# Talk:Manifold

## Untitled

Since I usually nitpick on discussion pages, I just wanted to say that this article is very coherently constructed, provides good examples, and covers the topic well for a wide range of readers. Thank you to all who contributed to it. —The preceding unsigned comment was added by 24.205.231.209 (talkcontribs) 20:36, 2006 November 12.

## "Euclidean space" or "coordinate space"?

A recent edit has changed, in the lead, "Euclidean space" into "coordinate space". I have reverted it for the following reasons: In higher mathematics, "Euclidean space" roughly means "metric affine space". But, for most people, it simply means the usual space of geometry over the reals, and is much more intuitive than "coordinate space". Thus the modification makes the lead unnecessary WP:TECHNICAL. Moreover, the edit suggests that one may consider manifolds over arbitrary fields, which is wrong. Apparently, the motivation of the edit was that that the Euclidean metric is not used in the definition of a manifold. But the coordinate space over the reals has also a natural Euclidean metric (the dot product) and has a further structure (of a vector space equipped with a basis) which is not used in manifold theory. Thus the version that I have reverted is not only too technical, but also less correct than the previous one. D.Lazard (talk) 09:55, 19 April 2013 (UTC)

The coordinate space over the reals has a natural Euclidean metric, which is not used in manifold theory, and, unlike an abstract Euclidean space, is has the coordinate structure, which is used in manifold theory. Why should the lead link the article about the structure which is not used, but avoid linking article about coordinate structure which is frequently used? Also, complex manifolds, algebraic manifolds, p-adic manifolds, and probably others do exists, so the main D.Lazard’s argument against my version is plainly wrong. If there will be no further objections, I’ll reinstate my version of the lead. Incnis Mrsi (talk) 10:23, 19 April 2013 (UTC)
I don't think of the new revision as an improvement. I don't think the distinction between a Euclidean space and real coordinate space is especially mathematically significant, and for most readers Euclidean space is likely to be clearer. The proposed revision leaves vague the main case of real coordinate space until after examples have been given (which I think defeats the purpose of those examples). For almost all mathematicians, the word "manifold" will mean "manifold over the reals" (probably evenly split between whether it has a differentiable structure or not), not a possibly p-adic manifold or complex manifold. If a mathematician means one of those things, then he will say "p-adic manifold" or "complex manifold". We shouldn't emphasize unusual cases in the lead of an article: things there should appear in proportion to their prominence. Sławomir Biały (talk) 12:18, 19 April 2013 (UTC)
Per WP:LEAD, the lead has two main functions: introduce the topic in an accessible way and summarize the content of the article. Euclidean space is a more familiar concept than coordinate space and is the better term to use in an accessible introduction. Euclidean space is a concept used throughout the article, while coordinate space is not. Even definitions of topological manifolds and scheme-theoretic analogs make reference to Euclidean space. So it is appropriate that Euclidean space be used in the lead. From a mathematical point of view, coordinate space is merely a representation of the geometric object called Euclidean space. It seems wrong to define a geometric concept, such as a manifold, in terms of a particular representation of another geometrical object, albeit a common one. --Mark viking (talk) 17:30, 19 April 2013 (UTC)
The real n-space has some structures which an Euclidean space has not. These are: the origin, n coordinates (or, dually, the standard basis), an orientation (or, the same, an order on coordinates or basis elements). You can assert that it is merely a representation of the geometric object called Euclidean space, but it means that you just do not understand that ℝn is an object on its own standing, of several structures which are not Euclidean. Could you provide citations for definitions of topological manifolds and scheme-theoretic analogs making reference to Euclidean space? Incnis Mrsi (talk) 17:46, 19 April 2013 (UTC)
The structure desired is primarily the local geometry, not the local coordinate system. Having a locally ${\displaystyle {\mathbb {R}}^{n}}$ coordinate system is a particular sense in which you've accomplished creating such a local geometry. This is a useful perspective to take when you seek to generalize the concept: You could make sheaves of all kinds of magmas. But you wouldn't if you were trying to produce the algebraic analogue of manifold semantics, because you're looking for the coordinate system you use to have laws corresponding to certain geometric properties. Effective featurelessness, cohesive behavior, and what else you like about Euclidean space are why they're chosen as building blocks of things, not coordinatizability, which you want for entirely its own reasons. Recent Advances in the Foundations of Euclidean Plane Geometry (Bruck 1953) breaks down the study of planar ternary rings in terms of generalizing the semantics of Euclidean space, because while coordinatizability is a given, without those semantics non-desarguessian planes can be taken for a formal aberration, not spaces in their own right. Geometric Algebra (Artin 1957) also takes the approach of deriving fields from affine spaces, not the reverse, because it's motivation for fields themselves, and it's the motive, which should not be hidden, for wanting things to be patched together from them. ᛭ LokiClock (talk) 18:28, 21 April 2013 (UTC)
