Talk:Differential geometry of surfaces/Archive 2

Recent edits by
This user did 51 edits in a row consisiting of creating and expanding a section "Function theory in two variables" and adding several items in section "References". I have reverted these edits for the following reasons.

Section "References" is already very long, and most (if not all) added items are not linked from the body of the article. So, such additions in a long and indiscriminate list are not useful.

The added section is not about surface nor about geometry. It consists of a content fork of articles about differential functions. It cannot be viewed as a recall of the background that is needed in the following sections, since it contains proofs, and the results of this section are not those that are needed in the following sections. So this new section is off-topic.

I see that while I was writing this post, my revert has been reverted, with an edit summary refering to WP:BRD. So, following the process described there, I'll restore the state of the article before mathsci's edits. Please, apply WP:BRD, and wait for a consensus here, before restoring the disputed edits. D.Lazard (talk) 16:25, 3 August 2020 (UTC)


 * D. Lazared, you've just seen a message that I've written above that you have ignored. Please don't do that: it's very impolite. You also placed me as a username here. That is also not appropriate and you should not do that again.


 * This article was started as 31 January 2008, transferred from the article on surfaces. I started contributing there on 2 February 2008 and have made 662 edits to the article.


 * I am unaware that the two first sections of the body of the article were not well written and that the lede does not properly summarise the content. I am also aware that the beginning chapters of the books of do Carmo and O'Neil were not dealt with there. For example, if you look at what is covered in UCLA lecture notes, there is much absent content.


 * I have checked wikipedia articles and am aware that diffeomorphisms are not treated properly there. Just look at the references or the segement on diffeomorphisms, a.k.a. inverse function theorem. :Is it true that you have made almost no edits to this article? Blanking content is vandalism.You have no idea what I am preparing at the moment and are simply edit-warring in an irrational way. Mathsci (talk) 17:05, 3 August 2020 (UTC)


 * Comment: I haven’t looked over the history of edits and don’t have much of an interest in the conduct discussion here, but I'll comment on the content.Regarding the content of the section "Function theory in two variables", the entire section should be removed. It's a textbook-like coverage of pedagogical prerequisites that falls outside of "differential geometry of surfaces" proper. Wikipedia does not need to cover all of the mathematical background necessary to learn the subject of every article; it is a policy that Wikipedia is not a textbook and does not aim to instruct. This material could be useful elsewhere (perhaps Wikiversity), but it is out of scope here. — MarkH21talk 21:19, 3 August 2020 (UTC)
 * This is a summary, not a textbook. All the proofs have been excised. In the case of symmetry of second derivatives, I have given two proofs there. In the case of the inverse function theorem, there was a very short proof given. The summaries in that article were inaccurate: there is a extremal principle, found in Tom Apostol; all other are essentially the same, although the other editors have not checked that the reference to "Differential manifolds" of Serge Lang are not to generalizations on Banach manifolds as claimed in that article. At the moment the very short section is a set of definitions and properties, relevant to 1 and 2 dimensions.


 * I've already said that the material from the books of O'Neill and do Carmo are missing, i.e. not available even in summary in wikipedia. (I haven't yet checked for Singer & Thorpe, which is used in the article.) Let's take the book of O'Neill.


 * Chapter 1, pages 3-42, is entitled "Calculus on Euclidean space". Just calculus, but 40 pages of it.


 * Chapter 2, "Frame Fields", pages 43-99. Some of these have already been covered by the Frenet formulas, but is about space curves not surfaces.


 * Chapter 3, "Euclidean geometry", pages 100-129, nothing yet about surfaces.


 * Chapter 4, "Calculus on a surface", pages 130-201. It is only at that point that anything new happens. Coordinate patches are introduced, diffeomorphisms, Monge patches are described in on surfaces (embedded in 3-space), all the partial derivatives in calculus arise, geographical patches on the sphere. regular parametrization, etc. Then a section on differentiable functions and tangent vectors. Differential forms on surfaces, exterior derivatives. (That is already covered in the article Closed and exact differential forms, quite of it I wrote.) Mappings of surfaces, derivatives, diffeomorphisms. Theorem 5.4, statement without proof of inverse function theorem in 2 variables (page 169). Integration of 2-forms, surfaces as 2-manifolds.


 * Chapter 5 is about "Shape Operators", pages 202-262. That is in the relevant section: I think I wrote it. 60 pages were condensed into a paragraph.


 * Chapter 6, "Geometry of surfaces in R3", pages 263-320. All of this already summarised in the article: the theory due to Gauss, with the sketches of the proof. Again I wrote that.


 * Chapter 7, "Riemannian geometry", pages 321-387. Gauss-Bonnet, connections, already in the article.


 * Chapter 8, "Global geometry of surfaces", pages 388-450. Constant curvature, Cartan-Hadamard theorem, etc, were already summarised in the article, including the paragraph on geodesics and the work of Birkhoff. Perhaps even a picture of Birkhoff. Much of the section on Global differential geometry of surfaces, was written by Katzmik, who is an expert in the area. His book is there, but there is no direct reference.


 * I will look through the book of do Carmo. The question is whether this article is aimed at a undergraduate readership or beyond that level. The uniformization theorem is proved in a sophisticated way using Liouville's equation (the books of Taylor and Berger). That works fine for genus higher than 0. On the sphere a different method works, but there is a discussion of Ricci flow. That section seems to have been written by me.


 * The main gaps are in the very elementary theory for undergraduates, especially coordinate patches, Monge patches, diffeomorphisms. Anyway all we do is summarise content from a good source. It is true that Lars Hörmander's book is a textbook, but only two pages have been used so far, right at the very beginning. Il ne faut pas exagérer. Mathsci (talk) 00:02, 4 August 2020 (UTC)
 * My point is not that WP articles shouldn’t be providing out-of-scope prerequisite background for the sake of plugging in knowledge gaps of undergraduate readers. Providing background to a limited degree is useful, but this material isn’t too far out of scope to be in this article. — MarkH21talk 00:37, 4 August 2020 (UTC)

Comment: The function theory of one/two/three real variables is not part of the differential geometry of surfaces. It is certainly a basic part of the toolbox used to study these geometric objects, and part of the basic language, but is not itself part of the differential geometry of surfaces. For example, the wikipedia page Manifold does not contain the definition of a topological space, nor a summary of the basic theorems and their proofs in the theory of point-set topology, despite these clearly being obvious prerequisites to understanding manifold theory. Just because there is a section of the standard textbooks on the theory which covers this material does not warrant its inclusion on this wikipedia page.

For one thing, having flicked through the other sections, I don't believe any of the content in the contested section is needed to understand anything else on the page. If the basic calculus had been written down so as to make clear how to, for example, derive an isothermal coordinate system or another specific coordinate system for surfaces which needs some calculus in two variables, then maybe there would be an argument, but the rest of the article is a summary of various concepts in surface theory which do not need the details of calculus of two variables to state or explain in words. The Wikiproject Mathematics is not meant to serve as a textbook, and it is expected that there is a basic level of understanding in some articles in order to avoid every single article consisting of vast seas of basic theory in order to understand more advanced concepts. It is not the scope of the differential geometry of surfaces article to present to people the basic function theory of two variables. This is possibly within the scope of the multivariable calculus page. For example, the multivariable calculus page is a bit sparse of examples and discussion of the basic concepts in that area, and the contested section would do well (with a bit of refactoring) as an extra section on that page. One could then redirect the reader to that article with a short link on this page. This would be my recommendation to resolve the dispute. Tazerenix (talk) 00:25, 4 August 2020 (UTC)


 * I will start comments on do Carmo's book.


