Talk:Metric tensor

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Distance

There are fundamental problems with the text On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function $d (p, q)$ whose value at a pair of points $p$ and $q$ is the distance from $p$ to $q$ .:

It is circular; presumably the author intended to say the length of a geodesic from $p$ to $q$ .
There may be no geodesic from $p$ to $q$ .
There may be multiple geodesics from $p$ to $q$ , of differing length.

I propose the text On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic. If there is a geodesic between two points $p$ and $q$ whose length is globally a minimum, then the distance $d (p, q)$ between the points is defined to be that length; otherwise $d (p, q)$ is defined to be the greatest lower bound of all the smooth curves connecting them. Equipped with this notion of length, a Riemannian manifold is a metric space with the distance function $d (p, q)$ , i.e., $d$ satisfies the identity, symmetry and triangle inequality axioms. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:42, 25 April 2022 (UTC)[reply]

I just saw that you had noticed the need for corrections before me; thank you for this. Hopefully the new version fixes the problems that you pointed out. Ebony Jackson (talk) 02:03, 26 August 2022 (UTC)[reply]

I believe that the wording of your recent edit is better. I had some concern about the use of the term infimum, but concluded that readers not familiar with it would also be unfamiliar with greatest lower bound. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:19, 26 August 2022 (UTC)[reply]

Yeah, I don't like these edits. They are wildly misleading, verging on being flat out wrong. For one, theorems like Hopf-Rinow theorem and polar coordinates guarantee there is one and only one unique (local) geodesic on compact Riemannian manifolds! (The "other" geodesic on a sphere does not count, because there exists a 0<rho that is shorter than it.) Recall what Hopf-Rinow says: if the closed and bounded subsets of M are compact, then its got an exponential function, and its geodesically complete! You're going to have to demonstrate some pretty wild and weirdo non-compact topological space of some strange kind that violates this theorem. (or are you trying to say something about sub-Riemannian geometry??) That's pretty advanced stuff, and it is incorrect to place this in the intro, which is supposed to stay simple and easy, so that beginners can understand it. I'm tempted to revert all this back to the original wording, except I'm to exhausted to start yet another edit war. Can someone else maybe restore the original text? 67.198.37.16 (talk) 01:30, 21 May 2023 (UTC)[reply]

@Ebony Jackson and 67.198.37.16: Did you read the pages you cited? Hopf-Rinow theorem, in particular, states may or may not be minima? Nor does polar coordinates in any way contradict the existence of longer geodesics. You need to carefully distinguish local and global properties and not read more into a theorem than what it actually says. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 08:49, 21 May 2023 (UTC)[reply]

@67.198.37.16: Hi, the Hopf-Rinow theorem and polar coordinates do not guarantee that between any two distinct points on a compact Riemannian manifold, there is "one and only one unique (local) geodesic" between them. In fact, it's not even true: on a sphere there are many geodesics from the north pole to the south pole, and there are no geodesics between them that are shorter. There are also some pretty simple non-compact manifolds, such as a sphere with one point removed, in which there is a geodesic between any pair of distinct points, even a geodesic for which there are no strictly shorter geodesics, but for some pairs of points there is no shortest curve between them. Reverting to a version prior to my August 25, 2022 edit would not be good since there were mathematical errors that were corrected there, as explained the edit summary. There may, however, be improvements that could be made to the present article. If you still feel that there is a sentence that is wrong or misleading, could you explain precisely which sentence it is? Thank you, Ebony Jackson (talk) 21:30, 22 May 2023 (UTC)[reply]

Recent edits by Sakelabc123

@Sakelabc123: Two recent edits (http://en.wiki.x.io/w/index.php?title=Metric_tensor&oldid=1103620937 and http://en.wiki.x.io/w/index.php?title=Metric_tensor&oldid=1103621138) by Sakelabc123 added the text The metric tensor is is (g= [E, F, F, G]) in the description below since and changed it to The metric tensor is [E, F, F, G] in the description below since; it is not clear whether since is extraneous or whether there is missing text. IAC, [E, F, F, G] should be the matrix ${\textstyle {\begin{bmatrix}E&F\\F&G\end{bmatrix}}}$ . --Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:11, 10 August 2022 (UTC)[reply]

Yes sure Sakelabc123 (talk) 04:47, 12 August 2022 (UTC)[reply]

NPOV: Algebra versus Differential Geometry

In Algebra and in Functional Analysis, the term metric tensor refers to a rank two tensor on the dual space V* of a vector space V, or to the equivalent real or complex bilinear function, with certain properties. In Differential Geometry, the term refers to a rank two doubly contravariant^[a] tensor field on a differentiable manifold M, or to the equivalent family of bilinear functions on the tangent spaces of M. Either the article should cover both related meanings, or there should be a separate article on the algebraic meaning with appropriate hatnotes. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:31, 25 August 2022 (UTC)[reply]

Notes

^ I am using the convention that the tangent space is contravariant and the cotangent space is covariant.

Surely there are other articles that already deal with this? There are articles on the inner product and bilinear forms that cover this. Also, although it is not wrong to call a bilinear form a "tensor", its also kind-of unusual. I'm pretty sure that if I went to the library, picked up some random book off the shelve dealing with Banach spaces or Hilbert spaces or Frechet spaces or topological vector spaces, none of them will even have the word "tensor" in them. (Am I wrong?) And if I pick a random book on general topology and Hausdorff spaces, etc. They'll say "metric" everywhere, but again, no tensors. I believe that the word "tensor" is more or less exclusive to Riemannian geometry (well, and a few other areas, e.g. category theory and tensor algebras, but these don't have metrics. Perhaps only in Banach algebras???) 67.198.37.16 (talk) 00:36, 21 May 2023 (UTC)[reply]

Hatnotes

@D.Lazard: The {{short description}} is too specific. Metric tensors exist in pseudo-Riemannian manifolds as well as in Riemannian manifolds. Should not the description be something like {{Short description|Structure that defines locally a distance on a (-pseudo-)Riemannian manifold}}?

