Talk:Differential geometry and topology

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This article is within the scope of WikiProject Mathematics.
Mathematics grading:	B Class	Top Importance	Field: Geometry and topology
"Intrinsic versus extrinsic" section needs expansion; needs more on history and examples. Consider splitting article into Differential geometry and Differential topology, with an overview here; more material on Differential topology needed. Tompw (talk) 20:10, 1 March 2007 (UTC) and Geometry guy 21:59, 20 March 2007 (UTC)

Contents 1 New To Advanced Math 2 where to put stuff? 3 Too technical? 4 Error? 5 Name 6 A differential geometer's view 7 Some recent additions

[edit] New To Advanced Math

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as differential geometry, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

• A good solution is to read the classicals, choose 5 or 6 classical authors and dig-dig-dig... also ask to everyone lets you. Attend any class related to. I'll also be glad to help to solve focused questions --kiddo 22:05, 18 November 2006 (UTC)

Someone really needs to add a coherent discussion about differential geometry and differential forms. There are many articles pointing to this article, but this article doesn't say anything of use at all! Phys

The case for having fundamental manifold material here is rather undermined by the existence of a differential manifold = manifold page duplicating some of it.

Charles Matthews 19:02, 12 Nov 2003 (UTC)

I think this page should now be used for a survey, with short overviews of all the major topics. Perhaps calculus on manifolds could be used as a way of organising the detailed discussion, of vector fields, tensor fields etc. That material exists, but somewhat scattered on various pages.

Charles Matthews 14:09, 13 Dec 2003 (UTC)

[edit] where to put stuff?

Tosha suggests that this article is not the right place to talk about vectors as derivations or bundles and their sections. But it seems that there is nowhere with a sophsiticated discussion of vector fields. we need one. but where? - Lethe 00:10, Jul 17, 2004 (UTC)

I simply think that good discussion vector fields sould be in vector field, but it is totally ok to mention it here. Tosha

Yet an other pace is tangent space Tosha 20:34, 19 Jul 2004 (UTC)

[edit] Too technical?

Is this article really too technical? I mean, what is someone searching for "differential geometry" expecting? What kind of information is most likely to help someone who visits this article?

We could add the usual boilerplate material for beginners that you find in the first pages of most differential geometry texts, i.e. something like "Calculus is concerned with functions on (subset of) the real line. Vector (or multidimensional) calculus extends this study to functions and vector fields on subsets of R^n, and maps between R^n and R^m. Differential geometry is a further generalization that extends concepts from calculus to functions on more general spaces that look like R^n on a small scale. These spaces are differentiable manifolds...". But honestly I think most people would be better served by keeping the level of technical detail approximately the same, but perhaps suggesting some references for beginners in a prominent place. --David Dumas 06:47, 11 July 2005 (UTC)

Differential geometry of curves is where I'd send someone who had no idea of the primary content. Charles Matthews 07:48, 11 July 2005 (UTC)

I would just like to agree with this comment. I think the label at the top of the page doesn't make much sense. What makes most sense is treating things at the lowest level of complexity necessary. If the topic is very advanced then the article should deal with the subject at the necessary level. This is what makes something like Wikipedia so excellent, optimally it allows one to work at these different levels and so be of use to many different people. The hyperlinked nature of wikipedia is optimal for this kind of learning, because rather than having to look up each item in a seperate book or have the entire story of a subject told to me over and over again I can follow links to read up on any topics I don't already know. On the other hand this does mean that it is important that the relavent links be clearly available in the text. Anyway, everybody seems to know this stuff already. User:Jabot the Scrob

[edit] Error?

