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I am trying to understand about principal G bundle given a Lie group $G$. For that, I started with the action of Lie groups on manifold $M$ and convinced myself that if the action is smooth, proper, and free then orbit space is a smooth manifold and the corresponding projection map gives a fiber bundle $(M,\pi, M/G)$. We call this bundle and any other bundle isomorphic to this bundle a principal $G$ bundle. Only to understand just this it took so much time for me as there are many books with different definitions of principal bundles.

I am comfortable with this definition of a principal bundle and would like to know more enough to start reading connections on principal bundles. So, I would like to ask for your suggestion regarding some reading materials online on principal bundles. I checked it but did not find it satisfactory. Any suggestions regarding references are welcome.

Please do not add husemoller fiber bundles as it uses some other definition of a principal bundle and not Kobayashi also. I am benifited very much with chapter on Lie group actions on manifolds in Lee’s Book Introduction to manifold. I wish he had published some notes on principal bundles as well. Please do let me know any references along these lines.

Edit: I am now able to understand roughly the concepts of connections, holonomy groups from Kobayashi. Any reference which can supplement this book is most welcome. Now I am reading curvature form and structure equation and do not really understand what they are saying.

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  • $\begingroup$ "Some other definition": all these definitions are easily seen to be equivalent. $\endgroup$
    – abx
    Commented Nov 17, 2017 at 19:43
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    $\begingroup$ I liked José Figueroa-O'Farrill's notes on gauge theory. You might find the first two lectures enlightening: empg.maths.ed.ac.uk/Activities/GT $\endgroup$
    – Piotr Achinger
    Commented Nov 17, 2017 at 21:17
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    $\begingroup$ I don't know if this is helpful or not, but I find it easier to figure out what a connection on a principal bundle is, if I view the principal bundle as a bundle of frames on a vector bundle and the connection on the principal bundle as a connection on the vector bundle acting on a "moving frame". That helps make the definition of a connection in, say, Kobayashi-Nomizu easier to understand. $\endgroup$
    – Deane Yang
    Commented Nov 26, 2017 at 1:05
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    $\begingroup$ Have you tried reading Chapter 8 of Volume II of Spivak's tomes on differential geometry (where (*) mentioned early on refers to an equation at the bottom of page 280 in Chapter 7)? It is somewhat formula-intensive, but does carefully address the passage between principal Ehresmann connections on principal bundles and the more classical Koszul connection viewpoint on rank-$n$ vector bundles, and has a nice Summary at the end of the chapter for passing among various viewpoints on connections. The end of the Wikipedia article on connections that @DeaneYang mentions ties up related loose ends. $\endgroup$
    – nfdc23
    Commented Dec 16, 2017 at 5:43
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    $\begingroup$ By the way, the published edition of Spivak's Volume II to which I referred (for the page number of (*), for example) is the LaTeX version, not the original published version (which probably had different page numbers). $\endgroup$
    – nfdc23
    Commented Dec 16, 2017 at 15:35

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"Foundations of Differential Geometry" by Kobayashi and Nomizu has a good introduction to principal bundles and connections on them. In fact, this is presented at the beginning and is used as the basis for their presentation of classical (in the present) differential geometry.

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  • $\begingroup$ I have mentioned not to mention kobayashi book as I am already reading that and not very comfortable with that way. So, asked for another reference. $\endgroup$
    – Praphulla Koushik
    Commented Nov 26, 2017 at 13:55
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Try the book by Michor, Kolar & Slovak titled Natural Operations in Differential Geometry, it's rather dense so if you're uncomfortable with Kobayashi & Nomizu you might find it doesn't work for you. It has a couple of chapters on vector, fibre, principal and jet bundles. It's also available as a free download. It's very comprehensive and best treated as a reference text.

Another book to try is Saunders The Geometry of Jet Bundles, this is more accessible but the notation is very heavy; he goes up to the variational bicomplex.

Also Moritas The Geometry of Differential Forms, I haven't read this apart from a brief look at its contents, but it looks very interesting.

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I like Chern's short and easy Vector bundles with a connection, in Chern, Global Differential Geometry, M.A.A., 1989. Chern focuses on vector bundles but does so by using simple examples of principal bundles. Chern gives some serious theorems that motivate learning the subject.

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  • $\begingroup$ I tried multiple number of times to read that article. Could not finish. I liked it though +1 $\endgroup$
    – Praphulla Koushik
    Commented May 31, 2020 at 8:03
  • $\begingroup$ Is there a link to this somewhere? $\endgroup$
    – warzasch
    Commented Jul 16 at 13:42
  • $\begingroup$ @warzasch: I don't think there is a link to this document. $\endgroup$
    – Ben McKay
    Commented Jul 16 at 15:49
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I realised one can not (should not) escape from reading Kobayashi and Nomizu’s book.

Other books that helped me to learn more about principal bundles are

  1. Differential Geometry: Connections, Curvature, and Characteristic Classes by Loring W. Tu
  2. Connections, Curvature, and Cohomology Volumes 1,2,3 by Werner Hildbert Greub, Stephen Halperin, James S. Vanstone, Ray Vanstone
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    $\begingroup$ I answered this because some one upvoted this question before couple of hours.. I thought it is good to to tell what helped me. $\endgroup$
    – Praphulla Koushik
    Commented May 31, 2020 at 7:06
  • $\begingroup$ Personally I like the three volume books "Connections, Curvature, and Cohomology" very much. However, these books are out of print. By the way, it seems the terms used in, say Volume I, of these books are not quite standard. $\endgroup$
    – Ho Man-Ho
    Commented Jul 26, 2020 at 10:03

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