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Just a proposal, would it help to add the following table and explanation to Ricci calculus (Raised and lowered indices) to illustrate how sup-/super-scripts and summation fit together in a way that relates "co-/contra-variance" to invariance?
Proposed table/text
| ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
This table summarizes how the manipulation of covariant and contravariant indices fit in with invariance under a passive transformation between bases, with the components of each basis set in terms of the other reflected in the first column. The barred indices refer to the final coordinate system after the transformation.
|
This is far clearer and briefer (to me at least...) than the main article Covariance and contravariance of vectors, and it fits in with the summary style of this article. It’s also another example for the manipulation of indices, including the Kronecker delta.
What do others think? Just a suggestion - the article is excellent and I don't want to touch it! As always I'm not forcing this in - it's for take or leave. Thanks once again to the editors here (although maybe/maybe not F= after all...). Maschen ( talk) 16:28, 15 August 2012 (UTC)
I can understand the removal from covariance and contravariance of vectors; very hasty at the time the image was added... However, it's been 3 months, no objections and one favour. I will take the liberty of adding it as planned long ago, better here than anywhere else (in a slightly extended form)... feel free to revert. Maschen ( talk) 10:15, 24 November 2012 (UTC)
This revert carries an edit summary that could be construed as a personal attack. The reverted edits introduced explicit notation giving the abstract tensors rather than only their components. We have stayed away from the abstract presentation in this article thusfar, but only as a matter of article style. While I do not object to the revert because of this, I do not in any way agree with the edit summary, and in particular with its inference about the editor. — Quondum 10:23, 2 January 2013 (UTC)
In the section on differentiation, for the partial derivative should we not have
My impression is that using the nabla is a preferred by some authors, is intuitive and fits into the notation.
Similarly, for the covariant derivative, seems to be notable. I notice that Penrose (in The Road to Reality) uses the nabla for the covariant derivative. Should we deal with these notations in the article? I have too little experience on what is notable here. — Quondum 12:03, 29 June 2013 (UTC)
The recently added notation seems to me to be too incomplete to be encyclopaedic. In particular, it omits crucial information from the notation that makes it pretty meaningless without explanatory text defining the family of curves that apply. It strikes me as made-up notation that an author (even MTW) might use by way of explaining something, not a notation that might see any use in other contexts. Does it really belong here? — Quondum 10:49, 10 August 2013 (UTC)
The operation referred to here as "sequential summation" doesn't make sense -- at least not as it's currently written.
Please explain how and why it's used, and why it's considered a tensor operation.
198.228.228.176 ( talk) 21:37, 6 February 2014 (UTC) Collin237
The earliest I can tell MTW use it is in chapter 4: Electromagnetism and differential forms, box 4.1 (p. 91). It only seems to be used in the context of p-forms (which are ... antisymmetric tensors). The authors only say "the sum is over i1 < i2 < i3 < ... in". So Quondum is correct so far. I don't know any other sources using this notation for this purpose, and it doesn't appear in Schouten's original work either (cited and linked in the article). But this summation seems to appear in a different notation which Quondum quotes above, in another reference by T. Frankel (which I don't have, and haven't seen it at the library).
Clearly, this convention of "sequential summation" exists so we shouldn't really remove it from the article. For now, let's just restrict to antisymmetric tensors. M∧Ŝ c2ħε Иτlk 08:27, 29 March 2014 (UTC)
Further notations appear to be introduced in this reference, specifically pp. 30–31. I don't understand German, but it appears to allow nesting of [], () and || on indices. My supposition is that the intention is that each of the inner nested index expressions is excluded from the higher-level symmetrization/antisymmetrization. Since this article covers a subset of exactly this type of notation, and this appears to be explicitly documented in this reference (and such exclusions make perfect sense), could someone with knowledge of German please verify my supposition so that we can include this? — Quondum 17:43, 29 March 2014 (UTC)
Does anyone know of conventions on the braiding of the free indices an expression in Ricci calculus? If so, this would be a useful addition to the article. The most obvious convention that might apply would lexicographic ordering be as in Abstract index notation#Braiding, but I do not know whether this extends to this context. — Quondum 00:09, 31 March 2014 (UTC)
This edit (with edit note I am referring to an expression where the x^{\mu} is in the denominator or x_{\mu} is in the denominator. I tried to clarify however I'm not the best at explaining. But I do think it is important enough to have.) appears to refer to a partial derivative. This is not a fraction, and has no numerator or denominator. In general the statement is also false, as the partial derivative only transforms covariantly (contravariantly) when the expression being differentiated is a scalar. This is handled under Ricci calculus#Differentiation, where I've added a mention of this special case. — Quondum 06:20, 26 August 2014 (UTC)
Yea that sounds good to me. Would you like to add it in or should I? Theoretical wormhole ( talk) 21:21, 26 August 2014 (UTC)
The present (early 2019) state of this article does very little to go beyond the subject as "a bunch of rules for operating on arrays of scalars". I think it would be useful to also provide connections with concepts from elementary linear algebra where convenient. For example (material from subsection "Upper and lower indices"):
Contravariant tensor components
An upper index (superscript) indicates contravariance of the components with respect to that index:
A vector corresponds to a tensor with one upper index . The counterpart of a tensor with two upper indices (a bivector) is less commonly seen in elementary linear algebra because it gets notationally cumbersome; many authors prefer to switch to tensor index notation when they need such objects.
