Talk:Riemann curvature tensor

Old comments

I started new page Curvature of Riemannian manifold, I think it is bit better than this one, I plan to remove from this page everything except things directly connected with curvature tensor and link it to page above.

Tosha 04:09, 13 May 2004 (UTC)

But please write Lie ... with a capital L, because Lie refers to the norwegian mathematician Sophus Lie. Hannes Tilgner

It will be hard for me, but I will try

Tosha 05:40, 15 May 2004 (UTC)

If you remove this article OK, but still I doubt that you catch the most general aspect of curvature in the sense of Nomizu, Kulkarni and other modern writers. Look at the reference http://www.EarningCharts.NET/ipm/ipmWaves.htm where you find more references. In the references there (look also at that one in Lecture Notes in Mathematics) you find a decomposition of the space of all curvature structures in terms of Lie and Jordan algebras. And you find how elegantly electrodynamics and gravitational waves fit into the curvature play, look at the basic work of Lichnerowics. As an ,algebraiker' I like to write the curvature structure in the following triple form, generalizing the concept of Lie triples (the book on Symmetric Spaces of Otmar Loos is a nice generalization of Lie theory): [x,y,z]=R(x,y)z. This concept generalizes the notion of a Lie triple to that one of a curvature triple, where only the Jacobiidentity is missing, but a reference to the bilinear form <,> is added in such a way, that R(x,y) is an element of the pseudoorthogonal Lie algebra. Note that the complete work of Ricci, Einstein and Weyl can be summarized as a decomposition of the space of curvature structures of Levi type (for Lie algebras). All this shows, that we do not yet understand this curvature space completely. Especially the gravitational wave aspect needs clarification. Hannes Tilgner

It seems that you want to include some basic identites with curvature plus Pseudo-Riemannian case (is it?) I think it is a good idea.

Tosha 05:40, 15 May 2004 (UTC)

Yes, I'm considering the following: Instead of writing a full publication in some mathematical journal (I have done that too often, it didn't pay out), use the Wikipedia for publication. By the rules of the scientific world, everything written down here, is published. Actually you can start with an outline of the idea, putting it step by step into a full scientific article. This cannot be done with a scientific journal. Writing an scientific article is time consuming. In this way everybody can see how - and immediately comment. The (my)problem is time - since I work hard on my webpage, mentioned above. Hannes Tilgner

Please have a look at Wikipedia:No original research. This is a good place for 'survey articles'; but not for new results. Charles Matthews 08:02, 17 May 2004 (UTC)

A new comment :) it would be a grave mistake to mistake Riemannian curvature for the curvature of a Riemann manifold, because Riemannian curvature does not require a Riemannian metric, it only requires a connection. 95.181.46.66 (talk) 16:58, 16 November 2010 (UTC)

the "typo"

Hey, I just reverted an edit. It was a subtle point (but perhaps worth writing down) but

$[\nabla _{u},\nabla _{v}]\neq \nabla _{[u,v]}$

and so the original formula was correct as written by Geometry guy. Cheers, Wesino 21:47, 14 April 2007 (UTC)

Merge proposal

The articles Riemann curvature tensor and Curvature of Riemannian manifolds cover much the same sort of material. I propose merging them. Does anyone have any objections? Silly rabbit 20:15, 21 May 2007 (UTC)

I agree to merge those two articles. But how about the opposite way? Currently, it is proposed to merge Curvature of Riemannian manifolds to Riemann curvature tensor. However, as Riemann curvature tensor is only one of the ways to define curvature on Riemannian manifold, I think Riemann curvature tensor should be a part of the other article. --Acepectif 08:43, 24 August 2007 (UTC)

Actually, I don't think we need to have each article for every kind of curvatures. For example, Weyl tensor article should better be a part of this article. --Acepectif 08:46, 24 August 2007 (UTC)

I think that it is valuable to have separate articles on the various types of curvatures (e.g., Weyl tensor). A great deal can be said about the Weyl tensor vis-a-vis conformal geometry, for example, that would be decidedly out of place in a general article on the curvature of Riemannian manifolds. The same sort of remark applies to the Ricci tensor, where people study things like the positivity of the eigenvalues (although in my field it's mostly the Weyl-Schouten tensor that is of interest). Again, much can be said which is not relevant here.

