# Talk:Differential geometry of surfaces

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## Introduction

I have restored introductory material describing the scope of the subject. It's good to see the forest before embarking on the detailed study of the trees … Arcfrk (talk) 22:38, 3 February 2008 (UTC)

Auguri, U r doing an excellent travail!--kiddo (talk) 17:52, 12 February 2008 (UTC)

## normal coordinates change ?

The following sentence :

== Taking a coordinate change from normal coordinates at p to normal coordinates at a nearby point q, yields the Sturm-Liouville equation satisfied by H(r,θ) = G(r,θ)½, discovered by Gauss and later generalised by Jacobi, Hrr = – K H. The Jacobian of this coordinate change at q is equal to Hr ==

is not clear. What is the link between a normal coordinate changes at p to q, and the equation Hrr = – K H. ? Why Hr is the Jacobian of this coordinate change ? Thank you for your explanations. 139.124.7.126 (talk) 17:06, 26 March 2008 (UTC)

This classical computation is discussed for example in Berger's book. I'll give you a detailed explanation myslef, if I have time. Mathsci (talk) 13:50, 18 April 2008 (UTC)

## Opener

Why would we have the opener sentence contrast with another subject, and moreover bad mouth it. Seems to be a shame, for there is much beauty in the differential theory of curves as well. Oded (talk) 06:16, 7 April 2008 (UTC)

I tend to agree with this comment. There are many kinds of questions that can be fruitfully studied in the curve situation that is different than the surface case. This is not a weakness but a strength. For example, curvature bounds and knot energies are related to knot type and that seems to be to be under the jurisdiction of "differential geometry of curves". I will remove that sentence for now, unless someone objects. --C S (talk) 15:43, 17 April 2008 (UTC)
Actually, it seems it would take a little work to remove the sentence, as it is so entangled in the lede. I will come look at it later. --C S (talk) 15:45, 17 April 2008 (UTC)
I completely agree. I have already said elsewhere that I do not like the lede, which was written by User:Arcfrk after he excised the main text, written almost wholly by me, from Surfaces. I also do not understand why he included the second paragraph, which has no relation to the contents of the article. Mathsci (talk) 13:47, 18 April 2008 (UTC)

I modified the lede accordingly. Oded (talk) 15:03, 18 April 2008 (UTC)

The first paragraph is very good now. (BTW nice to have you as a fellow WP editor!) Mathsci (talk) 15:44, 18 April 2008 (UTC)
thanks. Do I know you? Oded (talk) 16:18, 18 April 2008 (UTC)
Probably not. I was at MSRI during your "hot topic" week and have research interests in CFT, so I was very pleasantly surprised to see you here :) Mathsci (talk) 19:23, 18 April 2008 (UTC)
Actually, this should still be improved. The first sentence adopts the intrinsic viewpoint, while the second is explicitly assuming an embedding of the surface in Euclidean space. Oded (talk) 17:16, 18 April 2008 (UTC)

I tried to fix this. Hope it is better now.

It does not seem to me so. When comparing to differential geometry of curves it contains significantly less information with the same amount of introductory talk. Anybody wishing to update this article? —Preceding unsigned comment added by TomyDuby (talkcontribs) 15:30, 4 July 2008 (UTC)

What precisely do you mean? The differential geometry of curves is elementary high school material, while this article contains much harder concepts, often only covered in a final undergraduate year. Please try to be more specific. Mathsci (talk) 15:47, 4 July 2008 (UTC)
Thanks for your reply. Yes, I agree with you that the differential geometry of surfaces is a much harder concept. However, it should be possible to explain it reasonably clearly. What specifically I have in mind? For example the chapter called Definitions in differential geometry of curves is well written. I think that, to begin with, a similar chapter extended to surfaces should be included in this article. Ideally it should define a k dimensional surface in Rn of class Cr. Something along line of E Kreyszig, Differential Geometry, Dover Publications, ISBN 0 486 66721 9, section 24. TomyDuby (talk) 01:51, 5 July 2008 (UTC)
P.S. Is the stuff of differential geometry of curves really elementary high school material?
I learnt curves at grammar school. One of the problems is that this is not an article solely on embedded surfaces in Euclidean space. Another is that the article was originally ripped out of Surfaces by now disappeared editor User:Arcfrk. The lack of definitions dates from then; it seems like a good idea to define again embedded surfaces, with their induced metric, and then abstract 2-dimensional Riemannian manifolds, if this improves readability. Surfaces will always be harder than curves, because it is with surfaces that the story of curvature (and connections, not mentioned in this article) in Riemannian geometry starts. Curves play a vital role in understanding connections. As far as I can tell, most of the material is covered well. However, it might be helpful to use a modern text book like Pelham Wilson's CUP text to make the material a little more approachable towards the beginning of the article. One suggestion I have is to use xfig or some other software for drawing figures to illustrate what's going on. This applies particularly to curvature. Berger's panorama has lots of pictures. It might also be nice to expand the history with a picture of Gauss or some early texts. It seems like a very good idea to make this fundamental material as accessible as possible, as Willow and her collaborators have been doing at Emmy Noether. Mathsci (talk) 06:53, 5 July 2008 (UTC)

