Talk:Riemannian manifold

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this page. In geometric analysis, one considers things that are "almost Riemannian manifolds", but the case when you just so happen to have an honest-to-goodness smooth manifold with an honest-to-goodness non-regular metric is a very special case that shouldn't be the default assumption on this page. More often, the "almost Riemannian manifolds" considered in geometric analysis aren't even smooth manifolds! Indeed, if you look at the main source for the content on the previous versions of this page about the non-smooth case (Gromov), you will see that the results this page cites from it are actually not about smooth manifolds with non-regular metrics, but about much weaker spaces.

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say, at bare minimum, that a flat torus in R would have to possess a family of constant-length loops sweeping out the whole surface, orthogonal to anther family of constant-length loops sweeping out the surface. But this may not be significant enough to mention by itself, since I think it isn't obvious that even the standard embedding of the torus fails to satisfy this: it is only clear that the 'obvious' families of loops on the torus fail to satisfy this, only one of the two having constant length.

74: 53: 1265: 489: 1436: 679: 813: 1397: 166: 670: 1669:, in the sense that if two metrics are given and their corresponding metric's are isometric ('the same') in the sense of metric spaces, then the two metrics are isometric ('the same') in the sense of Riemannian manifolds. For this reason, often the word "metric" (and "isometry") in the Riemannian-geometric context does indeed have a somewhat blurred meaning, with (because of Myers-Steenrod) there being little danger involved. 1665:, and "metric" to refer to a Riemannian metric). A metric defines a metric (via a somewhat complicated procedure), and in this sense a Riemannian manifold "is" a metric space (among other things). However most metric's do not correspond to any metric, and so most metric spaces do not correspond to any Riemannian manifold. Notably the metric corresponding to a given metric fully characterizes the original metric due to the 1107: 1256: 1098: 849: 22: 311:

throughout the article where relevant instead of its own section. As far as I can tell, when working with continuous metrics, pretty much everything works exactly the same up until you get to connections, which falls flat on its face if the metric is not at least a few times differentiable. But I wouldn't trust my opinion on this without a second person backing this up.

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visually apparent three-dimensional Euclidean geometry; on the right, exactly the same kind of drawing but with vectors that don't 'look' orthonormal. The caption can explain that a declaration of an orthonormal basis defines an inner product, with the left image amounting to the standard Riemannian metric and the right image defining a different Riemannian metric.

1136:.) I also like the idea of having the flat torus, the embedded torus, and an animation connecting the two all in a multiple image. (Of course here the aspect ratios and colors don't match up.) Specifically, I just think the animation would be enlightening for people who don't know how to transform the square with sides identified into the embedded torus. 285:

to Bridson and Haefliger's book) that there is no modification required if the metric is only assumed to be continuous, or even Finsler. I may also try to rewrite that section a little to be a bit more direct, also removing the proof I had added; the Myers–Steenrod theorem should also be briefly discussed there.

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I also have a suggestion for another picture, for the purpose of illustrating the 'intrinsic idea' and a nonstandard Riemannian metric. Two copies of the same round sphere side by side; on the left, a drawing of orthonormal tangent vectors (ideally at multiple points along the sphere) relative to the

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embedding of the side-identified square in R, the difference in intrinsic geometry can be seen from the fact that the family of constant-y (or constant-x, depending on the embedding) loops on the square have nonconstant length once put through the embedding, while they clearly have constant length in

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Stylistically, I think the Kähler–Einstein problem is too close to the cscK problem for these to both be usefully mentioned/linked next to each other. (The cscK problem is a generalization of the Kähler–Einstein problem, worked on by the same research community and largely with the same techniques.)

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In other words, one could say that Riemannian manifolds form a (small) subclass of metric spaces, but nonetheless a metric is a very distinguishable object from the corresponding metric: one is a collection of inner products and the other is a real-valued function with point-pair input. Sometimes if

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I think the illustration of hyperbolic space and the animation of the torus with stereographic projection of SU(2) look very cool, but are far too non-transparent in meaning to be used here. I'm not even personally sure what they mean. (What is "an observer" in hyperbolic space? Is it based on a use

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It seems like this article could include links to more modern stuff. The Laplace–Beltrami operator, which is important in modern spectral theory of Riemannian manifolds. Einstein manifolds deserve a mention, and maybe Thurston geometrization. Kahler manifolds and connections to algebraic geometry.

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Thank you very much for these edits. Many of them are fixing some ill-advised edits (as I can see in hindsight) which I had made to the page back in 2020. I think the section on continuous metrics can be completely deleted; it can just be mentioned briefly in the metric space section (with reference

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Hi! Mathwriter2718 here. In the past few days I have made substantial modifications to this page. Most of my changes need no justification, but recently I have changed the page so that the metric is smooth (except for one section on the non-smooth case, which I am currently cleaning and shortening).

