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I have come across two seemingly disparate definitions of a principal $G$-bundle, and would like some comments on the merits of each one and how this links to the general use of principal bundles in geometry, since I have only encountered this concept recently.

The first is a definition given here: http://mathworld.wolfram.com/PrincipalBundle.html. Specifically, a principal bundle is defined as a special case of a fiber bundle, where the fibers are groups and the group acts freely on the fibers.

The second is one due to Dominic Joyce found in his book Riemannian Holonomy Groups and Calibrated Geometry: Let $M$ be a manifold, and $G$ a Lie group. A principal bundle $P$ over $M$ with fibre $G$ is a manifold $P$ equiped with a smooth projection $\pi : P \to M,$ and an action of $G$ on $P$, which we will write as $p \mapsto^{g} g \cdot p$, for $g \in G$ and $p \in P$. This $G$-action must be smooth and free, and the projection $\pi: P \to M$ must be a fibration, with fibres the orbits of the $G$-action, so that for each $m \in M$ the fibre $\pi^{-1}(m)$ is a copy of $G$.

Note that in the first definition there is no mention of fibrations - only fiber bundles - which makes the second definition slightly confusing in light of the first.

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  • $\begingroup$ I guess could be misleading, but what is exactly that disturbing you in the second definition? It's true that he speaks about fibration, but right after says that for each m the fiber is a copy of G, so indeed it's a fiber bundle. $\endgroup$ – Dac0 Jan 15 '17 at 7:45
  • $\begingroup$ I guess that he even brought the notion of a fibration in in the first place, when he could have just invoked the notion of a fiber bundle. It just seems a bit heavy-handed. $\endgroup$ – Andrew Whelan Jan 16 '17 at 12:26
  • $\begingroup$ I think it is and I wouldn't use this text as a first approach... $\endgroup$ – Dac0 Jan 16 '17 at 13:23
  • $\begingroup$ Thanks for reassurance - what text would you recommend for this purpose? $\endgroup$ – Andrew Whelan Jan 16 '17 at 14:33
  • $\begingroup$ Also, if you wish to leave a book suggestion and a general comment about principal bundles, I'll gladly accept it as an answer. $\endgroup$ – Andrew Whelan Jan 16 '17 at 14:47
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As I told you in the comment, both definitions are right, but the second as you pointed out is without any good reason a little bit heavy-handed. I'm not sure if you already have a back ground in Riemannian Geometry or not. If not I would suggest you to have a look to some standard book on Riemannian Geometry before starting. My suggestion would be Spivak.

After that you should propably go to Kobayashi, Nomizu Foundations of Differential Geometry (which is a standard in the field) and I would also suggest you Sharpe, Differential Geometry. After you gave a read on those probably you're ready for start with Riemannian Holonomy Groups and Calibrated Geometry.

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