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Why are Wikipedia mathematics articles so abstract?
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Why don't math pages rely more on helpful YouTube videos and media coverage of mathematical issues?
Mathematical content of YouTube videos is often unreliable (though some may be useful for pedagogical purposes rather than as references). Media reports are typically sensationalistic. This is why they are generally avoided.
Advice on dealing with questionable citations in lead[edit]
I'd like some advice on how to handle a problem which I encounter quite often in articles covering basic topics that are widely used in other fields.
The typical scenario goes like this: A is a central notion that was introduced a while ago and on which there are plenty of old and recent textbooks. A is now used in many fields outside of mathematics, and maybe in a trendy field such as machine learning. Some people keep adding references to recent textbook or articles on A in the lead.
Sometimes the references are research articles published in obscure journals, and in that case this is not really a problem (even though one might need to remove the same reference several times). But in some cases the references are legit — or at least "legit-looking" — textbooks, and then because Wikipedia does not have very clear guidelines regarding citations in the lead, I am not always sure what to do and end up losing time.
Maybe a concrete example will help: Have a look at the recent [as of 11/06/2024] history of the article Markov chain, more specifically at this diff and this one. Here we have two different IPs located in Romania who are actively monitoring the article and who seem extremely upset that a textbook by a Romanian author is not listed first to back-up:
the definition of a Markov chain;
the assertion that "Markov chains have many applications as statistical models of real-world processes";
the fact that Markovian and Markov can be used to refer to something that has the Markov property.
Of course, that makes me think that the person behind these IPs is either the author of said textbook; or someone who really likes this textbook.
The problem is that, as far as I can tell without reading it, this does indeed seem like a legitimate textbook on Markov chains. In fact, by some metrics it even seems to be a popular textbook: despite being fairly recent, it is already cited 900 times. That is of course impressive...But also not very surprising, considering that it has been the first reference of the Wikipedia article on Markov chains for a while.
(in fact, to try to get an idea of whether most people citing that book actually did so to reference specific properties and theorems, or simply to add a citation after their first use of the phrase "Markov chain". I am not going to copy and copy and paste excerpts, so as not to point fingers; but some authors seem to think that Gagniuc invented Markov chains, others that think that he recently discovered the game-changing fact that the rows of a stochastic matrix sum to 1, etc).
So, on the one hand I think that reference should be removed from the lead (and probably from the article altogether), because there are tons and tons of excellent textbooks on Markov chains, and I have some suspicions of self-promotion with this one (not to mention that I have no idea whether it is any good). On the other hand, this seems to be a legitimate reference (again, I have not read it) and so I can't really base myself on any clear Wikipedia policy to do so.
I would of course appreciate if someone could help me with this specific example (especially since it looks like some IP users are ready to engage in edit-warring). But I am mostly asking for general guidance here, because it is a problem I encounter regularly.
I'm inclined to agree that these fairly innocuous statements shouldn't be cited in the lead (per the guideline WP:LEADCITE) but instead in the body. The Gagniucs citation is particularly silly as it is used because it cites a 200+ page book without giving a page number. I suggest migrating citations out of the lead into the corresponding places in the body, leaving WP:LEADCITE in your edit summary; if you actually do run into any trouble (your idea about this doesn't seem entirely supported by data) then bringing the issue up here (and perhaps in parallel on the article's talk-page) and seeing if the angry IP pretending to be two different people engages. --JBL (talk) 00:28, 11 June 2024 (UTC)[reply]
This can also happen more innocently, when some random editor asks for a citation of some claim and then, to clear the citation needed tag, another editor does a search and finds a random citation that matches the claim. Especially in cases where a claim is a basic fact that everyone working in a subject knows but few bother to write down (because it is so basic), or when the terminology has shifted and the texts haven't been updated to match, finding the claim in a standard textbook rather than in a recent research work can sometimes be difficult. —David Eppstein (talk) 00:42, 11 June 2024 (UTC)[reply]
@David Eppstein I agree; but I think that it is usually possible to distinguish the situations I am referring to and the more innocent situations that you describe. For instance, Gagniuc's book is repeatedly cited to back-up statements that do not really need a reference, such as "Markov chains can be used to model many games of chance". So to me it really looks like someone — not saying it is Gagniuc; it could be, e.g, a student that worships him — tried to promote his book. Malparti (talk) 23:48, 11 June 2024 (UTC)[reply]
Oh, I agree in this case, but "Markov chains can be used to model many games of chance" is exactly the sort of obvious statement that you're likely to see editors demand citations for. For this sort of thing, expository articles rather than research articles or monographs might be a better fit; I found for instance "How long is a game of snakes and ladders?" Math. Gaz 1993 and "Snakes and Ladders and Intransitivity, or what mathematicians do in their time off" Intelligencer 2023. Also those editors may well argue that "many" is WP:PEACOCK and that we should provide specific examples (of which snakes and ladders is one). —David Eppstein (talk) 00:07, 12 June 2024 (UTC)[reply]
@David Eppstein Arf, yes you are right; I've actually taken part in unproductive debates on this topic on Wikipedia in another language, and would rather avoid this on en.wiki (all the more so since I tend to be a bit more on the WP:BLUE end of the spectrum than most editors). But thanks a lot for the references, they are indeed much better suited than the current ones so I'll add them to the article over the weekend; ideally, I should take the time to do the same thing for all other such citations... Malparti (talk) 00:19, 12 June 2024 (UTC)[reply]
@JBL Thanks, that is useful advice. In fact I don't think that migrating citations out of the lead is needed: in my opinion the body of the article already contains quite a lot of unnecessary citations... Malparti (talk) 23:56, 11 June 2024 (UTC)[reply]
Uncontroversial statements discussed later in the article don't need any citation in the lead section, and it can be more legible for readers to defer those footnotes until later. (Of course, it can also be fine to include footnotes in the lead, e.g. when linking to the original source where something was first described.) More generally, when trying to support uncontroversial widely known claims, there are often hundreds+ of sources that could be cited. If you have one easily available, I would recommend leaning on popular and widely cited textbooks or survey papers rather than more obscure sources. People shouldn't be trying to use Wikipedia for self promotion via citation spam. –jacobolus(t)01:55, 11 June 2024 (UTC)[reply]
I generally support moving citations out of the lead into the body or into ((refideas)), with a comment to the effect that the citation should be deferred to the body. Is there a template with the semantics this is the wrong place for a citation?
I'm a bit hesitant to complain about citation spam, because there are often articles whose contents are garbage but that contain useful citations. In fact, sometimes I use wiki as a search engine and go straight to the references -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:43, 11 June 2024 (UTC)[reply]
@Jacobolus"when trying to support uncontroversial widely known claims, there are often hundreds+ of sources that could be cited. If you have one easily available, I would recommend leaning on popular and widely cited textbooks or survey papers rather than more obscure sources." → Yes of course. But my problem is specifically in cases where there is already a source, and it looks like a legit textbook (here: Gagniuc's book is published by a reputable publisher, and very cited) but it still looks like there is something fishy going on. I see it quite frequently. Less frequently than the situation where someone adds an irrelevant paper published in an obscure journal (but, as I said this is not a problem because in that case it takes me very little time to see what is going on and to remove the citation); but still frequently enough that I am starting to feel like this is making me waste my time. Malparti (talk) 00:04, 12 June 2024 (UTC)[reply]
If the source clearly supports the claim and is in a highly cited legit textbook, then I wouldn't worry too much about including it in the article somewhere. The lead of Markov chain is currently absurdly overstuffed^{[1][2]} with gratuitous footnotes.^{[3][4][5][6][7]} In my opinion these should be either removed to the body of the article or consolidated into no more than a couple per paragraph, for legibility. If Gagniuc's book is clear and well written, I think it would be fine enough to include book among a list (inside a single footnote) of relevant survey sources supporting some particular claims in the lead, but it would also be fine to entirely defer those references to the body. Gagniuc's book should be moved down to the "References" section, and a specific page mentioned for each time it is used as a reference. A 20 page research paper is fair enough to cite as a whole unit for a list of claims, but vaguely waving at a 200+ page textbook is too much. –jacobolus(t)00:30, 12 June 2024 (UTC)[reply]
I concur with the sentiment expressed above that uncontroversial statements in the intro that are adequately elaborated upon in the main text don't need footnotes. In this particular case, that block of four citations in a row is just silly. Actually, that whole sentence has problems. The main text of the article doesn't say anything more about "cruise control systems in motor vehicles" (which sounds like a weirdly niche application to advertise up top), or queuing at an airport specifically, or currency exchanges. I'd cut that line after "of real-world processes" and replace the rest with a better summary. XOR'easter (talk) 16:07, 11 June 2024 (UTC)[reply]
@XOR'easter I agree that the article has many problems... In fact I think this is one of these articles on a popular topics that suffers a bit from "constant-growth" and needs to be trimmed on a regular basis; but that's a somewhat different issue. I might try to rewrite the lead over the weekend (but I'm pretty sure that if I remove oddly specific examples, they are going to be replaced by other ones in no time). Malparti (talk) 00:06, 12 June 2024 (UTC)[reply]
Well now, for what it's worth, the IP editor comments in an edit summary that Gagniuc is "the most reliable book on the subject, and the one that is part of ChatGPT training set." A different IP editor calls it the "top representative book on the subject". –jacobolus(t)00:21, 13 June 2024 (UTC)[reply]
For what it's worth: it seems that the references to Gagniuc's book were introduced here by (the now-banned) MegGutman. That user also wrote"After seeing the book on Wikipedia in 2017, I contacted Dr. Gagniuc for a collaboration proposal on a EU research project, which he kindly accepted. So, I'm personally involved.". I also found claims that Gagniuc's book is rubbish because it contains basic mistakes. I am going to skim through it to see about that for myself.