Covariant tensor components
A lower index (subscript) indicates covariance of the components with respect to that index:
A tensor with lower indices may correspond to a map that takes vectors as arguments. For example, the metric tensor corresponds to the dot product of vectors.
Mixed-variance tensor components
A tensor may have both upper and lower indices:
A matrix is usually a tensor with one upper and one lower index — this makes matrix–vector multiplication correspond to applying a linear transformation to the vector, and makes matrix multiplication correspond to a contraction of tensor indices — but there are matrices which rather have two indices of the same variance: the matrix of a bilinear form naturally has two lower indices, and the R-matrix of a quasitriangular Hopf algebra naturally has two upper indices.
Ordering of indices is significant, even when of differing variance. However, when it is understood that no indices will be raised or lowered while retaining the base symbol, covariant indices are sometimes placed below contravariant indices for notational convenience (e.g. on the generalized Kronecker delta).
Raising and lowering indices
By contracting an index with a non-singular metric tensor, the type of a tensor can be changed, converting a lower index to an upper index or vice versa:
The base symbol in many cases is retained (e.g. using A where B appears here), and when there is no ambiguity, repositioning an index may be taken to imply this operation.
Repositioning an index often corresponds to taking a transpose (or similar, such as a conjugate transpose) in matrix formalism. For example, that the dot-product may also be written corresponds to the fact that the two tensor expressions and are the same. A difference is that the transpose repositions all indices of a tensor, whereas raising or lowering acts on individual indices.
130.243.68.240 ( talk) 14:26, 30 April 2019 (UTC)
This article applies with the more general tetrad formalism, aside from Ricci calculus § Differentiation, which assumes a coordinate basis. We should be clear about the applicability, and it would be nice to make even the differentiation section general, though a suitable source would be needed. — Quondum 18:09, 7 May 2019 (UTC)
The article does not make clear that the Christoffel symbols are only defined in the context of a connection nor that multiple metrics may induce the same connection. I'm not sure whether that belongs in the lead, but it should be somewhere prior to reference to the Christoffel sysmbols and metrics. Shmuel (Seymour J.) Metz Username:Chatul ( talk) 20:19, 25 October 2020 (UTC)
even restriction to a pseudo-Riemannian manifold is unduly restrictive: Ricci calculus does not require a metric tensor; it merely accommodates it.and in another change removed the footnote
While the raising and lowering of indices is dependent on the metric tensor, the covariant derivative is only dependent on the affine connection derived from it.from the lede. If the article is to be more general then there should be discussion of the facts that
where Γαβγ is a Christoffel symbol of the second kind.
This derivative is characterized by the product rule and applied to the metric tensor gμν it gives zero:
where are the components of the connection. When are the components of the metric connection of a metric tensor then is a Christoffel symbol of the second kind.
This derivative is characterized by the product rule. When are the components of the metric connection of a metric tensor then the covariant derivative of that metric tensor is zero:
<math>...</math>
rather than {{
math}} because I find LaTeX easier to read and edit than HTML+wikitext; the article appears to use both. Is there a preferred style?
Shmuel (Seymour J.) Metz Username:Chatul (
talk)
08:19, 11 April 2021 (UTC)You're welcome to make changes directly – discussion can follow if need be; this is often more efficient than proposing them first unless you still need to make up your mind. Style choice is always tricky.
I tend to think that finding a link should be where a reader can easily refer back to it, rather than having to do a text search, so I incline to linking a term more than once in a large article, so once per piece (e.g. section) that might be referenced. However, this is one of those style things that preferences vary on, and I have no strong feelings on this.
