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Is there a good text that teaches principal bundle or frame bundle?

I am looking for a textbook that might serve as an introduction to topology of principal bundles or frame bundles, specially the global frame field and the connection on them.

Currently, the only book I know of in this regard is:

"Lectures on Differential Geometry" by S. S. Chern & W. H. Chen & K. S. Lam

I have been reading it, but I am occasionally frustrated by the lack of more details on the global section on frame bundles.

Thanks in advance!

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I suggest "Werner Greub, Stephen Halperin, Ray Vanstone - Connections, Curvature, and Cohomology. Academic Press (1973)". There are three volumes, you can find information on principal bundles on the second volume.

I started studying the subject from "Steenrod - The Topology of Fiber Bundles", and it was very good, but Greub's book helped me to grow more quickly on the subject, because it's very direct and easy to start at any point of the book.

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From the topological point of view I remember I learned and understood several important notions from : "The Topology of fibre Bundles" by Norman Steenrod.

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My personal favorites are "Topology, Geometry, & Gauge Fields: Foundations" by Gregory Naber and "Gauge Theory & Variational Principles" by David Bleecker. Naber's book does a lot of in-depth calculations and makes a solid attempt at explaining the motivation behind the various mathematical notions introduced, two things that the authors of more "sophisticated" books will not debase themselves with. Bleecker's book is nice because it's a cheap, compact, well-written paperback. It's perfunctory in places, but also goes deeper, especially with mathematical topics related to physics. I'll also add a suggestion that I have no personal experience with, which I just discovered by browsing Amazon: "Principal Bundles: The Classical Case" by Stephen Bruce Sontz. A look at the table of contents was enough to pique my interest, so I include it here for what it's worth.

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A principal bundle has a global section if and only if the bundle is trivial. But perhaps I misunderstood your complaint. I like Chern/Chen/Lam very much, but of course you can also consult Kobayashi/Nomizu.

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