I wonder who was the first to discover the notion of principal fiber bundle and its connection (gauge field in the physical language). Wikipedia cites the book by Steenrod (1951). But was he the first? I also heard that Serre and/or Grothendieck invented this or something similar?
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$\begingroup$ This seems like a question for HSM. $\endgroup$– LSpiceCommented Feb 5 at 19:43
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1$\begingroup$ see mathoverflow.net/questions/59941/… I would bet on Cartan $\endgroup$– Will JagyCommented Feb 5 at 19:46
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1$\begingroup$ en.wikipedia.org/wiki/Cartan_connection $\endgroup$– Will JagyCommented Feb 5 at 20:04
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Leaving aside the question of whether mathematics is discovered (as you imply) or invented, I suggest that you examine the paper:
Charles Ehresmann, "Les connexions infinitésimales dans un espace fibré différentiable" Séminaire N. Bourbaki, 1952, no. 24 153-158.
(N.B. Although this publication is dated 1952, the linked PDF typescript gives a date of March 1950; Ehresmann explains in a footnote added later that a more complete version of the paper - with the same title - was published in Colloque de Topologie (espaces fibrés) 1950. Bruxelles, Georges Thone et Paris, Masson, 1951 (Centre belge de Recherches mathématiques)).
Further to Will Jagy's comments on the question, Ehresmann explicitly states in the first sentence that he's generalising Cartan's definition of a connection:
Je me propose de préciser et de généraliser la notion d'espace à connexion de Cartan.
(Emphasis in the original.)
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$\begingroup$ OK, thank you, I looked through this paper and found a familiar formula at the end of sec. 5 on p. 14. In modern notation, it is F = dA + A * A. So Ehresmann had it. It is not yet clear for me whether Cartan also had it back in 1926. Probably not (?) $\endgroup$ Commented Feb 6 at 21:06
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$\begingroup$ Or probably Cartan had it for the tangent fiber bundle: R = d\omega + \omega * \omega is called the second Cartan structure equation. Tangent bundle is actually a principal bundle with the group SO(n). And Ehresmann generalized it for any group ? $\endgroup$ Commented Feb 6 at 22:33
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$\begingroup$ @AndreiSmilga The presentation by Charles-Michel Marie on the work of Charles Ehresmann on connections (marle.perso.math.cnrs.fr/diaporamas/connexions.pdf) explains in some detail the links between Cartan's and Ehresmann's work on connections. $\endgroup$ Commented Feb 8 at 6:55
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$\begingroup$ Yes, thank you. It seems that my guess above was correct. $\endgroup$ Commented Feb 28 at 15:25