Talk:Manifold

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Impressed with this article

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 (talk • contribs) 20:36, 2006 November 12.

No notion of distance is incorrect

The statement "we see that a manifold need not have any well-defined notion of distance" is incorrect (at least for what the article calls a 'topological manifold'). Taking a manifold as a Hausdorff, second countable space locally homeomorphic to R^n for some n, this shows that manifolds are locally compact; that means that they are regular, so by the Urysohn metrization theorem they are metrizable. Thus there is always a metric function for manifolds. — Preceding unsigned comment added by Forgetful functor187 (talk • contribs) 04:01, 9 November 2012 (UTC)

The fact that one may define a distance on any variety does not imply that it is unique. Well-defined does not means correctly defined but defined in an unambiguously way. --D.Lazard (talk) 07:25, 9 November 2012 (UTC)

I've deleted the relevant sentence. I think it gives a misleading impression (readers could be left wondering if disconnected metric spaces exist at all), the point it's making isn't valuable, and it's an unsourced statement. Jowa fan (talk) 23:38, 9 November 2012 (UTC)

Lede statement on sphere and torus

In the lede is the phrase "Examples include the plane, the sphere, and the torus, [...]". I would suggest that it should be written "Examples include the plane, the surface of a sphere, and the surface of a torus, [...]" as a sphere and torus each could be viewed as either a surface or as a solid, but only the surface view is what is intended here. — al-Shimoni (talk) 06:40, 11 December 2012 (UTC)

It was rather insightful to mark proposed wording with color:#600 (like {{!xt}}). I am not completely sure about “torus”, but in mathematics, “sphere” could be viewed only as a 2-manifold. A 3-dimensional subset of an Euclidean space is called “ball (mathematics)”. They can be sometimes confused by geographers, geologists, engineers and so, but not by mathematicians. “The surface of a sphere” is a nonsensical gibberish or, at best, a pleonasm. Incnis Mrsi (talk) 09:47, 20 April 2013 (UTC)

Great Article!

I never comment on Wikipedia, but I read just the first half and was already very impressed. I am a graduate student and took differential topology, and boy do I wish I looked at this article earlier. It's simplicity and clarity in explaining what can normally be very complicated concepts is a model for Wikipedia pages. I admittedly do not know the technical details about why this page was removed from being a featured article candidate, but it is by far one of the best written articles I have ever come across on wikipedia. Thus, I would like to thank the writers and contributors; you guys deserve kudos. — Preceding unsigned comment added by 146.201.205.212 (talk) 21:08, 23 January 2013 (UTC)

"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 ${\mathbb {R}}^{n}$ 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)