Rubbish or not — and irrespective of the identity of the person trying to promote the book — being referenced in this Wikipedia article seems to have paid off. So I think that show the importance of my initial question: how to deal with these kind of simulations without investing unreasonable amounts of time? If it was only about reading a few diffs and flipping through a textbook, it would already be annoying. But if each time some IP users pop up out of nowhere to reintroduce the reference, I need a simple protocol. Malparti (talk) 14:39, 13 June 2024 (UTC)[reply]
Update: what follows is only my opinion... but this book is worse than I thought. I'm not going to detail, as this would be a waste of time for everyone, but despite the book being called "from theory to implementation", there is not an ounce of theory — most of the basic concepts are not presented. The book is full of approximations, and the way things are written gives the impression that the author does not understand the basics... Not to mention the >100 pages of computer code which I doubt anyone is ever going to read (it is even hard for me to comprehend how Wiley could agree to print something like this in 2017). So my assessment is that the negative comments that I read were fully justified. I am going to remove this book entirely from the article, and replace it with more suitable references. Malparti (talk) 15:21, 13 June 2024 (UTC)[reply]
@Jacobolus I've just had a look at the recent history of the article, and things took a pretty absurd turn pretty quick. And because the user seems to know how to use different IPs, dealing with this is probably going to be a pain. The good thing is, that person is... Not very subtle. So it's pretty easy to see what's going on here; this may not always be the case... Malparti (talk) 19:06, 13 June 2024 (UTC)[reply]
I was about to file a request at WP:RPP until I saw that you already had; thanks. I also concur that zapping citations to Gagniuc would be a worthwhile cleanup job. It sounds like where they aren't irrelevant (e.g., citing the definition of a Markov chain), they should be replaced with pointers to more dependable references, even apart from WP:COI concerns. XOR'easter (talk) 19:28, 13 June 2024 (UTC)[reply]
@XOR'easter I am going to take care of this (running a search across Wikipedia for Gagniuc's work) before the end of the week, most likely over the weekend. Thanks again for your help — although I guess I will likely run into more trouble and will writing more here... Malparti (talk) 22:57, 13 June 2024 (UTC)[reply]
For those following along at home, the page has been protected for one week, and the two most aggressive IP addresses have been blocked for c. one day. (A third IP has not been blocked.) It seems reasonable to expect a resumption of similar behavior on other related articles once the blocks expire; that can be dealt with via a trip to either WP:AIV or WP:3RRN (or just a note to the blocking administrator Daniel Quinlan). --JBL (talk) 23:48, 13 June 2024 (UTC)[reply]
Despite Malparti warning that "it would be a waste of time for everyone" I took a look at the book myself. 60 pages of badly-worded boring worked examples with no theory before we even get to the possibility of having more than two states. As Malparti said, there is no theory, or rather theory is alluded to in vague and inaccurate form without any justification. For instance the steady state (still of a two-state chain) is first mentioned on 46 as "the unique solution" to an equilibrium equation, and is stated to be "eventually achieved", with no discussion of exceptional cases where the solution is not unique or not reached in the limit, and no discussion of the fact that it is never actually achieved, only found in the limit. Do not use for anything. I should have taken the fact that I could not find a review even on MR and zbl as a warning. —David Eppstein (talk) 23:53, 13 June 2024 (UTC)[reply]
I didn't have time to read much of it apart from discovering that it actually said nothing about one of the claims it was being used to support. But having had time since to evaluate it more, I have to agree: it's a sloppy book. The writing confuses urn draws with and without replacement, events probably happening versus definitively happening, etc. XOR'easter (talk) 23:26, 14 June 2024 (UTC)[reply]
This seems to have been a deletion without prejeudice to re-creation. About a year after this deletion, Dennis Cook's book An Introduction to Envelopes^{[1]} was published.
Although this topic was primarily the creation of Dennis Cook and some of his Ph.D. advisees, I believe some of his colleagues and students in his graduate courses have also influenced the topic. (In particular, the term "central subspace" was suggested by David Nelson.)
It appears to me that with the publication of the book, the time is ripe to think about re-creating the article, written in a more beginner-friendly way, perhaps under the title Envelope model (statistics) or Envelope (statistics).
One question. Why is this here in Wikiproject Mathematics? Isn't there a corresponding Wikiproject for Statistics, which would be more appropriate for this topic? PatrickR2 (talk) 05:36, 30 June 2024 (UTC)[reply]
In the past couple of days I spent some time researching the name "trammel of Archimedes", sometimes applied to the instrument for the several centuries previously and still often today called an elliptic trammel or elliptic compass (a "trammel" or beam compass is a wooden or metal rod or beam along which slide metal "trammel points", used to draw circles). This is a type of ellipsograph (tool for drawing ellipses). I learned that Archimedes had nothing to do with this tool, which may have been invented in the early 16th century by Leonardo da Vinci, and which operates on the same mathematical principle as a mechanism investigated by Proclus (5th century) based on the one Nicomedes (3rd century BC) used to trisect angles. Circa 1940 the name "trammel of Archimedes" showed up in the work of Robert C. Yates, apparently out of the blue (I speculate this may have been based on some confusion by Yates or whoever he got the name from between Nicomedes and Archimedes). Judging from searches of books/academic papers, the name "trammel of Archimedes" remained quite rare through the 20th century, but there have been a nontrivial number of people calling it that in the past couple of decades, perhaps partly under the influence of webpages like Wikipedia.
