Talk:G-structure on a manifold

Integrability condition versus other conditions
Although I stated that integrability conditions were the primary reason for higher-order G-structures, this really isn't the case at all. It is only the easiest way to see it. For example, a conformal structure induces a higher order G-structure because the associated prolongation of the principal bundle can be canonically embedded into the principal bundle of holonomic second-order frames. Gotta run right now, though. Silly rabbit 00:33, 15 November 2005 (UTC)

Chern
The article says that G-structures were introduced by Chern, but I can't find any evidence of this. Can anyone confirm? The 1966 article by Chern on G-structures references older papers by Cartan (1908 and 1953) and Bernard (1960). These appear to be in French, however, so I can't read them. -- Fropuff (talk) 17:09, 6 March 2008 (UTC)


 * It's fairly doubtful that Chern was responsible for introducing them. Do you have a full citation for the Cartan 1953 paper?  (The 1908 paper predates the innovation of the fibre bundle, so should probably be disqualified.) Silly rabbit (talk) 17:19, 6 March 2008 (UTC)


 * The one Chern references is: E. Cartan (1953), Pseudo-groupes continus infinis. Geometrie differentielle, Colloquium, Strasbourg, pp. 119–136, Centre National de la Recherche Scientifique, Paris. -- Fropuff (talk) 17:25, 6 March 2008 (UTC)


 * G-structures were introduced and extensively studied by Cartan. See Part 2 and Part 3 of Cartan's "Oeuvres Completes", Gauthier-Villars, 1955 Tiphareth (talk) 22:59, 7 March 2008 (UTC)

Merge proposal
The two articles Reduction of the structure group and G-structure are essentially about the same thing, so I think they should be merged. But it's probably worth discussing, fist, how the merged article should be articulated and which parts are redundant and less clear than the other article. Any thoughts? --Gro-Tsen (talk) 19:37, 10 August 2012 (UTC)
 * I think if it is merged Reduction of the structure group looks like it could be a section of G-structure, actually I wonder if the title should be more specific like G-structure on a manifold. Billlion (talk) 05:45, 11 August 2012 (UTC)
 * I agree, whatever the final article, the word "manifold" should probably appear in the title. --Gro-Tsen (talk) 17:11, 11 August 2012 (UTC)

As someone with a technical background and a member of WP:WPMERGE I've taken a look at doing this merge and found it to be over my head. ~Kvng (talk) 16:22, 22 June 2016 (UTC)
 * Page title changes prior to the merge, as per the consensus above. Klbrain (talk) 08:33, 25 June 2016 (UTC)

I'll leave the request for expert template above, as it is worth an expert looking at what was mostly a full-content merge to a new section. Klbrain (talk) 08:44, 25 June 2016 (UTC)
 * As a very inexperienced contributor to wikipedia, I am not sure how to proceed, but I am fairly sure that the merging of these two articles was not necessarily a good idea: the notion of "reduction of the structure group" is something that is defined for *any* principal bundle (and indeed, it is mentioned in the article on principal bundles), while a G-structure is a very specific kind of principal bundle (according to this article). Nielius (talk) 10:46, 20 July 2016 (UTC)

Merge cleanup proposal
It seems really necessary to me (mathematician working in a related field) to restructure this. I propose:


 * 1) Move "Examples" and "Integrability" out of "Reduction of the structure group" (since they clearly apply to G-structures on manifolds, and not general bundles)
 * 2) Rephrase "Reduction of the structure group" to make clear that it talks about general bundles
 * 3) Move "Examples" further up
 * 4) Bring a precise definition of G-structure before the examples, followed by a precise definition of "reduction of structure group"

If this is ok, I'll do it. --TurionTzukosson (talk) 10:29, 8 May 2018 (UTC)


 * I agree that a restructuring and clarification of concepts applicable to G-structures vs general bundles would would significantly improve this article. Your approach seems reasonable to me. Go for it. --Mark viking (talk) 16:43, 9 May 2018 (UTC)

Confused
The article in its present form gives a super short definition of G-structure (and a reference to the definition of principal G-subbundle is missing*). Then most of the discussion is about "Reduction of the structure group". This gives the impression that this is the same notion. However, the comment of Nielius in this talk page (20 July 2016) seems to hint that they are not. I highly recommend that there be a "Definition" section where the definition of G-structure is repeated and expanded a litte bit. If Wikipedia does not have a definition of the notion of principal G-subbundle it could be added here and to the article "Principal bundle".

(*): there is a ref to the artcile "Subbundle" but this is insufficient: a subbundle of a G-bundle is, I suppose, not a principal G-subbundle...

Arnaud Chéritat (talk) 13:07, 11 June 2022 (UTC)

Given its position in the article, the sentence "For a Lie group $$\displaystyle G$$ and a group homomorphism $$\displaystyle \phi \colon G\to GL(n)$$, a $$\displaystyle G$$-structure is a reduction of the structure group of the frame bundle to $$G$$." sounds like a property, buy I suspect this is intended to be the definition of G-structure. This should be clarified, and this definition should also appear at least in the header and I strongly suggest it also appears as the fist section of the article since the article is called "G-structure on a manifold". Arnaud Chéritat (talk) 08:44, 12 June 2022 (UTC)