This article is clearly deficient, but not for the reasons stated. It contains vastly more material than "Differential geometry of curves". The lack of definitions that occupy first chapters of many books on manifolds is not a shortcoming, in my opinion, since 1) Wikipedia is not a textbook, and proceeding in logical order, an article that treats the subject at this level of detail would have to be hundreds of pages long, and 2) they are not necessary to understand the main features and accomplishments of the theory of surfaces (indeed, these definitions were formulated long after the development of the theory). However, a major part of the article, Sections 3 through 8 ("Gaussian curvature of surfaces in E3" – "Surfaces of constant curvature"), lacks any sense of purpose or coherent structure. There are no clear prerequisites and they are not aimed at a particular class of readers. Certain parts (e.g. Gaussian curvature) are ad hoc, other sections contain odds and ends (Birkhoff curve shortening process, geodesic polar coordinates) presented in a technical manner that are not integrated with the rest of the article. I have a feeling that other editors are holding back because these sections cannot be mended, not because the material is "well covered". They had been written by Mathsci and tacked into the article "Surface" dealing with topological surfaces. Ever since I culled these sections out and moved them into a separate article (i.e. here), he's been following me around and, in turns, whines about how I desecrared his extraordinary work and slanders my contributions, in other words, behaves like an insolent child, rendering meaningful editing difficult and unnecessarily time-consuming. Perhaps, if Oded or R.e.b., whom he seems to respect, were to remove or trim these sections, that would lead to a way forward, but I am through. Arcfrk (talk) 22:03, 5 July 2008 (UTC)

Thanks for Arcfrk and Mathsci for their comments. I think that it is now time to do some work: to add a section / subsection with a decent definition of a surface from differential geometry point of view. I am willing to do it. Any objections / ideas?
TomyDuby (talk) 01:54, 17 July 2008 (UTC)
Pleased see the discussion below (following a discussion on the talk page of Wikiproject mathematics) and the history of the mainspace article. Work has already begun on rejigging the main article; please add further comments below where User:JackSchmidt has been leading the edits. (You might also like to look at the main article Surface.) I intend to do to work on Section 4 next with an informal introductory paragraph and illustrations in the near future. Cheers, Mathsci (talk) 08:56, 17 July 2008 (UTC)

## Assessment

Howdy, just to give an independent assessment of the article. The lead, overview, and history sections are very well written. Sections 3–7 are somewhat problematic as they do not contain much information about why the concepts are interesting (say in the opening sentence of each section). All other sections strike me as well motivated. All sections strike me as well written.

A few minor things:

• in section 3, I believe it is intended that the Taylor expansion is given up to elements of total degree 3. It might be nice to indicate this somehow. For instance, it was not clear to me why the coefficient on xy was omitted from the K=k1k2 definition, but probably the formula is intended to indicate the coefficient on xy was set to 0 by a rotation. There are 4 terms of degree 3, which make be a little verbose. Is there a standard O(x^3) notation for two variables? (done in a very simple way only)
• in section 3, can someone check if there is a factor of 4 left out? It seems like Fxx is 2*k1, not k1.
• in section 4, there is a footnote, but I think there is no <references/> tag.
• in section 4, I think this is covered in different notation in introductory calculus courses (second or third semester), and for instance mathworld has an "engineer" version at mathworld, equation (4). Notice that many of the symbols are the same, but mathworld provides the engineer with some critical information (basically being explicit about the parameterization). In the Stewart's Calculus book, I think one takes u=x, v=y, z=z(x,y), and the formula simplifies. I think including just this simplified case (like in section 3), would be useful to readers with an engineering background
• in section 4, I think there is a typo, 3=s, perhaps: "It is unknown at present, except in some special cases, whether every metric structure arises from a local embedding in E³ for some s in N."
• is section 5 two separate sections? It seems a second section has snuck in here using the first half to present the "remarkable theorem". Perhaps the remarkable theorem merits its own section?
• section 6 may be very useful for adding some motivation to section 3 (a one phrase summary of section 6 would probably be helpful in a "why is the gaussian curvature interesting")
• sections 1,3,4,5,6 uses E^n, sections 7,8,9 use R^n. Both are basically used without definition. It might be nice to define E^2 and E^3 in section 1 ("Euclidean space, E3," etc.), and E^n whenever it is first used (section 4 maybe), and change 7,8,9 to use E^n. (Though E^3 and E^n are still not explicitly defined, they seem clear from context)