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is contained in a coordinate chart; if you have this then you can immediately make a path with the same endpoints which is piecewise-smooth. (And then if you want this could be made smooth by rounding off corners.) This seems to be rather annoying but possible to get, based on starting with an open

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sorry for my very late response. I think it is best to put the flat torus image later on. The hyperbolic space images definitely deserve a place in this article later on as well. As for the sphere with the tangent plane, I like it, but that image is very 2007-esque. I just created this in Inkscape,

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The way the Riemannian metric changes over the sphere determines its intrinsic geometry. A tangent plane is shown, and its Euclidean structure visualized by plotting concentric circles. The corresponding circles on the sphere initially grow linearly in distance from the point of tangency, but for

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to say that the Gaussian curvature of the embedded torus is positive at a maximal-distance point from the origin. I think it is probably impossible to have a more elementary demonstration of this, given the C Nash–Kuiper embeddings (to which Theorema Egregium doesn't apply). It would be possible to

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The Riemannian manifold called a flat torus (left) looks like a square when rendered in Euclidean space, but traveling to the right wraps around to the left and top to the bottom. Attempting to embedded it in Euclidean space (right) bends and stretches the square in a way that changes the geometry.

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I reorganized the page, added some things, and deleted others. I think it reads better now. I plan to add a section on curvature and the geodesic equation. I've left the very last section exactly as is since it doesn't naturally go into the other one, but it'll be subsumed into the eventual part on

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In general the disjoint union of metric spaces is a metrizable topological space, but without any particular metric which is natural. (In the standard approach you would modify by truncation the metrics on the individual connected components.) So if you have a Riemannian manifold, without assuming

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I think it would make sense to move the whole Examples section to right before the 'infinite dimensions' section. At present the first four example sections don't require any advanced material, so it would be ok to split them out and keep those four where they are now, but in the end I think they

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Related, I think it would be good to have a reference for (and to include) the fact that 'piecewise' is superfluous to the definition – the reason for its presence being just that piecewise-smooth paths can be concatenated, making some technical arguments easier. I'm sure this is in some standard

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While we're on the subject, I ran into an annoying issue when writing about the distance function for this page. Lee's Riemannian Manifolds uses piecewise regular curves (which he calls admissible) in the definition of the Riemannian distance function. However, many papers in the field simply use

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As I have discussed above, describing the torus images leads away from the topic, as far as I can tell. The purpose of an image is to give the reader an object to help connect the words of the article. When the words of the caption are confusing, the image disconnects the reader from the article.

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I think the blue animation is the best bet, though I also really like the picture of the orange torus before and after identification. I think the blue animation's caption is much better for the lead and the orange torus's caption is better for the section on isometric embeddings. I would like to

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lists dozens of relevant papers by Ricci and cohorts, especially Bianchi as well as others. Ricci developed what is nowadays called tensor calculus (in Riemnnian manifolds). An amusing footnote on page 146 states that the Bianchi identity was originally discovered by Voss in 1880, independently

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A Riemannian manifold means that the metric is smooth. This is the default mathematical convention and other pages on Knowledge invariably agree with this. It is very unusual for the word "Riemannian manifold" to mean that there is no regularity at all on the metric, as was previously the case on

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thank you for adding your new excellently-written sections on Lie groups, homogeneous spaces, and symmetric spaces. However, I am concerned with how early in the article they are: they reference geodesic completeness, the Riemann and Ricci curvature tensors, scalar curvature, and the Levi-Civita

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It is necessary to introduce many examples early on, but then they should be discussed again in more details later. I'm not sure what the best way to do this is, but I can imagine several possible ways to do it, including a) having two sections about examples, one defining them, and a later one

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Yes, technically it is the preimage of the tessellation under the exponential map. I agree the animation is cool, and while it claims to be stereographic, the piecewise lines mesh is a bad illustration for this purpose. In case you didn't notice, I produced a better still of a stereographically

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Those are interesting counterexamples. I support having some light discussion of low regularity metrics on this page, but I think Finsler metrics and pseudo-Riemannian metrics are best discussed elsewhere. I think it makes sense to have the low regularity discussion be a few sentences sprinkled

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For now, I made your space forms subsection its own section and put the Lie groups, homogeneous spaces, and symmetric spaces subsection into a new section titled (maybe poorly) "Lie groups". Arguably, all four could also go under one heading that would be something like "Especially symmetrical

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with the distance function of a metric space, according to the article itself. It seems that the Riemannian metric is a kind of generalization of a distance metric but of course I don't know. The sentence is not sourced so I cannot look further into this issue. In my opinion this connection is

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With a little extra work, the flat torus could be clarified by drawing on it some curves, such as a loop which passes over the would-be edges of the square, and noting that the lengths of these curves (and the angles between them) are exactly as they visually appear. This way it's possible to

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Every metric space is paracompact, so it comes down to whether or not you believe that paracompactness is needed to get a metric space from a continuous Riemannian metric. It seems to me that it's not; if so, then paracompactness is equivalent to existence of a continuous Riemannian metric.