 * Chapter 1, "Curves", pages 1-51. Definitions from calculus, classical treatment. The exmaples on global properties are interesting. The article on differentiable curve is similar.


 * Chapter 2, "Regular surfaces", pages 52-119.


 * Appendix: "A brief review of continuity and differentiability", pages 120-135.


 * Chapter 3, "The geometry of the Gauss map", pages 136-216.


 * Appendix: "Self-adjoint linear maps and quadratic forms", pages 217-219.


 * Chapter 4, "The intrinsic geometry of surfaces", pages 220-314.


 * Appendix: "Proofs of the fundamental theorems of the local theory of curves and surfaces", pages 315-320.


 * Chapter 5, "Global differential geometry", pages 321-459.


 * Appendix: "Point-set topology of Euclidean geometry", pages 460-474.


 * Material from chapters 3,4, 5 and 6 have already been summarised in the article, starting from the Gauss map and the intrinsic geometry of surfaces. So it is the Chapter 2 and its Appendix that may need some tidying up. Monge patches, by a different name, are a still covered in the book. However, the name Monge appears in the article without any explanation at all. We know who is from a wikilink but that is all. Does that entry tell us what a Monge patch is? Some vague hints by his works on curvature, that's all. References to descriptive geometry.


 * Comment There is no section on calculus in 3 variables. I believe that the section of do Carmo's Appendix rom Chapter 2 entitled "A brief review of continuity and differentiability" covers much the theory as the preparatory segment. Perhaps it is more detailed. My question to those participating this discussion will be. The section on Monge patches and regular surfaces has not yet been written. It is not by democracy that this content will be written. It is by locating sources and then summarising the material accurately and succinctly. The idea that editors can blank content while it is in the process of being written is strange. In this context there is an section on Teichmuller theory. It is not explained at all in the body of the article: that is part of the article, a sort of vague essay, which probably needs to be rewritten. Uniformization is possible using Liouville's theorem as described in the article; this shows the metric can be changed conformally so that the curvature is constant. The section on surfaces of constant curvature were written by me, with material from the source book of Imayoshi & Taniguchi, "An Introduction to Teichmuller spaces". (Tromba from UC Santa Cruz wrote a short book on the topic.)


 * I suggest people read the sources that I have described before commenting on planned new content, starting with the content on "regular surfaces". If you look for regular surface, you will find a disambiguation point which then redirects to this article. That is why these loose ends are so problematic. Mathsci (talk) 01:48, 4 August 2020 (UTC)


 * First of all, the new section "‎Function theory in two variables" does look out of place; it is simply a background material needed from calculus. I looked at do Carmo's "Differential geometry of curves and surfaces" and O'Neill Elementary Differential Geometry. They do contain the materials that are quite similar to those in the section in question, but in much greater depth. This makes sense because they are textbooks, while this article is a Wikipedia article; as pointed out, we generally omit background materials in Wikipedia articles. The real question here, I think, is whether the coverage of calculus on Euclidean space is adequate in Wikipedia or not; maybe not. As Mathsci pointed out, Wikipedia currently does not give the definition of a regular surface (by definition, it is a two-dimensional differentiable manifold but it might make sense to give an explicit definition without a reference to differential manifolds). There would be a lot of overlaps between the article like that and many articles on specific calculus topics; I don't view that's particularly problematic. -- Taku (talk) 02:50, 4 August 2020 (UTC)
 * Where is the treatment about coordinate patches, Monge patches, etc? The section on "Curvature of surfaces in Euclidean space" does need some explanation. It starts off in mid-flight with the statement "The Gaussian curvature at a point in an embedded smooth curve given locally by the equation $$z=F(x,y)$$ ... ". Monge patches (or its variants) are the normal technology explained in do Carmo and O'Neill but so far omitted in the article. Again we look at coordinate patch and we arrive at atlas (topology), not helpful.


 * A parametrized surface is defined as a mapping $$f$$ of an open set $$U$$ into R3 such that $$\partial_x f$$ and $$\partial_y f$$ are linearly independent. A parametrization of a parametrized surface is one of the form $$ f \circ g$$ where $$g$$ is a diffeomorphism of $$V$$ onto $$U$$. A Monge patch is a special parametrized curve with $$f(x,y)=(x,y,F(x,y))$$. There are also two other kinds of Monge patches $$f(x,y)=(x,F(x,y),y)$$ and $$f(x,y) = (F(x,y),x,y)$$. The main result states that if $$f$$ is a parametrized surface on $$U$$, then for any point $$(x_0,y_0)$$ in $$U$$ there is an open neighbourhood on which the parametrized surface has one of the three types of Monge patch. This is just a corollary of the inverse function theorem in 2 dimensions applied to $$\partial_xf_i$$ and $$\partial_y f_j$$ for $$i<j$$. There are three possible pairs.


 * Your tacit assumption is that regular surface directs somewhere. But if you follow the wikilink you just get back to this article, which is no help. The article on surface is no better, just images of an apple, a water drop and the sun. The preliminary material that I've just sketched is fairly standard. Isometries of 3-space given by permutations of coordinates will reduce the case to the first Monge patch, $$(x,y,f(x,y))$$. Mathsci (talk) 04:32, 4 August 2020 (UTC)
 * Thanks to, this article contains now a definition of regular surfaces. This allowed me to fix the disambiguation page Regular surface, by creating the redirect , and replacing by it the first disambiguated link. I have also changed the second item of Regular surface, but this does not concern this article. D.Lazard (talk) 13:52, 9 August 2020 (UTC)

Why just regular surfaces? There are lots of surfaces like the focal surface where the most interesting parts of the geometry happen at places where the surface fails to be regular. The Gauss map is one of the key ingredients in the story of differential geometry of surfaces, this fails to be regular of surfaces on parabolic lines with higher singularities at special points as discussed in Banchoff, Gaffney, McCrory, "Cusps of the Gauss Map" is quite a key text. --Salix alba (talk): 22:26, 5 August 2020 (UTC)
 * The fact that there is a beautifully illustrated web book is a big plus. As you've said, it deals with the singular case and so parts of singularity theory. Mathsci (talk) 00:17, 6 August 2020 (UTC)

French article and R(X,Y)
I note that the French version of this article started out in 2011 as a literal translation of this article. Some bits are improved. Kuiper and Nash is mentioned. The article even gets a star as a featured article.

Gumshoe2's worries about $$R(X,Y)$$ seem puzzling. The definitions are standard. Perhaps I jumbled two words at one point.

