Talk:Torsion tensor

Understanding

I think I'm starting to understand this. Does this accurately describe torsion?:

Suppose you are in the day-to-day flat 3-D universe in a space ship just to the left of someone else in another spaceship, both of you facing in the same direction. If you both move forward that person remains just to your right.

• If you were not in flat space, but you were in torsion-free space, as you both move forward, the person next to you might move further away, but if you only took a small step forward, the other person would still be just to your right.
• In contrast, if the universe were flat but not torsion-free, as you moved forward, the other person would be just as far away, but might appear to orbit you, so that after some distance the other person would appear to be above you.

Is that an accurate description of torsion? —Ben FrantzDale 23:01, 28 April 2007 (UTC)

Yes, I think so, the section on "interpretation" goes into this in a more precise way. 67.198.37.16 (talk) 04:15, 21 April 2016 (UTC)

non-sense

The following makes no sense:

The components of the torsion tensor $T^{c}{}_{ab}$ can be derived by setting $X=e_{a},Y=e_{b}$ and by introducing the commutator coefficients given by $\gamma _{ab}^{c}e_{c}:=[e_{a},e_{b}]$. We finally obtain a component expression of the torsion tensor,

$T^{c}{}_{ab}:=\Gamma ^{c}{}_{ab}-\Gamma ^{c}{}_{ba}-\gamma ^{c}{}_{ba}$

It implies X=Y which is just silly. If I stretch my imagination a bit then γ might be meant to be simply [X, Y] which would make this the component version of the tensorial definition. Someone should fix or remove this. --MarSch 10:32, 20 May 2007 (UTC)

I agree that the statement was a little odd. The author clearly intended a and b to signify numerical indices of the frame (rather than abstract indices). This is the only interpretation where you can have X ≠ Y. Anyway, feel free to do with it as you like. Physicists, I believe, generally use Latin subscripts such as i,j,k,... (or sometimes Greek indices) for numerical indices as opposed to the letters from the beginning of the alphabet such as a,b,etc. I have preserved the notation, but perhaps it would be a good idea to bring it in line with prevailing conventions. Silly rabbit 11:48, 20 May 2007 (UTC)

Move to torsion (differential geometry)?

I propose moving this to torsion (differential geometry), currently a redirect here. That way the article can discuss torsion in a slightly broader context than just the torsion tensor (torsion form, intrinsic and extrinsic torsion, torsion of a prolongation, etc.) Silly rabbit 12:30, 20 May 2007 (UTC)

Probably a good idea. --MarSch 13:34, 20 May 2007 (UTC)

I agree to move this article to torsion (differential geometry). Then let's merge torsion of curves to the moved article. --Acepectif 03:20, 31 August 2007 (UTC)

Some other things to think about

• Torsion of almost complex structures: The Nijenhuis tensor.
• The torsion of a hermitian connection vanishes if and only if the manifold is Kähler.
• Analytic torsion and the Weitzenbock identity.
• Torsion of Cartan connections.
• Weyl geometries and projective connections.

Silly rabbit 16:12, 8 June 2007 (UTC)

Also torsion in supersymmetry. 67.198.37.16 (talk) 04:45, 21 April 2016 (UTC)

Does torsion only describe twisting?

My understanding is that expansion or contraction of a frame as it moves along a curve is also torsion. Is this wrong, or does the description of torsion need revision? 81.79.139.224 (talk) 07:29, 20 April 2008 (UTC)

I think that could be true for a general affine connection, but it seems unlikely for a metric connection, which would still preserve the metric and therefore also the length of the frame elements. siℓℓy rabbit (talk) 14:06, 6 August 2008 (UTC)

Apparently Cartan discusses the general case of a geometrical interpretation of torsion for an affine connection. I will try to add something about this over the next days, as soon as I decipher the Cartan paper (which could take some time). siℓℓy rabbit (talk) 18:00, 8 August 2008 (UTC)

Section Affine develompents seems incomplete

This section suggests that a connection is torsion-free if and only if the affine developments of loops are also loops. But it seems to me that a torsion-free (but not curvature-free) connection can also take loops into open curves. For ecample, the affine developments of great triangles on the sphere are open (see PhysicsForums, Torsion, affine development and Levi-Civita connection). Perhaps the statement is valid only for curvature-free connections. More precise description or citations will be appreciated. 89.135.19.87 (talk) 07:31, 22 November 2008 (UTC)

You're right. There is something wrong with the section. An easy way to see it is that geodesics develop to straight lines. But on the sphere the geodesics are great circles (and so are in particular closed). I'll see what I can do to mend it. siℓℓy rabbit (talk) 14:14, 22 November 2008 (UTC)

Missing equals sign in definition?