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I absolutely agree that the page would benefit from most of these things being added. When it comes to something like Kahler manifolds, I'm not sure where the line of things that really ought to have a drawn-out discussion on this page is drawn. But perhaps it should still be mentioned.

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A torus is obtained from a (Euclidean) parallelogram by identifying opposite edges. This is for many applications the natural Riemannian structure on a torus. But it cannot be visualized in the usual way, by an embedding into Euclidean space. The intrinsic geometries are different.

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Sorry if I changed too much or deleted something I shouldn't have. It seemed like some of the material was transplanted from books that obscured some of the point, and there was a lot of repetition and unclear statements. I tried to make sure that what I added is on a readable level.

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connection, and more, all before those terms have been discussed. What do you think about putting them all under a new section directly under the section about connections, geodesics, and curvature? Also, I hope you won't mind if I change the inline math to be in <math: -->

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I'm unsure what the torus is showing. In particular, the caption says a lot that is not in the image ("flat torus" in ordinary words is not a sensible combination) but omits critical aspects such as the meaning of the rotation. To me the caption requires too much background.

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How about "A torus can be given a flat Riemannian metric, so that in the picture shown, the lengths of all heavy segments in the mesh are equal. Here a flat torus is embedded conformally (preserving angles) but not isometrically (preserving lengths) into Euclidean space."

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I think there is probably some version of such a kind of picture (i.e. not necessarily exactly as I just described it) which is clear, efficient, and useful. Or instead, an analogous picture in the same spirit might serve as a good illustration of the hyperbolic plane.

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should contain some commentary to do with curvature and geodesics – especially the one on submanifolds. So my preference is to move them as you suggest, but keeping them all together — but I don't feel so strongly about it, you can reorganize however you think best.

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Indeed, I made that image! But since we're explicitly in the smooth category, I think we can minimize the need to mention it. My original idea was an isometric embedding into the 3-sphere, stereographically projected into R^3, but this is too confusing I think.

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There are very many Kähler–Einstein metrics found by PDE techniques which are genuinely new, previously unknown, examples of Riemannian manifolds. There is a large variety of them, inaccessible by other methods, and their existence theory is basically resolved

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Let M be a smooth manifold, not required to be paracompact or connected, with a Riemannian metric. Each connected component is a connected Riemannian manifold, not necessarily paracompact. Any topological space which is connected and locally path-connected is

2276:). In contrast, although everyone expects that there should be even more cscK metrics found by similar techniques, as far as I know this is still almost entirely conjectural. So I think it is actually a little inappropriate to mention them here separately as 857:

Your final statement "Thus the intrinsic geometry of a flat torus is different from that of an embedded torus." seems to be what you are trying to show. I would expect to see two tori or perhaps one torus and an explanation for why the second one cannot be

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Strictly speaking, Riemannian manifolds are only metric spaces when they are connected (presumably Riemannian manifolds are automatically paracompact). It seems like something more along the lines of "integration of the metric allows one to define a

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Furthermore a main section of the article is "Metric space structure", implying to a reader that a Reimannian manifold is a "metric space" and the distance function is discussed several times in the article. In my opinion the Riemannian metric

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I just updated the lead and in that update, I put in a version of the torus multiple image. Just wanted to say that I realize a consensus hasn't emerged here yet, and once a consensus reached, feel free to replace the multiple image I put in.

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I find the first one curiously unconvincing. The pentagon surrounding the foreground ('up close') has curved sides, but the one dead center ('further away') looks regular. Maybe the caption can guide the reader to particular comparisons?

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So "it" is not the blue checkered pattern image I guess? The orange one is clearly distorted. Why? I guess that the operations used to create the blue image are invalid so we must do something different and end up with the orange one.

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the word "metric" is used in Riemannian geometry you have to infer from context whether the metric or corresponding metric is meant; in practice this is usually easy. "Riemannian distance function" refers unambiguously to the metric.

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piecewise smooth curves. You can prove these different definitions both yield the same Riemannian distance function, and I know the proof, but I can't find a reference for it, and of course citing original research is not allowed.

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is flat, but a flat torus cannot be isometrically embedded in Euclidean space (but it can be conformally embedded, as here). The flat metric on the torus is that for which all of the heavy curved segments shown have equal

1119:(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. 1187:

the standard metric on the side-identified square. This isn't powerful enough to say that the induced metric isn't flat, but it's perfectly enough to say that what you're looking at is not an isometric embedding.