There are some disambiguations needed (most are routine, but some require someone more knowledgable):

• in section 1,
• polyhedra (to polyhedron)
• in section 5,
• in section 7,
• Cartan (Élie Cartan?)
• Alexandrov (Pavel Alexandrov or Aleksandr Danilovich Aleksandrov or?)
• Tits (Jacques Tits)
• von Mangoldt (Hans Carl Friedrich von Mangoldt)
• If one was feeling frisky, it would not hurt to update the other people links. Sometimes there is piping to a redirect, like for Gromov, and I think a direct pipe would be better.
• in section 8,
• hyperbolic plane (hyperbolic geometry?)

Hope this helps, JackSchmidt (talk) 07:14, 6 July 2008 (UTC)

I have started to make some additions to section 3 (pictures, emphasised equations). In this section it might be nice to add some introductory words on the meaning of Gaussian curvature for non-experts, possibly with pictures. This is done on pages 123-124 of Pelham Wilson's book, Curved spaces - my signed copy has an inscription reading, "For ******, who always shows good taste" :). Other introductory texts contain explanations like this. I couldn't see anything useful in Gaussian curvature, which could be similarly criticized. Why don't you try to add to section 3 a little more, possibly in this way? The aim should be to make this accessible to non-experts, in the way Willow does in her edits to mathematics articles. Labelled pictures illustrating Gaussian curvature - such as the figure on page 124 of PMHW - would seem a good alternative visual way of getting across the ideas. If you could produce such a picture, that would be great (my xfig is rusty). Please go ahead and make the changes you have suggested elsewhere, if you think they are uncontentious. Adding a second or third way of doing something is also great. Even an extra section at the end discussing connections (a la Singer-Thorpe) would be great. What I wrote was just a condensed account of an undergraduate course. Mathsci (talk) 07:38, 6 July 2008 (UTC)
I don't really know anything about geometry, not even basic 3D calculus, so I actually have no idea how to fix most of the problems (or whether some are in fact problems). I fixed the fairly straightforward things (but may have introduced historical errors; pointing out the wrong Cartan or Jacobi), and crossed them out. Can you check "the factor of 4" in section 3, the "typo 3=s" in section 4, and "who is Alexandrov" in section 7? JackSchmidt (talk) 08:14, 6 July 2008 (UTC)
Hello. I just corrected the statement about embeddings in E^3. In the analytic case this is Janet's theorem, discussed in the text. Isometric embedding in arbitrarily large euclidean spaces is always possible by Nash's theorem. Mathsci (talk) 08:25, 6 July 2008 (UTC)
I've done some more cleanup on section 3, adding a summary and a "spoiler" for the later sections. This is a first approximation. There could indeed be some missing factors of 2 or 4 - best to look in the Kreyszig, Eisenhart, etc, i.e. the old classics. I'll try to check this later. Mathsci (talk) 10:04, 6 July 2008 (UTC)
Excellent. I'm glad you handled the 3=s. The new intro looks good. The Alexandrov thing looks hard. Publishing in 1940 was a little slim, and western European math reviews were also a bit slowed. Neither Zentralblatt nor AMS's Math Reviews have any papers by Alexandrov in 1940. A.D. Alexandrov did work on polytopes, which seems to be what is intended in this article, but from 1940 to 1954 (MR63683), it is silence in the reviews. The factor of 4 is probably only important for internal consistency in that particular section. One should be able to calculate it for the surface z=xx+yy. I believe the k1*k2 definition gives K=1*1=1, and the one using derivatives should give (2*2-0*0)/(1+0^2+0^2)^2 = 4. I've been surprised at how differently I view basic calculus than differential geometers do, so I won't commit to 1 not being equal to 4. JackSchmidt (talk) 16:43, 6 July 2008 (UTC)
Thanks for making all your changes! I lectured the Aleksandrov comparison theorem for negatively curved surfaces in a working seminar here in France and gave a direct account in a UK graduate course (somewhere out there on the web) for the Poincare disc. I should add references (I think there are refs in the Orbifold article). One reference is "Intrinsic Geometry of Surfaces" by Aleksandrov and Zalgaller. It is also explained in Berger's book, in Bridson and Haefliger and in one of Jost's books. The correct wikilink is Aleksandr Danilovich Aleksandrov- I think he's mentioned in Bruhat-Tits building and certainly in CAT(k) space. You're probably right about the 4, but I haven't had a chance to look yet. (BTW I'm an analyst.) Keep up the good work. Cheers, Mathsci (talk) 18:15, 6 July 2008 (UTC)
I have started cleaning up section 4. I would like to find images illustrating geodesics and the geometry of the last section (I might have to make a diagram myself using xfig, unless there's already something free out there). I am not happy at the moment with the presentation of the Gauss differential equation, although this is how it appeared originally and how it can be found in numerous text books from the first half of the 20th century. At that time, with elliptic functions, this theory was drummed into the head of every university student in mathematics. I'll take another look at Berger's presentation. :-) Mathsci (talk) 10:49, 17 July 2008 (UTC)