Do you really contribute to this discussion, not to some new one? Euclidean space is a particular structure over the real numbers. Nobody wants to derive real numbers from affine spaces or such, but somebody can be interested in non-real-based manifolds. When one chooses “Euclidean space” instead of an (abstract) “coordinate space”, one loses all manifolds over non-Archimedean fields, as well as part of algebraic manifolds (those which are over finite fields, algebraic numbers, and possibly something else). Incnis Mrsi (talk) 18:48, 21 April 2013 (UTC)
Insisting on the lead of the article referring to abstract coordinate spaces in order to accommodate rather exotic objects like manifolds over non-Archimedean fields makes about as much sense in this article as it would to insist that the lead of the article integer should accommodate notions like p-adic integer. It's just not an appropriate focus for the lead of the article. Sławomir Biały (talk) 20:56, 21 April 2013 (UTC)
Partially answered at user talk: Sławomir Biały #Talk: Manifold because an off-topic starts here. The only correct point, although missed by the poster himself, is existence of two articles integer about the ring Z = O(Q) and ring of integers about O(whatever). We have differentiable manifold and topological manifold articles where the manifold structure is more specific. What should we have in the “Manifold” article? IMHO a WP:CONCEPTDAB-like article. Incnis Mrsi (talk) 06:26, 22 April 2013 (UTC)
I agree with Slawomir: Talking on "manifolds over non-Archimedan field" in the lead is confusing for most readers (including myself, see my post introducing this thread). Moreover, one may be interested by such extensions only if one has well understood the classical definition. IMO, the place of "manifolds over non-Archimedan field" is as a subsection of the section "Generalization", possibly with a {{main}} template, if a separate article is written. Emphasizing in the lead on such a particular generalization of the primary topic would break the WP:DUE policy. D.Lazard (talk) 10:20, 22 April 2013 (UTC)
Incnis - Euclidean space is not a structure over the real numbers. Euclidean space is a natural geometry for the real numbers, and the real numbers are a natural algebra for Euclidean space. Given the "what links here" articles are like Dimension (mathematics and physics), Dynamical system, or General relativity, it's unfair to suppose the reader takes this view of the real numbers, and implicitly make the transposition. Euclidean space might be their only reason for finding the real numbers so special. Besides, one isn't losing p-adic manifolds in the long term, one gains them in the first place by generalization of what you consider to be Euclidean space or the topology of the real numbers. What you lose is the analogy between generalizations and the original, because you've lost the original geometric notion. ᛭ LokiClock (talk) 13:47, 22 April 2013 (UTC)
What means “real numbers [are] so special”? Yes, they are intimately related with metric geometry, particularly with aforementioned Archimedean property and the concept of a complete metric space, but metric geometry is not a very important thing in the theory of manifolds. Topology is crucially important, differential calculus is important, linear algebra is important, and hence fields are important. A metric structure is important only in Riemannian geometry, a rather special part of the theory of manifolds. Incnis Mrsi (talk) 17:38, 22 April 2013 (UTC)
By setting coordinate spaces as your basic notion for manifolds, you're implicitly emphasizing the real numbers, and the properties that arise from them. Saying fields, linear algebra, and differential calculus are important begs the question - what makes fields give rise to linear and differential algebras, and why are linear algebras good? If you look at a sphere up close, you're thinking it's a manifold because of the plane geometry, of the behavior of the lines and points in the neighborhood. So maybe there are other geometries that behave closely to Euclidean space. Wondering if the real numbers aren't so special can lead you to finite fields as coordinate systems for geometries that are Euclidean for all intents and purposes, while also justifying the real numbers because it's the most complete option (pun intended). ᛭ LokiClock (talk) 19:07, 22 April 2013 (UTC)
Also, I want to clarify the original statement I made. Your concern about people being interested in manifolds over other fields is exactly what I was addressing - the reason you might call something with local field structure a manifold is precisely because they are the algebras for what can be thought of as generalized Euclidean spaces. The algebraic properties required to coordinatize a space are derived from the axioms of the geometry in the sources I cited. The difference between the axiomatic affine spaces studied in incidence geometry and the affine spaces which are vector spaces whose origins have been forgotten (by introducing new automorphisms) is that fields coordinatize spaces that satisfy more axioms of Euclidean geometry, special cases of the axiomatic affine spaces. And given that the first irrational numbers were derived geometrically, to show they must be included in a good number system, one would want to derive the reals from affine space. ᛭ LokiClock (talk) 16:19, 22 April 2013 (UTC)
Indeed, the first irrational was derived from Euclidean plane. You are correct in the point that real numbers are its natural algebra, and these are namely real numbers over which the structure of a quadratic/bilinear-symmetric form yields a useful geometry. Incnis Mrsi (talk) 17:38, 22 April 2013 (UTC)
Note that, in his book "Geometric algebra", Emil Artin showed (as far as I remember) that the axioms of Euclidean geometry allow to construct naturally a field, which is isomorphic to the reals, and, conversely, that the geometry which is constructed from the affine space over the reals, equipped with the dot product, is isomorphic to Euclidean geometry. To Incnis Mrsi: Nobody contest that manifolds may and have been generalized to other fields. The problem is the importance/notability of this generalization, which does not allow to mention it reasonably in the lead. D.Lazard (talk) 17:51, 22 April 2013 (UTC)

