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Let $(P,\pi,B,G)$ be a principal bundle with total space $P$, base $B$, projection $\pi$ and structure group $G$.

Now I am searching for a good reference (with proofs) for the following facts:

1) The fundamental vector fields on $P$ span pointwise the vertical space - or equivalently they generate the $C^\infty(P)$-module of smooth sections of the vertical bundle.

2) Let $\gamma \colon TP \to \mathrm{Lie}(G)$ a connection one-form. The horizontal lifts of vector fields span pointwise the horizontal space - or equivalently they generate the $C^\infty(P)$-module of smooth sections of the horizontal bundle.

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    $\begingroup$ The first chapter of the book of Kobayashi-N*** "Foundations of differential geometry" $\endgroup$
    – Guangbo Xu
    Commented Jul 16, 2010 at 9:28
  • $\begingroup$ Thanks, the exact references are: Kobayashi-N*** before Prop.I 5.1 and before Prop. II 1.2. Any further reference? $\endgroup$
    – student
    Commented Jul 16, 2010 at 10:11
  • $\begingroup$ For the second point, see mathoverflow.net/questions/34663/… $\endgroup$
    – student
    Commented Oct 2, 2010 at 14:56

2 Answers 2

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You can have a look at Sharpe's differential geometry, whose title makes it difficult to catch among all those books named the same, but is interested in Elie Cartan's point of view on geometric structures. It contains a lot of material on these topics, and tries to relate this to intuition in a nice manner.

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I would recommend the beautifully written book Differential Geometry : Connections, Curvature, and Characteristic classes by Loring Tu.

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