(unindent) The Gauss equation seems best to mention where it is used - in the proof of the Gauss-Bonnet theorem, so I have temporarily removed it. Mathsci (talk) 11:16, 17 July 2008 (UTC)

The following discussion has been copied from my talk page. Mathsci (talk) 13:37, 5 August 2008 (UTC)

The overview section is a little odd. The emphasis in the lead was on Gaussian curvature. The latter is undefined at the vertices of the cube. Perhaps a more appropriate example would be the unit sphere, and the discussion of polyhedra could be deleted. Katzmik (talk) 10:58, 5 August 2008 (UTC)

Furthermore, the comment in the subsection "riemannian connection" on characteristic classes is a little odd. I don't see how one can say that curvature paved the way for characteristic classes. The latter are topological in nature. There are numerous theorems expressing them as integrals of curvature, but that's already after the classes are defined. Also, certain characteristic classes would seem to have almost nothing to do with metrics, e.g. torsion classes such as Stieffel-Whitney. Katzmik (talk) 11:02, 5 August 2008 (UTC)

Well the overview needs completely rewriting because it was transplanted from another article, which explains the tags.
The examples section can clearly be expanded to cover what's usually treated in elementary text books.
Characteristic classes are often defined as invariant polynomials applied to curvature tensors. That they turn out to be homotopy invariants is usually a result of a theorem in mathematics, like the Novikov conjecture or Baum-Connes conjecture, etc. Their definition often uses a smooth structure. In my own field they come as tr (p (dp)2n) or tr((u*du)2n+1), the pullbacks of invariant forms on the grassmannian or the unitary group.
I'll add a reference to Chapter XII of Kobayashi & Nomizu Vol 2 on characteristic classes and will make "paved the way" more precise.
You're right about torsion classes - e.g. Borel cohomology (a la Calvin Moore) of a compact Lie group, aka the cohomology of the classifying space, needs all sorts of different things like this (one by the way being the classic paper of Cheeger and Simons). Cheers, Mathsci (talk) 11:24, 5 August 2008 (UTC)
I am not sure what you mean by your new reference to do Carmo's book on curves and surfaces. The only characteristic class I can see being defined there is the Euler class. I think it would be unusual to think of the Gauss-Bonnet theorem as a definition. Katzmik (talk) 11:36, 5 August 2008 (UTC)
P.S. Never mind, I see what you mean (I modified the text to make it clearer). I do think you mean the later book by do Carmo here, though. Katzmik (talk) 11:41, 5 August 2008 (UTC)
(ec)Click again. It's a reference to Chapter XII of Kobayashi and Nomizu, Characteristic Classes. That's what it says in the source file and that's what I see when I click. Mathsci (talk) 11:43, 5 August 2008 (UTC)
In the next sentence, you give a reference to do Carmo. That's the one that needs to be updated, I think. As far as overall structure of the page is concerned, it may be helpful to separate out local and global results, which are mixed up together at this stage. Katzmik (talk) 11:46, 5 August 2008 (UTC)
The 3 references, also mentioned in the section Reading guide, are put there because they are standard introductory graduate texts. Because this is intended as a starting introduction to differential geometry, for as wide an audience as possible (following a request from other mathematicians), that is how the new section on Riemannian connection has been written (from scratch). Simplifying the presentation slightly by assuming a local embedding in E3 is a standard approach adopted by the 3 standard introductory texts: the resulting formulas are invariantly defined. It's possible that I went too far at some stages. The method of Kobayashi for example applies equally well for isometric embeddings in E4. I will proof-read the article with local-global point of view in mind, but separating them does not seem a great idea. This is done in the otherwise excellent introductory book of Singer and Thorpe, but I don't think it's easy reading for non-experts. There are also plenty of other WP articles on the general theory, some of which can be seen in the template at the bottom of the page. Some of these are quite unreadable. The remit with this article was to make it as approachable as Differential geometry of curves, which is quite a tall order. I should have left the work in progress template up, because it still needs a lot more polishing (the parallel transport section is being revamped at the moment). All your suggestions for improvements are welcome and very helpful. Cheers, Mathsci (talk) 12:07, 5 August 2008 (UTC)
AS far as do Carmo's book is concerned, I was merely pointing out that the connection approach is barely mentioned there at all (only as an afterthought on page 442), so it cannot be said that the approach using connections is typical of do Carmo's first book. Katzmik (talk) 12:16, 5 August 2008 (UTC)
Covariant derivatives are defined on page 238 of do Carmo. I think you originally inserted connection into this sentence. I have modified it appropriately now.
BTW I am thinking about the point about abstract Riemannian 2-manifolds and embedded surfaces. The existence of the connection on the frame bundle is slightly hard to do without an embedding. That's why Singer & Thorpe's approach is tricky. It is true that an arbitrary surface can locally be embedded isometrically in E4, but it would be artificial to use this hard result. I will think some more about the problem. Either it could be assumed for simplicity that the surface was isometrically embedded in E3, with a statement to that effect in the introductory section of Riemannian connection. Or I could carefully modify the text so as to apply in general, proving the existence of the connection 1-form on the frame bundle indirectly (as on page 189 of Singer & Thorpe or on page 159 of Kobayashi and Nomizu, Vol I). I think the latter approach is better, but it would require quite a lot of work. (BTW when I've taught the Atiyah-Singer index theorem, I have included Gunther's short proof of the Nash embedding theorem to handle this problem. Nigel Hitchin told me later that it was a bit like using a sledge-hammer to crack a nut.) Cheers, Mathsci (talk) 12:40, 5 August 2008 (UTC)

