We introduce connections on principal bundles. We show that if  $ \pi \colon P \to M$ is a principal bundle and $ \rho \colon G \to \operatorname{GL}(V)$  is a smooth representation on a vector space, then the connection on $P$ induces a connection on the associated vector bundle $ \rho(P) := P \times_G V$.

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 $ \rho$ to be the identity representation then $E \cong \rho( \operatorname{Fr}(E))$. In this case 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

$$ \left\{ \text{connections on } P \right\} \leftrightarrow \left\{ \text{connections on } \rho(P) \right\}$$

need not be injective or surjective.

Comments and questions?