Wouldn't

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

just be twice

$-[X,Y]$

given the definition at Lie bracket of vector fields? I think that this line should instead say

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

—Preceding unsigned comment added by Rudminjd (talk • contribs) 17:33, 15 April 2009 (UTC)

This was an old question - but... No, the equation is correct, as stated. For one thing, without the $-[X,Y]$, the "tensor" $T'(X,Y)=\nabla _{X}Y-\nabla _{Y}X$ would not satisfy $T'(fX,Y)=fT'(X,Y)$, as it should, where $f$ is a differentiable function.

I remain confused. Apparently $\nabla _{X}Y-\nabla _{Y}X\neq [X,Y]$. But in my (possibly wrong) understanding, the article on Lie bracket of vector fields describes $[X,Y]$ as $\nabla _{X}Y-\nabla _{Y}X$. Could someone please explain the difference more clearly, for example by using index notation in a general coordinate vector base? 2A02:A210:2142:6C00:F804:B8B6:7CC8:55AD (talk) 22:28, 18 March 2020 (UTC)

The article Lie bracket of vector fields never says that $[X,Y]$ is $\nabla _{X}Y-\nabla _{Y}X$ (and I just looked at older versions too; it never said that.) It is true that $[X,Y]=L_{X}Y={\frac {1}{2}}(L_{X}Y-L_{Y}X)$ and it is true that the Lie derivative $L$ is a kind-of derivative. However, it is NOT the same as $\nabla$. The $\nabla$ is the affine connection, i.e. the ordinary derivative plus some extra stuff providing info on how adjacent fibers are glued together. The goal of subtracting out the $L_{X}Y$ part is to leave behind only gluing-ness, and remove the differential-ness. Well, that, and to get everything to transform properly under coordinate changes. 67.198.37.16 (talk) 05:02, 11 November 2023 (UTC)

Incorrect identity?

$d(trT)=0$

isn't true in general, and it isn't a consequence of any Bianchi identity!

Hep thinker (talk) 10:36, 13 July 2009 (UTC)

I concur. Since it is possible to define a connection with prescribed torsion, tr T is an essentially arbitrary one-form, at least locally. In particular, it is not closed. Sławomir Biały (talk) 15:58, 17 August 2009 (UTC)

Comment on Finsler-geometry

I think that characterizing Finsler geometry as a "non-metric situation" is non-sensical. A better contrast to Riemannian geometry would be that Finsler-manifolds are "not infinitesimally Euclidean" like Riemanian manifolds are in the sense that the norm induced into each $T_{x}M$ is given by an inner product.

As matter of fact there is a generalization of the concept of torsion (more adequate for Finsler-geometry) given by the Frölicher-Nijenhuis bracket

$T(X,Y):=[J,v](X,Y)$

where $J$ is the tangent structure and $v$ is the vertical projection of a nonlinear connection on $(TM,\pi ,M)$. Perhapt this should be incorporated into the article. —Preceding unsigned comment added by 82.181.89.137 (talk) 12:53, 13 August 2009 (UTC)