Sorry if I am being confused, but can't one write down the connection in terms of the sum of the exterior derivative and the Christoffel symbol? The latter has an explicit formula in terms of the metric coefficients, and we seem to be done. Katzmik (talk) 13:01, 5 August 2008 (UTC)

This is more or less what I was saying. Take the Grassmannian connection given by the embedding, look at it in local coords when it gives the Christoffel symbols and then notice that it solves the problem in general. Actually it's a good idea to give several points of view.
BTW in the section on Liouville's equation you should wikilink the Laplacian and provide a direct link for the existence of isothermal coordinates. It was a very good idea to include this material, but it could be more self-contained like the rest of the article from section 3 onwards. Isothermal coordinates and conformal structures on surfaces are discussed in on pages 376-378 of Taylor's PDEs, Vol I, or numerous other places (Chern). It wouldn't be a bad idea to add a separate section on this, but not without giving careful definitions and/or wikilinks. Mathsci (talk) 13:21, 5 August 2008 (UTC)

There is an error in the section on Hadamard's theorem. Who is vangoldt by the way? Never heard of him. The error is that the uniqueness of a minimizing geodesic is stated without the hypothesis of simple connectivity. Katzmik (talk) 13:44, 5 August 2008 (UTC)

Yes the mention of homotopy class got dropped when the previous section on Birkhoff was detached to its current location. For von Mangoldt, I think you just click on the wikilink. You can also look in Berger's book. Mathsci (talk) 15:23, 5 August 2008 (UTC)
OK, I was not aware of that. Berger usually knows what he is doing. The current phrase about the homotopy class is not very clear. Katzmik (talk) 15:26, 5 August 2008 (UTC)

## Levi-Civita connection

A more appropriate term for the canonical Riemannian connection is the Levi-Civita connection, a term traditionally reserved for this unique connection. Calling this connection THE Riemannian connection is technically speaking incorrect since a Riemannian connection may have torsion in general and is not unique. I see now on the Levi-Civita connection page that Christoffel discovered it before Levi-Civita, which I would like to know who to attribute such an assertion to. At any rate, this is commonly accepted terminology, and those textbooks that refer to it as the Riemannian connection are sloppy. It should be called either the canonical Riemannian connection, as Lawson and Michelsohn call it, or Levi-Civita. Katzmik (talk) 14:06, 5 August 2008 (UTC)

You've already brought up this argument elsewhere on silly rabbit's talk page when you interrupted our discussion there. There doesn't seem to be any need to repeat it here. It is the term used in most of the text books appearing as source books here. Compromises are possible, with an explicit discussion in the text (both are mentioned). However I don't think it is a very helpful way of participating in this project to bring this minor point up, where all sorts of compromise solutions are possible, while minutes later implying that the whole approach of textbooks by Isadore M. Singer & Thorpe, do Carmo and O'Neill are unfit for this encyclopedia. That seems like an extraordinary thing to suggest. Sorry, Mathsci (talk) 14:41, 5 August 2008 (UTC)
NO problem. I mentioned already that do Carmo's second book does attribute the connection to Levi-Civita, whereas his first book clearly calls it the Levi-Civita connection on page 442. I think you are simply in error if you think the connection is not commonly referred to by his name. I have not looked in the other two books you mentioned. If the question is to determine what the prevalent usage in differential geometry is today, we could raise this issue at wiki math project. At any rate, you have not responded to my point about Riemannian connections on an arbitrary bundle. Katzmik (talk) 14:48, 5 August 2008 (UTC)
Why not check all the references for the main space article? Mathsci (talk) 14:57, 5 August 2008 (UTC)

