Talk:Riemannian manifold

Reorganization

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 the geodesic equation.

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.

Gumshoe2 (talk) 09:00, 3 May 2020 (UTC)

Restructuring to emphasize the case when the metric is smooth

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

A Riemannian manifold means that the metric is smooth. This is the default mathematical convention and other pages on Wikipedia 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 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. Mathwriter2718 (talk) 19:22, 25 June 2024 (UTC)

I agree that this article should focus on the case where the metric is smooth. Thanks for your work. Mgnbar (talk) 20:17, 25 June 2024 (UTC)

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 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.

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 Schoen-Yau on the positive mass theorem with a C⁵ metric. Gumshoe2 (talk) 22:57, 12 July 2024 (UTC)

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 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. Mathwriter2718 (talk) 01:29, 15 July 2024 (UTC)

Why should I care?

Ok so I know Wikipedia math pages are famously mathy, but this important topic could use at a mention of where this thing is used and why. Johnjbarton (talk) 21:02, 4 July 2024 (UTC)

I believe physics is going to be the most rich source of applications. Mathwriter2718 (talk) 12:10, 5 July 2024 (UTC)

There are now many applications mentioned, but a section on applications is yet to be written. Mathwriter2718 (talk) 12:53, 16 July 2024 (UTC)

More modern stuff

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. (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.

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. Tito Omburo (talk) 22:03, 4 July 2024 (UTC)

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. Mathwriter2718 (talk) 12:07, 5 July 2024 (UTC)

Do you have a good reference for the history of the Italian school? Mathwriter2718 (talk) 12:54, 16 July 2024 (UTC)

If you can get it, the bibliography to Schouten's Ricci calculus 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 rediscovered by Ricci and sent to Padova who published it in 1889, and then independently discovered a third time by Bianchi in 1902. Tito Omburo (talk) 13:47, 16 July 2024 (UTC)

WikiProject Mathematics Discussion

There is a discussion concerning this article at Wikipedia_talk:WikiProject_Mathematics#Riemannian_manifold. Mathwriter2718 (talk) 11:54, 5 July 2024 (UTC)

A couple of example pictures, not sure if useful

An observer in hyperbolic 3-space, a Riemannian manifold, sees polyhedra of a three-dimensional hyperbolic tessellation as approximately Euclidean from up close, but increasingly distorted and shrunken the further away they are, because the metric is non-Euclidean (of negative curvature).

A torus can often be thought of as a surface in Euclidean space. But for many applications, the natural metric on a torus is flat. 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 Lie group $SU(2)$, also a Riemannian manifold, and then stereographically projected into Euclidean space.

I've selected these pictures possibly. Tito Omburo (talk) 15:41, 6 July 2024 (UTC)

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? Johnjbarton (talk) 15:54, 6 July 2024 (UTC)

Good point. Is it fair to say that the polyhedra look more distorted (rather than just their polygonal faces)? Tito Omburo (talk) 17:59, 8 July 2024 (UTC)

Unfortunately is it difficult for me to pick out the polyhedra. Maybe this topic needs comparisons between Riemannian and Euclidean images? Johnjbarton (talk) 18:30, 8 July 2024 (UTC)

Here's maybe a better pairing? I've also updated the first caption to be a little more geometrical. Tito Omburo (talk) 21:07, 8 July 2024 (UTC)

I think the honeycomb pair is clear and effective.

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. Johnjbarton (talk) 23:31, 8 July 2024 (UTC)

An observer in Euclidean space (left), observes a tiling of space by regular tetrahedra, and each tetrahedron is seen without distortion. In contrast, an observer in hyperbolic 3-space, a Riemannian manifold, will see a three-dimensional hyperbolic tessellation as approximately Euclidean from up close, but increasingly distorted and shrunken the further away they are, because the metric is non-Euclidean (of negative curvature).

A torus can often be thought of as a surface in Euclidean space. But for many applications, the natural metric on a torus is flat. 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 3-sphere, also a Riemannian manifold, and then stereographically projected into Euclidean space. The stereographic projection is not an isometry (it is conformal).

Ok, a better version of the second image (not animated). Tito Omburo (talk) 00:13, 9 July 2024 (UTC)

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):

• 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 appropriate.

This may be completely off base, but perhaps it sets the level I think the caption should aim for. Johnjbarton (talk) 01:47, 9 July 2024 (UTC)

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." Tito Omburo (talk) 01:53, 9 July 2024 (UTC)

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. Johnjbarton (talk) 15:46, 9 July 2024 (UTC)

For many applications, the natural Riemannian metric on a torus 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 length.

I'd hoped to make the point with one image, but here is a two image combo. Tito Omburo (talk) 21:16, 9 July 2024 (UTC)

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 Torus#Flat torus) which would create a torus from the almost-square parallelogram?

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.

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 drawn.