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I changed the titling a bit but I think this is a good fix for now. I think your suggestions in #2 are good (perhaps 2b the most?); as some more content is added we can think more about the best way to organize

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cannot be isometrically embedded in Euclidean space (right), because it is necessary to bend and stretch the sheet in doing so. Thus the intrinsic geometry of a flat torus is different from that of an embedded

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connectedness or paracompactness, you do still get paracompactness for free – since metrizable spaces are paracompact. Then second-countability comes out also if you assume there are countably many components.

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A torus representing a 2D Riemannian manifold in a 3D Euclidean space. The Riemannian metric is not flat as can be seen by the unequal lengths of the bold lines. For many applications, a flat metric would be

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In the lead an image should ideally be solely about the topic. It should be the simplest or most iconic image related to the topic. I would go with something like the tangent-plane-on-sphere in the lead.

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Better. The term "flat metric" is not defined in the article. The phrase "the lengths of all the heavy segments in the mesh are equal" keeps tripping me up. It is clearly not true in ordinary words.

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Purely as a side note, Riemannian manifolds are indeed often taken to be smooth, but there are notable counterexamples, such as Bridson and Haefliger's textbook which takes them to be continuous, or

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seems like an important topic for which there should be an article or section, but the only sections that I could find were for restricted contexts. Is there somewhere appropriate to point it? --

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I didn't see it in Lee, but I guess you could give a disconnected Riemannian manifold by picking basepoints on connected components and giving the set of basepoints the discrete metric.

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is about specifically the structure of a Riemannian manifold, basic theorems, the objects you can put on Riemannian manifolds, related structures, and examples. I myself haven't touched

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So each connected component is smoothly path-connected. Then, using the Riemannian metric, each connected component is metrizable, and so M itself is metrizable and hence paracompact.

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Worth noting that you need to assume that manifolds only have countably many components (which comes for free if you make the standard assumption that manifolds are second countable).

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I'm fine with manifolds being always paracompact. (I've seen claims that you get paracompactness for free if it's Riemannian, but I am skeptical. Happy to be convinced either way.)

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discussing them all again, b) having two sections about examples, one basic examples, one more advanced examples, c) discussing a few examples as each new concept is introduced.

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Agreed, although I don't think paracompactness is relevant to this. (But at least in the sources I am familiar with, it's always taken as part of the definition of "manifold.")

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Showing a square then calling it a "flat torus" without this being discussed in the article has this effect on me. It signals to me that the article is not going to be helpful.

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I think captions should start with a description of the scene/object. If I ignore your description I would make up this caption (as someone who does not understand the topic):

925:(after its opposite edges are identified). The process of realizing this torus in Euclidean space, however, involves strectching the sheet in ways that distort the metric. 1372:

In my view this topic was not focused on images for the lead, but on possible images and captions. I expected to see the tetrahedron tiling image as that one is convincing.

2061: 2034: 721:. It is not possible to isometrically (smoothly) embed a flat torus in three-dimensional Euclidean space. The torus shown is a flat torus isometrically embedded into the 1883: 1025: 965: 553: 2064: 2006: 394:

Also notably absent from the history is the Italian school (Bianchi, Ricci, etc), and the early contributions of Darboux, Monge, and others in the 19th century.

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In three dimensions, one can bend a rectangle into a torus, but doing this typically stretches the surface, as seen by the distortion of the checkered pattern.

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A torus is obtained from a (Euclidean) parallelogram by identifying opposite edges. This is for many applications the natural Riemannian structure on a torus.

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as approximately Euclidean from up close, but increasingly distorted and shrunken the further away they are, because the metric is non-Euclidean (of negative

520:. It is not possible to isometrically (smoothly) embed a flat torus in three-dimensional Euclidean space. The torus shown is a flat torus embedded into the 500:

as approximately Euclidean from up close, but increasingly distorted and shrunken the further away they are, because the metric is non-Euclidean (of negative

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I agree that there is confusion here. The situation is the following (just for talk page purposes I will use "metric" to refer to a metric in the sense of a

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Lee's choice seems to be rather unusual, I don't think I've seen it anywhere else. It might be best to just use other references for the distance function.

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You can make a disconnected Riemannian manifold a metric space by making each component a metric space via a silly procedure. This should be in Lee's

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By the way, in the near future I plan to add a short section to the body of the article on these Riemannian metrics constructed by analytic means.

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You left the vast majority of definitions bolded, and I don't see any difference between the definitions you unbolded and the ones you left bolded.

391:(Besse's Einstein manifolds in a good general source for a lot of this. I think Berger has a big survey book too.) Maybe the Yamabe problem too. 840:

The image on the left is an almost-square parallelogram, not a torus. I assume the text is implying the transformation shown in this image (from

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I see, thanks. I have redone your edit changing the definitions to italics but I have applied it to all of the definitions outside of the lead.

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Space forms should undoubtedly be discussed on the page and possibly also in the lead, but as a separate class of examples not to do with PDE.