Yes, it should be. There is another formulation that makes no reference to metrics, and that is this: if one has a solder form and writes θ for it, then the torsion tensor Θ is given by Θ = D θ (with D the exterior covariant derivative). For any given connection ω, there is a unique one-form σ on the tangent bundle TE that is vanishing in the vertical bundle, and is such that ω+σ is another connection 1-form that is torsion-free. The resulting one-form ω+σ is nothing other than the Levi-Civita connection. One can take this as a definition: since the torsion is given by $\Theta =D\theta =d\theta +\omega \wedge \theta$, the vanishing of the torsion is equivalent to having $d\theta =-(\omega +\sigma )\wedge \theta$, and it is not hard to show that σ must vanish on the vertical bundle, and that σ must be G-invariant on each fibre (more precisely, that σ transforms in the adjoint representation of G). Note that this defines the Levi-Civita connection without making any explicit reference to any metric tensor. Properly speaking, the metric is an example of a solder form. I am somewhat confused by exactly when and where one can write down a solder form, and thus don't quite know how to turn this into a fully general statement (and don't currently have any references that provide/clarify such details). 67.198.37.16 (talk) 21:30, 22 April 2016 (UTC)

This means that the article is wrong when it says that "torsion" in Finsler geometry is the same as the torsion of some affine connection. Where did all this word salad come from? --Svennik (talk) 22:15, 23 August 2023 (UTC)

I just looked. I don't see any place where the article says that it's the "same thing". 67.198.37.16 (talk) 04:12, 11 November 2023 (UTC)

The article suggests that "torsion" in Finsler geometry is generalised by "torsion" of a smooth manifold equipped with an affine connection. At least that's what the introduction to the article suggests in paragraphs 1 and 2. While that doesn't mean it's saying they're the same thing, it fails to provide evidence that those two notions are related in anything but name. And indeed, one discussion below suggests that there's no relationship between these at all. The article is then simply wrong. Svennik (talk) 19:42, 22 November 2023 (UTC)

Pseudoscience

TimothyRias (talk · contribs) moved Torsion field to Torsion field (pseudoscience), and then converted the redirect to point to Torsion tensor. (With similar changes to the talk pages.) I presume that this was because he thought that there was something scientific about torsion which made it distinct from the pseudoscience described in Torsion field (pseudoscience). I am not convinced that that is so. What evidence do you have that any such thing as torsion exists in the real world? JRSpriggs (talk) 09:22, 29 March 2011 (UTC)

!? The torsion tensor is a standard concept in differential geometry; its covered in all textbooks on differential or affine geometry. (also covered in newer books on general relativity, e.g. misner, thorne, wheeler, 1972) The torsion-field thing is an interesting bit of pseudoscience. Quite unrelated, except for two facts: first, that pseudoscience loves to mis-appropriate science terms, thus adding to the confusion, and, second, that anything that "spins" is like catnip to UFO enthusiasts. 67.198.37.16 (talk) 04:26, 21 April 2016 (UTC)

A very simple example in the real world is the torsion coefficients of non-orientable surfaces such as the Klein bottle and Boy surface - these manifolds look very pretty when made of glass. It is an open question whether the Universe as a whole has torsion. — Cheers, Steelpillow (Talk) 08:02, 21 April 2016 (UTC)

Twisting of reference frames

This section is confusing. Does the $\nabla$ represent a covariant derivative? If so, the two formulas

$\left.\nabla _{\partial /\partial t}{\frac {\partial }{\partial x}}\right|_{x=0}=0.$

and

$\left.T\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial t}}\right)\right|_{x=0}=\left.\nabla _{\frac {\partial }{\partial x}}{\frac {\partial }{\partial t}}\right|_{x=0}.$

are inconsistent with the definition

$T(X,Y):=\nabla _{X}Y-\nabla _{Y}X-[X,Y]$

given earlier in the article. This last formula implies that the torsion is skew symmetric. — Preceding unsigned comment added by 99.100.184.6 (talk) 12:22, 29 September 2012 (UTC)

They are consistent - while $T(X,Y)$ is always anti-symmetric in $X$ and $Y$, the article only maintains the validity of the (non-antisymmetric) equation

$T\left(X,Y\right)|_{x=0}=\left.\nabla _{\frac {\partial }{\partial x}}{\frac {\partial }{\partial t}}\right|_{x=0}$

with $X={\frac {\partial }{\partial x}}$ and $Y={\frac {\partial }{\partial t}}$, where $x$ and $t$ are as in the article. — Preceding unsigned comment added by 68.196.59.187 (talk) 17:58, 3 March 2013 (UTC)

And of course, $\left[{\frac {\partial }{\partial x}},{\frac {\partial }{\partial t}}\right]=0$ since they're coordinate frames (holonomic). 67.198.37.16 (talk) 04:28, 21 April 2016 (UTC)

Classic extremely bad exposition

In the (sub-)section Twisting of reference frames this passage appears:

"The case of a manifold with a (metric) connection admits an analogous interpretation. Suppose that an observer is moving along a geodesic for the connection. Such an observer is ordinarily thought of as inertial since she experiences no acceleration."

