Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Are there any good detailed historical sources about development of connections on vector/principal bundles over the last 100 years?

The best source I am aware of is Michael Spivak's 5 volume opus, but this is not detailed enough for the project I have in mind (I am intending to set this as a topics essay for my first year graduate differential geometry class, and I want to make sure I know enough myself first!).

share|improve this question
4  
There's a brief history with several references in Marcel Berger's "A Panoramic View of Riemannian Geometry". It looks similar to the information in Volume II of Spivak. –  Ryan Budney Feb 18 '11 at 23:52
2  
I don't know if it's possible to find the books of Robert Hermann, but I suspect that there is a lot history of connections in them. I don't know what was done before Elie Cartan, but Cartan certainly did a lot with connections on principal bundles. I don't think there was much before that. –  Deane Yang Feb 19 '11 at 0:56
2  
In a 1921 paper (digizeitschriften.de/dms/img/?PPN=GDZPPN002505908) Hermann Weyl was already writing in a modern way about conformal and projective equivalence of linear connections. I imagine the notion of affine connection was clearly formulated some decades earlier, though the notions of fiber and vector bundles were not clearly formulated as such until after their emergence in particular in Cartan's work. Something like the cocycle condition for the Schwarzian derivative (which appears explicitly in a paper of Cayley) could be seen as a precursor to the notion of connection. –  Dan Fox Feb 19 '11 at 12:20
1  
the concept seems to be due to Elie Cartan (numdam.org/numdam-bin/fitem?id=ASENS_1923_3_40__325_0) with later contributions by Ehresmann:see wiki (en.wikipedia.org/wiki/Ehresmann_connection) and essay( charles-michel.marle.pagesperso-orange.fr/pdffiles/connexions.pdf) –  SGP Feb 19 '11 at 13:20
1  
In the first paragraph of the cited paper of Cartan he says that he takes the terminology "affine connection" from H. Weyl. Of course the notion of a metric connection comes from earlier (Ricci, Levi-Civita, etc.), and was by then widely known due to Einstein's theory. The idea of monodromy (holonomy) was known from the study of ODEs. When was some link between the two first seen? When were the notions of connection and metric explicitly divorced and viewed as logically independent? The understanding of the link with monodromy seems important in this regard - this seems mainly due to Cartan. –  Dan Fox Feb 23 '11 at 7:16
show 1 more comment

2 Answers 2

up vote 29 down vote accepted

Here is a rough historical overview:

  • 1900: Ricci and his student Levi-Civita introduce the concept of a "tensor" in

MR1511109 Ricci, G.; Levi-Civita, T. Méthodes de calcul différentiel absolu et leurs applications. (French) Math. Ann. 54 (1900), no. 1-2, 125--201.

There they define an operation called "covariant differentiation" of a tensor which generalizes the usual differentition to curvelinear coordinates; the correction term is given by the Christoffel symbols. By that time there is no abstract notion of covariant derivative of a section or whatsoever nor the differentiation is considered together with the associated parallel transport. For more about this I would refer you to the book by Morris Kline on the history of mathematics. There is also an edited edition of this article available (Editor is Hermann who was already mentioned by Deane Yang). Whitney used the word "tensor" in connection with a group theoretic construction in 1938 (he obviuosly was influenced by what he knew about tensors; see end of the paper where he speaks about parallel transport which gives an impression on how people thought about thing like that in the 1930's). Later this was generalized to modules by Cartan and Dieudonné (see Weibel's history on homological algebra).

Meanwhile Einstein and his friend Grossmann used this "absolute differential calculus" (as it is also called) to give a mathematical footing to general relativity. Later Einstein had a lot of correspondence with Levi-Civita as well as Cartan on topics like parallel transport which can be found in their scientific correspondence.

  • 1917: Levi-Civita, T. Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana. (Italian) Palermo Rend. 42, 172-205 (1917).

Here Levi-Civita gives, following an indication of his teacher Ricci, a geometric interpretation of the covariant differentiation by means of a associated parallel transport. I only know of the italian version of this article but there is book of Levi-Civita (from the 1930s?) on tensor analysis where he gives a more or less plausible derivation of this concept; though I found it hard to follow in all details. Felix Klein by the way was not very happy with the derivation given by Levi-Civita and in some book (currently I can not recall the title but it is about geometry and also available in English) he derived the whole thing from a physical experiment with some peculiar machine which can be used to detect curvature. I think the books title must be something like "Higher Geometry". The idea seems to be due to Radon (1918); but seemingly published nowhere else. Anyhow the review by Blaschke (one of the leading person in differential geometry at this time) for the Zentralblatt is very illuminating (almost a prophecy).

  • between 1917 and 1920 (third German edition is published in 1919) Weyl published "Raum, Zeit, Materie" ("Space-time-matter"; available here: http://www.archive.org/details/spacetimematter00weyluoft). There he uses the Christoffel symbols and their transformation behaviour to define "connection". Around the same time as Levi-Civita, Schouten and Hesse made similar observations. This can be found in an easy to find article by acclaimed German mathematics historian Karin Reich. Unfortunately, this article is in German. Nevertheless you can see pictures of models of surfaces with parallel transport drawn on them, that Schouten produced. Later Weyl in connection with his physical studies introduced the concept of a Weyl connection (whose most peculiar property is that it does not preserve length); this was already mentioned in the comments. More can be found in Chapter I of this brilliant book (again German would be required): http://books.google.de/books?id=oZLiqDQGnjgC&printsec=frontcover&dq=Scholz+and+Weyl&hl=de&ei=IcVfTeSULILXsgaRz422CA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCoQ6AEwAA#v=onepage&q&f=false

  • Around 1923 Élie Cartan starts to study connections (his so-called projective connections) More on this can be found in the splendid article "Vector bundles and connections in physics and mathematics: some historical remarks" by Varadarajan.

Somewhere behind 1940 Norman Steenrod introduces the concept of a fibre bundle and around 1950 Charles Ehresmann introduces the concept of a connection on a fibre resp. principal fibre bundle. His article is very readable and a valuable historical source.

share|improve this answer
2  
Very nice answer. –  Deane Yang Feb 19 '11 at 15:01
    
Thank you, I really appreciate it, when it comes from you. –  Daniel Pape Feb 19 '11 at 15:24
    
These history related talks by Bergery. Bourguignon, Dumitrescu, Scholz and many others are also a real treasure (in French!): archivesaudiovisuelles.fr/412/liste_conf.asp?id=412 –  Daniel Pape Feb 19 '11 at 15:34
add comment

Try searching "A Historical Overview of Connections in Geometry". This is a wonderfully accurate account of the history of connections.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.