Similarly, should not the {{about}} be something like {{about|metric tensors on real (-pseudo-)Riemannian manifolds|Lorentzian metric tensors satisfying the Einstein field equations of general relativity|Metric tensor (general relativity)|metric tensors on complex Riemannian manifolds|Hermitian manifold#Formal definition|Metric tensors on a vector space|Inner product space}}?

Is it appropriate to add {{distinguish|Symplectic manifold#Definitio}}? Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:45, 11 December 2022 (UTC)[reply]

Short descriptions are aimed to help readers to decide whether the article could be interesting for them. Hatnotes are aimed to redirect readers to the right article when they are misled by an ambiguous title. So, technical accuracy is not needed, and may be confusing for non-experts.

So, I have removed the qualificative "Riemannian", that is not helpful as it does not appear in the beginning of the lead. I have also simplified the phrasing for removing technicalities. Finally I have removed two links to articles that have not their place in the hatnote, as they do not use the phrase "metric tensor". D.Lazard (talk) 16:29, 11 December 2022 (UTC)[reply]

I believe that the short description should still include local or locally.

What about ? I believe that I have seen the term metric tensor used for a symplectic 2-form, which is very different from the topic of the article. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:13, 11 December 2022 (UTC)[reply]

Why emphasizing on "local", when a metric tensor is a local data that allows defining a global distance (by integration along a geodesic)?

If a metric tensor is defined in Wikipedia, with another meaning than in this article, this must be linked in the hatnote. But this requires that the definition should be different from the definition of this article, and that the target of the link use the phrase "metric' tensori D.Lazard (talk) D.Lazard (talk) 21:14, 11 December 2022 (UTC)[reply]

A 270° arc of a great circle on a sphere is a geodesic, but it is not the shortest arc between its endpoints. It is, however, locally shortest.

In the most general case, there may be pairs of points in a connected manifold with no connecting geodesic, so defining a global distance takes a bit more work. You still define arc length by integration, but take the distance to be the infimum of arc length over all connecting paths. This, of course, is TMI for the short description. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 23:03, 11 December 2022 (UTC)[reply]

I notice that you reverted a change to {{about}} because Hermitian manifold does not contain the word tensor. Would you object to adding the word real to {{about}} and adding a {{distinguish|Hermitian manifold#Definitiions|text=Hernitian metric}} template? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:25, 13 December 2022 (UTC)[reply]

You also reverted adding locally to the short description. How about changing distance to arc length?

The documentation of {{distinguish}} says The "distinguish" template produces a hatnote to point out to our readers the existence of one or more articles whose title(s) is, or are, similar to the page in question. It is used in cases where the distinction between the titles is generally obvious and does not need further explanation. As nothing is similar but the word metric between the titles "Metric tensor" and "Hermitian metric", the template {{distinguish}} must not be used.

"Locally" is jargon and so must be avoided in hatnotes. Moreover, it does not help, here, for disambiguation, and there is thus no reason for adding it.

Similarly, the word "real" would suggest that the complex case is described elsewhere in Wikipedia. As this is not the case for the metric tensor, adding "real" would be confusing.

It seems that your concern is that the metric tensor could be defined in the case of Hermitian manifolds, but it is not defined in Wikipedia. For solving this issue, editing hatnotes is the wrong way. The right way is first to search for sources that define and use a metric tensor in this case. If there are not such sources, you must stop to try adding the topic to Wikipedia, per WP:OR. If there are reliable sources, you must add a section either here or in Hermitian manifold. I believe that this would be more convenient to add the section here, because of similar technicalities, which could be too technical there.

It is only when the generalization of the metric tensor to the complex case will be described in Wikipedia that adapting the hatnotes would eventually be possible. D.Lazard (talk) 18:59, 13 December 2022 (UTC)[reply]

Huh? The Hermitian metric defined in Hermitian manifold is the complex case. The terms real and complex are used in the literature. Google turned up discussions on StackExchange, but I can't use that as an RS. Hermitian manifold#Formal definition already exists; possibly it needs a discussion on nomenclature in the literature.. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:16, 13 December 2022 (UTC)[reply]

Again, the metric tensor is not defined in Hermitian manifold#Formal definition. If you want to link the hatnote to this section, you must edit the target article for adding to it a definition of this tensor. However, as this definition is essentially the same as in the real case, it seems nuch simpler to add to Metric tensor a section "Generalization to complex manifolds". Please, do not confuse the definition of a metric, and the definition of a tensor that specifies this metric. D.Lazard (talk) 21:13, 13 December 2022 (UTC)[reply]

See Symplectic geometry. This is a special type of non-singular metric in which each time dimension is paired with a spatial dimension (and visa versa). JRSpriggs (talk) 23:20, 14 December 2022 (UTC)[reply]

There's also sub-Riemannian geometry which is similar to Riemannian geometry, but strangely different. In particular, the metric goes really wonky, as do concepts of distance and concepts of geodesics. All this despite the fact that, algebraically, it sure "smells like" ordinary Riemannian geometry. The point is: it should not be mentioned in the hatnote. It's OK if the hatnote doesn't contain everything and the kitchen sink. As to real vs. complex, well, we've also got Kahler manifolds and almost-complex manifolds but these also should not go into any hatnotes. 67.198.37.16 (talk) 00:57, 21 May 2023 (UTC)[reply]

[1] I am using the convention that the tangent space is contravariant and the cotangent space is covariant.

[a]