In the complex analytic case, we consider the ordinary topology and all holomorphic functions, but as you probably know, functions agreeing in a neighborhood must agree (analytic continuation). This is what makes the tools of differential topology (in the real case) difficult to apply (no partions of unity!) and explains why the theory of complex analytic manifolds is much close in spirit to algebraic geometry. Greg Woodhouse 23:16, 22 March 2007 (UTC)

I agree entirely, but this whole article needs a substantial rewrite (which might reach the top of my list in a couple of weeks, after many weeks (months) working on the articles which underpin it), and I am a big fan of the sheaf-theoretic (am I allowed to say the following on wikipedia?) point of view. Geometry guy 23:44, 22 March 2007 (UTC)

[edit] Jet bundle

Does this section need to be a stub? I suppose more precise definition is possible. In particular, a convenient method of introducing tangent vectors is to define them as equivance classes of paths that agree in their first derivative. Of course the derivative itself isn't well-defined, but the chain rule immediately implies that if two paths agree in their first derivative in one chart, they will in any other. Jets are just defined as equivalence classes of paths agreeing in their first n derivatives. —The preceding unsigned comment was added by GregWoodhouse (talk • contribs) 23:39, 22 March 2007 (UTC).

I am really glad you have taken some interest in this article: it really needs substantial changes. Feel free to WP:be bold. Geometry guy 23:44, 22 March 2007 (UTC)
A section on jet bundles is problematic. There are too many possibilities (from R, to R, from M to itself, from R^n, to R^n, first order, higher order) and no clear way to decide which should be discussed. More useful, IMO, is a subsection about frame bundles. In particular, one can define the bundle of connections in term of the 2nd order frame bundle. Still to do: all the other bundles like tangent, cotangent, various tensor bundles, are associated bundles of F(M). Extra structure can be regarded as the reduction of a frame bundle; e.g. riemannian structure = reduction from GLn to O(n), projective structure = reduction of structure on F2(M), lots of other examples. The material about the tangent and cotangent bundles being jet bundles is useful, though. I've moved it into the tang/cotangent bundle sections. Rmilson 14:55, 24 March 2007 (UTC)
The argument in favor of incluing jet bundles is that they are fairly important in studying differentiable manifolds in general. But I agree we don't want to get involved in discussing every kind of bundle that might be used. I see that there is already an article on vector bundles and it looks pretty good, too. I'm not sure where to draw the line in this article, but it seems to me that the tangent and cotangent bundled can hardly be omitted, but it may well be reasonable to say that there are number of other vector bundles important in the study of differentiable manifolds, and link to the relevant article.

Greg Woodhouse 15:14, 24 March 2007 (UTC)

Why don't write an article about jet spaces? Jets of sections of bundles can be included, sections of jet bundles etc. ASlateff 128.131.37.74 00:18, 16 June 2007 (UTC)
See jet bundle and jet (mathematics). Silly rabbit 00:21, 16 June 2007 (UTC)

[edit] Assorted types of bundles

I see there is a section on frame bundles. There are a few types of bundles (obviously including the tangent and cotatengent bundles, and I think jet bundles, too) that are important to discuss in an article about differentiable manifolds per se. Frame bundles have more to do with connections and the study of differential geometry. Do we perhaps need a seperate article listing important examples of fiber bundles? Greg Woodhouse 15:03, 24 March 2007 (UTC)

[edit] Counterexample

A differentiable manifold is one where you can do calculus. Fine. But in the definition of Manifold, it states that a manifold is a thingy where each point has a local neighbourhood that looks like eucledian space (so there are c other points arbitrarily "close" to every point.).

Now - there's no Nondifferentiable manifold page, so can someone give an example of something that is a manifold but is not differentiable? I suppose that a plane crinkled up into a fractal shape might qualify. Or a plane with a crease in it? —The preceding unsigned comment was added by 203.0.101.131 (talk) 03:18, 10 April 2007 (UTC).