Point taken about merging the opposite way. Obviously this is a "slow merge". Silly rabbit (talk) 02:02, 23 February 2008 (UTC)

I don't think a merge is a really good idea. The articles do cover much of the same material but Curvature of Riemannian manifolds is meant to be an overview of the various ways of understanding curvature in the Riemannian setting. The Riemann curvature tensor is one way, but it is not the only one. Sectional curvature is an equivalent method. There are many other inequivalent methods. I think having a separate overview article (written in summary style) is helpful. -- Fropuff (talk) 06:03, 23 February 2008 (UTC)

You know, I am beginning to have the same feeling. However, the both articles could probably use some refocusing and trimming, with summary style in mind. Silly rabbit (talk) 14:03, 23 February 2008 (UTC)

I strongly disagree with the merge: you can define Riemannian curvature on any smooth manifold with a connection, but you obviously need a Riemannian metric for Riemannian manifold :) —Preceding unsigned comment added by 95.181.46.66 (talk) 17:00, 16 November 2010 (UTC)

Riemann

What is difference between a vector & a tensor?118.95.114.159 (talk) 13:06, 10 October 2008 (UTC)

If you do not know that, this article is much too advanced for you. Try reading Tensor first. As it says, "A tensor is an object which extends the notion of scalar, vector, and matrix.".

Thus at each event in our four dimensional space-time: A scalar (just a real number) is a rank 0 tensor; it has 4⁰ = 1 component. A vector is a rank 1 tensor; it has 4¹ = 4 components. A square matrix is a rank 2 tensor; it has 4² = 16 components. There are tensors of higher rank such as the one which is the subject of this article — the Riemann-Christoffel tensor is a rank 4 tensor. The Riemann-Christoffel tensor at each event is an array of 4⁴ = 256 real numbers. JRSpriggs (talk) 15:07, 10 October 2008 (UTC)

Deleted sentence

I removed the following sentence; it makes sense to experts, but is not sufficiently well-explained:

The transformation of a vector $\,V^{\mu }$ after circling an infinitesimal rectangle $\,dx^{\nu }dx^{\sigma }$ is: $\delta V^{\mu }=R_{\nu \sigma \tau }^{\mu }dx^{\nu }dx^{\sigma }V^{\tau }$.

Geometrical Meaning

I made a fix to this section a while back, but it has been reverted, perhaps for good reason. However, the reverter not only deleted the section I added to this page, but also did not make any record of what he did or why. Nor did he eliminate the original problems, which I will now attempt to explain.

First, the definition of the tau mapping can be a little mysterious if the tau of your font doesn't look like the Greek tau you get in the math-brackets of the equations that use tau. This was a problem for me in particular because I use the Times New Roman font, which I think is a standard font, at least it is not one that I remember ever having changed for my firefox browser. So, I put the definition of the mapping on its own line using a math-bracket for clarity.

Second, the tau symbol is used incorrectly (per the definition) except in the first equation that uses it. The subsequent two equations are incompatible with the definition. Therefore, we need a new definition or something else has to change. In the definition, the subscript of tau is the index to a point along the curve of transport. After the first equation, however, it is a designator of a side of a parallelogram.

My definition modified the original definition to use a curve-designator similar to the use of xt by the original author (I guess), who, I admit, uses it ambiguously as a point and as a curve-designator. Since the side of a parallelogram is a curve designator, that change rendered the uses consistent with the definition.

The t parameter is often used for time in physics. So, as it reads now, it looks like we have tau sometimes indexed by time and sometimes by a path.

I'll fix this whole thing properly in a new and improved way in a few days unless I get a response, preferably one that doesn't just wipe out my edits here.

