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In "The Topology of Fiber Bundles" by Cohen here, in page 39, he says:

We end this section by observing that if $P$ is a principal $G$ - bundle with a connection $\omega_A$, then any representation of $G$ on a finite dimensional vector space $V$ induces a connection on the corresponding vector bundle: $P \times_G V \to M.$

I supposed it is a kind of trivial. I was wondering how a connection on Principal Bundle easily induces a linear connection on the vector bundle $P\times_G V\to M$. I tried to induce a kind of connection on the vector bundle but I could not show it is either linear or is from zero forms to one forms and obeys Leibniz's rule.

P.S. I read the answer here but I could not find my answer in the references there.

Adam Chalumeau
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  • Can you write down here how you have tried to associate a connection on a fiber bundle $P\times_GV\rightarrow M$ given a connection on the principal bundle? Btw, what is your definition of connection on Principal bundle? –  Aug 11 '19 at 14:07

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