Anyway... I think this article would be improved by reorganizing it to discuss the general topic of ellipse drawing, so I proposed at Talk:Trammel of Archimedes § Requested move 1 July 2024 that it should be moved to the title Ellipsograph (which currently redirects there), with "Elliptic trammel" turned into a top-level section. Then we can add other sections about the pins-and-string method for drawing ellipses, as well as various other interesting ellipse drawing tools/methods, and some further discussion about how these tools were used in practice. –jacobolus(t)05:43, 1 July 2024 (UTC)[reply]
Discussion on links in the lead sentence of mathematics[edit]
That's an awkward ASCII way of writing $3\uparrow \uparrow \uparrow \uparrow 3$, which is the first term in a sequence whose 64th term is Graham's number. The particular number $3\uparrow \uparrow \uparrow \uparrow 3$ is mentioned in the lead section of the article Graham's number. --JBL (talk) 19:42, 4 July 2024 (UTC)[reply]
An article of mine seems to be not appearing on Google[edit]
Hello,
I wrote an article Deficiency (statistics) which was accepted but is still somehow hidden to the public since it does not appear on search engines like Google. Why is that? The article is about a term introduced by Lucien Le Cam in a famous paper called "Sufficiency and Approximate Sufficiency" in the Annals of Mathematical Statistics which was the starting point for Le Cam theory and he later extended in a book.--Tensorproduct (talk) 19:57, 4 July 2024 (UTC)[reply]
@Michael Hardy Thank you for the answer. Also thank you for your contribution to mathematics articles on Wikipedia! I saw your name as the initial author of a lot of articles about infinite-dimensional stochastic analysis. Thank you for your contribution.--Tensorproduct (talk) 20:38, 12 July 2024 (UTC)[reply]
Hi all, I have spent much time over the past week and a half editing the C-class article Riemannian manifold and I think it is ready for a reappraisal. I would also be very happy if others have ideas for how to improve the page or to make it more accessible and readable. I would love to have an image at the top of the page, but I couldn't think of a good one. Mathwriter2718 (talk) 20:37, 4 July 2024 (UTC)[reply]
It starts at the deep end. Shouldn't readers learn much earlier that Euclidean spaces and smooth surfaces embedded in them form Riemannian manifolds? —David Eppstein (talk) 22:25, 4 July 2024 (UTC)[reply]
Agreed. The content of the article is all important and should be there, but especially the lead could be a bit more general, especially given that a Riemannian manifold is quite an understandable concept (if not the details). For example a better first sentence or two might be something like
"in differential geometry, a Riemannian manifold is a (possibly non-Euclidean) geometric space for which traditional geometric notions of distance, angle, and volume from Euclidean geometry are defined. These notions can be defined through reference to an ambient Euclidean space which the manifold sits inside (and indeed any Riemannian manifold may be viewed this way due to the Nash embedding theorems) but the modern notion of a Riemannian manifold emphasizes the intrinsic point of view first developed by Bernhard Riemann, which makes no reference to an ambient space and instead defines the notions of distance, angle, and volume directly on the manifold, by specifying Euclidean inner products on each tangent space with a structure called a Riemannian metric. The techniques of differential and integral calculus can be used to transform this infinitesimal information into genuine geometric data about the manifold, and for example distance between points on the manifold along a path, the arc length, can be determined by integrating the infinitesimal measure of distance along the path given by the metric."
I also strongly recommend adding a section to the lead about applications, especially of Pseudo-Riemannian geometry to physics, and of the basic ideas of Riemannian geometry in design and engineering. Some of the technical stuff in the lead can be kept, but a good lead should have a little something for everyone. Tazerenix (talk) 01:51, 5 July 2024 (UTC)[reply]
This could still be less technical and more concise.
"First developed by Bernhard Riemann" seems oversimplified/imprecise. Maybe just in this very specific way? People were thinking about e.g. the sphere intrinsically many centuries before that (back to Menelaus of Alexandria if not before), and there are surely more general examples from the 18th or early 19th century. Where does Gauss fit in this story?
"Genuine geometric data" is a confusing phrase.
I recommend deferring mention of tangent spaces and Nash embedding theorems past the first paragraph, until such a space as they can be unpacked (briefly but) clearly where mentioned.
I recommend adding "locally", e.g. "... are locally defined", maybe with a wikilink to Local property. Though some more explicit phrase might be better, "in the vicinity of each point" or the like.
It would likely be clearer to keep the first paragraph more to the point, and contrast with an embedded-in-a-flat-ambient-space in a second paragraph.
I'd recommend trying to read some of Needham's Visual Differential Geometry when working to make relevant articles accessible. There are a lot of clear explanations and nice pictures there. –jacobolus(t)03:46, 5 July 2024 (UTC)[reply]
Maybe this is where I admit that differential geometry was my second least favorite undergraduate mathematics class. Too much focus on symbolic formalisms like Christoffel symbols, too little intuition. I like the material now but I didn't get it then. I should take a look at that book, I'd likely still get something out of it. (Least favorite was plug-and-chug differential equations.) —David Eppstein (talk) 07:06, 5 July 2024 (UTC)[reply]
@Tazerenix @Jacobolus @David Eppstein I definitely agree with adding applications. Physics, design/engineering, machine learning, and cartography all provide examples of applications. I propose the following lead spliced from Tazerenix's paragraph and the current lead and following Jacobolus's suggestions.
In differential geometry, a Riemannian manifold is a curvy space called a smooth manifold endowed with geometric information allowing many geometric notions such as distance, angles, length, volume, and curvature to be defined. These notions can be defined through reference to an ambient Euclidean space which the manifold sits inside. However, the notion of a Riemannian manifold emphasizes the intrinsic point of view as conceptualized by its namesake Bernhard Riemann, which makes no reference to an ambient space and instead defines geometric notions directly on the abstract manifold by specifying inner products on each tangent space. The tangent space at a point is the vector space of all vectors tangent to the manifold at that point, and it can be thought of as the Euclidean space best approximating the manifold at that point. An inner product is a measuring stick that defines Euclidean geometry on a vector space by specifying the length of each vector and the angles between each two vectors.
The choice of an inner product on each tangent space is called a Riemannian metric (or just a metric), and a Riemannian manifold is defined as a smooth manifold with a Riemannian metric. Riemannian geometry is the study of Riemannian manifolds. The techniques of differential and integral calculus are used to pull geometric data out of a Riemannian metric. For example, the length of a curve can be determined by integrating the infinitesimal measure of distance along the path given by the metric.
Formally, if $M$ is a smooth manifold, a Riemannian metric is a smoothly-varying family $g$ of positive-definite inner products $g_{p))$ on the tangent spaces $T_{p}M$ at each point $p$, and the pair $(M,g)$ is a Riemannian manifold. The requirement that $g$ is smoothly-varying is that for any smooth coordinate chart$(U,x)$ on $M$, the component functions of the metric
The reference to embedding belongs in a history section; Riemannian manifold is about intrinsic geometry.
The lead contains technical details that really should be defer to later in the article.
It does not address an issue raised by Jacobolus: I recommend deferring mention of tangent spaces and Nash embedding theorems past the first paragraph, until such a space as they can be unpacked (briefly but) clearly where mentioned.
I would cut it back to
In differential geometry, a Riemannian manifold is a space called a smooth manifold endowed with geometric information allowing many geometric notions such as distance, angles, length, volume, and curvature to be defined. Although Riemannian geometry has an intrinsic perspective, it was historically motivated by the study of surfaces in Euclidean geometry.
@Chatul I strongly disagree with cutting the lead down to those two sentences. Most critically, it makes no attempt to say what a Riemannian manifold is, instead merely saying a few properties it has. I think we ought to discuss the tangent space here. It can be explained in a non-technical way and it's a fundamental part of any description of a Riemannian manifold. I also think that clarifying embedded spaces vs abstract spaces as quickly as possible is a good idea, because omitting this is guaranteed to cause misunderstanding. That distinction should not merely be relegated to history. I also think including a single sentence mentioning applications in the lead is a good idea.
To make it less technical, my first thought is to shorten:
The techniques of differential and integral calculus are used to pull geometric data out of a Riemannian metric. For example, the length of a curve can be determined by integrating the infinitesimal measure of distance along the path given by the metric.
To:
The techniques of differential and integral calculus are used to pull geometric information, such as lengths of curves, distances between points, and volumes of shapes, out of a Riemannian metric.
