Talk:Riemannian manifold

Introduction: mention differentiability of vector fields?

Should the introduction mention that the vector fields X and Y have to be differentiable (otherwise there probably are choices for X and Y that would make fulfilling the smoothness condition of g impossible), or is this clear enough from context? 128.130.48.152 (talk) 11:32, 24 May 2013 (UTC)

I have just independently noticed (the differentiability of the fields X and Y is stated now) that the constraint of differentiability of X and Y is insufficient. For every nondegenerate g, one can always choose differentiable X and Y such that g(X, Y) is non-smooth. This implies that no manifolds meet this definition. Some repair is evidently needed. —Quondum 14:01, 29 January 2020 (UTC)

Smoothness is totally irrelevant to the actual definition of a Riemannian metric, especially when it comes to much modern work. Shouldn't the introduction just say that one assigns a positive-definite inner product to each tangent space?

Gumshoe2 (talk) 03:38, 3 May 2020 (UTC)

If one wants to define smoothness of the metric, the best way to do it (since it trivially extends to defining any desired regularity of the metric) is to use local coordinates. I'll make the edits

Gumshoe2 (talk) 03:41, 3 May 2020 (UTC)

Differential properties enabled by a metric tensor

I agree in with this edit of the lead. On a differentiable manifold without a metric tensor, a form of gradient of a scalar field (the index-lowered version, which is a co-vector field) is still well-defined, and the index-raised version is, in itself, not adding enough to merit a mention here, especially considering that it is not always clear whether "gradient" refers to the index-lowered or -raised version and this would need clarification. A similar (but trickier) argument applies to the divergence of a vector field: the n-form derivative of (n – 1)-form (e.g. the current density 3-form J_αβγ) is well-defined and is similar to a scalar; this is related to the divergence of a vector field using considerably less additional structure than a metric tensor. The subtleties and lack of evident added value brought by the metric tensor here make their inclusion in the lead unsuitable.

However, there is differential structure that is enabled by the metric tensor. The above two operations cannot be combined without it, and in particular the full structure of a metric tensor is necessary (AFAICT: this is WP:OR) for any form of the Laplace operator to be well-defined. Wave equations (or harmonic equations in the Riemannian context) are consequently only meaningful in the presence of a metric tensor. This seems like a worthwhile addition to the article, if it can be sourced. —Quondum 13:31, 14 August 2018 (UTC)

Yes, defining the Laplace operator would require the metric. JRSpriggs (talk) 23:54, 14 August 2018 (UTC)

The induced metric

I think that all the bullet points starting from the second one in the Examples section are actually about different ways that a metric can be induced and how to carry that process mathematically... I believe they would rather belong in a Immersed manifold/induced metric section, each in different subsections. — Preceding unsigned comment added by Vyrkk (talk • contribs) 21:53, 27 October 2019 (UTC)

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)

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)

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)

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)

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.

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)