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I guess the phrase "with sides identified" makes sense if you already understand the subject. Here is how I would rewrite the first sentence:

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important as it tells the non-math major about the relationship between this new thing in the article and a more familiar thing, "distance".

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I am not aware of any style guide saying that definitions on math articles are supposed to be italics and not bold. I didn't see anything on

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Unfortunately is it difficult for me to pick out the polyhedra. Maybe this topic needs comparisons between Riemannian and Euclidean images?

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Ok so I know Knowledge math pages are famously mathy, but this important topic could use at a mention of where this thing is used and why.

691:(left), observes a tiling of space by regular tetrahedra, and each tetrahedron is seen without distortion. In contrast, an observer in 1129:

I think the stereographic projection might be confusing, but I wonder what a bunch of cross-sections of the flat torus would look like.

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rediscovered by Ricci and sent to Padova who published it in 1889, and then independently discovered a third time by Bianchi in 1902.

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I think csck metrics where most attention goes right now (as opposed to csc metrics), so I support using csck metrics as the example.

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I really like the idea of having a multiple image like what is shown on the right for the lead. (See the very long lead discussion at

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this conversation has dried up, but I don't think we ever came to a clear consensus. What image do you support using for the lead?

2416:. I have essentially no sense of the difference between these pages; should any of the content presently on this page go there? 1273:

A torus naturally carries a Euclidean metric, obtained by identifying opposite sides of a parallelogram (left). The resulting

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My issue is that points in a non-paracompact space might not be joined by a smooth curve, although perhaps this is unfounded.

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However, the Definition section of the article confuses Riemannian metric and "distance" in the normal sense of the word:

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of Riemannian manifolds; at this point in time, most known examples of cscK metrics are just the Kähler–Einstein metrics.

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tags whenever possible, since in some browsers (including my own) the horizontal rendering for it is very distracting.

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A Riemannian metric is not to be confused with the distance function of a metric space, which is also called a metric.

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can often be thought of as a surface in Euclidean space. But for many applications, the natural metric on a torus is

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can often be thought of as a surface in Euclidean space. But for many applications, the natural metric on a torus is

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I'm also not clear on the connection to the article topic. (By the way I'm trying to be helpful not annoying ;-)

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Good point. Is it fair to say that the polyhedra look more distorted (rather than just their polygonal faces)?

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Is that to say that there's nothing in particular on either page that you think should be moved to the other?

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This was one consideration that led me to use a stereographic torus rather than a "standard" torus.

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Here's maybe a better pairing? I've also updated the first caption to be a little more geometrical.

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I agree that this article should focus on the case where the metric is smooth. Thanks for your work.

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This may be completely off base, but perhaps it sets the level I think the caption should aim for.

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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contains information about the lengths and angle between the vectors. The dot products on every

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I'm missing something key, perhaps the phrase "embedding into Euclidean space". You say:

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be smoothly isometrically embedded. The simplest way I know to see this is by using the

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Thus the intrinsic geometry of a flat torus is different from that of an embedded torus.

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The blue image also illustrates this, provided one understands the initial rectangle as

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tags vs templates on this page unresolved for now, and I will just not change anything.

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But it cannot be visualized in the usual way, by an embedding into Euclidean space.

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I don't think it is clear (either formally or intuitively) that the flat torus

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I complained about this sentence when it was in the lead; now it is the body:

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into Euclidean space. The stereographic projection is not an isometry (it is

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There's also the issue of what should go here and what should go instead in

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I'd hoped to make the point with one image, but here is a two image combo.

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I don't expect that this is controversial, but I want to justify it anyway.

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directly talk about the fundamental Riemannian notions of length and angle.

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is about the subject, its history, and big theorems and conjectures, and

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I believe physics is going to be the most rich source of applications.

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In the end I think the page should communicate the above in some way.

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Do you have a good reference for the history of the Italian school?

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The square with sides identified is a Riemannian manifold called a

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A Riemannian metric puts a measuring stick on every tangent space.

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further away points, the size grows as the sine of the distance.

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To me that is illustrated by the blue checkered pattern image.

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Restructuring to emphasize the case when the metric is smooth

2176:. It strikes me that constant sectional curvature metrics ( 2470:

I don't have a strong opinion, but I think it looks fine.

816:

For many applications, the natural Riemannian metric on a

740:

Ok, a better version of the second image (not animated).

467:

Wikipedia_talk:WikiProject_Mathematics#Riemannian_manifold

1949:{\displaystyle 0=x_{0}<x_{1}<\cdots <x_{k}=1} 1286:

of the exponential map? And why is the torus moving?)

695:, a Riemannian manifold, will see a three-dimensional 2042: 2015: 1962: 1891: 1853: 1033: 1006: 973: 946: 529: 2345:

LaTeX for consistency with the rest of the article.