But this is the only occurrence of the word "acceleration" that appears in the article. So this passage is more confusing than helpful. (And incidentally, the linked article on acceleration contains no reference to the word "geodesic".)

If there is a relationship between geodesics and the absence of acceleration, this article would be a good place to explain it, not merely refer to it mysteriously.Daqu (talk) 23:22, 5 October 2015 (UTC)

Concept from general relativity: geodesics are precisely those curves given by observers in free-fall, i.e. no acceleration. 67.198.37.16 (talk) 04:01, 21 April 2016 (UTC)

An object in free-fall does accelerate. --Svennik (talk) 22:01, 23 August 2023 (UTC)

Um, hey, by definition, free falling reference frames do not accelerate. They are inertial frames, and they have that name for a reason. See Equivalence principle if unclear on the concept. 67.198.37.16 (talk) 04:15, 11 November 2023 (UTC)

I never studied GR so I made a mistake. Thanks for the link. Svennik (talk) 19:45, 22 November 2023 (UTC)

Lie dragging and parallel transport are fundamentally different

I'm not convinced the section on Torsion tensor#Twisting of reference frames is correct. In particular, it makes the very serious conceptual mistake of confusing parallel transport with Lie dragging. While these are indeed two ways of picking up and moving a tangent vector to another base point, how they do this is different, and they produce different results. I'm not convinced at the moment that diff-geo torsion has any relationship to physical torsion or the torsion of curves. These concepts might share a name, but I'm not sure whether it's merely Cartan's fancy that they are related in any way. --Svennik (talk) 09:27, 23 August 2023 (UTC)

In the page 44 of the following book,

• Loring W. Tu (2017-06-15). Differential Geometry: Connections, Curvature, and Characteristic Classes. Graduate Texts in Mathematics. Vol. 275. Springer. ISBN 978-3319550824.

the author, Loring W. Tu, says,

There does not seem to be a good reason for calling $T(X,Y)$ the torsion.

I therefore think that "Twisting of reference frames" section, as well as the beginning of this article (both of which say that the torsion tensor has a relationship with (torsion of a curve in) the Frenet–Serret formulas) are incorrect.

Moreover, in the page 211 of the following book,

• Richard Sharpe (1997-06-12). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Graduate Texts in Mathematics. Vol. 166. Sprinver. ISBN 978-0387947327.

the author, Sharpe, says (when we consider $\nabla$ as (a part of) a Cartan connection)

One interpretation of the meaning of a torsion free geometry is that infinitesimal loops have no translation part to their holonomy.

That is, the interpretation of the "torsion (tensor)" corresponds not to "torsion" but to "translation".

In fact, the same authors computes the curvature (in Cartan's sense) of Riemann geometry in the page 68 of the following paper. Here, the torsion tensor corresponds to the translation part of the curvature in Cartan's sense.

• Richard Sharpe (2002). An introduction to Cartan Geometries. Proceedings of the 21st Winter School "Geometry and Physics". pp. 61–75.

--位相空間を中和 (talk) 14:16, 20 November 2023 (UTC)(I'm not native of English. So, I am sorry for my poor English.)

My two cents. I think this section has some value if we can make proper sense of it. I think starting with a concrete example might be helpful. Suppose we are in (flat) Euclidean space $\mathbb {R} ^{3}$, and we take the standard coordinate frame $e_{1},e_{2},e_{3}$. We put a metric connection on $\mathbb {R} ^{3}$ with torsion $T(X,Y)=X\times Y$ (Euclidean cross product). Now consider the parallel propagation of a vector $S=ae_{1}+be_{2}$ along the $x$-axis. The equations for parallel propagation are ${\dot {a}}-b={\dot {b}}+a=0$, and so a solution with initial condition $S(0)=e_{2}$ is $S=\cos x\,e_{2}-\sin x\,e_{3}$, and the orbit of the tip of the vector as it moves along the curve is the helix $x\,e_{1}+\cos x\,e_{2}-\sin x\,e_{3}$. This, at least, helps to justify the name "torsion", and it seems Loring Tu didn't think very hard about the question.