Using <math>...</math>
versus {{
math}} is all over the place on WP, and is complicated by different browsers and skins rendering things differently. I tend to try to keep the style in an article consistent, and if a style is established, to leave it as is. Inline <math>...</math>
has some issues of alignment, size and wrapping that can be problematic, and {{
math}} is not as neat standalone, nor is it as flexible. The style at the moment is {{
math}} when inline, and {{tag|math} on standalone lines. I would get a broader consensus from several editors before change this.
My knowledge of connections is primarily from WP. Now that I have separated the general connection from any more specific choice of connection, the latter should be edited freely. I inferred from Christoffel symbols that these apply to any metric connection and that a Levi-Civita connection is the special case defined as torsion-free (but I guess some people might reserve the term Christoffel symbols for a Levi-Civita connection); here we are using the same gamma symbols for the general connection. I would make it clear that there are distinct constraints: what constraint defines a metric connection, what constraint defines torsion-free, and that both constraints uniquely produce a Levi-Civita connection. Go ahead and edit this according to your understanding; I have a significant chance of unwittingly introducing some terminological or even a mathematical error. — Quondum 14:16, 11 April 2021 (UTC)
<math>...</math>
and {{
math}} were om different contexts. Does <math>...</math>
support the LaTeX environments
[a] for equations?One reference, MTW, is not entirely helpful here. To quote ( Gravitation: 208–209 ):
To quantify the contributions from ∇eβ and ∇ωα, i.e., to quantify the twisting, turning, expansion, and contraction of the basis vectors and 1-forms, one defines "connection coefficients":
- Γαβγ ≡ ⟨ωα, ∇γeβ⟩
and one proves that
⟨∇γωα, eβ⟩ = −Γαβγ.
This seems like a general definition that is metric- and torsion-independent (especially with "twisting, turning"). They then go on to present, without derivation or further constraints, a formula for Γαβγ that is dependent on the metric and is symmetric in β and γ, so they must have introduced the metric and tortionlessness assumptions into their derivation without mentioning it, even though the basis may be anholonomic. Very disappointing in their lack of rigour (introducing the restriction to the Levi-Civita connection without even mentioning the restriction).
Interestingly, they also say (
Gravitation: 210 ): In the holonomic case, the connection coefficients are sometimes called the Christoffel symbols.
This evidently adds a restriction to when the term applies. We would have to find a reference that explicitly gives formulae for the connection coefficients with torsion to get a better sense, but (because the statement is straight after the formula for the Levi-Civita connection coefficients) I assume that MTW use the term specifically to mean all of: metric, torsion-free, holonomic. —
Quondum
13:15, 12 April 2021 (UTC)
An example of where nomenclature varies is the term metric tensor; in some sources it implies torsion free, in others it doesn't.
As used in the article,
Shmuel (Seymour J.) Metz Username:Chatul ( talk) 02:28, 16 April 2021 (UTC)
is equivalent to
, i.e., vanishing torsion. Shmuel (Seymour J.) Metz Username:Chatul ( talk) 13:49, 16 April 2021 (UTC)
Should the article show the more general formula for vanishing torsion in a nonholonomic basis, i.e., with a Lie bracket?
Notes
The exterior derivative is a notable operator expressible in Ricci calculus, so it seems appropriate to include it. It might not be defined by this name in most texts, since pretty much every derivative can be constructed from a covariant derivative. However, derivatives that are independent of the connection should be shown independently, for example, the Lie derivative (already present). Though I have not seen this defined in a text, I expect the exterior derivative of any totally antisymmetric covariant tensor with components Xα...γ to be Xα...γ,δ in any coordinate basis. Expressions like this occur (e.g. in Maxwell's equations), but the name "exterior derivative" is not often used. — Quondum 16:42, 9 April 2021 (UTC)
To the question "Should the more general formula with Lie brackets be shown here?", yes, this should be more general, and I support this. Phrasing things only in terms of special cases (e.g. a coordinate basis) ends up with people not being aware of the more general treatment (e.g. that anholonomic bases exist). I would tend to add the expression for the torsion tensor (possibly under Ricci calculus § Notable tensors) and reference that categorizing connections. My original inclusion of the simpler expression for a holonomic basis was just a quick fix. — Quondum 02:26, 19 April 2021 (UTC)
![]() | This article is rated B-class on Wikipedia's
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Daily pageviews of this article
A graph should have been displayed here but
graphs are temporarily disabled. Until they are enabled again, visit the interactive graph at
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Just a proposal, would it help to add the following table and explanation to Ricci calculus (Raised and lowered indices) to illustrate how sup-/super-scripts and summation fit together in a way that relates "co-/contra-variance" to invariance?
