In some places that I have seen, given the covariant derivative $\nabla$ of a linear connection on a vector bundle $\pi: E\rightarrow M$ (dim(M)=k), an Ehresmann connection is defined in the following way:
Find $k$ local sections $\sigma_i$ of $E$ such that $\nabla(\sigma_i)=0$ and then compute the horizontal space by taking images of $T_xM$ by $D\sigma_i: T_xM \rightarrow T_{x,\sigma_i(x)}E$.
But as far as I know such sections would only exist when curvature vanishes? So what am I missing here?