In the case of a Riemannian metric, given a geodesic parametrized by arclength $u$, we can introduce a coordinate system such that the metric has the form

$$du^{2}+\sum _{i}a_{i}(u)^{2}\,(dx^{i})^{2}+O(u^{2},x^{2})$$

where the vector fields $\partial /\partial x^{i}$ are Jacobi fields along the geodesic. Then the vector fields $X_{i}=a_{i}^{-1}\partial _{x^{i}}$ are parallel with respect to the Levi-Civita connection (modulo $O(u)$). On the other hand, for a connection with torsion $T$, we have

$$\nabla _{\partial _{u}}X_{i}=T(\partial _{u},X_{i}).$$

So, roughly speaking, if the tidal effect in any direction from an observer is purely compressive/expansive, then the torsion represents the additional (shearing/rotation) force needed to keep a rod in that direction parallel. That seems like the gist of the section, although it is somewhat inartful (and arguably inaccurate) in its translation into "intuitive" language. Tito Omburo (talk) 17:48, 5 February 2024 (UTC)

> it seems Loring Tu didn't think very hard about the question.

Michael Spivak says the same thing

no one seems to have a good explanation for the term "torsion"

in

• Michael Spivak. A Comprehensive Introduction to Differential Geometry. VOLUME TWO (Second Edition ed.). Publish or Perish, Incorporated. ISBN 978-0914098805, p.234.

>we are in (flat) Euclidean space $\mathbb {R} ^{3}$, and we take the standard coordinate frame $e_{1},e_{2},e_{3}$

In this case, $\nabla _{X}Y$ is equals to directional derivative, $X^{i}{\partial x^{i}}Y^{j}$. Hence, The torion is 0.

Could you clarify the citation for your opinion?

--Idutsu (talk) 09:51, 13 March 2024 (UTC)

As I said, you give Euclidean space the metric connection with torsion $T(X,Y)=X\times Y$. (Thus the Christoffel symbols are $\nabla _{e_{i}}e_{j}=e_{i}\times e_{j}$), then the parallel transport is precisely as I have described. More generally, if we give the connection $\nabla _{e_{i}}e_{j}=\tau e_{i}\times e_{j}$ for a constant τ, then the development of the unit circle is a curve of unit curvature and torsion τ. It's well-known that torsion gives a screw dislocation of the development, analogously to the torsion of curves, notwithstanding Spivak and Tu, who seem not to have really thought about it. Tito Omburo (talk) 13:03, 13 March 2024 (UTC)

>It's well-known that torsion gives a screw dislocation of the development,

If it is well known, could you clarify the citation, in order to show that your opinion is not original research?

All content must be verifiable. The burden to demonstrate verifiability lies with the editor who adds or restores material, especially when the editor says the experts, Tu and Spivak "seem not to have really thought about it".

--Idutsu (talk) 13:40, 13 March 2024 (UTC)

This is already a cited fact in the article, and is even supported by the Sharpe citation someone gave above. Not sure what the problem is. Certainly not original research. Might I suggest that you first try to understand the example I gave before making such suggestions? Thanks, Tito Omburo (talk) 14:12, 13 March 2024 (UTC)

>This is already a cited fact

One problem is in the following part, where you do not gave any citation:

> The torsion tensor is related to, although distinct from, the torsion of a curve, as it appears in the Frenet–Serret formulas, which quantifies the twist of a curve about its tangent vector as the curve evolves

I read Kobayashi-Nomizu, which you refered in other parts of this article, but the book does not say about the above fact.

---Idutsu (talk) 07:49, 14 March 2024 (UTC)

I agree that this sentence is problematic, perhaps not for the reason you think. The torsion of a curve gives a displacement out of the osculating plane. It has more to do with translation (i.e., dislocation) than with twisting (rotation). (E.g., [1]). Tito Omburo (talk) 09:57, 14 March 2024 (UTC)

This is the sentence which you edit and you yourself says it is problematic?