We could also move the paragraph starting "Formally" out of the lead, or just the description of smoothness, though I think that the audience for this page includes many people for having a formal description in the lead would be very useful. Mathwriter2718 (talk) 14:42, 5 July 2024 (UTC)[reply]
This first paragraph is a bit bloated I think. I would just mention "Riemannian metric" in the first paragraph, and defer discussion of tangent spaces etc. to a subsequent paragraph. –jacobolus(t)17:35, 6 July 2024 (UTC)[reply]
Fwiw, I think tazernix lede is good as is. There's no need to complicate matters with endless debate as to the merits of this or that. Tito Omburo (talk) 17:37, 5 July 2024 (UTC)[reply]
However my intention was to start a conversation! I'm sure there are ways of improving my attempt as others have commented on. I think the most essential point is to find just the right first sentence. Everything after that is natural. One needs to find the right word to convey to the reader that Riemannian manifolds can be curved, folded, that they are manifolds, but have the same geometric information as rigid figures from Euclidean geometry, which is the main interaction the lay person or those uneducated in geometry understand. We've had "curvy", "non-Euclidean", "smooth manifold", "space" etc. My attempt tried to rely on peoples lay knowledge of "Euclidean" although even that may be a bit esoteric for the first sentence. Happy for people to keep debating it! Tazerenix (talk) 23:55, 5 July 2024 (UTC)[reply]
I think some kind of picture(s) would help a lot, even if it's just showing a funky 2-dimensional surface immersed in Euclidean space. –jacobolus(t)00:10, 6 July 2024 (UTC)[reply]
Aren't there some nice pictures available of what hyperbolic 3-manifolds look like "from within"? Seems like this conveys the idea of "intrinsic geomtery" in a fairly striking way. Tito Omburo (talk) 00:19, 6 July 2024 (UTC)[reply]
I would suggest two images: a Klein bottle, since this is a surface which can be smoothly immersed in 3-space but which is not an embedded surface (though here the intrinsic uniformizing geometry is flat!), and an image like File:Order 5 dodecahedral honeycomb.png with words to the effect of: "An observer inside hyperbolic 3-space will see polygons in a hyperbolic tessellation up close as almost Euclidean, while polygons further away become distorted because of the non-Euclidean Riemannian metric." Tito Omburo (talk) 10:37, 6 July 2024 (UTC)[reply]
@Tazerenix I'm really glad a discussion about this page is happening. Yet another possibility for a first sentence emphasizes that it is a generalization of Euclidean geometry:
I don't think leading with "generalization of Euclidean geometry" gives the right impression either. I would make a first paragraph more along the lines of:
I'm not sure if "infinitesimal" is the best word – it might be confusing or ambiguous – and neither infinitesimal nor differential (mathematics) seems like quite the right Wikilink to employ, and e.g. differential form may be be unhelpfully advanced for less prepared readers. Do we have a clear lay-accessible article about these general kinds of concepts? Anyway, I'd then defer discussion of more precise definitions of Riemannian metric, tangent space, etc. to after the first paragraph. The second or third paragraph can also discuss the difference between extrinsic vs. intrinsic definitions, embedding theorems, and so on. –jacobolus(t)21:00, 6 July 2024 (UTC)[reply]
We can argue about wordsmithing but I like the general focus of this version on local geometry, and I especially like the suggestion of flat maps of the earth early in the lead as an analogy. —David Eppstein (talk) 22:12, 6 July 2024 (UTC)[reply]
I can see why this paragraph feels compelling, but I don't endorse this usage of "locally". Specifically, the first sentences are not true: a Riemannian manifold does not have the same local metric space structure or Riemannian metric structure as flat Euclidean space, and indeed it is impossible to have a map of a part of Earth that preserves these structures. The thing that is true is that Riemannian manifolds look "infinitesimally" like flat Euclidean space. But "locally" should mean on a neighborhood. Riemannian manifolds that look locally like flat Euclidean space are called flat. Mathwriter2718 (talk) 14:29, 7 July 2024 (UTC)[reply]
I'm afraid the phrasing "locally the same metrical structure as flat Euclidean space" is unsalvageable even if qualifiers are added to it, because it's not nearly true, it's very false. The phrasing "infinitesimally the same geometry as flat Euclidean space" would be true. Mathwriter2718 (talk) 20:59, 7 July 2024 (UTC)[reply]
Maybe it would be better to say something like it resembles the plane, and has the same structure in the infinitesimal limit. –jacobolus(t)22:00, 7 July 2024 (UTC)[reply]
I agree with Mathwriter2718 that this isn't accurate. The metrical structure of a sphere, on even the finest scale, has notably different metrical structure from flat Euclidean space: for instance the sectional curvature (as a Riemannian-geometric notion) in any small region of the sphere is exactly one, never getting closer to zero (the curvature of Euclidean space).
The basic fact is this: a Riemannian metric gives each tangent space an inner product, and any inner product space is (up to isometry) the same as a Euclidean space. I can see why some might phrase this as "infinitesimally Euclidean geometry" but I think it's a clunky way to view it and could lead to confusion. Taking a somewhat broader view, "infinitesimally Euclidean geometry" could naturally be formalized by saying that every tangent cone of a Riemannian manifold (viewed as a pure metric space) is isometric to a Euclidean space. That's true, but it fails to distinguish Riemannian manifolds. I think it would be better to simply say exactly what a Riemannian metric is: an inner product on each tangent space. Personally, I believe that would be as simply-put as possible. Gumshoe2 (talk) 21:09, 7 July 2024 (UTC)[reply]
"I think it's a clunky way to view it" – It's a sort of hand-wavy view, but it's not detailed enough to be clunky. By comparison, the business about tangent spaces is a very "clunky" way of expressing this idea, a formal definition for a new concept duct taped together from other abstractions previously defined and already at hand. It's not a requirement to define it this way, and most students do not have a clear intuition about the concept of the tangent space for a long time after being introduced to it, but it was convenient for the other proofs people wanted to make.
"sectional curvature (as a Riemannian-geometric notion) in any small region of the sphere is exactly one" – this is not so. The way you "zoom in" on a small portion of the sphere is by expanding the sphere until the portion of interest fills your view (or equivalently, imagine yourself and your natural scale of measurement to be shrinking and shrinking). In the limit as the sphere becomes infinitely large or you become infinitely small, you are left looking at a completely flat surface, indistinguishable in any way from part of a plane. [For a physical example, we don't yet know if the large scale structure of spacetime is flat or not, and we could well imagine the universe being "spherical" or "hyperbolic", but if so the curvature is so slight that it appears flat to within our capacity to measure. The curvature of a spherical, flat, or hyperbolic universe would be very very nearly the same, and you'd need a whopping big length scale to say it had sectional curvature of 1.] –jacobolus(t)21:40, 7 July 2024 (UTC)[reply]
Each of the textbooks I have at hand (Petersen, do Carmo, Kobayashi–Nomizu) define a Riemannian metric as a choice of inner product on each tangent space. What alternative do you have in mind?
I didn't realize you had zooming/rescaling in mind, since the proposed opening paragraph above didn't mention it. If that paragraph is to be used, I think that would have to be clarified. Regardless, I think it is a curious notion to put up front, since even in the most basic examples of Riemannian manifold – namely surfaces in 3-space – this idea of rescaling and recovering a Euclidean space in the limit is not of major importance, nor is it terribly immediate from the actual definitions. (And I don't know of any textbooks where it is emphasized.) The notion of tangent space is more immediate, both in the intuitive visual sense and in the formal setup. (Nor, at least in the formal version of saying that every tangent cone is a Euclidean space, does it even characterize the spaces in question. There are geometric spaces which equally have 'infinitesimally Euclidean geometry' which are not Riemannian manifolds.)
In terms of an opening line, I hardly think it's necessary to mention tangent space and inner product, but something along the lines of tazerenix's, something like
Riemannian manifolds are certain geometric spaces in which Euclidean notions of length and angle are generalized.
Thanks everyone for your interesting comments in this discussion.
From my experience, math undergraduates find the tangent space intuitively clear soon after it is introduced.
In my opinion, the definition of a Riemannian metric as an inner product on each tangent space is extremely elegant and not clunky.
@Jacobolus you suggest that there is an alternative definition of a Riemannian manifold. I have never heard of an alternative definition, so if one exists, I would be extremely interested.
If you "take the limit approaching a point" by choosing smaller and smaller coordinate neighborhoods, you will not approach Euclidean space. But if you zoom in and rescale as you do the limiting process, you will approach Euclidean space. If someone just said "take the limit approaching a point", I would expect that they meant the first construction. The second construction is just as valid, but I think it is more unusual in the math world.
Here is yet another ordered set of words for our collective consideration:
In differential geometry, a Riemannian manifold is a geometric space equipped with, at each point, a copy of the Euclidean space most closely approximating it near that point.
I also like the idea of having an opening line that avoids the notion of tangent space or entirely. Here is an appropriate modification of what I said earlier:
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined.
Provide definitions that are understandable by the uninitiated
Not be wrong, although vagueness is fine
Either your In differential geometry, a Riemannian manifold is a geometric space equipped with, at each point, a copy of the Euclidean space most closely approximating it near that point. or your In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. seem fine.
"a copy of the Euclidean space most closely approximating it near that point" seems confusing (and possibly meaningless) to me, but I think your second version is good. Ideally there would be a more descriptive label than "geometric space" but I'm not sure what it would be.
