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Let $(P,\pi,M)$ be a principal $G$-bundle, and $A \in \Omega^1(P,\mathfrak g)$ is a connection 1-form, and $f:N \rightarrow M$ be a smooth map between manifolds. Then how to rigorously show that the pullback $f^* A$ is again a connection 1-form on $f^*P$

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This is a standard result in differential topology and I recommend that you take a look at a suitable text book. – Andrew Stacey Apr 29 2011 at 6:51
Which textbook are you referring to? – Hu Yi Chen Apr 29 2011 at 7:02
I think I saw this a couple of days ago in "Foundations of differential geometry", Vol. 1, Chapter 2 by Kobayashi-Nomuzu. – Gunnar Magnusson Apr 29 2011 at 7:53
And it is an easy exercise. – Johannes Ebert Apr 29 2011 at 8:45
I think math.stackexchange.com (or one of the other sites listed in the FAQ) is a more appropriate place for your question, but I think that if you are comfortable with the definitions you can work it out yourself. – S. Carnahan♦ Apr 29 2011 at 10:02

closed as too localized by Andrew Stacey, Johannes Ebert, S. Carnahan♦ Apr 29 2011 at 10:00

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