Then I revert your edits. Do you agree on it?

--Idutsu (talk) 11:24, 14 March 2024 (UTC)

No. The problematic wording was not added by me, or in that edit. Which removed some other, equally problematic content, and provided other useful and mathematically correct context for the article. Tito Omburo (talk) 12:11, 14 March 2024 (UTC)

>the notion of torsion is a manner of characterizing the amount of slipping or twisting that a plane does when rolling along a surface or higher dimensional affine manifold.

Sharpe (p.381.) only shows the relationship between the connection and "rolling without slipping or twisting" only for the Levi-Civita connection, whose torsion tensor is 0.

Your sentence claims that this relationship holds even for the connection with non-zero torsion.

So, you have to clarify the citation for this sentence.

--Idutsu (talk) 12:23, 14 March 2024 (UTC)

I have edited the sentence in accord with my observation above. I assume you are now satisfied? Tito Omburo (talk) 12:14, 14 March 2024 (UTC)

We edit concurrently...

>I have edited the sentence

Which sentence you are talking about?

> (A) The torsion tensor is related to, although distinct from, the torsion of a curve, as it appears in the Frenet–Serret formulas, which quantifies the twist of a curve about its tangent vector as the curve evolves

>(B) the notion of torsion is a manner of characterizing the amount of slipping or twisting that a plane does when rolling along a surface or higher dimensional affine manifold. Idutsu (talk) 12:26, 14 March 2024 (UTC)

We were discussing only (A) before. Do you wish to now dispute (B), which is a different sentence than what you objected to earlier? Sharpe is not a good source on torsion, because he discusses almost exclusively the torsion-free case. Tito Omburo (talk) 12:33, 14 March 2024 (UTC)

>Do you wish to now dispute (B),

Sure. I think (A) is problematic also but I agree that (A) is not your original sentence.

>Sharpe is not a good source

If you think so, you have to clarify the citation because, even if you are sure something is true, it must have been previously published in a reliable source before you can add it.

As I said before, all content must be verifiable. The burden to demonstrate verifiability lies with the editor who adds or restores material. Idutsu (talk) 12:40, 14 March 2024 (UTC)

Did you read the rest of the sentence after "Sharpe is not a good source". He does not treat the case of nonzero torsion except in passing. Why do you think it is a good source for the present article? Nothing I have written contradicts Sharpe, by the way. Verifiability can be demonstrated by sources other than one book. Tito Omburo (talk) 12:48, 14 March 2024 (UTC)

>Verifiability can be demonstrated by sources other than one book.

Then, could you show the "sources other than one book"?

>Nothing I have written contradicts Sharpe, by the way.

It is not important whether your sentence is true or not.

Even if it is true, you can add your sentence only if you clarify the source.

--Idutsu (talk) 13:00, 14 March 2024 (UTC)

You brought up Sharpe, as if it was the only source cited. In fact, it is not even cited in the article. So I am confused what you think this source does not verify, or what you think should be verified by another source. Tito Omburo (talk) 13:03, 14 March 2024 (UTC)

Could you read the following rule of Wikipedia: The burden to demonstrate verifiability lies with the editor who adds or restores material, in this case, you.

Sharpe is the only source (for "rolling") which I can show you.

However, this is not problem. I am not required to show the sources, because I am not "the editor who adds or restores material".

---Idutsu (talk) 13:10, 14 March 2024 (UTC)

What currently fails WP:V? You do have a burden of saying what you think is not supported. Tito Omburo (talk) 13:16, 14 March 2024 (UTC)

We need the third opinion. Do you agree on it?

--Idutsu (talk) 13:28, 14 March 2024 (UTC)

Probably should post at WT:WPM. But it seems like this could be easily settled if you would just say what needs a source. Tito Omburo (talk) 13:32, 14 March 2024 (UTC)

Sure. Then I will add the topic to it.

To help the third one to understand our discussion, I summarize our discussion in my view point ( which is recommended in Wikipedia:Third opinion).

If you disagree my summary, could you write another summary in your view point?