The first examples of topological manifolds that do not admit a smooth structure occur in dimension 4. The book by Freedman and Quinn "Topology of 4-manifolds" (Princeton University Press) gives examples. It'll require some work to do a good job of writing up such examples. Rybu 10:02, 10 April 2007 (UTC)

[edit] Definition of classification

When I made my point about 3-manifolds not being classified I was not making a point about the isomorphism problem for hyperbolic groups. The point is that there is no list of hyperbolic 3-manifolds. So if by classification all you mean is that there are procedures to tell any two 3-manifolds apart, sure there is that. But I don't think that's what people commonly mean by classification. If they do, it's the weakest sense of the word. My point is there is no list of hyperbolic 3-manifolds together with all the data one would need to distinguish them. A proper classification would say something more along the lines of exactly what the set of volumes are, together with an efficient procedure for constructing all hyperbolic 3-manifolds of a given volume.—Preceding unsigned comment added by rybu (talk • contribs)

I'm a little confused. Your second sentence and part of what you wrote in your edit (about there being no list of 3-manifolds without redundancies) is simply wrong. Once you have procedures to generate all 3-manifolds and distinguish any two such 3-manifolds, you can generate a list with no redundancies. Of course, the known procedures for generating such a list are incredibly "stupid" in the sense of being extremely complicated in the number of operations and not practical at all. But this really is what a lot of people mean by "classification", just a list of all 3-manifolds with no redundancies. Efficiency is separate issue.
A procedure to construct a list, and a list are two different things. For example, the Fibbionaci sequence a(n+2)=a(n+1)+a(n) with the initial data vs. the closed form solution (p^n - (1-p)^n)/root(5) (p the 'golden ratio') are two different things. There is a major step between one and the other. In particular, the latter tells you a lot more about the Fibionacci sequence because it gives you asymptotic data. This is my point. Admittedly, all lists tend to be given by procedures but some are more informative and efficient than others. The current procedures for constructing hyperbolic 3-manifolds leaves a lot to be desired.Rybu 12:10, 11 April 2007 (UTC)
I agree with the sentiment, but the issue is terminology. A list, in this mathematical context, is an enumeration (bijection with natural numbers), and to say one has a classification of 3-manifolds, one is supposed to have an algorithm for such an enumeration. --C S (Talk) 12:28, 11 April 2007 (UTC)
In fact, if you are just interested in hyperbolic 3-manifolds, you can go through this list and also cull out only the hyperbolic ones (using say Manning's paper and a solution to the word problem given by geometrization). And there, once you have your presentation of these hyperbolic 3-manifolds, as say triangulations, that is all the data needed to distinguish them (using the procedure in Sela's paper). Doing this efficiently is a different matter, again.
What you describe as a "proper classification" sounds good :-), but that's certainly not what is usually meant when one hears 3-manifolds (or even hyperbolic ones) are or are not classified.
I think you'll find a lot of people talk about these things. I seem to run into these conversations pretty often. Rybu 12:10, 11 April 2007 (UTC)
You misunderstand me. Of course people talk a lot about these things, but in my experience people understand and say the statement "3-manifolds have been classified". That is what I meant. People may rebut this by saying something like it's not a good classification, but generally, it's understood what classification (with no qualifiers) means, as lame as it sounds. --C S (Talk) 12:28, 11 April 2007 (UTC)
This whole thing reminds me of the great annoyance by the 3-manifold theorists after Perelman's work. Many people started saying now 3-manifolds are classified (including many 3-manifolds people), but the 3-manifolds folk understood how useless this was, whereas most people not in the subject seemed to not understand. Oh well. The usual example people give is that surfaces are classified but there are many interesting questions about surfaces. A better example is that natural numbers are classified :-), but we there's still a lot we don't know a lot about them!
I usually tell people it depends on what you mean by 'classified' and then let them know these problems with hyperbolic 3-manifolds. Rybu 12:10, 11 April 2007 (UTC)
Anyway, I don't know how much is appropriate to go into in this article. I think it's relevant to at least mention the result on classification, but an extensive caveat is probably not. --C S (Talk) 11:54, 11 April 2007 (UTC)
Certainly an extensive digression into what classification should mean, complexity, etc, would be over the top but some indicators (or appropriate links) to the subtle issues that come up should be given.
Well, feel free to add them! --C S (Talk) 12:28, 11 April 2007 (UTC)

I tried to fix it up in the footnote. I think rather than assuming that the reader knows what "classification" means, we should say what it means! Also is there the same issue for simply connected n-manifolds with n≥5? I was only aware of a classification of s.c. 5-manifolds. Is there an "in principle" (i.e., impractical) algorithm in general (as for 3-manifolds) or is the situation more complicated? Geometry guy 16:54, 12 April 2007 (UTC)

[edit] Higher dimensional classification

What is the following sentence trying to express?