Proposed table/text
| ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
This table summarizes how the manipulation of covariant and contravariant indices fit in with invariance under a passive transformation between bases, with the components of each basis set in terms of the other reflected in the first column. The barred indices refer to the final coordinate system after the transformation.
|
This is far clearer and briefer (to me at least...) than the main article Covariance and contravariance of vectors, and it fits in with the summary style of this article. It’s also another example for the manipulation of indices, including the Kronecker delta.
What do others think? Just a suggestion - the article is excellent and I don't want to touch it! As always I'm not forcing this in - it's for take or leave. Thanks once again to the editors here (although maybe/maybe not F= after all...). Maschen ( talk) 16:28, 15 August 2012 (UTC)
I can understand the removal from covariance and contravariance of vectors; very hasty at the time the image was added... However, it's been 3 months, no objections and one favour. I will take the liberty of adding it as planned long ago, better here than anywhere else (in a slightly extended form)... feel free to revert. Maschen ( talk) 10:15, 24 November 2012 (UTC)
This revert carries an edit summary that could be construed as a personal attack. The reverted edits introduced explicit notation giving the abstract tensors rather than only their components. We have stayed away from the abstract presentation in this article thusfar, but only as a matter of article style. While I do not object to the revert because of this, I do not in any way agree with the edit summary, and in particular with its inference about the editor. — Quondum 10:23, 2 January 2013 (UTC)
In the section on differentiation, for the partial derivative should we not have
My impression is that using the nabla is a preferred by some authors, is intuitive and fits into the notation.
Similarly, for the covariant derivative, seems to be notable. I notice that Penrose (in The Road to Reality) uses the nabla for the covariant derivative. Should we deal with these notations in the article? I have too little experience on what is notable here. — Quondum 12:03, 29 June 2013 (UTC)
The recently added notation seems to me to be too incomplete to be encyclopaedic. In particular, it omits crucial information from the notation that makes it pretty meaningless without explanatory text defining the family of curves that apply. It strikes me as made-up notation that an author (even MTW) might use by way of explaining something, not a notation that might see any use in other contexts. Does it really belong here? — Quondum 10:49, 10 August 2013 (UTC)
The operation referred to here as "sequential summation" doesn't make sense -- at least not as it's currently written.
Please explain how and why it's used, and why it's considered a tensor operation.
198.228.228.176 ( talk) 21:37, 6 February 2014 (UTC) Collin237
The earliest I can tell MTW use it is in chapter 4: Electromagnetism and differential forms, box 4.1 (p. 91). It only seems to be used in the context of p-forms (which are ... antisymmetric tensors). The authors only say "the sum is over i1 < i2 < i3 < ... in". So Quondum is correct so far. I don't know any other sources using this notation for this purpose, and it doesn't appear in Schouten's original work either (cited and linked in the article). But this summation seems to appear in a different notation which Quondum quotes above, in another reference by T. Frankel (which I don't have, and haven't seen it at the library).
Clearly, this convention of "sequential summation" exists so we shouldn't really remove it from the article. For now, let's just restrict to antisymmetric tensors. M∧Ŝ c2ħε Иτlk 08:27, 29 March 2014 (UTC)
Further notations appear to be introduced in this reference, specifically pp. 30–31. I don't understand German, but it appears to allow nesting of [], () and || on indices. My supposition is that the intention is that each of the inner nested index expressions is excluded from the higher-level symmetrization/antisymmetrization. Since this article covers a subset of exactly this type of notation, and this appears to be explicitly documented in this reference (and such exclusions make perfect sense), could someone with knowledge of German please verify my supposition so that we can include this? — Quondum 17:43, 29 March 2014 (UTC)
Does anyone know of conventions on the braiding of the free indices an expression in Ricci calculus? If so, this would be a useful addition to the article. The most obvious convention that might apply would lexicographic ordering be as in Abstract index notation#Braiding, but I do not know whether this extends to this context. — Quondum 00:09, 31 March 2014 (UTC)
This edit (with edit note I am referring to an expression where the x^{\mu} is in the denominator or x_{\mu} is in the denominator. I tried to clarify however I'm not the best at explaining. But I do think it is important enough to have.) appears to refer to a partial derivative. This is not a fraction, and has no numerator or denominator. In general the statement is also false, as the partial derivative only transforms covariantly (contravariantly) when the expression being differentiated is a scalar. This is handled under Ricci calculus#Differentiation, where I've added a mention of this special case. — Quondum 06:20, 26 August 2014 (UTC)
Yea that sounds good to me. Would you like to add it in or should I? Theoretical wormhole ( talk) 21:21, 26 August 2014 (UTC)
The present (early 2019) state of this article does very little to go beyond the subject as "a bunch of rules for operating on arrays of scalars". I think it would be useful to also provide connections with concepts from elementary linear algebra where convenient. For example (material from subsection "Upper and lower indices"):
Contravariant tensor components
An upper index (superscript) indicates contravariance of the components with respect to that index:
A vector corresponds to a tensor with one upper index . The counterpart of a tensor with two upper indices (a bivector) is less commonly seen in elementary linear algebra because it gets notationally cumbersome; many authors prefer to switch to tensor index notation when they need such objects.
