Definition talk:Differentiable Structure

Any reason not to separate out pre-differential structure into its own page? --prime mover 13:01, 7 May 2011 (CDT)

Smooth DiffStr
The definition under smooth diff str. appears to cover real-analytic manifolds. These are only a subclass of the general smooth manifolds. --Lord_Farin (talk) 15:04, 30 November 2012 (UTC)


 * True, I'll move stuff about in a sec, and change that. I've removed maximality for the moment, I think if everything that follows is to be rigorous we'll need maximality, but not too sure. ---Linus44 (talk) 15:33, 30 November 2012 (UTC)


 * Normally I'd object to your rigorous cutting of this definition page, but the whole subject is sufficiently f.. messed up that I don't mind it being built up carefully and with rigour from the start. I'll try to track your movements and implement house style where necessary; I'm sure you'll know or learn soon enough how to take care of that yourself. --Lord_Farin (talk) 15:38, 30 November 2012 (UTC)


 * @PM: Do you agree that there is site-structural benefit for a transclusion-subpage setup here? --Lord_Farin (talk) 15:39, 30 November 2012 (UTC)

Transclusion-subpage paradigm is IMO essential here - the subject appears to be of a complexity approaching e.g. Continuity. Recommend we look to that as an example of something we need to aim for. I have started by setting up a Definition:Manifold (Topology) in the appropriate format.


 * Ok the definitions as they stand I think are workable. I don't have anything like the time it'd take to write out the elements of differential geometry carefully, but it's a modest improvement on the background info for Stokes I hope. --Linus44 (talk) 15:59, 30 November 2012 (UTC)


 * I'd say so as well. On a side note, isn't a DiffStr more usually called a maximal atlas, and a PreDiffStr an atlas? I like the term atlas because it is more convenient to talk about a $C^k$ atlas, a real-analytic atlas, a complex-analytic atlas, etc. etc. Also, it allows to discuss topological manifolds (I admit to not knowing much about those, but still) later on. --Lord_Farin (talk) 16:02, 30 November 2012 (UTC)


 * Yep I think the terminology you suggest is usual (and in this case a differentiable structure I think is usually an equivalence class of compatible atlases); pre-differentiable structure I think I took from Warner's book on the subject; I've no attachment to his terminology. --Linus44 (talk) 16:11, 30 November 2012 (UTC)

Class
As with "dimension" of a manifold, I recommend that the "class" of a differentiable structure should be extracted and made into a definition of its own, to emphasise that it is an instance of a "differentiability class". As it stands, when you get to "differentiable manifold", the definition of its "class" is some distance away, further away than perhaps is desirable. --prime mover (talk) 12:56, 1 December 2012 (UTC)