2335:

Lie groups, homogeneous spaces, and symmetric spaces

1699:

under which (connected) Riemannian manifolds become

1058:

https://www.emis.de/journals/em/images/pdf/em_24.pdf

101:, a collaborative effort to improve the coverage of 1547:instead. I am reverting this edit for two reasons: 2055: 2028: 2000: 1948: 1877: 1048: 1019: 988: 959: 645:I think the honeycomb pair is clear and effective. 547: 438:If you can get it, the bibliography to Schouten's 465:There is a discussion concerning this article at 484:A couple of example pictures, not sure if useful 2252:are constructed intrinsically using tools from 1956:such that the restriction of the path to each 203:This page has archives. Sections older than 8: 2430:Based on absolutely nothing, I decided that 2117:I agree with both of your proposed changes. 2036:of the interval for which the image of each 1027:isometric embedding of the flat torus into 967:isometric embedding of the flat torus into 47: 2047: 2041: 2020: 2014: 1983: 1970: 1961: 1934: 1915: 1902: 1890: 1852: 1040: 1036: 1035: 1032: 1011: 1005: 980: 976: 975: 972: 951: 945: 528: 2360:I strongly prefer to avoid <math: --> 2228:constant scalar curvature Kähler metrics 2159:Shmuel (Seymour J.) Metz Username:Chatul 2063:is contained in a coordinate chart; see 563:I've selected these pictures possibly. 1505:I agree that it's good, I would add it. 725:, also a Riemannian manifold, and then 555:, also a Riemannian manifold, and then 213:when more than 10 sections are present. 49: 19: 2381:Let's leave the issue of <math: --> 370:on applications is yet to be written. 2262:I think there should be more variety. 1553:Knowledge:Manual_of_Style/Mathematics 7: 1301:projected torus with a smooth mesh. 95:This article is within the scope of 38:It is of interest to the following 2510:High-priority mathematics articles 461:WikiProject Mathematics Discussion 14: 2246:constant scalar curvature metrics 207:may be automatically archived by 115:Knowledge:WikiProject Mathematics 2244:...many special metrics such as 2155:Constant scalar curvature metric 1847:Take as given a continuous path 1263: 1254: 1105: 1096: 1049:{\displaystyle \mathbb {R} ^{3}} 996:which is visualized on the page 989:{\displaystyle \mathbb {R} ^{3}} 677: 668: 362:There are now many applications 164: 118:Template:WikiProject Mathematics 82: 72: 51: 20: 1543:definitions on this page to be 940:mention that there really is a 135:This article has been rated as 2254:partial differential equations 2103:textbook, but not sure which. 1995: 1963: 1869: 1866: 1854: 1443:of two vectors tangent to the 1139:What do you think about this? 542: 536: 1: 2274:Yau-Tian-Donaldson conjecture 1449:3-dimensional Euclidean space 109:and see a list of open tasks. 2505:C-Class mathematics articles 2180:) could also be mentioned. 2056:{\displaystyle U_{\alpha }} 2029:{\displaystyle U_{\alpha }} 1539:you recently changed a few 727:stereographically projected 557:stereographically projected 2526: 2299:Sounds fair enough to me. 2237:in context the sentence is 2234:for two separate reasons: 2212:should be pinged on this. 2127:17:27, 5 August 2024 (UTC) 2113:17:19, 5 August 2024 (UTC) 1620:Distance metric confusion. 