Talk:Riemann curvature tensor

message: 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)
replies: text: It will be hard for me, but I will try 
 Tosha 05:40, 15 May 2004 (UTC)

text: 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)
thread_title: Old comments

message: Hey, I just reverted an edit.  It was a subtle point (but perhaps worth writing down) but 
:[\nabla_u,\nabla_v] \ne \nabla_{[u,v]}
and so the original formula was correct as written by Geometry guy. Cheers, Wesino 21:47, 14 April 2007 (UTC)
thread_title:  the "typo" 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Geometrical Meaning 

message: 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:
 
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)
replies: text: 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)
replies: text: 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)
replies: text:  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)
thread_title:  Bianchi identity 

message: 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)
thread_title:  Historical comment. 

message: 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_XY}-\nabla_{\nabla_YX}))Z
:=(\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]})Z.
--IkamusumeFan (talk) 09:33, 9 October 2014 (UTC)
replies: text: The section assumes that X and Y commute, so [X,Y]=0.  Sławomir Biały (talk) 11:35, 9 October 2014 (UTC)

text: 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)
replies: text: 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)

text: 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)
thread_title:  Calculation in the Geometric meaning section 

message: 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)
thread_title:  Symmetries and identities - undefined bracket operation 

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

message: 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)
thread_title:  About the authorship of Bianchi identities 

message: Many visitors to this page will be students trying to acquaint themselves with the mathematics needed for problems involving general relativity. As such as student (self-study), this page appears to me to be admirably clear but situated on a plane a bit too high for my reach. It needs concrete examples with numbers, vectors, results, and definitions of all symbols. Without that connection to palpable entities, the symbols just float in some ethereal space where their meaning can be perceived only by the literati. Gpalmernc (talk) 00:43, 30 April 2024 (UTC)
thread_title:  Examples needed