If it's possible to make the lead section both accurate and generally accessible, I think it's good to skip formal definitions until sections in the body. Gumshoe2 (talk) 17:41, 10 July 2024 (UTC)[reply]
"A copy of the Euclidean space most closely approximating it near that point" is my best attempt to describe a Riemannian metric both intuitively and accurately. I really think "Riemannian metric" should be defined in the lead. The page for Euclidean space defines a Euclidean space as a finite-dimensional real inner product space. Indeed, each tangent space of a Riemannian manifold is a finite-dimensional real inner product space. Now it is hopefully clear why I claim that out of all the finite-dimensional real inner product spaces one could associate with a point of a manifold, the tangent space equipped with the metric at that point is the best approximation. Mathwriter2718 (talk) 19:10, 10 July 2024 (UTC)[reply]
I haven't thought very deeply about it but there are surely many ways that the class of Riemannian manifolds might be precisely characterized instead of points + quadratic forms in a tangent space. For example, as the limit of some discrete triangular-mesh approximations; as points along with some full description of intrinsic n-dimensional curvature at each point; flipping the embedding theorem around, as something isometric to a sub-manifold of Euclidean n-space; ...; anything you might come up with would surely have its own complications as a definition, and might be inconvenient, but the standard definition (for this or any other class of mathematical objects) is an arbitrary cultural choice.
There are two different kinds of questions the start of an article like this could be trying to answer. (1) What sort of a thing is a Riemannian manifold? how is it different from other objects? what are examples? how does it relate to other concepts it is used with? what can be done with one? etc. (2) How do mathematicians formally define Riemannian manifolds? what other abstract concepts is that definition built on? what theorems can we prove about it and specifically how? and so on.
I don't personally see how to make a first paragraph from the perspective of #2 which is both technically precise at all accessible to people who haven't spent many years of diligent effort learning about a large number of prerequisite abstractions amounting to most of an undergraduate pure math degree. However, if we start (just in a first paragraph or two) with something a bit more handwavy and written in plain language, I think we can give some reasonable approximation of an answer to #1 type questions which can be understood by, say, high school students. So I hope we'll keep trying. –jacobolus(t)02:40, 8 July 2024 (UTC)[reply]
I think it would be good to remove the details of proofs. They are pretty irrelevant for the page and the claims seem to have appropriate textbook citations. (I think I'm guilty of adding at least one of the proofs, some time ago!) Gumshoe2 (talk) 21:14, 7 July 2024 (UTC)[reply]
This conversation has slowed down, so I am going to propose yet another lead (not just the first paragraph, but the whole section), attempting to compromise between all of the perspectives I have heard. I think it's really good to at define the terms "Riemannian manifold", "Riemannian metric", and "Riemannian geometry" in the lead. I am also throwing in an image from the ongoing discussion at Talk:Riemannian_manifold#A_couple_of_example_pictures,_not_sure_if_useful; please discuss the image there.
The square with sides identified is a Riemannian manifold called a flat torus (left). Attempting to embedded it in Euclidean space (right) bends and stretches the square in a way that changes the geometry. Thus the intrinsic geometry of a flat torus is different from that of an embedded torus.
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. These notions can be defined through reference to an ambient Euclidean space which the manifold sits inside. However, the idea of a Riemannian manifold emphasizes the intrinsic point of view as conceptualized by its namesake Bernhard Riemann, which defines geometric notions directly on the abstract space itself with no reference to an ambient space.
A Riemannian manifold is defined as a smooth manifold equipped with, at each point, a copy of the Euclidean space most closely approximating it near that point. The techniques of differential and integral calculus are used to pull geometric data out of the Euclidean approximations. Formally, if $M$ is a smooth manifold, a Riemannian metric (or just a metric) $g$ is a smoothly-varying family of inner products on the tangent spaces of $M$, and the pair $(M,g)$ is a Riemannian manifold.
The sentences in the first paragraph after the first sentence should be deferred. Putting them there puts emphasis in a misleading direction, and is not really the point of the subject. From the second paragraph, I don't think «smooth manifold equipped with a copy of Euclidean space at each point» is a good explanation for non-experts. It's too abstract and confusing, and doesn't really give an idea why you would want to "equip" a space with a bunch of other spaces (the word "equip" in plain English also has a sense of "put provisions in a backpack" or "pick up a sword" or something). –jacobolus(t)17:42, 10 July 2024 (UTC)[reply]
I think this is a definite improvement over the current lead, but I agree with jacobulus' comments. But it's not very clear to me what content an ideal opening paragraph would contain.
Also, I would remove "smoothly-varying." Many introductory textbooks do make this part of the definition, but it is not required and many Riemannian manifolds, especially in Riemannian convergence theory, have less regularity. I also wonder if it would be more clear to say that a Riemannian metric is a "choice of inner product for each tangent space" rather than a "family of inner products on the tangent spaces." Gumshoe2 (talk) 17:52, 10 July 2024 (UTC)[reply]
I would get rid of "smoothly-varying" from any lede written for a general audience, but it's fine as a general assumption throughout the article (and obviously should be stated explicitly in the body). Someone should probably write a section about generalizations like relaxed smoothness (important for geometric analysis, an example that occurs to me is plane wave vacua) or relaxed boundary conditions (e.g., manifold with boundary/singularities). Tito Omburo (talk) 20:42, 10 July 2024 (UTC)[reply]
@Tito Omburo @Gumshoe2 @Jacobolus @Chatul Based on your feedback, I have a new lead for your consideration. I tried to reorder things to emphasize @Jacobolus's perspective #1 (see the discussion a bit above). I still strongly feel that this lead cannot be complete without a description of a Riemannian metric. I don't think this has to be a formal definition, but at this time I think it's the best option. I won't repeat the lead's image to save space.
Formally, if $M$ is a smooth manifold, a Riemannian metric (or just a metric) $g$ is a choice of inner product for each tangent space of $M$. The pair $(M,g)$ is a Riemannian manifold. The techniques of differential and integral calculus are then used to pull geometric data out of the Riemannian metric.
The geometric notions that a Riemannian manifold has could be defined through reference to an ambientEuclidean space which the manifold sits inside. However, the idea of a Riemannian manifold emphasizes the intrinsic point of view, which defines geometric notions directly on the abstract space itself without referencing an ambient space.
A couple of tweaks: smooth manifold should be linked, and I would pull the parenthetical remark about whom it is named after (as it distracts from the main point and breaks up the flow of the sentence) to the end of the paragraph rather than placing it first. —David Eppstein (talk) 22:35, 10 July 2024 (UTC)[reply]
I think this is pretty good, enough so that I think it would be worth making the edit. My only major comment is that in the second paragraph I think it would be worth noting that Riemannian geometry is also of purely mathematical interest, having applications in other mathematical fields such as geometric topology, algebraic geometry, and statistics, and it has inspired modern developments in group theory and graph theory. (My goal being to avoid the impression that Riemannian manifolds are more of applied than pure interest.) Gumshoe2 (talk) 22:44, 10 July 2024 (UTC)[reply]
Non-math major here. I thought both of these versions were good. The only place I held up and wondered was the list of examples. When I read sphere and ellipsoid I see their 3D representation, but I believe the manifold refers to the surface only? Would it be correct to segment the example list into 2D and 3D manifolds? Johnjbarton (talk) 01:04, 11 July 2024 (UTC)[reply]
Then my suggestion is to give the simple examples and mention in a trailing phrase the general case. eg.
Examples of Reimannian manifolds include the 2D ellipsoid or sphere, which can be thought as surfaces in 3D space, as well as higher dimensional manifolds such as n-dimensional Euclidean or hyperbolic spaces.