1523:17:07, 5 August 2024 (UTC) 1501:15:26, 5 August 2024 (UTC) 1483:13:54, 5 August 2024 (UTC) 2480:11:54, 15 July 2024 (UTC) 2466:04:14, 15 July 2024 (UTC) 2452:04:11, 15 July 2024 (UTC) 2426:03:46, 15 July 2024 (UTC) 2404:01:20, 15 July 2024 (UTC) 2375:00:21, 15 July 2024 (UTC) 2355:00:00, 15 July 2024 (UTC) 2327:18:39, 14 July 2024 (UTC) 2309:23:19, 14 July 2024 (UTC) 2295:18:37, 14 July 2024 (UTC) 2222:15:03, 14 July 2024 (UTC) 2204:13:57, 14 July 2024 (UTC) 2190:10:40, 14 July 2024 (UTC) 2167:07:56, 14 July 2024 (UTC) 2095:23:50, 14 July 2024 (UTC) 2080:20:10, 14 July 2024 (UTC) 1839:19:31, 14 July 2024 (UTC) 1823:18:13, 14 July 2024 (UTC) 1806:15:49, 14 July 2024 (UTC) 1792:15:16, 14 July 2024 (UTC) 1778:14:39, 14 July 2024 (UTC) 1756:18:49, 13 July 2024 (UTC) 1741:18:40, 13 July 2024 (UTC) 1727:18:30, 13 July 2024 (UTC) 1713:18:21, 13 July 2024 (UTC) 1686:17:26, 13 July 2024 (UTC) 1656:16:59, 13 July 2024 (UTC) 1430:17:05, 16 July 2024 (UTC) 1415:16:58, 16 July 2024 (UTC) 1389:16:07, 16 July 2024 (UTC) 1368:12:52, 16 July 2024 (UTC) 1341:02:30, 13 July 2024 (UTC) 1311:17:46, 10 July 2024 (UTC) 1296:17:25, 10 July 2024 (UTC) 1225:17:24, 10 July 2024 (UTC) 1209:17:14, 10 July 2024 (UTC) 1174:16:21, 10 July 2024 (UTC) 1149:13:39, 10 July 2024 (UTC) 1087:12:55, 10 July 2024 (UTC) 1070:12:38, 10 July 2024 (UTC) 935:09:47, 10 July 2024 (UTC) 917:02:29, 10 July 2024 (UTC) 453:13:47, 16 July 2024 (UTC) 434:12:54, 16 July 2024 (UTC) 380:12:53, 16 July 2024 (UTC) 321:01:29, 15 July 2024 (UTC) 306:22:57, 12 July 2024 (UTC) 280:20:17, 25 June 2024 (UTC) 265:19:22, 25 June 2024 (UTC) 134: 67: 46: 2226:I changed the link from 1612:12:08, 8 July 2024 (UTC) 1594:04:40, 8 July 2024 (UTC) 1571:01:43, 8 July 2024 (UTC) 1491:Much better, go for it. 1182:In that spirit, given a 886:23:26, 9 July 2024 (UTC) 871:22:39, 9 July 2024 (UTC) 835:21:16, 9 July 2024 (UTC) 803:15:46, 9 July 2024 (UTC) 789:01:53, 9 July 2024 (UTC) 774:01:47, 9 July 2024 (UTC) 750:00:13, 9 July 2024 (UTC) 659:23:31, 8 July 2024 (UTC) 640:21:07, 8 July 2024 (UTC) 621:18:30, 8 July 2024 (UTC) 607:17:59, 8 July 2024 (UTC) 589:15:54, 6 July 2024 (UTC) 573:15:41, 6 July 2024 (UTC) 479:11:54, 5 July 2024 (UTC) 420:12:07, 5 July 2024 (UTC) 404:22:03, 4 July 2024 (UTC) 358:12:10, 5 July 2024 (UTC) 343:21:02, 4 July 2024 (UTC) 141:project's priority scale 2250:Kähler–Einstein metrics 1580:. This is explained in 1134:WikiProject_Mathematics 1126:It's a very nice image! 697:hyperbolic tessellation 498:hyperbolic tessellation 241:09:00, 3 May 2020 (UTC) 225:the geodesic equation. 98:WikiProject Mathematics 2390:Riemannian manifolds". 2057: 2030: 2002: 1950: 1879: 1667:Myers-Steenrod theorem 1456: 1402: 1050: 1021: 990: 961: 853: 822: 734: 560: 549: 505: 210:Lowercase sigmabot III 28:This article is rated 2058: 2031: 2003: 1951: 1880: 1878:{\displaystyle \to M} 1438: 1399: 1051: 1022: 1020:{\displaystyle C^{2}} 1000:, though there is no 991: 962: 960:{\displaystyle C^{1}} 851: 815: 712: 559:into Euclidean space. 550: 548:{\displaystyle SU(2)} 511: 491: 294:positive mass theorem 2040: 2013: 1960: 1889: 1851: 1766:Riemannian Manifolds 1031: 1004: 971: 944: 527: 121:mathematics articles 2440:Riemannian geometry 2436:Riemannian manifold 2432:Riemannian geometry 2414:Riemannian geometry 1473:what do you think? 2053: 2026: 1998: 1946: 1875: 1457: 1403: 1046: 1017: 986: 957: 854: 823: 735: 693:hyperbolic 3-space 561: 545: 506: 494:hyperbolic 3-space 329:Why should I care? 296:with a C metric. 