So could Gumshoe2 explain why the definition of the curvature operator $$R(X,Y) =[\nabla_X,\nabla_Y] - \nabla_{[X,Y]} $$ is problematic? The definition in the text is verbatim that of Boothby (the usual technique that assignment $$ (X,Y,Z)\mapsto R(X,Y)Z$$ is trilinear over $$C^\infty(S)$$). It explains why $$R(X_p,Y_p)Z_p$$ can be defined at a point. The same formalism can be found in do Carmo, Helgason, etc. Mathsci (talk) 02:00, 10 August 2020 (UTC)
 * The material is standard in Riemannian geometry. I believe the use of it here is inappropriate and doesn't add value. I'd be happy to hear the opinions of others and defer to consensus. I'm also rewriting that section now, maybe we'll both be happy with the results. Should be done soon Gumshoe2 (talk) 02:28, 10 August 2020 (UTC)
 * It's clear how Hitchin has written his notes. It's undergraduate level, with a quirky but interesting "topics" feel. After the introduction, there's a chapter as topology of surfaces, then a chapter on Riemann surfaces (8 pages of comments on complex function theory!) and then a chapter on surfaces. It's elementary but Hitchin sneaks in his special way of viewing curvature on pages 64-65. It is just routine vector calculus, nothing fancy. Mathsci (talk) 03:16, 10 August 2020 (UTC)


 * Here is my edit of the section on covariant derivatives. Gumshoe2 (talk) 05:52, 10 August 2020 (UTC)

Multiple issues
This article has multiple issues. I open a discussion on the main ones in following subsections, except for the out-of-scope section "Function theory in two variables", which is discussed above.

Some of these issues could be fixed by a bold edit. However, as a discussion on the content of this article has just started, it is better to wait for knowing if there are objections, ar to a consensus. D.Lazard (talk) 09:37, 4 August 2020 (UTC)


 * Ordinary mathematics editing is happening on elementary topics. Impeccable sources are being used. Please assume good faith. Differential geometry of surfaces does involve calculus in 2 variables. That cannot be avoided. I don't think a confrontational method of discussion is helpful. Please spend your time thinking about the source material: it is just the topic of differential geometry of 2-manifolds, part of which involves calculus in 2 variables. There is no need to belittle the topic by ridiculing the words "differential geometry" below. This is just undergraduate material which has not yet been presented properly. The sources have been gathered (do Carmo, O'Neill, Hitchin, Calabi, Pelham Wilson, Pressley); now that material has to be summarised in the standard way. Almost all texts have pictures to illustrate examples. Existing images on commons can be used to accompany the text. Mathsci (talk) 19:22, 4 August 2020 (UTC)

Article's title
The present title is pleonasmic, as a surface is always a geometric object. This may be confusing for some readers who searching information on differentiable surfaces. Therefore the article must be moved to Differentiable surface. Note also that has already be moved to Differentiable curve. D.Lazard (talk) 09:37, 4 August 2020 (UTC)
 * The topic is differential geometry. Mathsci (talk) 17:25, 4 August 2020 (UTC)
 * Is that a suggestion for renaming this article with the title of an existing article? This thread is aimed for discussing whether the title of the article is adapted to its content and follows the rules of MOS:TITLE. D.Lazard (talk) 17:45, 4 August 2020 (UTC)
 * The subject of differential geometry is well established. It's standard material for third year undergraduates (in the U.S. honors, probably). It has been like that for 12 years and is unlikely to change. The subject is an established area of mathematics. The books in this area use the words "differential geometry". They books use the terms "curves" and "surfaces". So on wikipedia, we find out the sources (mostly books) and that is what we record. Mathsci (talk) 18:23, 4 August 2020 (UTC)

After some thinking, two things appears to me. Firstly, Differentiable surface is not a good title because the phrase is not enoughly used in the literature. Secondly it appears that the intended meaning of the present title is "Surfaces in differential geometry". So "differential geometry" is here for disambiguating "surface". So, we have to follow the standard rules of WP:Disambiguation, that is to use a parenthetical disambiguation. So, I suggest to rename this article Surface (differential geometry) (which already redirects here). This will be coherent with the names of other articles on surfaces such as Surface (mathematics), Surface (topology). D.Lazard (talk) 09:16, 6 August 2020 (UTC)


 * "Differential geometry of curves and surfaces" is a widely understood phrase, at least in the US; it means curves and surfaces in three-dimensional Euclidean space. The phrase "Differentiable surface" is inherently ambiguous, like the phrase "differentiable manifold," and I wouldn't know what precisely it means. I disagree that the topic here is "surfaces in differential geometry" as this page does not cover (and I think should not cover) surfaces in Riemannian manifolds, or surfaces in higher codimension or abstract two-dimensional Riemannian manifolds, except for a few errant subsections which I think should be migrated elsewhere. I also disagree that the current title is redundant, as surfaces also appear in a different way in, for instance, algebraic geometry. The current title is the only phrase I know of which is widely understood (at least in US) to refer to the article content. The most accurate title would be "Geometry of surfaces in three-dimensional Euclidean space," as this is what precisely I believe the page ought to be about. But it's too wordy and, in the sociological sense, "Differential geometry of surfaces" carries essentially the identical meaning (again, at least in US). Gumshoe2 (talk) 06:09, 7 August 2020 (UTC)
 * At the moment, in some kind of free for all, you have written an unsourced essay in the body of this article. Please could you now find some time to find sources (mostly books) which will justify your personal essay? The material is straightforward, as is matching the content to the texts of do Carmo, Hitchin, Pressley, Wilson, O'Neill, Struik, etc. I don't agree with you about the sociologocal sense or books on differential geometry. Books like "Lectures on Classical Differential Geometry" (Struik) may seem old-fashioned, but they are still quite readable (and short!). Graduate student lecture notes on this material have also changed very little, at Harvard, MIT, Penn. UCLA, Berkeley, Chicago, Princeton, etc. They are stable. They have better pictures now and sometimes they have better ways of presenting material. But you know the rules for wikipedia: please provide sources. Thanks, Mathsci (talk) 08:21, 7 August 2020 (UTC)
 * I'm not sure what you're responding to about books like Struik's. If anyone wants to add material to the overview on two-dimensional Riemannian manifolds, they should feel free to do so. Anyway, everything I say there is utterly standard and can be referenced to any book whatsoever on the topic. I will add specific references when I find the time, for now I was just trying to write a readable, relevant, standard "overview" for the page. At the moment I'm focused on the rest of the page, which has many errors and confusions. For instance the sentences "On the surface $$S$$, the functions $$f$$ from $$V$$ onto $$U$$ are diffeomorphisms by the inverse function theorem. Thus their inverse functions $$f_U^{-1}$$ provide coordinate charts and their compositions $$f_{U^\prime}\circ f_U^{-1}$$ provide a smooth structure in $$S$$" are totally garbled, the definition of the first fundamental form is completely incorrect, etc. Gumshoe2 (talk) 08:45, 7 August 2020 (UTC)
 * Hi. I assume that you've written other articles on wikipedia elsewhere: history, geography, art, music, etc. The rule is that no matter what we write on wikipedia, we have to provide reliable sources. So if for example statements are made in a paragraph, that would require reference with page numbers for the statements. At the moment you've added content without sourcing. It's easy to correct—that's why I mentioned the sources above. Thanks, Mathsci (talk) 10:31, 7 August 2020 (UTC)

The article is getting a little unwieldy and it might be an idea to limit its scope. There is plenty enough material for an article which is Differential geometry of surfaces in 3D leaving general material of Riemannian manifolds to a seperate article. This need not be actually reflected in the article title but in the description of the article in the lead.--Salix alba (talk): 07:26, 12 August 2020 (UTC)
 * I would strongly agree with splitting the material. Surfaces in 3D is a major topic which is of its own interest, both pedagogically and in research, not just as a prelude to Riemannian geometry in two dimensions, which is only a partial generalization Gumshoe2 (talk) 13:22, 12 August 2020 (UTC)

History section
In any case this must not be a subsection, since it is important by itself.

