Talk:Differentiable manifold

Equivalence

I have my doubts about compatibility of atlasses being an equivalence relation. It is not clear if the relation is transitive. For instance compatibility of charts means:

$(U,\phi ),(V,\psi )$ are C-compatible charts, hence

$\psi (U\cap V)\ {\stackrel {\psi ^{-1}}{\longrightarrow }}\ U\cap V\ {\stackrel {\phi }{\longrightarrow }}\ \phi (U\cap V)$ is of differentiability class C

$(V,\psi ),(W,\chi )$ are C-compatible charts, hence

$\chi (V\cap W)\ {\stackrel {\chi ^{-1}}{\longrightarrow }}\ V\cap W\ {\stackrel {\psi }{\longrightarrow }}\ \psi (V\cap W)$ is of differentiability class C

The question is: are

$(U,\phi ),(W,\chi )$ also C-compatible charts?

As a consequence of the above

$\chi (U\cap V\cap W)\ {\stackrel {\chi ^{-1}}{\longrightarrow }}\ U\cap V\cap W\ {\stackrel {\psi }{\longrightarrow }}\ \psi (U\cap V\cap W)\ {\stackrel {\psi ^{-1}}{\longrightarrow }}\ U\cap V\cap W\ {\stackrel {\phi }{\longrightarrow }}\ \phi (U\cap V\cap W)$ is of differentiability class C, but what about

$\chi (U\cap W)\ {\stackrel {\chi ^{-1}}{\longrightarrow }}\ U\cap W\ {\stackrel {\phi }{\longrightarrow }}\ \phi (U\cap W)$?

Madyno (talk) 22:41, 30 September 2017 (UTC)

A bit late: differentiability and differentiability class are local properties, The article does not discuss equivalence of charts, but only equivalence of atlases. If you define equivalence of charts at a point, it is easy to see that the notion is transitive. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:08, 21 November 2022 (UTC)

Merger proposal

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

There is consensus to merge. Both the opinions held by the opposer and by the proponents are aligned with policy. The opposer argues that since the methods applied are different, the analytic manifolds are not better understood in the context of differential manifolds. Proponents rebut this claim and refer to content duplication. Overall, the arguments in favour of the merge have convinced more editors (considering SilverMatsu's oppose vote as stricken by his last comment) Felix QW (talk) 12:35, 13 October 2022 (UTC)

---

I propose to merge Analytic manifold into Differentiable manifold. The content in the Analytic article (as it stands) doesn't seem to merit an article. As far as article size is concerned, this one may be getting up there, but can definitely take a small addition. Horsesizedduck (talk) 15:41, 17 July 2021 (UTC)

• Strongly oppose. Every analytic manifod can be complexified for forming a complex analytic variety. So the methods of study of real analytic manifold are those of Analytic geometry, which are completely different from those of Differential geometry. If a merge should be done, it must be with Analytic variety, not Differentiable manifold. D.Lazard (talk) 18:14, 17 July 2021 (UTC)
• oppose As an example, Stein space. From this example, if they are need to merge, Analytic space.--SilverMatsu (talk) 22:47, 20 July 2021 (UTC)

Proposal Move an analytic manifold to a Real analytic manifold and merge them into differentiable manifold and also an analytic manifold redirects to an analytic variety.--SilverMatsu (talk) 13:40, 3 August 2021 (UTC)

Is it possible to write a complex complex manifold simply as an analytic manifold ? Looking at the Mathematics Subject Classification, it says:Several complex variables and analytic spaces (analytic variety). I forgot to reference (It was a reference that says holomorphy is regular) to it, but it was written about Cousin problem as follows:the difficulty of the Cousin problem is due to the lack of a unified theory of the property of the singular point in the analytical variety, so we introduce the Stein manifold (domain of holomorphy) as an analytic submanifold that can solve the Cousin problem. Also, as a related topic, "Several complex variables" and "analytic variety" article (category) names may be merged into "Several complex variables and analytic variety" or "Several complex variables and analytic spaces".--SilverMatsu (talk) 23:37, 6 August 2021 (UTC)

I found a reference that explains that it is a real analytic manifold that is closely related to a differentiable manifold. Onishchik, A. L. (2001) [1994], "Analytic manifold", Encyclopedia of Mathematics, EMS Press