90:Mathematics portal 34:content assessment 2270:Calabi conjecture 1697:distance function 1643:is to be confused 1405:How about this? 1196:Theorema Egregium 386:More modern stuff 217: 216: 155: 154: 151: 150: 147: 146: 2517: 2062: 2060: 2059: 2054: 2052: 2051: 2035: 2033: 2032: 2027: 2025: 2024: 2007: 2005: 2004: 2001:{\displaystyle } 1999: 1994: 1993: 1975: 1974: 1955: 1953: 1952: 1947: 1939: 1938: 1920: 1919: 1907: 1906: 1884: 1882: 1881: 1876: 1531:Bold definitions 1267: 1258: 1109: 1100: 1055: 1053: 1052: 1047: 1045: 1044: 1039: 1026: 1024: 1023: 1018: 1016: 1015: 995: 993: 992: 987: 985: 984: 979: 966: 964: 963: 958: 956: 955: 842:Torus#Flat torus 681: 672: 554: 552: 551: 546: 212: 196: 168: 160: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 2525: 2524: 2520: 2519: 2518: 2516: 2515: 2514: 2495: 2494: 2337: 2151: 2043: 2038: 2037: 2016: 2011: 2010: 1979: 1966: 1958: 1957: 1930: 1911: 1898: 1887: 1886: 1849: 1848: 1844:path-connected. 1622: 1533: 1447:sitting inside 1282: 1281: 1280: 1279: 1270: 1269: 1268: 1260: 1259: 1123: 1122: 1121: 1120: 1112: 1111: 1110: 1102: 1101: 1034: 1029: 1028: 1007: 1002: 1001: 974: 969: 968: 947: 942: 941: 923:already a torus 707: 706: 705: 704: 689:Euclidean space 687:An observer in 684: 683: 682: 674: 673: 525: 524: 492:An observer in 486: 463: 388: 331: 248: 222: 208: 197: 191: 173: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 2523: 2521: 2513: 2512: 2507: 2497: 2496: 2493: 2492: 2491: 2490: 2489: 2488: 2487: 2486: 2485: 2484: 2483: 2482: 2472:Mathwriter2718 2444:Mathwriter2718 2410: 2396:Mathwriter2718 2393: 2392: 2391: 2387: 2383: 2362: 2347:Mathwriter2718 2336: 2333: 2332: 2331: 2330: 2329: 2315: 2314: 2313: 2312: 2311: 2301:Mathwriter2718 2283: 2282: 2281: 2264: 2263: 2259: 2258: 2257: 2239: 2238: 2232:Yamabe problem 2224: 2214:Mathwriter2718 2196:Mathwriter2718 2174:Yamabe problem 2172:This links to 2150: 2147: 2146: 2145: 2144: 2143: 2142: 2141: 2140: 2139: 2138: 2137: 2136: 2135: 2134: 2133: 2132: 2131: 2130: 2129: 2119:Mathwriter2718 2100: 2087:Mathwriter2718 2082: 2068: 2050: 2046: 2023: 2019: 1997: 1992: 1989: 1986: 1982: 1978: 1973: 1969: 1965: 1945: 1942: 1937: 1933: 1929: 1926: 1923: 1918: 1914: 1910: 1905: 1901: 1897: 1894: 1874: 1871: 1868: 1865: 1862: 1859: 1856: 1845: 1810: 1809: 1808: 1784:Mathwriter2718 1770:Mathwriter2718 1762: 1761: 1760: 1759: 1758: 1689: 1688: 1674: 1670: 1638: 1637: 1630: 1629: 1621: 1618: 1617: 1616: 1615: 1614: 1604:Mathwriter2718 1578:Mathwriter2718 1563:Mathwriter2718 1560: 1559: 1556: 1532: 1529: 1528: 1527: 1526: 1525: 1510: 1506: 1503: 1489:Mathwriter2718 1475:Mathwriter2718 1458: 1432: 1394: 1393: 1392: 1391: 1376: 1373: 1360:Mathwriter2718 1333:Mathwriter2718 1316: 1315: 1314: 1313: 1272: 1271: 1262: 1261: 1253: 1252: 1251: 1250: 1249: 1248: 1247: 1246: 1245: 1244: 1243: 1242: 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Omburo 1648:Johnjbarton 1582:MOS:NOTBOLD 1555:about this. 1493:Johnjbarton 1466:Johnjbarton 1462:Tito Omburo 1441:dot product 1422:Johnjbarton 1407:Tito Omburo 1381:Johnjbarton 1352:Johnjbarton 1348:Tito Omburo 1324:Johnjbarton 1320:Tito Omburo 1303:Tito Omburo 1217:Tito Omburo 1166:Johnjbarton 1079:Tito Omburo 927:Tito Omburo 909:Johnjbarton 878:Tito Omburo 863:Johnjbarton 827:Tito Omburo 795:Johnjbarton 781:Tito Omburo 766:Johnjbarton 742:Tito Omburo 651:Johnjbarton 632:Tito Omburo 613:Johnjbarton 599:Tito Omburo 581:Johnjbarton 565:Tito Omburo 445:Tito Omburo 396:Tito Omburo 335:Johnjbarton 112:Mathematics 103:mathematics 59:Mathematics 2499:Categories 2230:(cscK) to 1600:Dedhert.Jr 1586:Dedhert.Jr 1545:italicized 1537:Dedhert.Jr 1275:flat torus 1117:flat torus 998:Flat torus 290:Schoen-Yau 731:conformal 701:curvature 522:Lie group 502:curvature 364:mentioned 2458:Gumshoe2 2442:at all. 2418:Gumshoe2 2367:Gumshoe2 2341:Gumshoe2 2319:Gumshoe2 2287:Gumshoe2 2278:examples 2210:Gumshoe2 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