Talk:Differential geometry/Archive 1

Archive 1

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

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)

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 $\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ and decide differentiability--kiddo 22:17, 18 November 2006 (UTC)

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)

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)

Split?

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)

I think it should one page for both, it is usually hard to say the difference...--Tosha 19:44, 23 August 2007 (UTC)

And...

kumod kumar?--kiddo 01:02, 16 July 2007 (UTC)

Probaby, somebody inadvertently added this when pushing some buttons, or they thought it was funny to add nonsense. Anyway, I removed it. Next time you see something similar, please remove it. -- Jitse Niesen (talk) 13:31, 22 July 2007 (UTC)

I just wanted to tell all the regular reader of this article —Preceding unsigned comment added by 61.246.218.1 (talk) 07:08, 3 September 2007 (UTC)

What does "m" mean?

Could anyone tell me what the lower case "m" symbol means in the "Finsler geometry" section? MiNombreDeGuerra 21:43, 14 June 2007 (UTC)

Weisstein, Eric W. "Finsler Metric." From MathWorld--A Wolfram Web Resource. [[1]] has a simple definition of Finsler Metric. According to it: $F(x,my)=|m|F(x,y)$ for any real $m$ and any $x,y$ from the tangent space T$M$. Note the absolute value around $m$ to preserve the non-negative value of the Finsler metric.

TomyDuby 03:26, 10 September 2007 (UTC)

tortuous sentence

Likewise, the problem of computing a quantity on a manifold which is invariant under

differentiable mappings is inherently global, since any local invariant will be trivial in 

the sense that it is already exhibited in the topology of Rn

This is terribly written. I've studied differentiable manifolds for years and I don't understand this statement. It needs to be rewritten. 24.59.111.68 (talk) 20:17, 18 November 2007 (UTC)

Bizarre and unfortunate organization of differential topology and geometry articles

I find it incomprehensible that there are no separate articles addressing each of differential topology and geometry individually. Each field is huge and merits its own article.

Nothing wrong with there being an article like this one that compares and contrasts these two closely related, but overwhelmingly different, fields (that indeed borrow from each other at times).

But it is absurd that the two fields don't each have their own article. (Furthermore, this article gives the shortest possible shrift to differential topology.)

To put it another way, Geometry guy's comments above are spot-on.Daqu (talk) 17:16, 19 December 2007 (UTC)

Split

I, too, thought that it makes little sense to forcibly marry such two major distinct disciplines, and given the overwhelming support, I have boldly split the old article Differential geometry and topology into Differential geometry (which preserves full history) and Differential topology (which is almost a stub). The essay on the relations between them went to Differential topology. Given the marked improvements in the fundamental articles treating manifolds and differential calculus in the past year, I hope that now we are finally in a position to improve this top level and top priority article. Arcfrk (talk) 07:56, 30 January 2008 (UTC)

Technical requirements

I feel that this section may have outlived its usefulness. It basically consists of a compendium of definitions that are now available in a better form (both in terms of completeness and the quality of presentation) elsewhere on Wikipedia. Does it make sense to delve into technicalities too much in a top level article, such as this one? Should this section be removed or severely trimmed? Arcfrk (talk) 04:27, 31 January 2008 (UTC)

I have moved the section here, in case there is anything salvageable. Arcfrk (talk) 05:16, 4 February 2008 (UTC)

Technical requirements

The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent bundles, differential forms, exterior derivatives, integrals of p-forms over p-dimensional submanifolds and Stokes' theorem, wedge products, and Lie derivatives. These all relate to multivariable calculus; but for geometric applications, differential geometry must be developed in a way that makes good sense without a preferred coordinate system. The distinctive concepts of differential geometry can be said to be those that embody the geometric nature of the second derivative: the many aspects of curvature.

A real differentiable manifold is a topological space with a collection of diffeomorphisms from open sets of the space to open subsets in Rⁿ such that the open sets cover the space, and if f, g are diffeomorphisms then the composite mapping f o g ⁻¹ from an open subset of the open unit ball to the open unit ball is infinitely differentiable. We say a function from the manifold to R is infinitely differentiable if its composition with every diffeomorphism results in an infinitely differentiable function from the open unit ball to R. Of course manifolds need not be real, for example we can have complex manifolds.