I'm also not clear on the connection to the article topic. (By the way I'm trying to be helpful not annoying ;-) Johnjbarton (talk) 22:39, 9 July 2024 (UTC)

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. Tito Omburo (talk) 23:26, 9 July 2024 (UTC)

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

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

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

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

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. Johnjbarton (talk) 02:29, 10 July 2024 (UTC)

The blue image also illustrates this, provided one understands the initial rectangle as already a torus (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. Tito Omburo (talk) 09:47, 10 July 2024 (UTC)

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 mention that there really is a $C^{1}$ isometric embedding of the flat torus into $\mathbb {R} ^{3}$ which is visualized on the page Flat torus, though there is no $C^{2}$ isometric embedding of the flat torus into $\mathbb {R} ^{3}$ (see https://www.emis.de/journals/em/images/pdf/em_24.pdf pages 7-9). Mathwriter2718 (talk) 12:38, 10 July 2024 (UTC)

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. Tito Omburo (talk) 12:55, 10 July 2024 (UTC)

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.

It's a very nice image!

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

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 WikiProject_Mathematics.) 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.

What do you think about this? Mathwriter2718 (talk) 13:39, 10 July 2024 (UTC)

(I suggest taking a still image out of the blue gif rather than the orange square).

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

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

Johnjbarton (talk) 16:21, 10 July 2024 (UTC)

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

In that spirit, given a specific embedding of the side-identified square in R³, 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 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.

I don't think it is clear (either formally or intuitively) that the flat torus cannot be smoothly isometrically embedded. The simplest way I know to see this is by using the Theorema Egregium 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 C¹ Nash–Kuiper embeddings (to which Theorema Egregium doesn't apply). It would be possible to say, at bare minimum, that a flat torus in R³ 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. Gumshoe2 (talk) 17:14, 10 July 2024 (UTC)

This was one consideration that led me to use a stereographic torus rather than a "standard" torus. Tito Omburo (talk) 17:24, 10 July 2024 (UTC)

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

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 of the exponential map? And why is the torus moving?) Gumshoe2 (talk) 17:25, 10 July 2024 (UTC)

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 projected torus with a smooth mesh. Tito Omburo (talk) 17:46, 10 July 2024 (UTC)

@Tito Omburo @Johnjbarton @Gumshoe2 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. Mathwriter2718 (talk) 02:30, 13 July 2024 (UTC)

@Tito Omburo @Johnjbarton @Gumshoe2 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? Mathwriter2718 (talk) 12:52, 16 July 2024 (UTC)

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.

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.

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. 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. Johnjbarton (talk) 16:07, 16 July 2024 (UTC)

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

How about this? Tito Omburo (talk) 16:58, 16 July 2024 (UTC)

LGTM Johnjbarton (talk) 17:05, 16 July 2024 (UTC)

The dot product of two vectors tangent to the sphere sitting inside 3-dimensional Euclidean space contains information about the lengths and angle between the vectors. The dot products on every tangent plane, packaged together into one mathematical object, are a Riemannian metric.

@Tito Omburo @Johnjbarton @Gumshoe2 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, what do you think? Mathwriter2718 (talk) 13:54, 5 August 2024 (UTC)

@Mathwriter2718 Much better, go for it. Johnjbarton (talk) 15:26, 5 August 2024 (UTC)

I agree that it's good, I would add it.

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 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.

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. Gumshoe2 (talk) 17:07, 5 August 2024 (UTC)

Bold definitions

@Dedhert.Jr you recently changed a few bolded definitions on this page to be italicized instead. I am reverting this edit for two reasons:

1. 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 Wikipedia:Manual_of_Style/Mathematics about this.
2. 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.

Mathwriter2718 (talk) 01:43, 8 July 2024 (UTC)

@Mathwriter2718. This is explained in MOS:NOTBOLD: "Avoid using boldface for introducing new terms. Instead, italics are preferred". Dedhert.Jr (talk) 04:40, 8 July 2024 (UTC)

@Dedhert.Jr 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. Mathwriter2718 (talk) 12:08, 8 July 2024 (UTC)

Distance metric confusion.

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

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

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

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

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 is to be confused 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 important as it tells the non-math major about the relationship between this new thing in the article and a more familiar thing, "distance". Johnjbarton (talk) 16:59, 13 July 2024 (UTC)

I agree that there is confusion here. The situation is the following (just for talk page purposes I will use "metric^S" to refer to a metric in the sense of a metric space, and "metric" to refer to a Riemannian metric). A metric defines a metric^S (via a somewhat complicated procedure), and in this sense a Riemannian manifold "is" a metric space (among other things). However most metric^S's do not correspond to any metric, and so most metric spaces do not correspond to any Riemannian manifold. Notably the metric^S corresponding to a given metric fully characterizes the original metric due to the Myers-Steenrod theorem, in the sense that if two metrics are given and their corresponding metric^S'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.