## surfaces and connections

If we are going to be serious about this acticle being actually about surfaces rather than manifolds, then the emphasis on connections is blown completely out of proportion. Note that on a surface, once we know what the geodesics are, we automatically know what parallel transport is. Namely, the tangent vector to the surface is parallel, whereas the unique (up to sign) normal vector is, well, also parallel. Any other vector is a linear combination of the two. It will be parallel if and only if the coefficients in the tangent/normal decomposition are constant. This remark should probably replace the entire section on connections, riemannian or L-C. Katzmik (talk) 14:13, 5 August 2008 (UTC)

Except this is not what is said in any standard textbook. What you say in fact does not seem to be correct. Parallel translation requires an ordinary differential equation to be solved and this does not seem to be what you are writing, although every book on the subject mentions it. Are you sure you know what you're talking about? Mathsci (talk) 14:29, 5 August 2008 (UTC)
Wiki is anything but standard, so that should not be a problem :-) To get parallel transport along an arbitrary curve, use a piecewise geodesic approximation. I am certainly proud to have been able to startle you :-) Katzmik (talk) 14:33, 5 August 2008 (UTC)
P.S. You certainly need to solve a differential equation to construct geodesics. Katzmik (talk) 14:34, 5 August 2008 (UTC)
Presumably what you wrote above refers to parallel transport along geodesics. You are correct that this species of parallel transport is an almost trivial construction. However, the article should also define parallel transport along arbitrary curves, in my opinion. siℓℓy rabbit (talk) 14:51, 5 August 2008 (UTC)
I addressed that in my comment above about PL approximation. In a philosophical aside, I would point out that the discussion above captures very well the difference between a Riemannian geometer and a differential geometer :-) Katzmik (talk) 14:54, 5 August 2008 (UTC)
Oh, I missed that. Except not every curve is piecewise a geodesic. There is nevertheless a nice theorem due to Francesco Severi in which one looks at the geodesic tangent to a given curve. Unfortunately, in order to be an interesting geometrical statement, one needs to allow oneself to think of "infinitesimally near points". siℓℓy rabbit (talk) 15:00, 5 August 2008 (UTC)
For any smooth curve joining point A to point B one can define parallel transport along the curve by approximating it by a PL geodesic curve, and using the tangent/normal decomposition I mentioned above. It is immediate from the Gauss-Bonnet theorem that the process will converge to give a well defined notion of parallel transport from A to B. Again, this only works for surfaces. For this reason I prefaced my remark by the proviso that one should decide how seriously one wants this to be an article about surfaces. Katzmik (talk) 15:07, 5 August 2008 (UTC)
Ahh... I see. siℓℓy rabbit (talk) 15:09, 5 August 2008 (UTC)

One disadvantage of this method is that it requires the solution of a second order equation (the geodesic equation), whereas parallel transport along a given curve requires the solution of only a first order equation. If one wants the surfaces to be as general as possible from the point of view of smoothness, this may be significant. Katzmik (talk) 15:13, 5 August 2008 (UTC)

Unless you can provide a WP:RS, this counts as original research. It would also be giving undue weight to include it in the informal introduction, since this is not found in the majority of text books. However, if a source can be found, it can be certainly be included as a comment at an apporiate stage. Mathsci (talk) 16:14, 5 August 2008 (UTC)
Appendix 1 of Arnold's book on Mathematical Methods in Classical Mechanics contains the following assertion:

Finally, parallel translation of a vector along any smooth curve of the surface is defined by a limiting procedure, in which the curve is approximated by broken lines consisting of geodesic arcs.