As the history is not fundamental for understanding this article, the section would better placed near the end of this article.

However, readers intersested in history would probably search such a section either in Differential geometry (which has a poor history section) or in Surface (mathematics), which has no history section. As, until the end of the 19th century, there was no clear distinction between surfaces and differentiable surfaces, I suggest to move this section to Surface (mathematics).

In summary, three possibilities, with my preference for the last one:
 * 1) Doing nothing, or simply upgrading from a subsection to a section
 * 2) Moving toward the end of the article
 * 3) Moving to Surface (mathematics), with the addition of some comments on other types of surfaces (when they were distinguished from differential surfaces) D.Lazard (talk) 09:37, 4 August 2020 (UTC)
 * The history section predates me. It is essay-like and not properly sourced. It was put in place by Arcfrk in 2008, twelve years ago. Mathsci (talk) 18:26, 4 August 2020 (UTC)

Definition and basic properties
The surfaces that are the subject of this article are never clearly defined. The first sentence of the article suggests that surfaces of class C2 are not considered here. In the first paragraph of section "Overview", a circular definition is sketched, where a differential structure on the surface is supposed to be known before defining it.

Not only a definition, but also basic the basic properties of differentiable surfaces are lacking: existence of tangent planes, definition of a metric structure, existence of a normal when a metric structure exists, different ways to specify a surface in $$\mathbb R^3$$ (graph of a bivariate function, parametrization, implicit equation), and condition under which the resulting surface is differentiable, etc. Without a section "Definition and basic properties", the article is much too WP:TECHNICAL for non-specialists of differential geometry. D.Lazard (talk) 09:37, 4 August 2020 (UTC)


 * Standard undergraduate material for final years (honors in the US). Is lectured everywhere. Many of the standard text books are listed in references of the article. Some are old-fashioned (Eisenhart, Struik, Kreyszig) and some have been more user friendly. The books of Pelham Wilson and Andrew Pressley are recent undergraduate texts. Similarly do Carmo and O'Neill. The notes of Nigel Hitchin and Eugenio Calabi are again standard. They are not technical, but need to discuss "regular surfaces", "inverse function theorem" and so on. Coordinate charts are also terms that are used. Hitchin's lectures give a very approachable to the material which is non-technical. Formulas will inevitably appear, sometimes involving third derivatives. The sources often allow that presentation to be streamlined so that the geometric reasoning becomes clear. That is particularly true for the first and second fundamental form, which are easily interpreted in terms of motions along curves. Summarising the common presentation is not particularly difficult; look at the introductory books of Pressley and Wilson or Hitchin's notes. Mathsci (talk) 18:51, 4 August 2020 (UTC)


 * I know that there are standard textbooks that give the definitions and basic properties, but not all reader have their content in mind, even if they have read these books. An encyclopedia must be self contained, and should be open to those that have never had a course of differential geometry, or have forgot it. Specially, it must be open to specialists of other areas who need to work with surfaces. D.Lazard (talk) 09:28, 7 August 2020 (UTC)

Lead and "Overview" section
These are two sections for giving a summary of the subject. As both are rather long, it is unclear why these sections are there. Moreover, none provide a summary of the content of the article, and both include general considerations that are not developped in the article.

Thus these sections must be completely rewritten. D.Lazard (talk) 09:37, 4 August 2020 (UTC)
 * The creation of the article started with Arcfrk; there were multiple problems with the lede, the overview, the history, definitions, concepts not properly explained. Wikipedia acts by incremental modifying content. In this case the whole foundation, starting with the vague discussion of principal curvature, was shaky.


 * But sources are there. They just have to be summarised carefully, with a knowledge of the material. Any account has to start with regular surfaces. Eugenio Calabi gave lectures which are still available. The same material is available in elementary text books like those of Pelham Wilson, Pressley, do Carmo and O'Neill. Here is the pdf file of Calabi's 1994 lectures. Almost all the writers cover the same material, with some minor changes in the order. The starting point is regular surfaces, with the same definition for all writers.  Another example is the presentation by Nigel Hitchin from his Oxford lectures. There is an appendix with the same proof of the inverse function theorem. Mathsci (talk) 14:11, 4 August 2020 (UTC)
 * The problem of these sction is not the lack of sources. I agree that there are plenty of them. The problem is to give an introduction that is accessible for non specialists and summarizes the main results. This is not correcty done with the present sections. Their authors does not matter, as Wikipedia is a collective work.
 * About "regular surfaces": as the term is used in reliable sources, it must be defined in WP. As a regular surface is a differentiable surface, it must be defined in this article. I cannot understand that you have not yet introduced their definition in your hundreds of edits of this article. D.Lazard (talk) 18:08, 4 August 2020 (UTC)
 * Before your blanking, I was in the process of preparing the material. It is elementary content, but does require care. I prepared the material on the symmetry of second derivatives and the inverse function theorem. I decided that the article needed cleaning up, so that is what I set about doing in. I've given the diff explaining how Arcfrk created the article. As far as "differential geometry" is concerned, functions are usually taken to be smooth, so the phrase you used—differentiable surface—would never be used: I would recommend that you read the lecture notes of Hitchin. You asked a question about my edit history. The plan mostly is to continue editing using the good source, probably starting with Hitchin's lectures. I recuperated Calabi's 1994 lectures from the wayback machine. There is another article on differential geometry and surfaces. It has a lede, a summary, 3 pictures, several images and not too many formulas. Mathsci (talk) 21:37, 4 August 2020 (UTC)
 * This is an encyclopedia. So we cannot limit the context to differentiable geometry. The general mathematical definitions of a surface is given in Surface (mathematics) (apparently you do not know of this article, as you always refer to Surface). As there are several different definitions, the use of "surface" in the article must make clear the definition that is used. As "differentiable surface" appears there, it must be defined in this article.
 * WP:Wikipedia is not a textbook. So, it is not convenient to follow closely the presentation given in one or two textbooks, as it seems to be your project.
 * Both terms "differentiable surface" and "regular surface" are widely used in the literature. So they must both appear and be defined in this article. As far as I can understand from a quick look on the literature, they have commonly a slightly different meaning: a differentiable surface is simply a differentiable manifold of dimension 2; a regular surface is a subset of $$\mathbb R^n$$ that is diffeomorphic to a differentiable manifold of dimension 2, or, equivalently, a differentiable surface embedded (not immersed) in $$\mathbb R^n.$$ It is possible that some author give slighty different definitions, so we must give the most common definitions, not the one of a specific author. An example of a differentiable surface that is not a regular surface is the immersion of the Klein bottle in $$\mathbb R^3.$$ As this is undoubtly an example for "differentiable geometry of surfaces", the article must be written in a way that is compatibl with it. Presently this is far to be the case. D.Lazard (talk) 10:03, 5 August 2020 (UTC)
 * I am not going to discuss this in full flow. This is not a WP:FORUM. The sources are clear enough and that's what I'm using. Instead of allowing editing to proceed slowly and carefully, you are making disruptive and impatient edits. Orientability, the cross product of vectors, the second fundamental form are all in the sources, but will be added at the appropriate time. There is no rush, no pressure. Compris? Mathsci (talk) 12:31, 5 August 2020 (UTC)


