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)

[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)

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