Covariant tensor components
A lower index (subscript) indicates covariance of the components with respect to that index:
A tensor with lower indices may correspond to a map that takes vectors as arguments. For example, the metric tensor corresponds to the dot product of vectors.
Mixed-variance tensor components
A tensor may have both upper and lower indices:
A matrix is usually a tensor with one upper and one lower index — this makes matrix–vector multiplication correspond to applying a linear transformation to the vector, and makes matrix multiplication correspond to a contraction of tensor indices — but there are matrices which rather have two indices of the same variance: the matrix of a bilinear form naturally has two lower indices, and the R-matrix of a quasitriangular Hopf algebra naturally has two upper indices.
Ordering of indices is significant, even when of differing variance. However, when it is understood that no indices will be raised or lowered while retaining the base symbol, covariant indices are sometimes placed below contravariant indices for notational convenience (e.g. on the generalized Kronecker delta).
Raising and lowering indices
By contracting an index with a non-singular metric tensor, the type of a tensor can be changed, converting a lower index to an upper index or vice versa:
The base symbol in many cases is retained (e.g. using A where B appears here), and when there is no ambiguity, repositioning an index may be taken to imply this operation.
Repositioning an index often corresponds to taking a transpose (or similar, such as a conjugate transpose) in matrix formalism. For example, that the dot-product may also be written corresponds to the fact that the two tensor expressions and are the same. A difference is that the transpose repositions all indices of a tensor, whereas raising or lowering acts on individual indices.
130.243.68.240 ( talk) 14:26, 30 April 2019 (UTC)
This article applies with the more general tetrad formalism, aside from Ricci calculus § Differentiation, which assumes a coordinate basis. We should be clear about the applicability, and it would be nice to make even the differentiation section general, though a suitable source would be needed. — Quondum 18:09, 7 May 2019 (UTC)
The article does not make clear that the Christoffel symbols are only defined in the context of a connection nor that multiple metrics may induce the same connection. I'm not sure whether that belongs in the lead, but it should be somewhere prior to reference to the Christoffel sysmbols and metrics. Shmuel (Seymour J.) Metz Username:Chatul ( talk) 20:19, 25 October 2020 (UTC)
even restriction to a pseudo-Riemannian manifold is unduly restrictive: Ricci calculus does not require a metric tensor; it merely accommodates it.and in another change removed the footnote
While the raising and lowering of indices is dependent on the metric tensor, the covariant derivative is only dependent on the affine connection derived from it.from the lede. If the article is to be more general then there should be discussion of the facts that
where Γαβγ is a Christoffel symbol of the second kind.
This derivative is characterized by the product rule and applied to the metric tensor gμν it gives zero:
where are the components of the connection. When are the components of the metric connection of a metric tensor then is a Christoffel symbol of the second kind.
This derivative is characterized by the product rule. When are the components of the metric connection of a metric tensor then the covariant derivative of that metric tensor is zero:
<math>...</math>
rather than {{
math}} because I find LaTeX easier to read and edit than HTML+wikitext; the article appears to use both. Is there a preferred style?
Shmuel (Seymour J.) Metz Username:Chatul (
talk)
08:19, 11 April 2021 (UTC)You're welcome to make changes directly – discussion can follow if need be; this is often more efficient than proposing them first unless you still need to make up your mind. Style choice is always tricky.
