Talk:Differentiable manifold

From Wikipedia, the free encyclopedia

Contents

[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)

[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)

Retrieved from "http://en.wikipedia.org../../../d/i/f/Talk%7EDifferentiable_manifold_16bb.html"