 * There is no universal definition of "differentiable surface" or "regular surface" and any careful author would have to define their use of it at the beginning of their own article or book. So I think it would be appropriate to give one choice of definition here and to mention that conventions vary. Appropriate unambiguous technical language would use "smooth embedded submanifold" or "smooth immersion" or "smooth embedding," sometimes swapping "smooth" out for some different regularity. But I think that language would be inappropriate in this article Gumshoe2 (talk) 06:13, 7 August 2020 (UTC)

Theorema Egregium
There's apparently some edit disagreement between myself and. My edit in question is here. It seems to me that mathsci has misunderstood the meaning of the theorema egregium, based on their comments there. Maybe some community members can weigh in

(It also seems to me that the "tangential derivative" section should be rewritten so as to not suddenly dip into Riemannian geometry) Gumshoe2 (talk) 23:36, 9 August 2020 (UTC)


 * I was going to ask you a related quetion. My username is Mathsci not mathsci and has been for some time (2006?). I extracted the material on $$LN-M^2$$ and Brioschi; it was not there before me.


 * The treatment of Hitchin on tangential derivatives is short and elementary but the punchline is left dangling in his notes. However, the framework of covariant derivatives does make it clear why without making any computations Gaussian curvature is intrinsic. Thus without calculation, it is possible to see that an isometry between open sets preserves Gaussian curvature. In the article there are several ways of computing Gaussian curvature which give alternative accounts of the theorema egregium, e.g. geodesic polar coordinates. Mathsci (talk) 02:36, 10 August 2020 (UTC)


 * I believe you are incorrect. The framework of covariant derivatives, at least as you presented it, shows that the scalar curvature of a Riemannian metric is a scalar invariant. That's an interesting fact, but it's not the theorema egregium, which relates the determinant of the shape operator to the scalar curvature.


 * That is, the theorema egregium states that the determinant of the shape operator is invariant under isometries of the surface. It seems that your paragraph is about the scalar curvature being invariant under isometries of the surface. The link between these is the Gauss formula, which was not part of your presentation.


 * (Also, if this were somehow the theorema egregium, it would make Marcel Berger's quote about the nonexistence of simple geometric proofs somewhat strange) Gumshoe2 (talk) 02:48, 10 August 2020 (UTC)


 * The usual setting is just $$(R(X,Y)Z,W)$$ and the computation is of $$(R(E_1,E_2)E_2,E_1)$$. In Boothby's book there is a section entitled, "The Theorema Egregium of Gauss." There he states that if $$S_1 $$ and $$S_2$$ are regular surfaces and if $$f$$ is an isometry between these, then corresponding points have the same Gaussian curvature. The proof on pages 368-370 is the standard one. Marcel Berger mentions various proofs in his Panoramic book. do Carmo (1976), Klingenberg (1995), O'Neill (1966), Stoker (1989), Thorpe (1984), Sternberg (1983) and last but not least Boothby (1986). So nothing has really changed. Boothby's proof is one of the standard proofs of the Theorema Egregium and the statements about isometries are exactly as I wrote. I checked the other books on Berger's list: Klingenberg, same as Boothby but being explicit about tangent vectors for $$(R(X,Y)Y,X)$$; and Sternberg, who gives a treatment for manifolds of codimension 1 in Eucidean space. Mathsci (talk) 04:28, 10 August 2020 (UTC)


 * Ok, I'm looking in Boothby's book now, the revised second edition. On page 366 it indeed says that the Theorema Egregium says: isometric surfaces have the same Gaussian curvature. No problem so far. At the beginning of the proof on page 368, the Gaussian curvature is defined as the determinant of the shape operator, $$\frac{LN-M^2}{EG-F^2}.$$ As said on page 369, the work is then to show that
 * $$LN-M^2=\big\langle\nabla_{E_1}\nabla_{E_2}E_2-\nabla_{E_2}\nabla_{E_1}E_2-\nabla_{[E_1,E_2]}E_2,E_1\big\rangle.$$
 * Such work was done in your previous edit here, which I deleted for the reason "Deleted section which gives a proof of theorema egregium which is the same as the one discussed in previous subsection, except in language of covariant differentiation. There is also no need for full computations of a proof on such a page as this." But such work was not done in the edit that I linked to in the first post in this thread; you had nothing there to address the formula
 * $$LN-M^2=\big\langle\nabla_{E_1}\nabla_{E_2}E_2-\nabla_{E_2}\nabla_{E_1}E_2-\nabla_{[E_1,E_2]}E_2,E_1\big\rangle;$$
 * you said only that it's enough to see that the right-hand side is isometry invariant. So I still believe that the deleted paragraph incorrectly claims to address the Theorema Egregium. Gumshoe2 (talk) 04:57, 10 August 2020 (UTC)
 * If you're interested in an elementary proof which is similar to the book of Boothby, then Thorpe's undergraduate book is fine. Title, "Elementary Topics in Differential Geometry." Chapter 23 on isometries gives a down-to-earth account of covariant derivatives and all of that. Pages 220-230. Thorpe starts out by telling us, "As inhabitants of the earth, we are (or, at least, we were until the invention of air and space vehicles) forced to deduce the geometry of the earth from measurements made on the earth. We can measure distance along curves, and by taking a derivative with respect to time, we can measure velocity and speed. The geometry which can be derived from such measurements is called intrinsic geometry." It's very readable and covers the same material as Boothby. It is labelled as an undergraduate book.


 * I don't know why you're having problems with Boothby et al. Probably the treatment in Thorpe is a little clearer since they keep emphasising "intrinsic" ad infinitum. Exercise 23.13 is also helpful in Thorpe. To the naked eye, Hitchin is just borrowing Thorpe's naive treatment of covariant derivatives. Why not? But remember that Marcel Berger recommended these texts, including both Boothby and Thorpe. (Klingenberg and Sternberg are heavy going.) Mathsci (talk) 05:33, 10 August 2020 (UTC)
 * I have no issue with Boothby or Hitchin. I'm saying that your edit of the wiki page failed to contain the key point of what is in those texts Gumshoe2 (talk) 05:48, 10 August 2020 (UTC)
 * You are editing too rapidly at the moment and are making errors. Please read Thorpe: then you will understand why isometries are needed to understand covariant derivatives. Make cumulative changes instead of bulldozing into the whole article. There is no point. I am quite willing to have the material on covariant derivatives added in an added amplified form but would prefer that the isometries come before that. Mathsci (talk) 06:20, 10 August 2020 (UTC)
 * What errors have I made? Gumshoe2 (talk) 06:21, 10 August 2020 (UTC)
 * I have added back in my material after your deletion. It's fine with me if you want isometries first, although it wouldn't be my choice. There are many ways to present this material and no single one of them is the definitive correct version. Gumshoe2 (talk) 06:30, 10 August 2020 (UTC)
 * (ec) You are editing in good faith but often too rapidly. The material on isometries is elementary and was added straightforwardly with examples. I made a typing error when starting the section on isometries: you blanked the section.