The classification of n-manifolds for n greater than three is known to be impossible, even up to homotopy equivalence.

I thought that it's about impossibility of classifying the fundamental groups, and it appeared so from the next couple of sentences, but when I incorporated a manifest reference to the fundamental group there, it quickly got reverted with a note about 'misleading argument'. So now I am confused:

  • Given a finite presentation of a group, are there further algorithmic obstructions to classifying four manifolds with that fundamental group (choose your favorite way of interpreting 'classifying')?
I think the point was that the fundamental group is one obstruction. There are in general further complications such as the classification of higher-order forms such as cubic forms, things like intersection forms and torsion linking forms..Rybu 08:56, 21 April 2007 (UTC)
  • How does the undecidability of the isomorphism problem imply the impossibility of classification? It does not seem obvious on the surface.
Say you had a classification of 4-manifolds up to homotopy-equivalence. Then there is a procedure to determine if \pi_1 of a manifold M is trivial: it is trivial iff the manifold is homotopy-equivalent to a simply-connected 4-manifold. But simply-connected 4-manifolds we have a homotopy-equivalence classification by the H_2 intersection form. So find the simply-connected 4-manifold with the same H_2 form as M and ask your 'classification' whether or not the two manifold are homotopy-equivalent. So now you have an algorithm to determine if pi_1 M is trivial. Here is how you construct a procedure to determine if any finitely-presented group is trivial: take a connect sum of copies of S^1xS^3, as many as you have generators in your finitely-generated group. Now \pi_1 of this connect sum is a free group. Every relator in your finitely presented group is a word in \pi_1 (your connect sum) so represent it by an embeded curve, and do surgery on that curve. If you do that for all relators, you get a 4-manifold whose fundamental group is your initial finitely-presented group (this pi_1 computation is where we're using the fact that we're in dimension 4). This is the contradiction we were looking for, because there are known finitely-presented groups for which the word problem is algorithmically undecidable. Rybu 08:56, 21 April 2007 (UTC)

By the way, I think that the fact that certain problems are algorithmically unsolvable deserves a bit more than the casual 'since there is no algorithm' clause in the article. At least, there should be a link! Arcfrk 07:38, 21 April 2007 (UTC)

If you look at the above definition of classification comments in talk, you'll see a conversation on this already. I've been thinking of putting up a 'classification (mathematics)' Wikipedia page where we talk about what such a word typically means and all the side issues that it brings up like algorithmic complexity ie: some classifications are effectively useless because it's impossible on a silicone-based computer to handle the simplest cases. I'll put in some comments to your questions above.Rybu 08:56, 21 April 2007 (UTC)

[edit] Some improvements, but mostly not

I do not like the current revision for a number of reasons. Although there are some improvements (mostly cutting down on the wordiness of it), a number of inaccuracies have been reintroduced which I had previously eliminated.

  • First of all, it is rather misleading to say that a topological manifold can be given a differentiable structure locally. What is more accurate is that one can "do calculus" in an individual chart. As the introduction goes on to illustrate, problems can arise when you go from one chart to the other: that is, there are problems even locally, to say nothing about globally.
  • Second of all, much of the article is dedicated to various notions of differentiability and smoothness (e.g., C^k, real analytic, holomorphic, and so forth). I suggest that it should be made clear from the very beginning that the article plans to treat several different notions of smoothness. It may well be that for most authors, differentiable means smooth. Nevertheless, the article must clearly state its scope up front. So, if there is another, more agreeable way to do this than by saying that differentiability may mean different things in different contexts, then please use an alternative wording. However, omission of these details from the lead is not a good idea.
  • Thirdly, most of the "cleanup" involved removal of motivation. The first two paragraphs were intended to be a somewhat informal introduction to the notion of a differentiable manifold. There is a school of thought that believes that the lead should be as short as possible, but it is unacceptable to remove all motivating discussion from the article.