I believe that the second sentence, These notions can be defined through reference to an ambient Euclidean space which the manifold sits inside., is inappropriate. Manifolds are intrinsic and not dependent on any particular embedding. For most applications there is no natural embedding. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:57, 10 July 2024 (UTC)[reply]
The next sentence explicitly states that Riemannian manifolds are intrinsic and not dependent on a particular embedding. The point of these sentences is actually to clear up the misconception that geometric spaces should be thought of as embedded, which many readers will have by default. Mathwriter2718 (talk) 19:01, 10 July 2024 (UTC)[reply]
I believe "can be defined through reference..." is accurate, because of the Nash embedding theorem. There may be no canonical embedding but one can still define a Riemannian manifold to be a manifold equipped with a distance and smoothly embedded in some Euclidean space with distance equal to the geodesic distance in that space. It would not be as good a definition as the intrinsic one and it does not match most of the literature but it would still define the same class of objects. The lack of a canonical choice of embedding is not really a problem. All that said we should emphasize the intrinsic approach here both because that's the way our sources treat it and because it's better. —David Eppstein (talk) 02:03, 11 July 2024 (UTC)[reply]
Just a thought. It might be natural to change that paragraph to say something in the spirit of:
Any surface in three-dimensional Euclidean space has an automatically induced Riemannian structure. Although Nash proved that every Riemannian manifold arises as a submanifold of some (higher-dimensional) Euclidean space and although some Riemannian manifolds are naturally exhibited or defined as such submanifolds, in many contexts Riemannian metrics are more naturally defined or constructed directly, without reference to any Euclidean structure. For example, natural metrics on Lie groups can be defined by using group theory to transport an inner product on a single tangent space to the entire manifold; many metrics with special curvature properties such as constant scalar curvature metrics or Kähler–Einstein metrics are constructed as direct modifications of more generic metrics using tools from partial differential equations.
It could also be valuable to mention that even as fundamental a Riemannian manifold as hyperbolic space has no known natural isometric embedding into Euclidean space. (Natural meaning that internal metric symmetries are represented by symmetries of the ambient space, as for the sphere.) Gumshoe2 (talk) 03:39, 11 July 2024 (UTC)[reply]
How about Any regular surface in three-dimensional Euclidean space has an automatically induced Riemannian structure. Although Nash proved that every Riemannian manifold arises as a submanifold of some (higher-dimensional) Euclidean space and although some Riemannian manifolds are naturally exhibited or defined as such submanifolds, in many contexts Riemannian metrics are more naturally defined or constructed directly, without reference to any Euclidean structure. For example, natural metrics on Lie groups can be defined by using group theory to transport an inner product on a single tangent space to the entire manifold; many metrics with special curvature properties such as constant scalar curvature metrics or Kähler–Einstein metrics are constructed as direct modifications of more generic metrics using tools from partial differential equations.? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 08:33, 11 July 2024 (UTC)[reply]
Anyway, I would not endorse putting that last sentence in the lead, I think it is too vague and too far afield and belongs later in the article body where it could be explained in more detail. But I do like the other parts of it. I suggest this:
Any smooth surface in three-dimensional Euclidean space, such as an ellipsoid or a paraboloid, has an automatically induced Riemannian structure. The same is true for any submanifold of Euclidean space of any dimension. Although Nash proved that every Riemannian manifold arises as a submanifold of Euclidean space, and although some Riemannian manifolds are naturally exhibited or defined in that way, the idea of a Riemannian manifold emphasizes the intrinsic point of view, which defines geometric notions directly on the abstract space itself without referencing an ambient space. In many contexts, Riemannian metrics are more naturally defined or constructed using the intrinsic point of view. For example, there is no known way to place the hyperbolic plane into Euclidean space that perseveres its internal symmetries.
It worries me slightly that we are using the word "submanifold" here to mean variously "submanifold" and "Riemannian submanifold", but I'm not qualified to declare if this is actually confusing or not. If it is, try this:
Any smooth surface in three-dimensional Euclidean space, such as an ellipsoid or a paraboloid, has an automatically induced Riemannian structure. The same is true for any submanifold of Euclidean space of any dimension. Such a submanifold is called a Riemannian submanifold of Euclidean space. Although Nash proved that every Riemannian manifold arises as a Riemannian submanifold of Euclidean space, and although some Riemannian manifolds are naturally exhibited or defined as one, the idea of a Riemannian manifold emphasizes the intrinsic point of view, which defines geometric notions directly on the abstract space itself without referencing an ambient space. In many contexts, Riemannian metrics are more naturally defined or constructed using the intrinsic point of view. For example, there is no known way to realize the hyperbolic plane as a Riemannian submanifold of Euclidean space that perseveres its internal symmetries.
The phrase "automatically induced structure" is needlessly confusing for non-experts. It would be better to say something more like, "Any smooth surface in three-dimensional Euclidean space, such as an ellipsoid or a cone, is a Riemannian manifold, inheriting its infinitesimal [?] definition of distance from the ambient space." –jacobolus(t)18:19, 11 July 2024 (UTC)[reply]
I agree that the phrase "automatically induced structure" is confusing for non-experts. Cones in the sense of the page you linked are not smooth manifolds, so I won't replace paraboloid with them. How about this?
Any smooth surface in three-dimensional Euclidean space, such as an ellipsoid or a paraboloid, is a Riemannian manifold with its Riemannian metric coming from the way it sits inside the ambient space.
We need to distinguish the intrinsic metric from the Euclidean metric, but I would prefer to talk about geodesic distance than infinitesimal distance. —David Eppstein (talk) 19:00, 11 July 2024 (UTC)[reply]
This is only something to distinguish if you're talking about metrics as in metric spaces – there are two natural metric space structures on a surface in R^{3}, one intrinsic and one Euclidean. But metric spaces are currently not mentioned at all in these proposals for the lead, except indirectly in the one mention of "distance." And for metric instead as shorthand for Riemannian metric, there's no ambiguity (there is only one natural Riemannian metric on the surface).
Possibly it could be worth adding a line or two to the lead explicitly about the metric space structure induced by a Riemannian metric, along with a warning about the resulting double/inconsistent meaning of "metric." Gumshoe2 (talk) 19:23, 11 July 2024 (UTC)[reply]
The current lead (a big improvement) says:
A Riemannian metric is not to be confused with the distance function of a metric space, which is also called a metric.
This is not helpful. If you don't know the topic, you will be confused. This sentence also does not summarize the article. The closest thing I could find was:
The distance function ... called the geodesic distance, is always a pseudometric (a metric that does not separate points), but it may not be a metric.
which is a clear as mud.
IMO the article should have a paragraph explaining the difference and the sentence should summarize the paragraph, not tell us we are confused. Johnjbarton (talk) 03:12, 13 July 2024 (UTC)[reply]
For the purposes of the lead, personally I see no issue with just saying that a surface automatically inherits a Riemannian metric (though I am not attached to "automatically induced structure" in particular) since there are details further down the page. But it is also pretty elementary to just say how it works: the inner products defining the induced Riemannian metric are just given by restricting the inputs of the usual dot product to vectors tangent to the surface. Gumshoe2 (talk) 19:17, 11 July 2024 (UTC)[reply]
The best reference I have is (see pages 2-3 in linked pdf):
The square with sides identified is a Riemannian manifold called a flat torus (left). Attempting to embedded it in Euclidean space (right) bends and stretches the square in a way that changes the geometry. Thus the intrinsic geometry of a flat torus is different from that of an embedded torus.
Formally, if $M$ is a smooth manifold, a Riemannian metric (or just a metric) $g$ is a choice of inner product for each tangent space of $M$. The pair $(M,g)$ is a Riemannian manifold. The techniques of differential and integral calculus are then used to pull geometric data out of the Riemannian metric. A Riemannian metric is not to be confused with the distance function of a metric space, which is also called a metric.
Any smooth surface in three-dimensional Euclidean space is a Riemannian manifold with a Riemannian metric coming from the way it sits inside the ambient space. The same is true for any submanifold of Euclidean space of any dimension. Although Nash proved that every Riemannian manifold arises as a submanifold of Euclidean space, and although some Riemannian manifolds are naturally exhibited or defined in that way, the idea of a Riemannian manifold emphasizes the intrinsic point of view, which defines geometric notions directly on the abstract space itself without referencing an ambient space. In many instances, such as for hyperbolic space and projective space, Riemannian metrics are more naturally defined or constructed using the intrinsic point of view. Additionally, many metrics on Lie groups are defined intrinsically by using group actions to transport an inner product on a single tangent space to the entire manifold, and many special metrics such as constant scalar curvature metrics and Kähler–Einstein metrics are constructed intrinsically using tools from partial differential equations.
I personally am quite happy with this. I previously opposed adding the last sentence (suggested by Chatul) because I thought it was too technical, but now I think it's a good example. I did remove a few words adding detail though.
I think this introduction is still simultaneously too technical and too vague.
There are many objects which are "geometric space[s] on which many geometric notions such as distance, angles, length, volume, and curvature are defined" but which are not Riemannian manifolds. The examples are helpful, but I think we should also try to make the first paragraph give a bit clearer plain-language impression of what the word means. The later description "choice of inner product for each tangent space of M" doesn't cut it, as many potential readers may not know the meaning of "inner product" or "tangent space".
