I'm looking for a textbook on the differential geometry of fibre bundles containing a not too brief discussion of the following topics:

  • Principal and associated bundles, (reduction of) structure groups.
  • Ehresmann connections and their curvature
  • Other common definitions of a connection on a bundle and various ways of organizing that information (connection forms etc.)
  • Holonomy, mondromy and gauge groups
  • Yang-Mils functionals
  • Foliations and their holonomy
  • Jet bundles

Is there such a textbook? (By which i mean a book that contains exercise problems).

If not, where can i find problem sets for these topics?

  • $\begingroup$ Kobayashi & Nomizu, Foundations of Differential Geometry Vol. 1 for principal and associated bundles $\endgroup$ – Donyarley Sep 24 '15 at 17:13

A standard great reference textbook with exercises is definitely Husemöller's "Fiber bundles", especially Part I and III for your needs. It covers most of your topics (I don't think there is a book covering all of your topics in a great way, so I am convinced that this one should be the perfect fit, as it covers most of them in a great way). There is also good stuff to say about the lecture notes of Koszul, but they are without exercises.

  • $\begingroup$ I've actually considered it before but from what I've seen in the syllabus it seems very homotopically/topologically oriented (which is probably another angle i'd need to read about evetually). Would you recommend it to someone who wants to get comfortable with the math behind gauge theory? What would you say are the prerequisites? $\endgroup$ – Saal Hardali Sep 23 '15 at 9:10
  • $\begingroup$ In particular do I need to know homotopy theory (homotopy (co-)limits, (co-)fibrations etc.) before tackling it? $\endgroup$ – Saal Hardali Sep 23 '15 at 9:17
  • 1
    $\begingroup$ Not really. All you need from that kind of stuff is covered in the first chapter, which is almost nothing and particularly nothing deep (you would need this in most fiber bundle literature). However you should know about cohomology but that holds definitely true for all fiber bundle books. $\endgroup$ – Daniel Valenzuela Sep 23 '15 at 10:49

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