• Strongly agree. I disagree with the relevance of D.Lazard's comments. I don't think his second sentence follows from the first. For instance, the same reasoning whould apply to real and complex vector spaces, but one would not say that the study of complex vector spaces subsumes the study of real vector spaces (even though for some specific purposes it does). Anyway, the actual content on the Analytic manifold page is exactly like the actual content on the Differentiable manifold page and not at all like the content on the Analytic variety page. (That makes perfect sense, since an analytic manifold is nothing but a special species of differentiable manifold.) In fact in some sense the merge is already accomplished, since nearly all information on the analytic manifold page already appears on the differentiable manifold page. Gumshoe2 (talk) 02:46, 21 July 2021 (UTC)
• Agree. Just to provide a counterweight, every $C^{k}$ manifold for $k\geq 1$ admits a unique smoothing to a $C^{k+1}$-manifold, and indeed a unique smoothing to a $C^{\infty }$-manifold and a real analytic $C^{\omega }$-manifold. For the same reason that $C^{k}$ manifolds don't have a distinguished place in the literature, real analytic manifolds don't either. Whilst it is true that there are things you can do with real analytic structures you can't do with merely smooth or differentiable sturctures (such as ask for a complex analytic structure as D.Lazard noted), the same can be said of $C^{k}$ manifolds, for which the lack of smoothness also changes how they are studied. But we don't give $C^{k}$ manifolds their own page, and I don't see why we should give analytic manifolds their own page, especially since the page for analytic manifolds doesn't contain any particularly notable information about them other than what is already contained in this article. I'm not sure the single sentence of novel content "real analytic manifolds are close in nature to complex analytic manifolds, and can be studied using tools from analytic geometry and have links to algebraic geometry" is enough to justify the separate pages.

I think if we want to keep analytic manifold as its own page, then we should also have pages for smooth manifold and C^k manifold, but 90% of the content of such pages would be identical and equal to what is on this page.Tazerenix (talk) 11:10, 10 August 2021 (UTC)

Perhaps I could suggest that we include in the lead explicitly the different types of differentiable manifolds, just after where we state differentiable manifold. For example, in the paragraph:

In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the various charts to one another are called transition maps. When these transition maps are k-times differentiable, the manifold is called a $C^{k}$-manifold. When the transition maps are infinitely differentiable, the manifold is called a smooth manifold or $C^{\infty }$-manifold. When the transition maps are analytic functions the manifold is called a real analytic manifold or $C^{\omega }$-manifold. In particular a topological manifold with no differentiable structure is also known as a $C^{0}$-manifold. Depending on the level of differentiability, the tools used to study such manifolds may differ: the study of $C^{k}$-manifolds is closer to the study of topological manifolds, and the study of real analytic manifolds is closer to the study of analytic varieties.

This way we don't mislead anyone in the lead that there is only one monolithic type for differentiable manifolds, and the distinctions can be expanded upon in the body?Tazerenix (talk) 11:19, 10 August 2021 (UTC)

Even in the case of real analytic manifold, coherence of the structure sheaf always holds, but what about differentiable manifold ? But I no longer oppose to merging analytic manifold into differentiable manifolds. Because perhaps in order to make a stand-alone article, I think we need to explain from the analytic set which is more weaker definition , but the structure sheaves of real-analytic spaces need not be coherent. --SilverMatsu (talk) 09:01, 23 February 2022 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

I'm very late to the party, but (1) I don't see anything resembling consensus to merge in the above discussion, which was roughly evenly split among participants, (2) In the two years since then, nobody with the expertise to perform the intended merge has done so, and (3) I oppose the merge, on more or less the same grounds as Lazard. More strongly, it appears that if any merge might be justified, it is with complex manifold, not with differentiable manifold. —David Eppstein (talk) 20:53, 28 June 2024 (UTC)

Implementing the merger

@Horsesizedduck, D.Lazard, SilverMatsu, Gumshoe2, and Tazerenix:Since some ideas on implementing the merger have already been floated in the merger discussion itself, I leave the implementation of the merger to participants and other interested editors. Further discussion can take place in this thread. Felix QW (talk) 12:44, 13 October 2022 (UTC)

Possibly move some text from the lead to the body while merging? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:10, 21 November 2022 (UTC)

Phi redux

@D.Lazard: User D.Lazard recently reinstated an inconsistent use of \varphi with the comment The phi that appears in the text (in {{math}} templates) is rendered similar to \varphi, not \phi}. However, the rendering of φ in {{math|φ}} ("φ") on my browser is "${\textstyle \phi }$", not "$\varphi$". Also, the {{Annotated image}} still uses \phi.