There is neither a proof and does not give a reference to a proof. Please could you provide a reference which contains a proof. So far it does not to be contained in any standard textbook, apart from the mention in Arnold. That would indicate that it being given WP:UNDUE weight. The argument is anyway circular, since parallel transport is actually used to prove Gauss-Bonnet (eg in Singer & Thorpe, etc). If a proper citation can be found then it can certainly be included in the article in an appropriate context. Mathsci (talk) 17:18, 5 August 2008 (UTC)

## Move out proposal

I agree with Katzmik that in the present text of the article, the theory of connections and parallel transport are completely blown out of proportion. While most books on differential geometry of surfaces do mention parallel transport, typically, in the context of Gauss–Bonnet theorem, this is at best a small part of the general theory of surfaces. The corresponding section seems to be a highly technical Ersatz for Riemannian connection in Riemannian geometry. I propose to move the full section there, and replace it with a one or two paragraph geometric description pertinent to surfaces. A further remark concerning the level of generality: although parallel transport along geodesics is only a special case, it is easier to explain and is by far the most important case needed in the theory of surfaces. Arcfrk (talk) 15:59, 5 August 2008 (UTC)

Please see the comments on WP mathematics. Mathsci (talk) 23:39, 6 August 2008 (UTC)

An odd and dismissive comment was made above about a book by Vladimir Arnold, one of the greatest mathematicians living today. Judging by author citations in mathscinet, his influence is greater than that of another great mathematician, Michael Atiyah. That a statement of Arnold's should be secondguessed by a wiki editor calls for a discussion, in my opinion. Katzmik (talk) 13:09, 6 August 2008 (UTC)

Was there a particular reason for mentioning the influence of Michael Atiyah here or are you just trying to WP:HARRASS me?
I made no value judgement on Arnold or his book. I wrote that one unreferenced sentence in the appendix of his book did not cover the remarks you had added in your inadequately sourced edit. Please read WP:RS and WP:V. The fact that your method does not appear to be in any standard textbook on the differential geometry of surfaces also suggests WP:UNDUE weight. Mathsci (talk) 13:52, 6 August 2008 (UTC)
Please also see the comment on your talk page. Mathsci (talk) 23:40, 6 August 2008 (UTC)

## Surfaces of constant Gaussian curvature

I don't quite understand why tangent developable, cone (geometry) and developable surface were removed from the section. It seems logical to enumerate the different posibilities in the same place, the fact that they don't meet the lie group criteria, would suggest rewording rather than removal. --Salix alba (talk) 16:30, 16 August 2008 (UTC)

I second that. Classification of surfaces of constant Gaussian curvature in R3 is a highlight of the subject, well worth exposing well. On the other hand, the standardly embedded torus obviously doesn't have zero Gaussian curvature. Arcfrk (talk) 21:28, 16 August 2008 (UTC)

## Error in Surfaces of constant Gaussian curvature

I think there's an error where the article claims that the surfaces of revolution obtained by revolving e^t or cosh(t) or sinh(t) have constant gaussian curvature -1. This would contradict Hilbert's theorem of no complete -1 curvature surfaces in E^3. The surfaces obtained are negatively curved, but not of constant negative curvature.

## error in discussion of constant curvature

In the discussion of simply connected surfaces of constant curvature, the claim is made that in each of the three cases, the surface is a quotient of the isometry group by the maximal compact subgroup SO(2). This does not apply in the case of the sphere. Katzmik (talk) 14:08, 20 August 2008 (UTC)

## euclidean geometry

To get a torus out of a plane one needs more than just a free abelian subgroup of rank 2. It needs to be discrete. Thus, horizontal translations by 1 and \sqrt{2} do not define a lattice. Katzmik (talk) 10:07, 24 August 2008 (UTC)

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## rotman

Hi,

Rotman has certainly made a crucial contribution. One should certainly mention also Nabutovsky and Sabourau, in my opinion. Katzmik (talk) 12:24, 10 September 2008 (UTC)

Greetings. I have added this in the Rotman footnote. Please could you add all the references explicitly to your article systoles of surfaces? That is probably where all the detail should be. This last section is only intended to be a short summary. Cheers, Mathsci (talk) 12:51, 10 September 2008 (UTC)

## Lie groups and Erlangen

Now that the lead deals with Lie groups, it should also contain at least an allusion to Felix Klein and the Erlangen program, in such a way that the remark about Lie groups should not appear in a vacuum. Katzmik (talk) 13:56, 10 September 2008 (UTC)

Lie groups are already used at length in the constant curvature section. That is what the lede refers to. This is an introductory article on the differential geometry of surfaces - please cite a precise source on the differential geometry of surfaces and the parts of it you think are suitable for inclusion here. Mathsci (talk) 17:14, 10 September 2008 (UTC)
There is in fact a photo of Klein in the article and various historical references to the development of hyperbolic geometry, including some to him. There is no harm in adding something about the Erlangen programme - it's mentioned in some of the references in the bibliography. That would seem to be a natural place for it. When editing Robert Fricke's BLP recently I did read about the terrible illness in 1883-4 that stopped Klein in his tracks. All of these intellectual leaps, against the flow of old-school geometers who villified Bolyai and Labatchevsky, are probably of interest to a general non-expert readership. Mathsci (talk) 17:32, 10 September 2008 (UTC)

## Geodesic Curvature

I'm slightly confused. The geodesic curvature is defined to be the dot product of the normal to the surface with the acceleration of a curve. One bullet point says that the acceleration of a geodesic is normal to the surface and another says that the geodesic curvature vanishes.