We should be saying more clearly that the Riemannian metric is an instantaneous/infinitesimal/local/whatever definition of distance and angle, and that by integrating, we can also see larger-scale behavior, with concepts like geodesics along which we measure length, area/volume of shapes, geodesic circles, geodesic polygons, parallel curves/surfaces, ... The phrase "pull geometric data out" is too vague.
The paragraph "Riemannian geometry, the study of Riemannian manifolds, ..." should be moved to the bottom of the lead section.
I'd drop the disclaimer about metric spaces. A Riemannian manifold is a type of metric space, and this is distracting and seems unnecessary here.
The paragraph about embeddings seems too into the weeds and sort of disjointed. Some of this should be saved for the article body. –jacobolus(t)15:22, 12 July 2024 (UTC)[reply]
I don't think that replacing the current definition to say that a Riemannian metric defines "infinitesimal distances" would achieve your goal of making this less technical and less vague. Mathwriter2718 (talk) 15:38, 12 July 2024 (UTC)[reply]
My concrete proposal was above, "A Riemannian manifold is a geometric space which locally, in the vicinity of each of its points, has the same metrical structure as flat Euclidean space – in the same way that spatial relationships in a small portion of a globe's surface can be modeled using a flat map – ..." –jacobolus(t)15:59, 12 July 2024 (UTC)[reply]
"There are many objects which are "geometric space[s] on which many geometric notions such as distance, angles, length, volume, and curvature are defined" but which are not Riemannian manifolds."
What other such objects do you have in mind? But I do agree that it would be ideal to have a more direct way of saying what a Riemannian metric is and not just what it defines. I don't have a non-clunky way to say it at the moment, but I think it would be nearly precise to say that on a smooth manifold you can look at all the possible (smooth) curves, and a Riemannian metric is an internally consistent way of assigning them lengths, along with angles between them when two of them intersect. (The other notions of distance, volume, and curvature are of course secondary.) It could be said that this assignment is based on infinitesimal information, with lengths defined by integration in the same way arclength is computed in standard calculus.
In the end, I think it is impossible to fully meet the combined requirements of: (1) using plain language (no inner product, no tangent space), (2) having a description which uniquely distinguishes the class of Riemannian manifolds, (3) being correct. But I think each one is good to aspire to. Gumshoe2 (talk) 16:29, 12 July 2024 (UTC)[reply]
For one thing, there are pseudo-Riemannian manifolds. But also plenty of other more exotic metric spaces can have "many geometric notions ..." defined, including the listed ones.
The essence of a Riemannian manifold is not only that these notions can be defined, but that they are locally the same as Euclidean space. –jacobolus(t)17:38, 12 July 2024 (UTC)[reply]
Pseudo-Riemannian manifolds don't have a distance function; for them, distance is only defined for certain pairs of points. Perhaps there are other kinds of spaces out there which have all of these objects, but I think pretty much any mathematician would see 'geometric space on which distance, angles, length, volume, and curvature are defined' and immediately think 'Riemannian manifold.' Which is why I think it is ok here, if not ideal.
Taken at face value, your description of Riemannian manifolds is not accurate, it only applies to flat Riemannian manifolds. The 'zooming' procedure you described before as what you have in mind also does not pick out Riemannian manifolds in particular, so I don't see any reason to prefer it to the present suggestions. (And I see a good reason to not prefer it, which is that it's nonstandard.) Gumshoe2 (talk) 19:05, 12 July 2024 (UTC)[reply]
Pseudo-Riemannian manifolds have a notion of "distance" in the same physical sense that spacetime does, which is frankly the most important and real sense. Distances have to be broken into "timelike" vs. "spacelike" or similar, but that's not really a problem. The most unambiguous formalized concept, of "squared distance", is in fact a much more natural and better one to use than its square root, and is treated as secondary because of historical inertia. Features of Lorentzian geometry and the hyperbolic number system are precisely analogous to those of Euclidean geometry and the "circular" complex number system, and we can certainly talk about angle measure, (timelike or spacelike) distance, "circles", curvature, and so on.
[It's quite a tangent here, but not teaching students about the Lorentzian plane, hyperbolic numbers, etc. starting in high school or early in undergraduate school blinkers them and significantly limits their understanding not only of such spaces but also of the geometry in Euclidean space (etc.) which are intimately intertwined with pseudo-Euclidean concepts and models. Focusing on Riemannian manifolds and Euclidean tangent spaces to the exclusion of pseudo-Riemannian manifolds and pseudo-Euclidean tangent spaces is a serious pedagogical blunder.]
"Taken at face value ... only applies to flat Riemannian manifolds." – My explicit example is of a globe, so your characterization clearly can't be right, but I clearly am expressing the idea in a way which is unclear or confusing to you personally, so we can probably do better. I expect there's enough brainpower in this discussion that we can collectively come up with some description which is accurate enough for your taste while still being clear and accessible. Does anyone have ideas? –jacobolus(t)19:36, 12 July 2024 (UTC)[reply]
Sorry if the above sounds combative/defensive. I just want to make sure we give readers a reasonably clear idea of the purpose and nature of the subject. For example, "tangent space" more or less means "space of infinitesimal motions at a point", and "inner product space" (i.e. finite-dimensional vector space with a positive definite quadratic form) is jargon for "has the same geometry as the space of Euclidean translations". But to a reader who doesn't already know that, those jargon words are not meaningful. –jacobolus(t)20:00, 12 July 2024 (UTC)[reply]
No problem, as long as my own posts here are understood in the same spirit ;)
Within the context of Riemannian manifolds, "locally the same as Euclidean space" is pretty much a meaningful and precise definition of the flat ones. The problem with your description is that it doesn't say anything about the 'zooming in' procedure that you explained earlier as what you have in mind.
But even the 'zooming in' procedure does not characterize Riemannian manifolds, see e.g. either Reifenberg subsets of Euclidean space or certain metric spaces. It is very possible that in some contexts some form of 'approximately Euclidean on zoomed-in scales' does actually characterize Riemannian manifolds (I would find it very interesting if so) but it would be nonstandard and would likely count as original research by wiki standards. (Possibly also by any standards.)
I don't see any way to avoid the actual fact of the matter, which is (in seemingly every standard account) inner products on tangent spaces. It would be great to have a way to say it without jargon, but I think it would be a big mistake to try to do so by reformulating it in some nonstandard/conjectural way. Gumshoe2 (talk) 20:42, 12 July 2024 (UTC)[reply]
I've read all of the comments in this thread. You both have some very interesting points, but in the end, I cannot sign off on a version of "locally the same as Euclidean space" because many or most reasonable readers will not understand "locally" to mean with zooming. I think there is a glimmer of hope for this strategy, perhaps one can think of a Riemannian manifold as locally looking like a Euclidean space up to the first order, with the second order giving curvature?
I think such a strategy has a long road to being written up, validated as correct, being more transparent than the current lead, achieving consensus over it, and steering clear of original research. I encourage those who are interested to keep working on it. But right now it seems like there is a consensus among the editors besides @Jacobolus around a lead that is a small perturbation of the current one, and I would really like to make a "version 1" edit of the lead to the live page soon. Mathwriter2718 (talk) 21:13, 12 July 2024 (UTC)[reply]
I'd encourage you to make the edit whenever you feel like it. I think we all agree that it's an improvement over the current lead, and we can keep discussing even after an edit.
The reason I think explicitly (briefly) mentioning cartography is helpful is that (a) it's a subject much more familiar to a wide range of readers than inner products or tangent spaces, and (b) these concepts and problems concretely arose because Euler, Lagrange, Lambert, Chebyshev, Jacobi, Gauss, etc. were directly working in cartography/geodesy. Map projections were quite directly in mind (not sure about Riemann per se), which is why we ended up with names like "chart", "atlas", and "geodesic".
The point here being that if we "map" a small part of a Riemannian manifold, the flat map is very nearly accurate, and gets more and more accurate as we shrink the area being mapped. So that e.g. a geodesic looks locally like a straight line, a tiny enough circle has π as its ratio of circumference to diameter, a tiny enough geodesic triangle has interior angles almost exactly summing to π/2, and so on. We get closer and closer to Euclidean geometry. The Riemannian metric is a way of formalizing this idea, using the name "tangent space" for the space of infinitesimal motions (or if you prefer, the space of velocities).