I don't want to get into a reversion war, but I do want the problem to be fixed. I'll take it up through complaint channels if my changes are reverted again without comment or the fixes are still needed.

I have added my name to the WikiProject Physics table, in case its absence had anything to do with the reversion. Thinkor (talk) 13:21, 6 February 2010 (UTC)

I don't see any indication that you have ever added a section to the page. Rather the only edit I see you ever having made to this page is this one. This edit is problematic for two reasons. First of all, it changes from a well-established notation for the parallel transport (which is exactly that used by the very authoritative text of Kobayashi and Nomizu) to a misleading notation that seems to suggest that the parallel transport map depends only on the endpoint. (Parallel transport depends only on the endpoint if and only if the connection is flat.) Secondly, the formatting changes violate the manual of style for mathematics, specifically the section that discourages inline LaTeX formulas that render as PNG images (for a litany of reasons). Sławomir Biały (talk) 16:36, 10 February 2010 (UTC)

Okay. My main interest is in having the text rendered more understandable. I was unaware of the discouragement of inline LaTeX formulas. If the original mystifying (to me) notation was consistent with Kobayashi and Nomizu, there should be a reference somewhere to what that notation is. I'm interested in this subject, but I'm not an expert. I think the articles should be written with non-experts at least somewhat in mind. My idea, perhaps naive, is that if I want to find out about GR, I should be able to go to the main article, read everything it references, etc., right down to the bottom of the hierarchy, and eventually understand GR.

Thinkor (talk) 00:54, 15 February 2010 (UTC)

Bianchi identity

The following subject of a recent edit war is a good addition to the encyclopedia, but it seems terribly out of place in this article:

[The Bianchi identity] may be written concisely:

$d^{\nabla }R=0.$

Here $R$ is regarded as an alternating map on pairs of vector fields, taking values in endomorphisms of vector fields. Thus $d^{\nabla }R$ is an alternating map on triples of vector fields, so the Bianchi identity may be written out:

$(d^{\nabla }R)(u,v,w)=$

$\nabla _{u}(R(v,w))+\nabla _{v}(R(w,u))+\nabla _{w}(R(u,v))-R([u,v],w)-R([v,w],u)-R([w,u],v)=0.$

Returning to the case of a torsion free connection on a tangent bundle (such as the Levi-Civita connection), we may regard $R$ as a trilinear map from triples of vector fields to vector fields. In this case the Bianchi identiy takes the form:

There is already the article curvature form that addresses more general kinds of curvature forms (and their associated Bianchi identities to some extent), as well as a few other articles (such as connection (vector bundle) and exterior covariant derivative) that might benefit from a more thorough general discussion of Bianchi identities. This article, however, is about the Riemann tensor (associated to the Levi-Civita connection), not about connection forms and more general sorts of Bianchi identities. It seems that we should keep the notation as familiar as possible without getting side-tracked with discussion of tensor-valued forms and exterior covariant derivatives. Sławomir Biały (talk) 01:11, 5 August 2011 (UTC)

Thanks for resolving that edit war (even if I lost it!). I agree with what you are saying but it is extremely confusing if you are looking up the Bianchi identity and Google or Bing take you to this page and you get something which claims to be the Bianchi identity but isn't.

Perhaps the best way to resolve it is to qualify in this article that this is the Bianchi identity for the Levi-Cevita connection:

`On a Riemanninan manifold the Bianchi identity (often called the second Bianchi identity or differential Bianchi identity) involves the covariant derivative:'

Then the word `Bianchi identity' could link to one of the other articles you mention and the co-ordinate free definition (above) could be included in that article.

