289:, 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.
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396:, 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.
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263:. 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
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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
932:
Cartan 1926 Acta Math. paper title is actually "Les groupes d’holonomie des espaces généralisés". Title in
Knowledge article became "Espaces à connexion affine, projective et conforme". I do not know whether the title or the ref is wrong.
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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.
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Oh, I see what the problem is. Now that I read it more carefully, I can see that the stub is about connections on
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Perhaps I'm wrong, since I'm not well-versed in this, but based on the context, I would expect it should say
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Knowledge. If you would like to participate, please visit the project page, where you can join
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While awkward, the original is correct. The differential operator being applied is
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508:{\displaystyle \nabla _{X}fY=(\nabla _{X}f)Y+f\nabla _{X}Y}
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Dubious tag on Formal
Definition as a Differential Operator
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773:{\displaystyle \nabla _{X}fY=Y\nabla _{X}f+f\nabla _{X}Y}
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702:{\displaystyle df(X)=X(f).}
431:{\displaystyle \nabla _{X}}
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