While "up to the first order" is more or less synonymous with the formal definition, I don't think it's really that much more accessible. –jacobolus(t)21:43, 12 July 2024 (UTC)[reply]
One way to adjust the sentence might be to say
"In differential geometry, a Riemannian manifold, is a geometric space on which the geometric notions of length and angle, and subsequently distance, volume, and curvature are defined. These notions of distance and angle are specified infinitesimally in the form of a Riemannian metric, and the techniques of differential and integral calculus are used to link this infinitesimal data with notion of lengths, volumes, and curvature as they are commonly understood."
It specifies what a Riemannian manifold is a bit more uniquely, whilst still emphasizing the most important geometric concepts defined on them. Tazerenix (talk) 13:22, 13 July 2024 (UTC)[reply]
I like something like that - although in the second sentence I think you mean to switch "length" and "distance". Putting this in terms of the present second paragraph on the page:
Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport.
I think it would make sense to edit to:
Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. These inner products represent infinitesimal measures of length and angle. By integration, these infinitesimal measures define measures of surface area and volume, including (non-infinitesimal) lengths of curves. A distance function, giving the structure of a metric space, is constructed from curve lengths using the calculus of variations. By contrast, differential calculus is employed to use the inner products to define curvature and parallel transport. Special curves known as geodesics can be constructed by the calculus of variations used to define the distance function, or as solutions of certain ordinary differential equations constructed from the metric using differential calculus.
I think that with the new reference to surface area and geodesics, this covers all of the major bases in terms of key objects. (Most notably, minimal surfaces are missing, but presently they are not at all present on the page itself.) Happy to take alternative suggestions or further edits. Gumshoe2 (talk) 17:36, 13 July 2024 (UTC)[reply]
Agreed, maybe "Curvature and parallel transport are constructed by differential calculus from the inner products" is an improvement. I am also not satisfied with the awkwardness of "can be constructed by the calculus of variations used to define the distance function"; my intention was to make clear that it comes from the same calculus of variations problem that defines the distance function, the distance function being the minimal values in the optimization and the geodesics being (locally) the minimizers. Gumshoe2 (talk) 18:07, 13 July 2024 (UTC)[reply]
I think this lede is really good. Since most contention seems to focus on the third paragraph, how about something like this
A Riemannian metric on a smooth manifold gives a local way (in each tangent space) of measuring lengths. Formally, a Riemannian metric $g$ is a choice of inner product for each tangent space of $M$ (usually assumed to be smooth as well). The pair $(M,g)$ is a Riemannian manifold. Integration of the metric leads to a distance function (in the sense of metric spaces), whereas differentiation of the metric leads to notions of curvature and parallel transport.
I also like it, along with your edit. But I think it is not even necessary to use any symbols, which I think is preferable when possible. (Formally, a Riemannian metric is a choice of inner product for each tangent space (usually assumed to be smooth as well). The pair of manifold with metric is a Riemannian manifold.) It would also be more accurate to replace "local" in the first sentence with "infinitesimal." Gumshoe2 (talk) 16:34, 12 July 2024 (UTC)[reply]
There is a new article Quasilinearization which was restored from a deleted form and has been moved directly to main. I know that using linear approximations is very common in optimization and similar problems, and it is of course everywhere in science (first order expansions). I don't know if there are other articles on this, hopefully someone in the applied math area has a better feel for what is already on Wikipedia. For certain I think Quasilinearization can do with better and wider context, but perhaps there is more that should be done. Over to others. Ldm1954 (talk) 19:27, 6 July 2024 (UTC)[reply]
I have recently expanded the article Cube, one of them is the Cube#In architecture. However, one source says that Kaaba is a nearly cube building [2], which I have not included in the article. If that's the case, should this be included elsewhere, the Square cuboid, or keep it in the article Cube but quote what is the source saying? I don't want to have a conflict because of my editing. More opinions are extremely needed. Dedhert.Jr (talk) 13:03, 8 July 2024 (UTC)[reply]
@MrOllie I see. Then I guess six-faced dice and Rubik's cube should not included as well in the pre-planned section "In popular culture". I do not get why someone reverted about the architecture one, like, do we actually have a manual of style in Wikipedia about those? Why do articles like Isosceles triangle also mention the architecture, or ubiquitous shape like Mobius strip in popular culture? Dedhert.Jr (talk) 13:09, 8 July 2024 (UTC)[reply]
Because there are not many pieces of architecture that incorporate a proper Möbius strip, so when one does it becomes more unusual and interesting than a piece of architecture that incorporates a cube, a cylinder, or a hemisphere. —David Eppstein (talk) 05:44, 9 July 2024 (UTC)[reply]
It is possible that there are particular buildings that are notable for being exactly cubical, but one would want to see extremely good sourcing for that to pass WP:DUE. I don't think there's anything wrong in principle with noting, somewhere in Cube, the ubiquity of cubes and including mention of some examples like dice. --JBL (talk) 18:18, 9 July 2024 (UTC)[reply]
The Kaaba (literally "cube") is a pretty notable example of a cubical structure. If someone wants to make a general point about the popularity of cube-shaped buildings with examples, Mukaab and Cube Berlin [de] have wiki pages, and there is surely discussion to be found about other examples in some architecture journal or another. –jacobolus(t)22:11, 9 July 2024 (UTC)[reply]
@Jacobolus Another interesting building. I previously wrote Genzyme building and the interior building of Duchess Anna Amalia Library, and sources were supplied. If these can be written again, buildings from the West and the Arabic may be split in the pre-planned section "In architecture". Do you think this is fine? I also can't put them in popular culture because of my reasoning from the very beginning. Dedhert.Jr (talk) 06:03, 11 July 2024 (UTC)[reply]
I don't think an article cubehas to have an architecture section, but there's certainly enough that has been written about this topic that it could be supported. I probably wouldn't make it longer than a single unified section. (Does the Borg Cube count as a building? :P ) –jacobolus(t)06:11, 11 July 2024 (UTC)[reply]
@MrOllie I agree with this conclusion but not the reasoning. A list of cube shaped things is only notable if references make it so. Specific examples of notable cubes could nevertheless be listed with due discussion. The Kaaba for example was discussed as having a singular appearance, quite the opposite of ubiquity. Johnjbarton (talk) 18:37, 9 July 2024 (UTC)[reply]
In 2021, a talk page user pointed out that the definition of the musical isomorphisms on the page musical isomorphism is needlessly complicated. Indeed, since at least 2020, the text itself begrudgingly admits that the second description it gives is "somewhat more transparent" than the first one it gives:
This is referred to as lowering an index. Using angle bracket notation for the bilinear form defined by g, we obtain the somewhat more transparent relation
$X^{\flat }(Y)=\langle X,Y\rangle$
for any vector fields X and Y.
But the problem is actually much more significant than this. Indeed, the definitions as stated are mathematically invalid, as the vector field $X$ is not an element of the tangent bundle $TM$, which consists of individual vectors. Immediately after this is a parallel discussion on the sharp isomorphism, which suffers from exactly the same defects. Mathwriter2718 (talk) 02:59, 11 July 2024 (UTC)[reply]
I don't see a major issue, just change "vector field" to "vector" and "covector field" to "covector."
Actually, the whole article is really just about linear algebra in a single vector space with inner product – setting the context as Riemannian metrics and bundles is completely unnecessary. It's kind of a fake generality since the musical isomorphisms on a Riemannian manifold are just defined point by point, and for each point you have a single vector space with inner product in question. It would be just like defining the determinant as taking a map $\Omega \to \mathbb {R} ^{n\times n))$ and returning a map $\Omega \to \mathbb {R}$; it's technically more general than the determinant as a map $\mathbb {R} ^{n\times n}\to \mathbb {R}$ but not in any important way. Gumshoe2 (talk) 03:54, 11 July 2024 (UTC)[reply]
I absolutely see where you're coming from, but the phrase "musical isomorphism" really means in the setting of Riemannian metrics and bundles. I'll try today to make the discussion be more clearly the generalization of the linear algebra isomorphism though. Mathwriter2718 (talk) 11:28, 11 July 2024 (UTC)[reply]