W

More or less off-topic, but why is there no Bianchi identity? Various formulations of the identity (in local coordinates, for instance) seem to fit to that article. (I guess maybe it should be called Bianchi identities.) -- Taku (talk) 15:57, 5 August 2011 (UTC)

Someone who is merely trying to learn about general relativity would be confused by references to torsion which does not exist in the real world and is not a function of the metric tensor. JRSpriggs (talk) 12:04, 6 August 2011 (UTC)

Would anyone mind if I made the following edit. It is very little different to the current sentence, but avoids confusing anyone who is not from a GR background:

On a Riemannian manifold one has the covariant derivative $\nabla _{u}R$ and the Bianchi identity (often called the second Bianchi identity or differential Bianchi identity) takes the form: — Preceding unsigned comment added by 2.124.32.5 (talk) 15:04, 6 August 2011 (UTC)

Historical comment.

The curvature tensor was invented by Riemann in his 1854 Habilitationvortrag "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen".

As Marcel Berger wrote at page 214 of his book "A Panoramic View of Riemannian Geometry":

Enthusiast of mathematical history will note that Riemann made a small mistake in his Habilitationschrift (sic) of 1854, in characterizing flat manifolds; see a detailed historical account in Di Scala 2001.'

More precisely, Riemann claimed that if the sectional curvature vanish at n(n-1)/2 independent 2-planes at each point of the manifold then the manifold is flat. Several counterexamples are provided in:

A.J. Di Scala, On an assertion in Riemann's Habilitationsvortrag,l'Enseignement Mathematique 47, p. 57-63, (2001). — Preceding unsigned comment added by Holonomia (talk • contribs) 08:14, 18 March 2012 (UTC)

Parallel transport and covariant derivative: sign error?

To me it seems as though the equation after "The parallel transport maps are related to the covariant derivative by" needs a minus sign to conform to http://en.wikipedia.org/wiki/Parallel_transport#Recovering_the_connection_from_the_parallel_transport The vector at x0 should be subtracted, not be subtracted from.

Calculation in the Geometric meaning section

There was a calculation mistake in the previous edit. I give the detailed reasoning as follows. Let

$W=\tau _{sX}^{-1}\tau _{tY}^{-1}\tau _{sX}\tau _{tY}.$

Then

$\tau _{tY}\tau _{sX}WZ=\tau _{sX}\tau _{tY}Z.$

Thus

$({\frac {d}{ds}}{\frac {d}{dt}}\tau _{tY}\tau _{sX})WZ+\tau _{tY}\tau _{sX}({\frac {d}{ds}}{\frac {d}{dt}}WZ)+\tau _{tY}{\frac {d}{ds}}\tau _{sX}({\frac {d}{dt}}WZ)+{\frac {d}{dt}}\tau _{tY}\tau _{sX}({\frac {d}{ds}}WZ)={\frac {d}{ds}}{\frac {d}{dt}}\tau _{sX}\tau _{tY}Z.$

Since s=t=0,

$({\frac {d}{ds}}{\frac {d}{dt}}\tau _{tY}\tau _{sX})Z+{\frac {d}{ds}}{\frac {d}{dt}}WZ={\frac {d}{ds}}{\frac {d}{dt}}\tau _{sX}\tau _{tY}Z.$

We have

${\frac {d}{ds}}{\frac {d}{dt}}WZ|_{s=t=0}=({\frac {d}{ds}}{\frac {d}{dt}}\tau _{sX}\tau _{tY}Z-{\frac {d}{ds}}{\frac {d}{dt}}\tau _{tY}\tau _{sX}Z)|_{s=t=0}$

$=(({\frac {d}{ds}}\tau _{sX}{\frac {d}{dt}}\tau _{tY}Z+{\frac {d}{ds}}\tau _{s{\frac {d}{dt}}\tau _{tY}X}Z)-({\frac {d}{dt}}\tau _{tY}{\frac {d}{ds}}\tau _{sX}Z+{\frac {d}{dt}}\tau _{t{\frac {d}{ds}}\tau _{sX}Y}Z))|_{s=t=0}$

$=({\frac {d}{ds}}\tau _{sX}{\frac {d}{dt}}\tau _{tY}Z-{\frac {d}{dt}}\tau _{tY}{\frac {d}{ds}}\tau _{sX}Z-({\frac {d}{dt}}\tau _{t{\frac {d}{ds}}\tau _{sX}Y}-{\frac {d}{ds}}\tau _{s{\frac {d}{dt}}\tau _{tY}X})Z)|_{s=t=0}$

