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:Topological Manifold 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)

Ok I'll switch round the terminology; I think "differentiable structure" is particular to Warner so I won't bother with redirect's. --Linus44 (talk) 22:06, 5 December 2012 (UTC)

Actually I will since they're automatic. --Linus44 (talk) 22:07, 5 December 2012 (UTC)

You're doing a great job; to complete the paradigm, "coordinate system" could be changed into "chart". --Lord_Farin (talk) 22:51, 5 December 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)

Mind you, the class of a DiffStr is not unique. A $C^1$ manifold is also $C^0$. Still, the notion should have a separate definition, of course. --Lord_Farin (talk) 18:37, 1 December 2012 (UTC)

The way I understand, it, the differentiability class of a mapping is the maximum order of differentiability it has. Thus a mapping which is $C^1$ is not $C_0$ and a mapping which is $C^2$ is not $C^1$ or $C^0$. --prime mover (talk) 20:01, 1 December 2012 (UTC)

So you'd say that when writing "Let $f$ be a continuous real-valued function" you exclude that e.g. $f$ can be the identity because the identity is $C^\infty$? The definition you propose to me simply doesn't correspond to the way it is used. --Lord_Farin (talk) 20:12, 1 December 2012 (UTC)

No, not at all. saying that "$f$ is continuous" does not say anything about its differentiability class. --prime mover (talk) 20:29, 1 December 2012 (UTC)

Archetypal use of this terminology is e.g. "Let $\Omega$ be a manifold with $C^1$ boundary" (where $C^1$ boundary means: a boundary that is a $C^1$ manifold). Apologies for oversimplification. --Lord_Farin (talk) 20:38, 1 December 2012 (UTC)

Aha - I'm on the same page now. Yes - we may indeed need an "also defined as" - but the "exclusive" definition is the one we have in at the moment. Maybe we need a different notation, one to indicate "diff order at most n" and a separate one meaning "diff order at least n". --prime mover (talk) 20:44, 1 December 2012 (UTC)