Just my two cents. (P.S. I have reverted the recent edits, but kept the Mathworld link.) Silly rabbit (talk) 23:06, 4 March 2008 (UTC)

Right. There is, of course, a multitude of conventions in the differential geometry literature concerning what "smooth", "regular", and so on means, but his article has been carefully thought through, and it is much more explicit about the technical points and internally consistent than many sources, including Mathworld. As a matter of fact, I think that references to Mathworld do more harm than good: after chasing links for a few minutes, I've discovered that at Mathworld, "differentiable manifold" (which appears as an explicit link in some, but not all differential geometry articles) redirects to "smooth manifold" (which is taken to mean C^\infty), hence the notion of a Ck manifold simply falls through the cracks, whereas "topological manifold" and "manifold" are used interchangeably, and the corresponding articles have in fact be written by two different people. Mathworld is also rather cavalier about defining submanifolds (either by neglect or by ignorance, they are defined to be embedded submanifolds), and it cites a fairly random selection of texts, which is another reason not to use it as a reference. Arcfrk (talk) 00:03, 5 March 2008 (UTC)
I'm sorry, I should have posted here on the talk page as to my motivation for the changes I made. I cut out the other notions of differentiability because a differentiable manifold--what this article is about--is defined in the smooth sense. It seems unnecessary to clutter this particular article with other notions when it distracts from the overarching concept. I feel this is especially so in an article about something at this level of mathematics. It is my belief that anyone who has gotten to such topics as this understands that there are several notions of differentiability, and by noting early on that the sense we mean (without the trite reference to the others) is smooth we can establish the convention without any losses at all to the content. The local differentiable structure is really quite reasonable terminology. "Do Calculus" seems too informal, and with the links to the appropriate articles that I put in, one can navigate any extra information which is not present in the immediate article. Since wikipedia is such an interactive encyclopedia, it seems in the best interest of the articles to utilize this interdependence with those links so one can eliminate the unnecessary while still providing access to supplementary material. It is the very great virtue of encyclopedias that each article is not self-contained; it allows each one to be concise and has information in other locations to provide more info. I think we should utilize this since it is available to us. (I.e. best of both worlds: conciseness and fullness.) I do agree that the local charts' problems are something that should be noted, and I apologize for that change. My main issue with the first paragraphs is the informality is just far too much. The style is a major issue which needs addressing, even at the expense of some of the motivation. The full article on manifolds already provides a truly excellent explanation of the motivation for the theory of manifolds, and the links to that are very helpful.

En Suma: I think most of the changes I made were reasonable given the fact that this is not an isolated article. They do not allow one as full a view of the topic from this article alone (as it stands now), but given the other resources the changes are justified.

Guardian of Light (talk 05:44, 10 March 2008 (UTC)

I would object to the removal of the various notions of differentiability as "clutter." If you would carefully read the article, a great deal of attention is paid to various sorts of different kinds of differentiable structures, and various ways the manifold concepts is generalized to include such alternative ideas of differentiability. (E.g., real-analytic is fundamentally different from smooth which is fundamentally different from C^k.) The fact that there are many different notions of differentiability, and attendant notions of "differentiable manifold," needs to be stated up front not to patronize the reader, but to indicate the scope of the article. Secondly, the introduction to this article was modeled on that of the manifold article. It primarily emphasizes how the charts for a differentiable manifold are different from those of a topological manifold. It may be a bit too wordy or informal, but it is absolutely vital to indicate that a differentiable manifold really is something more than just a regular manifold. In an earlier version of the lead, it gave the misleading impression that any topological atlas is also a differentiable atlas. Somehow it needs to be said that this is not the case. If there is a more elegant way to do it, then I would welcome such a presentation. But removing the existing discussion does not seem to be the way to go about doing it. Silly rabbit (talk) 18:15, 10 March 2008 (UTC)
I'll give a shot at simply style revision and perhaps some minor extraneous elminations. eg. "A differentiable manifold is a type of manifold (which in turn is a type of topological space)" I'm replacing with ""A differentiable manifold is a type of manifold" since the manifold article has what a manifold is. Guardian of Light (talk) 21:21, 10 March 2008 (UTC)

I would say it's looking much better. Thanks for all the effort, and for taking my comments to heart. Silly rabbit (talk) 21:55, 10 March 2008 (UTC)