--Idutsu (talk) 13:47, 14 March 2024 (UTC)

Ok, but I suggest posting to WT:WPM rather than WP:3O. Tito Omburo (talk) 13:48, 14 March 2024 (UTC)

Added to WT:WPM --Idutsu (talk) 14:20, 14 March 2024 (UTC)

Discussion between Tito Omburo and Idutsu

We are discussing about the recent edit of Tito Omburo at "Lie dragging and parallel transport are fundamentally different".

The summary of the discussion is as follows.

[Idutsu's opinion]
In my view point, Tito Omburo's edits are original researchs. He/She is writing about the intuitive meaning behind the torsion tensor $T(X,Y)$ in (for instance) the first part of this article:

>the notion of torsion is a manner of characterizing the amount of slipping or twisting that a plane does when rolling along a surface or higher dimensional affine manifold.

However, the experts of this topic say there are no explanation for it:

> Michael Spivak [1] : no one seems to have a good explanation for the term "torsion"
> Loring W. Tu [2] : There does not seem to be a good reason for calling $T(X,Y)$ the torsion.

I therefore think that the verifiable sources for his/her edits are definitely required.

However, he/she seems to think that they are not required. He/She says his/her edits are correct because

>it seems Loring Tu didn't think very hard about the question.
> Spivak and Tu, who seem not to have really thought about it

I have already said to him/her about WP:VERIFY and WP:BURDEN three times...

Note that the book of Sharpe [3] are talking about the related topic (that is, "slipping or twisting") at p.381, but this part of the book discusses only in the case of Levi-Civita connection, whose torsion tensor is 0. The book, therefore, cannot be used as the source of his/her edits.

• [1] Michael Spivak. A Comprehensive Introduction to Differential Geometry. VOLUME TWO (Second Edition ed.). Publish or Perish, Incorporated. ISBN 978-0914098805, p.234.
• [2] Loring W. Tu (2017-06-15). Differential Geometry: Connections, Curvature, and Characteristic Classes. Graduate Texts in Mathematics. Vol. 275. Springer. ISBN 978-3319550824. p.44.
• [3] Richard Sharpe (1997-06-12). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Graduate Texts in Mathematics. Vol. 166. Sprinver. ISBN 978-0387947327. p.381

--Idutsu (talk) 14:17, 14 March 2024 (UTC)

• This is a very misleading summary of the discussion. Firstly, none of the sources Idutsu mentions are even referenced in the article. As far as I can tell, everything they are concerned about currently has a citation. I have attempted to press them for exactly what needs to be verified further, and received no coherent explanation.

Secondly, it is unclear how the sentence they object to is remotely connected with Spivak and Tu's (erroneous) contention that there is "no good reason for calling T(X,Y) the torsion." Whether there is a good reason or not, it is called the torsion, so this is a red herring.

Thirdly, even Cartan gave a reason for calling it the torsion, basically to do with twisting of reference frames. It is difficult to find secondary sources for this particular explanation, and indeed the article prior to my edits has a lot of unreferenced material along these lines. I have trimmed this out, in favor of the more modern view in geometric elasticity, that the torsion is a dislocation vector in the affine development (this is discussed in Kobatashi and Nomizu as well as some other references in the article). I'd like to find references that explain Cartan's view more coherently (in discussing a simple three-dimensional example (Cartan 1922): "The space F thus defined admits a six-parameter group of transformations; it would be our ordinary space as viewed by observers whose perceptions have been twisted. Mechanically, it corresponds to a medium having constant pressure and constant internal torque.") Tito Omburo (talk) 14:49, 14 March 2024 (UTC)

I now find you add the reference([2]) after our discussion. I have not realize it.

Could you specify the page of it which you refer?

>Cartan gave a reason for calling it the torsion

Then you can use the Cartan's paper as a reference. : Sorry, I misunderstood what you are saying.

