# Reference for some elementary facts about principal bundles

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