 * Isometries are needed in the article and almost certainly should appear in that order (as happens in standard books). Covariant derivatives are a little bit more sophisticated. So far the best treatment seems to be that of Thorpe: it's aimed at undergraduates. If you want to write in more detail about covariant derivative please feel free. But please find sources for the content instead of writing an essay. Mathsci (talk) 06:48, 10 August 2020 (UTC)
 * I fully appreciate the need for references, but if you want references for specific claims please point me to them instead of deleting my contributions. I plan to reference everything in the end, but for now I'm focused on the content itself. I deleted the isometry section only when it consisted of two sentences, one of which was vague and one of which was wrong. I didn't make any further edits to it when you put the "under construction" tag on it. I completely agree with you that there should be a section for isometries. Gumshoe2 (talk) 07:01, 10 August 2020 (UTC)

' The standard treatments of covariant derivative are easy to find and clean. Helgason's 1962 book on DG, do Carmo's book on DG, Klingenberg and Thorpe, etc. The algebraic starting point is $$\cal{D}=C^\infty(M)$$ and their derivations, the space of vector fields $$\cal{X}$$. Levi-Civita proved the existence and uniqueness of a covariant derivative which is symmetric and compatible with the Riemannian metric. Existence is easy when an isometric embedding in Euclidean space is given: in the case of a regular surface all that is needed is that unit normal vectors exist. The algebraic properties are:


 * $${\cal X}$$ acts naturally on $$\cal{D}$$ by derivations
 * $$\cal{X}$$ is a left $$\cal{D}$$ module
 * if $$X$$ and $$Y$$ are in $$\cal{X}$$, so too is $$[X,Y]$$
 * for each $$X$$ in $$\cal{X}$$ there is a uniquely defined operator $$\nabla_X$$ map of $$\cal{X}$$ to $$\cal{X}$$ such that
 * $$\nabla_{fX + gY}=f\nabla_X + g\nabla_Y$$
 * $$\nabla_X$$ is a derivation: $$\nabla_X(fY)=(Xf) Y + f \nabla_X Y$$
 * $$\nabla_X(\lambda Y+\mu Z) = \lambda \nabla_X Y + \mu \nabla_X Z$$
 * $$\nabla_X Y - \nabla_Y X = [X,Y]$$
 * $$ X(Y,Z)=(\nabla_X Y,Z) + (Y ,\nabla_X Z)$$

Here $$(X,Y)$$ is the inner product which takes value in $$\cal{D}$$. Uniqueness follows because $$2(\partial_YX,Z)= X(Y,Z) + Y(Z,X) -Z(X,Y) -([X,Z],Y) - ([Y,Z],X) + ([X,Y],Z).$$ It is routine to that this formula for $$(\partial_Y X,Z)$$ recaptures all the properties of $$\nabla_Y$$ listed above. Mathsci (talk) 18:51, 10 August 2020 (UTC)
 * As I said before- in the edit of yours I posted at the beginning of this thread, there was nothing to link these standard properties to $LN − M^{2}$. This can be done by a computation, and it was done in an edit of yours at a different time. But in the edit I posted at the beginning of the thread, it was not addressed and so I maintain that it was inaccurate to say that it gave a non-computational proof of Theorema Egregium. Gumshoe2 (talk) 19:09, 10 August 2020 (UTC)
 * Nobody is discussing the Theorema Egregium content today. As I wrote just above, I've noticed that you've completely changed the material on covariant dervatives. What you've written seems to be a stream of consciousness essay, none of it properly sourced. In principle it is possible ypur content to reconcile with standard material above, but so far that's not happened. Contrary to what you've written in the article, the fundamental theorem of Riemannian geometry is not hard to proof. Better to give careful references instead of giving a point of view. I have more or less worked out how to make the improvements to the section on covariant derivatives. Remember, sometimes less is more. Mathsci (talk) 21:51, 10 August 2020 (UTC)
 * ?? This whole thread has been about the theorema egregium. I have not said or implied anywhere that the fundamental theorem of Riemannian geometry, which I am very familiar with, is hard to prove. Please direct me to the statements in the "covariant derivative" section that you would like references for. Gumshoe2 (talk) 21:58, 10 August 2020 (UTC)

'Comment: I am not aware of the technical details given here. However, 's edits are generally confusing (unclear hypothesis specifications, use of definitions before given them, etc). Their arguments in the discussion are all of the form "somebody's books says that", and they do not answer when the other user shows that this is a misinterpretation of the source. On the other hand, explanations are clear and convincing, and their edits are of a great help for non-specialists trying to understand the subject (which 's edits are never). So, placing me from the point of view of Wikipedia, and following its basic principles, I totally support 's edits. D.Lazard (talk) 08:33, 10 August 2020 (UTC)
 * Helgason et al use the framework of commutative algebra and its modules for setting up covariant derivatives. I am surprised you are not familiar with this material or more importantly the sources. So far on this article talk page, you have made no attempt to discuss sources. Your own commentary of my editing history doesn't make any sense. Mathsci (talk) 21:51, 10 August 2020 (UTC)
 * I believe it isn't a good or useful idea to present the material on this page using the language of modules and any general theory of connections. If there is a community consensus to do so, I will be happy to follow it. Gumshoe2 (talk) 22:02, 10 August 2020 (UTC)
 * I directed those comments at D.Lazard not you. I've already made clear comments to you above. I've talked to you about Thorpe and "less is more". Mathsci (talk) 22:39, 10 August 2020 (UTC)
 * This page is WP:not a forum, so, discussing the content of the sources is irrelevant. This is the choice of the sources that is relevant, and there is a section in this page for that. Apparently, your are unable to understand what I am saying ("doesn't make any sense"). Wikipedia being a collective work, it is problematic if some editors cannot understand others, are unable to understand Wikipedia rules, and refuse to focus a discussion on the discussed subject. This is what is generally called WP:disruptive editing. D.Lazard (talk) 09:00, 11 August 2020 (UTC)
 * I agree with : Wikipedia must be accessible to the largest possible audience. This means that when one can avoid using more general theories than needed we must do that. This does not mean that we have to not mention these general theories, because readers who know them must understand the relationship between what they read and what they know. D.Lazard (talk) 09:00, 11 August 2020 (UTC)
 * D.Lazard, you know yourself that to edit wiipedia you have to use WP:reliable sources. We cannot create content without having books and then summarising them. Gumshoe2 has been been creating content without any sources. That is a no-no on wikipedia. However, the five pillars of wikipedia still apply. If you don't access to the sources and don't use them, how can you edit? The Any material that is challenged or likely to be challenged must be supported by a reliable source is another question. Is any content that you have personally created on wikipedia intended for a general readership?  Certainly not multi-homogeneous Bézout theorem: not material for a general readership. The book "Elementary topics in differential geometry" by John A. Thorpe is a WP:RS]. Please read it. Mathsci (talk) 10:08, 11 August 2020 (UTC)
 * You wrote "We cannot create content without having books and then summarising them". No, this is not this way that Wikipedia works, at least because there are many results that are worth for Wikipedia and do nor appear in books.
 * You misinterpret completely the policy WP:NOR which says . This does not mean that sources must be used for editing Wikipedia, but sources must be provided when an edit is likely to be challenged. Moreover, as sources are generally textbooks that contain author's opinions and specific choices, following sources too directly may lead to not follow other Wikipedia rules, especially WP:Wikipedia is not a textbook.
 * Are you unable to understand that "intended for a general readership" and "accessible to the largest possible audience" are far to be the same thing? No mathematical article (except sometimes the most elementary ones) is really intended for a general readership.
 * If you have concerns with some of my edits, discuss them on the relevant talk page. Not here. Otherwise, this must be considered as WP:HOUNDING. D.Lazard (talk) 11:00, 11 August 2020 (UTC)
 * Wikipedia is edited according to the usual five pillars. Content is created by summarising material using WP:RS. Recently Lie brackets of vector fields were introduced with sourcing from Singer & Thorpe. That's how I normally edit.