At every point of the manifold, there is the tangent space at that point, which consists of every possible velocity (direction and magnitude) with which it is possible to travel away from this point. For an n-dimensional manifold, the tangent space at any point is an n-dimensional vector space, or in other words a copy of Rⁿ. The tangent space has many definitions. One definition of the tangent space is as the dual space to the linear space of all functions which are zero at that point, divided by the space of functions which are zero and have a first derivative of zero at that point. Having a zero derivative can be defined by "composition by every differentiable function to the reals has a zero derivative", so it is defined just by differentiability.

A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point. Such a mapping is called a section of a bundle. A vector field is differentiable if for every differentiable function, applying the vector field to the function at each point yields a differentiable function. Vector fields can be thought of as time-independent differential equations. A differentiable function from the reals to the manifold is a curve on the manifold. This defines a function from the reals to the tangent spaces: the velocity of the curve at each point it passes through. A curve will be said to be a solution of the vector field if, at every point, the velocity of the curve is equal to the vector field at that point.

An alternating k-dimensional linear form is an element of the antisymmetric k'th tensor power of the dual V^* of some vector space V. A differential k-form on a manifold is a choice, at each point of the manifold, of such an alternating k-form -- where V is the tangent space at that point. This will be called differentiable if whenever it operates on k differentiable vector fields, the result is a differentiable function from the manifold to the reals. A space form is a linear form with the dimensionality of the manifold.

“Intrinsic versus extrinsic” section should be first.

I believe it will be much better if the “Intrinsic versus extrinsic” section goes first in this article. —Preceding unsigned comment added by 72.87.242.181 (talk) 18:29, 6 September 2008 (UTC)

Global analysis

Global analysis redirects here, but this article doesn't even contain the term. Being ignorant myself, could someone add a section about what part of differential geometry is global analysis? Rain74 (talk) 18:54, 23 November 2008 (UTC)

Introduction unfocused, tangential or essential?

Without knowledge of this field, I found the introduction pretty clear until the sentence starting with "Grigori Perelman's..." The remaining sentences don't offer any details to better help me frame this topic, distinguish it from other topics, or appreciate any subtleties. It seems tangential. If these details are essential for distinguishing this branch of mathematics, then you should make that clear in the introduction. The specifics would probably better be in a subsection, more fully fleshed out or referenced. Currently, this reads as if specialists stuck a few details onto the opening paragraph. It's unfocused. —Preceding unsigned comment added by 98.30.194.163 (talk) 06:56, 13 May 2010 (UTC)

Needs some general discussion of Euclidean Diff Geom

After the lead (intro) the article delves right into branches of differential geometry. I believe that some discussion of basic results from "Euclidean differential geometry", or the differential geometry considered part of multivariable calculus (for instance, curvature, Frenet–Serret formulas, 3-manifolds and so on.). At least a mention of the relationship of these topics of Vector calculus seems worthwhile, although perhaps most of the actual material would fit better in other pages (multivariable calculus).

Actually, it appears the article that covers the topic that I am thinking of is differential geometry of curves. This should be linked here, and perhaps even the first branch of differential geometry (as it is historically, and often pedagogically since Multivariate Calc is often taken before a course that covers Riemannian manifolds). Brent Perreault (talk) 19:14, 20 December 2012 (UTC)

There is no history on the development of the subject

I find no history on the development of the subject of differential geometry, of curves and manifolds. A cursory history won't suffice: only a detailed history may do justice to the subject.

It needs to cover everything, even the minor nuances. Detailed bibliography is also required in support of key sentences and comments. Would like to help develop this section if a bibliography list is provided. Bkpsusmitaa (talk) 03:32, 3 August 2015 (UTC)

I don't agree that ANY article in a encyclopedia should EVER "cover everything". First: it is logically impossible. Second: writing must target its audience. At the two ends of the knowledge spectrum, a reader with no differential calculus nor topological knowledge will find specialist writing incomprehensible and the specialist will find the necessarily crude and imprecise language used for the lay public almost useless. Both will be frustrated.(Sorry, if I'm stating the obvious, but it seems it needed to be stated.) Third: history of a subject is at most a minor part of the information about a subject (except, of course historical subjects). This article (as of Feb 1, 2016) is certainly NOT detailed, and it follows that its history sub-section should be the same in that regard.216.96.79.179 (talk) 16:30, 1 February 2016 (UTC)