We introduce connections on principal bundles. We show that if  $ \pi \colon P \to M$ is a principal bundle and $\sigma$ is a representation of $G$ on a vector space $V$, then the connection on $P$ induces a connection on the associated vector bundle $ P \times_G V$.

The proof uses the technical result that $V$-valued forms on $P$ that are both horizontal and equivariant can be identified with $E$-valued forms on $M$.

If $E$ is a vector bundle of rank $k$ then its frame bundle $\operatorname{Fr}(E)$ is a principal $\operatorname{GL}(k)$-bundle. If we take $ \sigma$ to be the identity representation then we show how a connection on $E$ induces a connection on $ \operatorname{Fr}(E)$. This establishes a bijective correspondence between connections on $E$ and connections on $ \operatorname{Fr}(E)$. In general however the map

$$ \left\{ \text{connections on } P \right\} \leftrightarrow \left\{ \text{connections on } P \times_G V \right\}$$

may be neither injective or surjective.


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