 * The article Algebraic geometry, however, does not seem to be in a state that is accessible to a general readership. D.Lazard made lots of edits to the section on computational algebraic geometry, but none of them seem to be sourced. WP:RS and WP:V are the usual rules on wikipedia; normally content cannot be checked without inline references. In this case there are none. There is also no mention of the Fields medallist Caucher Birkar. In a related area that I know fairly well, linear algebraic groups, the article seems OK, except that, for the most part, only one source has been used. Not a major problem: Humphreys' book is listed as a reference and the usual theory of Borel subgroups is mentioned. There is a stub on Borel & parabolic subgroups. Mathsci (talk) 23:46, 11 August 2020 (UTC)

I've changed the section in the article from


 * Taking $x$ and $y$ coordinates of a surface in $E^{3}$ corresponding to $F(x,y) = k_{1}x^{2} + k_{2}y^{2} + …$, the power series expansion of the metric is given in normal coordinates $(u, v)$ as


 * This extraordinary result — Gauss's Theorema Egregium — shows that the Gaussian curvature of a surface can be computed solely in terms of the metric and is thus an intrinsic invariant of the surface, independent of any embedding in $ds^{2} = du^{2} + dv^{2} + K(u dv – v du)^{2} + …$ and unchanged under coordinate transformations. In particular isometries of surfaces preserve Gaussian curvature.

to
 * Gauss's Theorema Egregium, the "Remarkable Theorem", shows that the Gaussian curvature of a surface can be computed solely in terms of the metric and is thus an intrinsic invariant of the surface, independent of any isometric embedding in $E^{3}$ and unchanged under coordinate transformations. In particular isometries of surfaces preserve Gaussian curvature.


 * This theorem can expressed in terms of the power series expansion of the metric, $E^{3}$, is given in normal coordinates $ds$ as

There were a number of problems with the previous version.
 * It starts with a formula rather than a plain English statement. A reader unfamiliar with the particular notation used will struggle. The text description can be understood by a wider audience.
 * Just using embedding is too week, especially if the reader comes from a background in topology where embedding need not have any metrical properties.
 * The notation $(u, v)$ is novel for this article as it stands. I looked back up the article and I can't clearly see where it is defined.
 * The use of a Monge form parameterization $ds^{2} = du^{2} + dv^{2} + K(u dv – v du)^{2} + …$ does not serve a purpose here, as we don't use anything introduced in the rest of the section.
 * The Monge parameterization kind of defeats the result which is independent of the parameterization used.
 * We should try of consistency of notation in the article. Its starts using $$\mathbb R^3$$ and switches to $ds^{2}$. I've not changed this yet.

I've not looked at the previous discussion, but following WP:SUMMARY style. Here we should just summarise the result, whether we include a proof and how we present that is really a discussion for the Theorema Egregium page.--Salix alba (talk): 07:14, 12 August 2020 (UTC)
 * There are some problems with both versions. Firstly, it should reference the previous discussion of the Theorema Egregium in section 3.4. More importantly, the certain Monge parametrization is actually relevant for making the link to Gaussian curvature (as explained in chapter 3 of Berger's book). This wasn't clearly explained at all in the previous version of that section. In your version, although it is an improvement, the "extrinsic to intrinsic" nature of the theorem is unclear since the only presence of "extrinsic" is via the presence of $K$ in the final formula. A reader would guess that $K$, as used there, must "really" mean the scalar curvature of the first fundamental form, with the Gaussian curvature appearing due to the Gauss equation -- but this just reduces things to the previous proof of the Theorema Egregium; it ends up just being a nice way to encode the scalar curvature of a two-dimensional Riemannian manifold using Taylor's theorem, which is interesting but not the Theorema Egregium. The proper explanation, which links the given Taylor formula with the Theorema Egregium, uses the special Monge parametrization to do a Taylor approximation from the Monge-induced quantities to the normal coordinate quantities. The Gaussian curvature is naturally part of the Monge-induced quantities, and this directly makes the crucial extrinsic-to-intrinsic link. (I haven't yet edited any of the page's material on geodesics, I will try to get to it soon) Gumshoe2 (talk) 13:43, 12 August 2020 (UTC)
 * Marcel Berger presents two proofs of the Theorema Egregium in his "A Panoramic View" book. The second it one he describes as being rarely presented. He starts on page 41 with a Monge graph presentation on (1.9) so $$(x,y)\mapsto (x,y,F(x,y))$$. The curve $$c(t)=(x(t),y(t),F(x(t),y(t)))$$ must have $$\ddot{c}$$ normal. Berger's computations give formulas (1.10) and (1.11) on page 41: $$\ddot{x}$$ and $$\ddot{y}$$ can be written as quadratic forms involving $$\dot{x}$$ and $$\dot{y}$$. Page 61 continues the Monge graph discussion with the formula (1.13) for the Gaussian curvature. On pages 123-124 Berger then gives his second proof of the Theorema Egregium. He wants to compute coordinates up to second term. By rotating he shows that $$F(x,y)

=k_1 x^2/2 + k_2 y^2 + \cdots $$. Then "after some pain" he shows that with $$ x = u - k_1(k_1 u^2 + k_2 v^2)/24 + \cdots$$ and $$y=v-k_2(k_1 u^2 + k_2 v^2)/24$$, he gets (3.14):
 * $$ds^2 = du^2 + dv^2 - {K\over 12}(udv -vdu)^2 + \cdots$$