---Idutsu (talk) 14:58, 14 March 2024 (UTC)

Page 5 in that reference. Another further item is that Bill Thurston wrote a stack exchange post regarding the "rolling without slipping" interpretation of torsion. It is mostly good (probably not as a reference, so I've left it as an external link), although I think he is slightly wrong that torsion doesn't make sense in two dimensions. For example, one can take the two-dimensional Euclidean plane with a connection defined by $\nabla _{e_{1}}e_{2}=e_{1}=-\nabla _{e_{2}}e_{1}$. Then the development of the unit circle is no longer a circle (it is not closed). Tito Omburo (talk) 15:17, 14 March 2024 (UTC)

Thank you, I read them tommorow. (I am in Japan and the current time is midnight)...

---Idutsu (talk) 15:23, 14 March 2024 (UTC)

I read the materials which you refered and am convinced that what you are saying is true.

In my understanding, you wants to say three things.

• (A) the torsion is "slipping or sliding"
• (B) the torsion is closure failure of sliding along infinitesimal circle.
• (C) the torsion relates Frenet–Serret

(A) is satisfied because

• (1) The Cartan displacement (a.k.a. development) is equal to "rolling without slipping or sliding" in the case of the Levi-Civita connection
• (2) So, you (and Thurston) call the Cartan displacement "rolling" even for general connection.
• (3) The torsion of the Levi-Civita connection, which is "rolling without slipping or sliding", is 0. Hence the torsion corresponds with "slipping or sliding".

(B) is satisfied because of (1), (2) and

• (4) the torsion is closure failure of the Cartan displacement along infinitesimal circle

(C) is written in the post of Thurston.

The only (small) mistake of you is the following part:

> a closed curve that begins and ends at the same point

You should say "infinitesimal" closed curve

> It is difficult to find secondary sources

(1) is written in Sharpe p.381 and (4) is written at page 3. in the paper of Hehl and Obukhov which you refer.

(2) and (3) are simple observations.

So, how do you think you write (A) and (B) with

• the above references for (1) and (4)
• explicit mentioning about (2) and (3) to clarify the reasoning behind (A) and (B)

I failed to follow the discussion of Thurston about Frenet–Serret (sorry...) but I think you can use the Thurston's post as the reference of (C).

--Idutsu (talk) 16:58, 15 March 2024 (UTC)

There is no mistake. Any curve that is null homotopic will do. It is not necessary that it be infinitesimal, because the translation part of the holonomy is zero. (The theorem is that if the torsion is zero, the development of any closed null homotopic curve is closed. This follows by the Ambrose-Singer theorem. One can still have momodromy for non-trivial elements of the fundamental group, but that is all.) Tito Omburo (talk) 19:04, 15 March 2024 (UTC)

Intro paragraphs off-topic.

The first two paragraphs in this article are about some definition of torsion contrary to WP:LEAD. The lead should be about a "tensor" of type "torsion".

A further symptom of this problem is the first main heading, "The torsion tensor". The article title should never need to be a section heading. Johnjbarton (talk) 15:54, 14 March 2024 (UTC)

Is this better? Tito Omburo (talk) 16:12, 14 March 2024 (UTC)

Sorry, but no, the linked edit makes no difference. The material in the first two paragraphs do not belong in the introduction.

The lede (first sentence) should define the topic "torsion tensor"; the remainder of the intro should summarize the article. No part of the intro should be unique to the intro other than perhaps the opening sentence. Johnjbarton (talk) 17:09, 14 March 2024 (UTC)

The details of development are discussed (with examples) later in the article. My impression was that the lead is supposed to summarize the article for someone with very little math background. Tito Omburo (talk) 17:18, 14 March 2024 (UTC)

Yes, the summary should assume as little background as reasonable to the topic. But the summary needs to be about the topic 'tensor' not about an adjective related to the topic, 'torsion'. The connection, beyond perhaps a sentence in the intro, belongs in the article.

I see you have made subsequent edits that address my complaint, thanks! Johnjbarton (talk) 17:36, 14 March 2024 (UTC)

I think the article history might explain why there were some conflicting uses of torsion hanging around. Originally, this seems to have been an umbrella article for all the different types of torsion in differential geometry. I note that torsion (differential geometry) is actually a redirect here. Perhaps such an umbrella article could yet be written, but it might be inherently fraught. Tito Omburo (talk) 17:42, 14 March 2024 (UTC)

How about now? Tito Omburo (talk) 17:29, 14 March 2024 (UTC)