Should the composition of f and g in the technical requirements section actually be fg^-1? if i'm wrong, then maybe whatever i'm missing could be made clearer? halio

• Isn't wrong. It is perfectly clear, to go and decide differentiability--kiddo 22:17, 18 November 2006 (UTC)

[edit] Name

Should this page just be Differential topology as we also have a page specific to Differential geometry? --Salix alba (talk) 08:10, 17 July 2006 (UTC)

• In fact this two topics were fused by someone who must be re-thinking why did that. He (or they) must explain, right?--kiddo 22:12, 18 November 2006 (UTC)

[edit] A differential geometer's view

In my view, there should be a Differential geometry and topology article, but it should only a short survey and overview of the common ground between the two areas, perhaps with some history. Differential topology is mainly concerned with the global topology of smooth manifolds and topics such as transversality. Differential geometry is mainly concerned with structures on manifolds, and the relation between local structure and global topology. They are really very different subjects, and deserve their own articles. Both subjects make extensive use of calculus on manifolds, and I agree with the comments below that there should be a Calculus on manifolds page which gives an overview (with links) of all the important notions, such as differential forms, vector fields, Lie derivatives, etc.

This article would then have links to all three articles: Differential geometry, Differential topology, and Calculus on manifolds.

I think it is also important to link to a smooth manifolds article. At present some suitable material is present in the differentiable manifold page, but the term differentiable manifold is rarely used these days, and manifolds which are merely C¹ or C^k, instead of smooth, are a minority interest. In my opinion, the differentiable manifold article should just survey the definitions and different degrees of differentiability of a manifold, with the rest (including examples) in a separate Smooth manifolds article, linked from here.

I could try to do some of this, but it is not a straightforward job, so help and encouragement would be welcome if others think this is a good plan. Geometry guy 17:43, 8 February 2007 (UTC)

(PS Sorry if it is impolite to put this at the top of the talk page, but I wanted to address the above class B, Top importance header directly.)

(I moved your comment to the bottom of the page in keeping with common practice). All good points. I am in general agreement with you. Three separate articles with a common tie-in on this page seems like the ideal situation. But as you say we have a ways to go before we get there. -- Fropuff 19:45, 9 February 2007 (UTC)

Great! I hope if the underlying articles are improved, then this structure will be widely accepted. Geometry guy 02:11, 10 February 2007 (UTC)

As someone who uses a lot of differential geometry / topology in my work, I'm in agreement with Geometry guy. I think it would make a lot of sense / clarify the discussion somewhat. --Bongoherbert 15:02, 28 February 2007 (UTC)

[edit] Some recent additions

Since I've made some recent additions to the differential geometry/topology page, maybe I should add a note here. I did notice the discussion of the page split proposal (which obviously must happen), but also saw little movement in it. So I thought I'd add some things anyway, in anticipation of the split.

I also noticed that the French version of this page is a translation basically word for word of the English page, and that someone has commented that the whole page is worthless rubbish (or something like that). Now someone called GeometryGuy has sent me a note about this, but I can't figure out how to reply to someone in wikipedia. So I'm writing this note instead.

In my opinion, differential topology is clearly a sub-topic of differential geometry. There should be one page for each. Then there should be a link each to the other.

I also believe that differential geometry, considered by its subject matter, should be divided into 3 areas: (a) differentiable manifolds, (b) differentiable manifolds with connections, and (c) differentiable manfiolds with metrics. Structures (c) include structures (b) because you can make a canonical connection (e.g. Levi-Civita) out of a metric.

Alan U. Kennington 00:10, 24 February 2007 (UTC)

The wikipedia categories also give differential topology as a subcategory of differential geometry, but I think we should be careful not to push this to far. In my view, they are distinct overlapping subjects, whose common theme concerns smooth manifolds, smooth functions, etc. In differential geometry, the emphasis is on local structures (i.e., additional geometric structure) and the relationship between such structures and global topology, whereas differential topology concerns global features such as the algebraic topology of smooth manifolds.

I do not agree with your division into three areas: it is too limiting. Where do symplectic manifolds fit, for example? There is much more to differential geometry than just connections and metrics.

Anyway, I'm glad you support the idea to split this article. At the moment I am looking at some of the lower-level articles to make sure the foundations are in good shape before making changes to this one. In particular, I would like there to be a solid smooth manifolds article in place, and this needs a lot of work, as the current differentiable manifolds article is a cut-and-paste job. Geometry guy 00:32, 24 February 2007 (UTC)