 * Some of this material also appears in the book of Berger & Gostiaux. The presence of $$-K/12$$ instead of $$K$$ is not explained. I haven't checked Berger's calculation: it does not seem to be a classic treatment. Mathsci (talk) 19:46, 12 August 2020 (UTC)

Vector fields
See WP:::BRD. Vector fields on a smooth surface or manifold are standard material. The brief discussion has been removed without explanation. Could somebody please explain what's going on? I.M. Singer uses the definition given (Singer & Thorpe). He is a world expert who knows how to write. Mathsci (talk) 20:33, 12 August 2020 (UTC)
 * I gave an explanation with the edit. The material already exists on other pages, and in contexts where I believe it is more natural and important. Gumshoe2 (talk) 20:44, 12 August 2020 (UTC)

Deleting section on isometric embedding
I'm deleting the following section:
 * ===Isometric embedding problem===
 * A result of Jacobowitz and Poznjak shows that every metric structure on a surface arises from a local embedding in $F(x,y) = k_{1}x^{2} + k_{2}y^{2} + …$. Apart from some special cases, whether this is possible in $E^{3}$ remains an open question. [Han and Hong ref]. In 1926 Maurice Janet proved that it is always possible locally if $E^{4}$, $E^{3}$ and $E$ are analytic; soon afterwards Élie Cartan generalised this to local embeddings of Riemannian $F$-manifolds in $G$ where $n$. To prove Janet's theorem near (0,0), the Cauchy–Kowalevski theorem is used twice to produce analytic geodesics orthogonal to the $E^{m}$-axis and then the $m = 1⁄2(n2 + n)$-axis to make an analytic change of coordinate so that $y$ and $x$. An isometric embedding $E = 1$ must satisfy
 * Differentiating gives the three additional equations
 * with $F = 0$ and $u$ prescribed. These equations can be solved near (0,0) using the Cauchy–Kowalevski theorem and yield a solution of the original embedding equations.
 * with $u_{x} ⋅ u_{x} = 1, u_{x} ⋅ u_{y} = 0, u_{y} ⋅ u_{y} = G$ and $u_{xx} ⋅ u_{y} = 0, u_{xx} ⋅ u_{x} = 0, u_{xx} ⋅ u_{yy} = u_{xy} ⋅ u_{xy} − 1⁄2G_{xx}$ prescribed. These equations can be solved near (0,0) using the Cauchy–Kowalevski theorem and yield a solution of the original embedding equations.
 * with $u(0,y)$ and $u_{x}(0,y)$ prescribed. These equations can be solved near (0,0) using the Cauchy–Kowalevski theorem and yield a solution of the original embedding equations.

It seems out of place, and (after modification) belongs much better on some other page, perhaps Nash embedding theorem or on geodesics in Riemannian geometry, in the context of the Cartan formulation. Jacobowitz and Poznjak's results seem to be of major interest only to specialists. Of course Janet's result could and perhaps should be mentioned somewhere on this page. Gumshoe2 (talk) 15:19, 12 August 2020 (UTC)
 * Perhaps you should discuss this before deleting this kind of content. The content here concerns specifically surfaces. Most of it is classic. The content of the original article in 2008 was there at its time of creation by Arcfrk in January 2008. (Arcfrk correctly reports the expression $$-K/12$$ instead of $$K$$ in Berger's Panoramic book.) The proof of the Nash embedding theorem has been simplified (by Günther in 1989), but is not relevant to this article. Mathsci (talk) 21:10, 12 August 2020 (UTC)
 * My point is that, although it was presented for surfaces, the same argument is now (since the 1930s) understood for arbitrary dimension. So I believe it ought to be presented or given in some form in another article. I preserved the deleted bit here so that someone can easily copy it over somewhere. Gumshoe2 (talk) 21:16, 12 August 2020 (UTC)

Standard references
The list of references is rather long, and the citations of the standard material are referenced to a hodge-podge of different books and notes, sometimes without continuity from one sentence to the next. After some digging around on Google Scholar, it seems like the following books are by far the most widely cited references for regular surfaces: These also seem to cover a number of different levels of difficulty, also ranging from quite old to new. I am also not aware of any major topics on regular surfaces, at least as contained on the page at the moment, which are not covered by these books. So I suggest that, as much as possible, these books be used as the main references. Does this seem reasonable? Gumshoe2 (talk) 19:05, 10 August 2020 (UTC)
 * do Carmo "Differential geometry of curves & surfaces"
 * Eisenhart "A treatise on the differential geometry of curves and surfaces"
 * Gray, Abbena, & Salamon "Modern differential geometry of curves and surfaces with Mathematica®"
 * O'Neill "Elementary differential geometry"
 * Struik "Lectures on classical differential geometry"
 * The problem is that you are not using inline references. In your edits, most of the material you are creating is impossible to verify. For scholarly research, please use mathscinet not google scholar: google scholar is no use for mathematics. (Do you have access to mathscinet?) For differential geometry of surfaces, there are standard references which cannot be ignored. Not all of those references have been used. Berger and Gostioux is one example not featured. But book of Klingenberg's is another. But five references is not reasonable. Not actually citing references is also not a good idea. How can any other wikipedia editor heck what you;re writing. How could any layperson trying to learn wikipedia verify the stuff you've written or find adequate references? In this case there are no "standard references". Mathsci (talk) 20:19, 12 August 2020 (UTC)
 * I'm not suggesting to have only five references, just that the main citations go to a smaller group of widely cited sources. I have access to mathscinet. But it seems to me that this material is important for many people who are not research mathematicians, so I disagree with its relevance here. I'm a little confused by why you're so concerned with my citations. For instance, most of the section on Gauss-Bonnet has only a single reference, to Levi-Civita's 1917 paper, which is useless for almost everybody; much of the previous material, before I started editing, such as on the Taylor approximations of geometric quantities, had no references whatsoever. Anyway, you added the tag saying "This section does not cite any sources" on the covariant derivatives section, which is obviously false. Are there any particular statements you're concerned with? Gumshoe2 (talk) 20:41, 12 August 2020 (UTC)
 * The reference for computations of geodesic triangles and Gauss-Bonnet was Eisenhart (2004) with a page number. At some stage an editor changed  as an abbreviation for the Eisenhart reference. The reference then pointed to Levi-Civita. The 1900 paper on convariant derivatives by Ricci and Levi-Civita has been translated (with commentaries) by Robert Hermann. Ditto the 1917 paper. I don't know which editor added details about Euler and Gauss. Struik doesn't mention covariant derivatives in his book; and Hitchin gets just one ambiguous page. do Carmo (1992) seems cleaner than do Carmo (1976), even for two dimensions. Affine connections and vector fields are standard, but the definitions and statements of facts at the moment seem vague and incomplete. Too touchy feely.
 * In general non-specialized people might want to find shorter books that are readable as well as being accurate. Equally well they might also like long books like Marcel Berger's "A Panoramic View". Or they might want to find books more oriented to history. Mathsci (talk) 05:29, 13 August 2020 (UTC)
 * I'm a little confused by the answer. Are there any particular statements in this article you're concerned with? Gumshoe2 (talk) 05:45, 13 August 2020 (UTC)