On my browser the rendering of φ in {{mvar|φ}} ("φ") is "${\textstyle \phi }$"; I don't know how it renders in other browsers.

I suggest that the article use consistent markup for φ, and at first glance seems most suitable. Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:19, 9 December 2022 (UTC)

So, the only solution is to convert to latex all formulas containing phi. Also, there are other special symbols that must be converted to latex (∘, ∈, etc.) as they are not correctly displayed on some browsers (on my browser, ∘ is so small that it is hardly distinguished from a dot). D.Lazard (talk) 19:01, 9 December 2022 (UTC)

An article about Charts

In my opinion there should exist a short article about charts, their notation and usage in physics as a modern view of reference systems and coordinates. --- DrQsiris (talk) 18:24, 23 May 2024 (UTC)

Chart (topology) redirects to Atlas (topology). Since one cannot define one without defining the other, it seems not useful to have two separate articles. However, the article is rather short and would deserve to be expanded. Are youwilling to do this task? D.Lazard (talk) 21:14, 23 May 2024 (UTC)

One can certainly define a chart without an Atlas. That can be important at singularities. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:23, 24 May 2024 (UTC)

What do you mean in terms of singularities?

In general, any single chart automatically defines an atlas and any atlas just consists of a number of charts. Gumshoe2 (talk) 14:06, 24 May 2024 (UTC)

Not if you're using the definition in Atlas (topology) § Formal definition of atlas: An atlas for a topological space $M$ is an indexed family $\{(U_{\alpha },\varphi _{\alpha }):\alpha \in I\}$ of charts on $M$ which covers $M$.

Consider A disk with a tail $M=\{(x,y,0)\in \mathbb {R} ^{3}:x^{2}+y^{2}<1\}\cup \{(0,0,z):z\in [0,1]\}$. It cannot be covered with a family of charts $\{(U_{\alpha },\varphi _{\alpha }):\alpha \in I\}$ where each $U_{\alpha }$ is open in $\mathbb {R}$. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 01:33, 26 May 2024 (UTC) -- Revised 06:24, 26 May 2024 (UTC)

This example shows only that not all topological spaces are manifolds, since a manifold is, by definition, a topological space that can be covered by charts. Also, every chart $(U,\varphi )$ is evidently an atlas for $U.$ D.Lazard (talk) 07:18, 26 May 2024 (UTC)

It shows that there can be a chart in a space that is not part of an atlas in that space.

$\{(U,\varphi )\}$ may be an atlas for U but $(U,\varphi )$ is not. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 23:13, 26 May 2024 (UTC)

The article did a confusion between charts (which do not imply coordinates) and "local coordinate systems" that are pairs of a chart and a coordinate system on its image. I have fixed this.

As far as I know, a reference system is called a frame (mathematics). In other words, a frame is a set of data that defines coordinates; for example, a basis in case of a vector space; an origin and an orthonormal basis in case of an Euclidean space; a set of 4 points not included in a plane for barycentric coordinates in 3D space; etc. A frame can also be viewed as a (possibly partial) function from a space to $\mathbb {R} ^{n}$ (viewed as a set of tuples).

IMO, we need an article on frames rather than an article on charts, but this requires an editor having the competence (knowledge of available sources) and the time for that. D.Lazard (talk) 07:58, 26 May 2024 (UTC)

A chart is a local coordinate system. A Moving frame in Mathematics is something quite different, and is a function that assigns to each point a Frame (linear algebra) on the tangent space, although in Physics a Frame of reference is equivalent to a chart. None of these is equivalent to a neighborhood. E.g., if $\{(U_{\alpha },\varphi _{\alpha }\}$ is a chart then $U_{\alpha }$ is a neighborhood but not a chart.

for concreteness, let M be a differentiable manifold of dimension n The we have

Atlas

$\{(U_{\alpha },\varphi _{\alpha }):\alpha \in I\}$

Chart

$(U_{\alpha },\varphi _{\alpha })$

Frame

$f_{\alpha }:x\in U_{\alpha }\rightarrow (V_{1}\in T_{x}^{M},\ldots ,V_{n}\in T_{x}^{M})$

Neighborhood

$U_{\alpha }$

There is a lot of overlap between, e.g., differentiable manifold and Differential geometry. I'm not sure how much information belongs where. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 23:13, 26 May 2024 (UTC)

Notes