I obviously understand why the acceleration should be normal to the space, but then why is the dot product of the normal with the acceleration 0?

81.148.106.221 (talk) 11:07, 16 May 2010 (UTC)

## Weingarten Equations

I tried calculating the Weingarten equations and I get fG-Fg in the top right of the matrix, whereas what is written is fE-eF.

Jjw19 (talk) 21:53, 16 May 2010 (UTC)

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## Smooth surface

If someone is looking for the definition of a smooth surface will be redirected to this article. However from the context the meaning of smooth surface obvious is not...Theodore Yoda (talk) 14:51, 27 February 2013 (UTC)

I agree that it is confusing. The wiki link to smooth in the lead links to smooth manifold, which redirects to differentiable manifold, which gives the definition "A smooth manifold or C-manifold is a differentiable manifold for which all the transition maps are smooth. That is, derivatives of all orders exist; so it is a Ck-manifold for all k." In contrast, Surface#Surfaces in geometry for which this article is the main article, defines "It is also possible to define smooth surfaces, in which each point has a neighborhood diffeomorphic to some open set in E²." This is also the definition given in the Overview section. An [external reference] defines a smooth surface in R3 to be both diffeomorphic to an open set in R2 and for that diffeomorphism to have derivatives to all orders. I do not know which is most popular/established definition, but this would be good to sort out and once sorted out, add a citation. --Mark viking (talk) 18:09, 29 April 2013 (UTC)
Looking at Diffeomorphism it is defined as "an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth." So it looks like its virtually the same thing.--Salix (talk): 19:21, 29 April 2013 (UTC)
Thanks for pointing that out. That is what it says in the lead of Diffeomorphism. But the next section Diffeomorphism#Definition only requires an r times continuously differentiable map and r can be less than infinity. And the next section then talks about diffeomorphisms as being smooth maps again. At the encyclopedia of mathematics article on differentiable manifolds, a differentiable manifold is only required to have class k (Ck) differentiability. It is still confusing to me, but it seems that there is some evidence for "being homeomorphic to E²" and "maps are differentiable to all orders" being different things. --Mark viking (talk) 20:19, 29 April 2013 (UTC)
Differentiable manifold nicely explains the different types. The articles are quite careful about specifying it being a Ck-diffeomorphism or Ck-manifolds if you only have a finite number of derivatives. Smooth is infinitely differentiable. Sometimes the prefix is dropped if the class your working in is clear. If you just have homeomorphism then you just have a topological manifold with no differential structure (some times called C0 as you have continuity). --Salix (talk): 21:12, 29 April 2013 (UTC)

## 1825-1827

Our lede currently says, "One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss (1825–1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space." First, the (1825-1827) formulation after a person's name usually means their lifespan. So using it here to date Gauss's papers is clumsy. It is also imprecise, if Gauss studied it in depth before 1825, which is likely. Also, the dates link to nothing. If you click them, nothing happens. I propose changing the wording to "One of the fundamental concepts investigated is Gaussian curvature, first described in two papers (1825 and 1827) by Carl Friedrich Gauss, who showed that curvature is an intrinsic property of a surface, independent of its isometric embedding in Euclidean space" and removing or repairing the wikilink from the dates. --Anthonyhcole (talk · contribs · email) 15:08, 6 March 2014 (UTC)

Fixed, I guess. D.Lazard (talk) 18:09, 6 March 2014 (UTC)

## Definition of mean curvature utilizes undefined quantities E,F,G introduced later ?

Early in this article, section 'Curvature of surfaces in E^3', the definition of mean curvatures K.sub.m = (ET + GR -2FS) / (1+P^2+Q^2)^2 utilizes quantities not defined up to that point. I believe that (E,F,G) are the parameters of the first fundamental form introduced later in section "Line and area elements", or possibly the (e,f,g) of the second fundamental form.

I don't want to tamper with the article, but would the latest editor of this section or some other dispassionate soul kindly replace E,F,G with 1,0,1 (special case of the introductory discussion) or else define the quantities before they're used? / bruce_bush_nj /

## Assessment comment

The comment(s) below were originally left at Talk:Differential geometry of surfaces/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

 The lead needs some work, and there could be more on extrinsic curvature, but this is comfortably B-class. Geometry guy 12:12, 13 April 2008 (UTC)

Last edited at 12:12, 13 April 2008 (UTC). Substituted at 02:00, 5 May 2016 (UTC)