Talk:Affine connection

mergefrom connection (affine bundle)

The article connection (affine bundle) consists of a cut-n-paste of a bunch of standard eqns from a standard textbook, with zero explanation or commentary. I propose that all of that content should be moved here, where at least we get some explanations. 67.198.37.16 (talk) 18:43, 24 October 2020 (UTC)

The current article seems fine. The stub connection (affine bundle) on the other hand is unreadable. Standard textbooks, e.g. those of Kobayashi & Nomizu or Helgason, do not use the notation of the stub. I do not agree that the content from there should be merged. It is quite clear that the editor deliberately used notation that is not standard. An affine connection defines a way for tangent vector fields X to act on sections ξ of a fixed vector bundle, i.e. ∇_Xξ. That is clear in the article but not the stub. Why mention "jets"? The compatibility conditions for affine connections are described in both sources but in the stub are almost undecipherable. Similarly, the Cartan formulation in terms of 1-forms is well presented in Helgason and K & N, but completely opaque in the stub. The editor who created most of this content was User:Gsard. He created his own wikipedia page Gennadi Sardanashvily but died in 2016. Mathsci (talk) 00:39, 25 October 2020 (UTC)

Oh, I see what the problem is. Now that I read it more carefully, I can see that the stub is about connections on affine bundles, which the author gave the unfortunate name "affine connection". Which... is not a good name. It is probably safe to say that the connection on the affine bundle is a kind of an affine connection, but it is not at all the same thing as "the affine connection" defined here. The non-standard notation is then a by-product of trying to distinguish between a tangent bundle, and the associated affine bundle derived from it. There might be some pile of commonalities, but without a reference at one's fingertips and some hard work, stating them/merging them is impossible. So I'm now retracting the merge tags. Heh. 67.198.37.16 (talk) 21:46, 25 October 2020 (UTC)

Dubious tag on Formal Definition as a Differential Operator

Perhaps I'm wrong, since I'm not well-versed in this, but based on the context, I would expect it should say $\nabla _{X}f(Y)=Xdf(Y)+f(\nabla _{X}Y)$, instead of df(X)Y as the article currently states, since below it states that this criteria implies that the connection depends on X at only the very point under consideration, and depends on Y in a neighborhood of that point. Doesn't that only hold if the df is applied to Y? Also, describing this property as being analogous to the product rule suggests to me that the df is applied to the Y, since that is more similar to the product rule. Ramzuiv (talk) 02:14, 3 March 2021 (UTC)

While awkward, the original is correct. The differential operator being applied is $\nabla _{X}$. Most straightforwardly the Leibniz rule is written $\nabla _{X}fY=(\nabla _{X}f)Y+f\nabla _{X}Y$. The covariant derivative for functions is defined to be $\nabla _{X}f=X(f)=\partial _{X}f$. These are different notations for the derivative of $f$ in the direction of $X$. Written in terms of $df$, the gradient of $f$, you can write $df(X)=X(f).$ Edited: originally had Leibniz rule as $\nabla _{X}fY=Y\nabla _{X}f+f\nabla _{X}Y$.Zephyr the west wind (talk) 22:36, 6 May 2022 (UTC)

This should be $\nabla _{X}(fY)=(Xf)Y+f\nabla _{X}Y$, as in the books of Helgason or Kobayashi & Nomizu. Vector fields X are a left module over the ring of differentiable functions f. Mathsci (talk) 23:43, 6 May 2022 (UTC)

Changed it now - thanks. Zephyr the west wind (talk) 23:56, 6 May 2022 (UTC)

Ref title wrong

Cartan 1926 Acta Math. paper title is actually "Les groupes d’holonomie des espaces généralisés". Title in Wikipedia article became "Espaces à connexion affine, projective et conforme". I do not know whether the title or the ref is wrong. Arnaud Chéritat (talk) 21:33, 15 June 2022 (UTC)