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^S: one is a collection of inner products and the other is a real-valued function with point-pair input. Sometimes if the word "metric" is used in Riemannian geometry you have to infer from context whether the metric or corresponding metric^S is meant; in practice this is usually easy. "Riemannian distance function" refers unambiguously to the metric^S.

In the end I think the page should communicate the above in some way. Gumshoe2 (talk) 17:26, 13 July 2024 (UTC)

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 distance function under which (connected) Riemannian manifolds become metric spaces." (and a link to where this is done in more detail in the article) Tito Omburo (talk) 18:21, 13 July 2024 (UTC)

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.") Gumshoe2 (talk) 18:30, 13 July 2024 (UTC)

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.) Tito Omburo (talk) 18:40, 13 July 2024 (UTC)

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. Gumshoe2 (talk) 18:49, 13 July 2024 (UTC)

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 Riemannian Manifolds. Mathwriter2718 (talk) 14:39, 14 July 2024 (UTC)

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). Mathwriter2718 (talk) 15:16, 14 July 2024 (UTC)

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. Tito Omburo (talk) 15:49, 14 July 2024 (UTC)

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 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. Gumshoe2 (talk) 18:13, 14 July 2024 (UTC)

My issue is that points in a non-paracompact space might not be joined by a smooth curve, although perhaps this is unfounded. Tito Omburo (talk) 19:31, 14 July 2024 (UTC)

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 path-connected.

Take as given a continuous path $[0,1]\to M$; you need $0=x_{0}<x_{1}<\cdots <x_{k}=1$ such that the restriction of the path to each $[x_{i},x_{i+1}]$ 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 cover $U_{\alpha }$ of the interval for which the image of each $U_{\alpha }$ is contained in a coordinate chart; see here for instance.

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. Gumshoe2 (talk) 20:10, 14 July 2024 (UTC)

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 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. Mathwriter2718 (talk) 23:50, 14 July 2024 (UTC)

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.

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 textbook, but not sure which. Gumshoe2 (talk) 17:19, 5 August 2024 (UTC)

I agree with both of your proposed changes. Mathwriter2718 (talk) 17:27, 5 August 2024 (UTC)

Redlink

The redlink Constant scalar curvature metric 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? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 07:56, 14 July 2024 (UTC)

This links to Yamabe problem. It strikes me that constant sectional curvature metrics (space forms) could also be mentioned. Tito Omburo (talk) 10:40, 14 July 2024 (UTC)

I think csck metrics where most attention goes right now (as opposed to csc metrics), so I support using csck metrics as the example. Mathwriter2718 (talk) 13:57, 14 July 2024 (UTC)

@Gumshoe2 should be pinged on this. Mathwriter2718 (talk) 15:03, 14 July 2024 (UTC)

I changed the link from constant scalar curvature Kähler metrics (cscK) to Yamabe problem for two separate reasons:

• in context the sentence is

...many special metrics such as constant scalar curvature metrics and Kähler–Einstein metrics are constructed intrinsically using tools from partial differential equations.

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.) I think there should be more variety.

• 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 (Calabi conjecture and Yau-Tian-Donaldson conjecture). 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 examples of Riemannian manifolds; at this point in time, most known examples of cscK metrics are just the Kähler–Einstein metrics.

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. Gumshoe2 (talk) 18:37, 14 July 2024 (UTC)

Sounds fair enough to me. Mathwriter2718 (talk) 23:19, 14 July 2024 (UTC)

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. Gumshoe2 (talk) 18:39, 14 July 2024 (UTC)

Lie groups, homogeneous spaces, and symmetric spaces

@Gumshoe2 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 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> LaTeX for consistency with the rest of the article. Mathwriter2718 (talk) 00:00, 15 July 2024 (UTC)

I strongly prefer to avoid <math> tags whenever possible, since in some browsers (including my own) the horizontal rendering for it is very distracting.

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 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. Gumshoe2 (talk) 00:21, 15 July 2024 (UTC)

1. Let's leave the issue of <math> tags vs templates on this page unresolved for now, and I will just not change anything.
2. 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 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.
3. 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 Riemannian manifolds".

Mathwriter2718 (talk) 01:20, 15 July 2024 (UTC)

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 it.

There's also the issue of what should go here and what should go instead in Riemannian geometry. I have essentially no sense of the difference between these pages; should any of the content presently on this page go there? Gumshoe2 (talk) 03:46, 15 July 2024 (UTC)

Based on absolutely nothing, I decided that Riemannian geometry is about the subject, its history, and big theorems and conjectures, and Riemannian manifold 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 Riemannian geometry at all. Mathwriter2718 (talk) 04:11, 15 July 2024 (UTC)

Is that to say that there's nothing in particular on either page that you think should be moved to the other? Gumshoe2 (talk) 04:14, 15 July 2024 (UTC)

I don't have a strong opinion, but I think it looks fine. Mathwriter2718 (talk) 11:54, 15 July 2024 (UTC)