I am looking for a textbook that might serve as an introduction to principal bundles, curvature forms and characteristic classes, and perhaps towards 4-manifolds and gauge theory.

Currently, the only books I know of in this regard are:

  • "From Calculus to Cohomology" (Madsen, Tornehave)
  • "Geometry of Differential Forms" (Morita)
  • "Differential Forms in Algebraic Topology" (Bott, Tu)

I have been reading both "Calculus to Cohomology" and "Geometry of Differential Forms," but am occasionally frustrated by the lack of thoroughness. Both are at the perfect level for me, and cover almost exactly what I'm looking for, but I really prefer textbooks which are as thorough as possible, ideally to the extent of, say, John Lee's books (which I adore). Meanwhile, Bott and Tu is a little advanced for me right now.

Of course, I don't mean to be picky, but I also can't believe that the three I've listed are the most thorough accounts of the subject.

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    $\begingroup$ Maybe Scorpan's "Wild World of Four-Manifolds" will help, even it is not a textbook, I think it is nicely-written. $\endgroup$ – gary Sep 13 '11 at 23:23
  • $\begingroup$ @gary: Yes, I've heard of that one, too, and am very interested in reading it. However, I am, as you mentioned, primarily in the market for textbooks. $\endgroup$ – Jesse Madnick Sep 13 '11 at 23:25
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    $\begingroup$ I've been in a similar boat and just used the three you mentioned as well as random notes floating around the internet. I've searched long and hard for a text that covers Chern-Weil theory and I really think Morita is the best I've found. It's a shame though that it is riddled with typos and some errors. $\endgroup$ – Eric O. Korman Sep 14 '11 at 1:22
  • $\begingroup$ Morita is quite good,but I agree it's not as thorough as one would like. $\endgroup$ – Mathemagician1234 Mar 30 '12 at 6:24

You might find the following useful: G.L. Naber, Topology, Geometry, and Gauge Fields: Foundations, 2nd.. It has a specific aim and purpose though: it's oriented towards those who want to learn the math foundations for gauge theory within a rigorous setting. Maybe pure math students might like a more broader approach.

  • $\begingroup$ Don't forget Topology, Geometry and Guage Fields, Interactions (2nd ed). I'm a big fan of Naber's work $\endgroup$ – ItsNotObvious Sep 14 '11 at 19:28
  • $\begingroup$ I've just ordered Naber's book, which looks very promising. Thanks. $\endgroup$ – Jesse Madnick Dec 28 '11 at 4:40
  • $\begingroup$ Naber has not only written several outstanding textbooks,he also has several wonderful sets of lecture notes on various topics ranging from first year undergraduate to graduate level at his website: pages.drexel.edu/~gln22 His algebraic topology notes will be of particular use to serious students. $\endgroup$ – Mathemagician1234 Mar 30 '12 at 6:39

C.H. Taubes, Differential Geometry: Bundles, Connections, Metrics and Curvature (Oxford Graduate Texts in Mathematics) might also be helpful. ... It hasn't been released yet, but given the author's fame and stature I think it might be a good pick. ...

  • $\begingroup$ It would be nice if you mentioned the author(s) as well. It is a bit cumbersome to follow the links to find out what book exactly you're talking about. $\endgroup$ – t.b. Sep 15 '11 at 10:48
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    $\begingroup$ The problem with Taubes' book-although it's very well written indeed-is that it's not really a textbook at all,it's a set of lecture notes. It has ZERO exercises. Not that that's a BAD thing,of course-some of the best sources there are on differential geometry (and advanced mathematics in general) are lecture notes (S.S.Chern and John Milnors's classic notes come to mind).But for coursework and something you want to pay considerable money for-a set of exercises FROM THE AUTHOR to test your understanding really isn't too much to ask,is it? $\endgroup$ – Mathemagician1234 Mar 30 '12 at 6:30
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    $\begingroup$ @Mathemagician1234 : I think you don't know the difference between lecture notes and a textbook. Saying something is just "lecture notes" implies that it is poorly edited and slapdash. It is insulting and unfair to claim this for Taubes's book, which is quite polished and nice. $\endgroup$ – Adam Smith Mar 30 '12 at 6:53
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    $\begingroup$ @Adam As usual,you jumped all over my comment without carefully reading it. How could I say it is "well written indeed" if it's 'poorly edited and slapdash'? It depends on your definition-clearly I didn't mean "chicken scratchings on the back of a cafeteria napkin"-lecture notes. The classic "texts" of S.S.Chern and Ira Singer/John Thorpe are lecture notes-they do not have clear chapter delineations, they don't have exercise sets,etc.-they are hardly slapdash or poorly edited.That being said-they are also more difficult to use as classroom texts because of these omissions.THAT'S the point. $\endgroup$ – Mathemagician1234 Mar 31 '12 at 3:22
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    $\begingroup$ @AdamSmith Did we read the same book? You are a much better mathematician than I, so probably better qualified to judge, but I found many typos and errors when I tried to work through it. $\endgroup$ – Potato Jul 9 '15 at 23:58

If you want to focus on four manifolds, don't forget the classic of Donaldson and Kronheimer, Geometry of four manifolds. It may be a tad on the advanced side, but does contain some information specifc to 4 dimensions not available in the other books you listed.

  • $\begingroup$ It's a little advanced for me, yes, but I do look forward to reading it someday. Thanks, Willie. $\endgroup$ – Jesse Madnick Dec 28 '11 at 4:41

There are, by now, many excellent sources. A few off the top of my head:

  • Naber (2 volumes), mentioned above
  • Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes
  • Nakahara, Geometry, Topology and Physics
  • Choquet-Bruhat, DeWitt-Morette, Analysis, Manifolds and Physics (2 volumes)
  • Bleecker, Gauge Theory and Variational Principles
  • Marathe, Topics in Physical Mathematics
  • Sontz, Principal Bundles: The Classical Case
  • Greub, Connections, Curvature and Cohomology (3 volumes)

For a high level overview see also the survey article Gravitation, Gauge Theories and Differential Geometry. I have quite a few more, but they may not be mathematical enough for your tastes. If you'd like them, just let me know.

  • $\begingroup$ This is a good list! And thank you for the offer, though I think I'm good for now. I did ask this question 7 years ago, after all, and have learned a little bit since then :-) $\endgroup$ – Jesse Madnick Dec 8 '18 at 1:24

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