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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?

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    $\begingroup$ The condition on the vanishing of the covariant derivatives should be a punctual one, not a local one and it might be formulated so to involve only one section. That is, the condition should read "Given $p \in \pi^{-1}(x)$ with $x \in M$, find a local section $\sigma$ of $E$ over a neighbordhood of $U \ni x$ such that $\sigma(x) = p$ and $(\nabla \sigma_i)(x)=0$. Then compute the horizontal space by taking the image of $D\sigma : T_xM \to T_{x, p}E$". In this way, we can infer nothing about the Frobenius integrability of the horizontal distribution i.e. about the vanishing of the curvature. $\endgroup$
    – Jordan Payette
    Jul 23, 2015 at 13:33
  • $\begingroup$ This seems more right I will reflect on it for a while thanks. $\endgroup$
    – Sina
    Jul 23, 2015 at 14:17

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