I tend to think that finding a link should be where a reader can easily refer back to it, rather than having to do a text search, so I incline to linking a term more than once in a large article, so once per piece (e.g. section) that might be referenced. However, this is one of those style things that preferences vary on, and I have no strong feelings on this.
Using <math>...</math>
versus {{
math}} is all over the place on WP, and is complicated by different browsers and skins rendering things differently. I tend to try to keep the style in an article consistent, and if a style is established, to leave it as is. Inline <math>...</math>
has some issues of alignment, size and wrapping that can be problematic, and {{
math}} is not as neat standalone, nor is it as flexible. The style at the moment is {{
math}} when inline, and {{tag|math} on standalone lines. I would get a broader consensus from several editors before change this.
My knowledge of connections is primarily from WP. Now that I have separated the general connection from any more specific choice of connection, the latter should be edited freely. I inferred from Christoffel symbols that these apply to any metric connection and that a Levi-Civita connection is the special case defined as torsion-free (but I guess some people might reserve the term Christoffel symbols for a Levi-Civita connection); here we are using the same gamma symbols for the general connection. I would make it clear that there are distinct constraints: what constraint defines a metric connection, what constraint defines torsion-free, and that both constraints uniquely produce a Levi-Civita connection. Go ahead and edit this according to your understanding; I have a significant chance of unwittingly introducing some terminological or even a mathematical error. — Quondum 14:16, 11 April 2021 (UTC)
<math>...</math>
and {{
math}} were om different contexts. Does <math>...</math>
support the LaTeX environments
[a] for equations?One reference, MTW, is not entirely helpful here. To quote ( Gravitation: 208–209 ):
To quantify the contributions from ∇eβ and ∇ωα, i.e., to quantify the twisting, turning, expansion, and contraction of the basis vectors and 1-forms, one defines "connection coefficients":
- Γαβγ ≡ ⟨ωα, ∇γeβ⟩
and one proves that
⟨∇γωα, eβ⟩ = −Γαβγ.
This seems like a general definition that is metric- and torsion-independent (especially with "twisting, turning"). They then go on to present, without derivation or further constraints, a formula for Γαβγ that is dependent on the metric and is symmetric in β and γ, so they must have introduced the metric and tortionlessness assumptions into their derivation without mentioning it, even though the basis may be anholonomic. Very disappointing in their lack of rigour (introducing the restriction to the Levi-Civita connection without even mentioning the restriction).
Interestingly, they also say (
Gravitation: 210 ): In the holonomic case, the connection coefficients are sometimes called the Christoffel symbols.
This evidently adds a restriction to when the term applies. We would have to find a reference that explicitly gives formulae for the connection coefficients with torsion to get a better sense, but (because the statement is straight after the formula for the Levi-Civita connection coefficients) I assume that MTW use the term specifically to mean all of: metric, torsion-free, holonomic. —
Quondum
13:15, 12 April 2021 (UTC)
An example of where nomenclature varies is the term metric tensor; in some sources it implies torsion free, in others it doesn't.
As used in the article,
Shmuel (Seymour J.) Metz Username:Chatul ( talk) 02:28, 16 April 2021 (UTC)
is equivalent to
, i.e., vanishing torsion. Shmuel (Seymour J.) Metz Username:Chatul ( talk) 13:49, 16 April 2021 (UTC)
Should the article show the more general formula for vanishing torsion in a nonholonomic basis, i.e., with a Lie bracket?
Notes
The exterior derivative is a notable operator expressible in Ricci calculus, so it seems appropriate to include it. It might not be defined by this name in most texts, since pretty much every derivative can be constructed from a covariant derivative. However, derivatives that are independent of the connection should be shown independently, for example, the Lie derivative (already present). Though I have not seen this defined in a text, I expect the exterior derivative of any totally antisymmetric covariant tensor with components Xα...γ to be Xα...γ,δ in any coordinate basis. Expressions like this occur (e.g. in Maxwell's equations), but the name "exterior derivative" is not often used. — Quondum 16:42, 9 April 2021 (UTC)
To the question "Should the more general formula with Lie brackets be shown here?", yes, this should be more general, and I support this. Phrasing things only in terms of special cases (e.g. a coordinate basis) ends up with people not being aware of the more general treatment (e.g. that anholonomic bases exist). I would tend to add the expression for the torsion tensor (possibly under Ricci calculus § Notable tensors) and reference that categorizing connections. My original inclusion of the simpler expression for a holonomic basis was just a quick fix. — Quondum 02:26, 19 April 2021 (UTC)