## Figure 1 does not illustrate text

The text refers to semicircles (verbally and mathematically) while the diagram illustrates shorter arcs. Not too serious, but it detracts. — Preceding unsigned comment added by Pierreva (talkcontribs) 03:08, 21 October 2013 (UTC)

That's true. I've replaced semicircles with arcs in the prose. Thanks, --Mark viking (talk) 03:57, 21 October 2013 (UTC)

I was going to suggest that simple change, then I noticed the reference to the interval (-1,1), and the following two paragraphs both only make sense in the context of semicircles. I'm afraid it is the graphic that is the smallest change target. — Preceding unsigned comment added by Pierreva (talkcontribs) 18:11, 22 October 2013 (UTC)

## Animated GIF

I've removed the animated GIF of "boy's surface" because of the distraction it causes. This action is consistent with MOS:ACCESS but counter to the wishes of User:Slawekb

https://en.wikipedia.org/w/index.php?title=Manifold&diff=581403804&oldid=581391247 : (The image and its caption accompany the text of the lead. If you don't like this particular image of Boy's surface, then find another one.)

Any opinions on this apart from the two of us?

-- Catskul (talk) 23:09, 14 November 2013 (UTC)

Presumably the onus is on you to make a more suitable image and convert the gif to video, per the guideline. The manual of style should not be used to dictate what kinds of informative content to have in articles. This image has informative value. Sławomir Biały (talk) 00:35, 15 November 2013 (UTC)
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
As I understand it there is no onus on editors to replace offending content. If, for example, a statement is unsourced, an editor is not obligated to find a source which negates the content before removing. While the content I am attempting to remove has value, it is neither critical to the article nor mentioned anywhere in the text.
Despite my belief that replacement is not a requirement, consensus is needed. So in an attempt to achieve consensus, I suggest replacement of the original with the following:
-- Catskul (talk) 16:30, 15 November 2013 (UTC)
Aesthetically, that image is not really an improvement over what is there now. I will work on a better replacement when I get the time. Sławomir Biały (talk) 00:08, 16 November 2013 (UTC)
I'm going to add in an opinion here. The point that some 2d manifolds cannot be embedded in 3d space without self-intersection is not central to the concept of a manifold; it is an essentially topological result that applies only for an arbitrarily constrained choice of embedding space for a some manifolds. I'd say that a far more significant (or more generally applicable) points topologically are that manifolds can be closed or can be non-orientable, which are not even mentioned in the lead. I'd think that it would be sufficient to mention one or two illustrative cases such a the Klein bottle without mentioning properties of selected embeddings. —Quondum 13:01, 16 November 2013 (UTC)

I think the idea that there are 2-manifolds that are not realizable as surfaces in Euclidean space is actually quite important for understanding why there is a mathematical notion of "manifold" at all. Certainly, one can study manifolds without worrying about their embedding properties (although there are people who build their careers entirely on the latter), but to someone with no idea what a manifold even is, I think it is very important to realize that they do represent a significant generalization of the elementary notion of a surface. Sławomir Biały (talk) 13:31, 17 November 2013 (UTC)

The topological aspects of manifolds are important, I agree, but is only one of what they are useful for. The same point about embedding can be illustrated within a Klein bottle without challenging the reader nearly as much. Emphasizing the topological aspects at the expense of the geometric aspects is also not ideal. For example, the real projective plane also represents elliptic geometry, which does not come through at all. All these issues are complex enough that more than a mention in the lead can hint to the reader that this article is hard work to understand. —Quondum 21:32, 17 November 2013 (UTC)