$=(\nabla _{X}\nabla _{Y}-\nabla _{Y}\nabla _{X}-(\nabla _{\nabla _{X}Y}-\nabla _{\nabla _{Y}X}))Z$

$=(\nabla _{X}\nabla _{Y}-\nabla _{Y}\nabla _{X}-\nabla _{[X,Y]})Z.$

--IkamusumeFan (talk) 09:33, 9 October 2014 (UTC)

The section assumes that X and Y commute, so [X,Y]=0. Sławomir Biały (talk) 11:35, 9 October 2014 (UTC)

I notice the assumption. Hence it was not a mistake. Thanks for reminding.

One may consider removing the commuting assumption, since the general case admits the quadrilateral construction. --IkamusumeFan (talk) 17:01, 9 October 2014 (UTC)

I agree overall with the edit. I don't think there is much to be gained by assuming commutativity, and the risk of confusion is very real. But if we do decide to omit the assumption, then we should justify where the extra term comes from, possibly by adding some version of your derivation in the article. Sławomir Biały (talk) 17:23, 9 October 2014 (UTC)

I made the calculation clearer. I tried to make it as short as possible, but it is still quite lengthy. So I do not know whether the derivation fits the Wiki style. Maybe assuming the commutativity is just fine. Thanks.--IkamusumeFan (talk) 19:00, 9 October 2014 (UTC)

Link to empty page

The link ("second covariant derivative") in the sentence "The curvature formula can also be expressed in terms of the second covariant derivative defined as" links to an empty topic. Is the empty page in work, or does it need to link somewhere else? Thanks. BornRightTheFirstTime (talk) 19:02, 22 November 2014 (UTC)

See Wikipedia:Red link for an explanation of the value of red-links.

If you know of an existing article which would be an appropriate target for this, please feel free to change the link. JRSpriggs (talk) 11:03, 23 November 2014 (UTC)

Symmetries and identities - undefined bracket operation

In the section symmetries and identities a bracket operation of the following syntax appears:

$<T,w>$

which is not defined or referenced anywhere in the article (where T is apparently a (0,2) tensor, and w a (0,1) tensor - or maybe w is a scalar field? this is also never specified in the article). Please add a definition or external reference to clarify what this bracket operation is supposed to mean. Thank you! — Preceding unsigned comment added by 138.16.124.81 (talk • contribs) 21:00, 28 November 2014‎ (UTC)

The only places where I see the "bracket operation" used in the article, it is the inner product of two contravariant vector fields using the metric. So

$\langle x,y\rangle =g_{\alpha \beta }\,x^{\alpha }\,y^{\beta }\,.$

JRSpriggs (talk) 22:16, 28 November 2014 (UTC)

Informally-

Bravo to writers of this section! Terrific. Electricmic (talk) 17:16, 31 December 2019 (UTC)

Metric signature

It may be worth adding a comment on how there are two options for sign conventions in GR, which affects, say, the formula for Riemann in terms of the covariant derivatives.

About the authorship of Bianchi identities

In the symmetries section there is a claim that Ricci came up with the algebraic identity while Bianchi came up with the second. I've found no evidence of this claim, which by the way is repeated in Curvature_of_Riemannian_manifolds#Symmetries_and_identities (actually I think it was first written there, but I don't understand very well the revision history). Well, in any case, there is no evidence provided for the claim, and in this mathoverflow link there seems to be evidence that it's the other way around. For the moment I left the "citation needed" tag. Might come later to check. If you know something, please share the sources.

I also rolled back the 17:17, 17 December 2021 edit by Theodornak. The Bianchi identities should indeed involve antisimmetrization, not simmetrization. The expressions with simmetrization are also zero (trivially), but that is not related to the Bianchi identities. --JackozeeHakkiuz(talk) 05:32, 28 March 2022 (UTC)