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> [!definition] Definition. ([[flat connection]])
> We call a [[connection on a vector bundle|connection]] $A$ on a [[vector bundle]] $E$ **flat** if its [[curvature form]] vanishes: $F(A)=0$. A **flat vector bundle** is a [[vector bundle]] together with a choice of flat connection.
^definition
> [!basicexample] The **trivial connection** (or **product connection**).
> Consider the [[product bundle|trivial product bundle]] $E=B \times \mathbb{R}^{m}$. [[section|Sections]] $s \in \Gamma(E)$ are smooth maps $B \to \mathbb{R}^{m}$, that is, $\Gamma(E)=\Omega_{B}^{0}(E)=C^{\infty}(B; \mathbb{R}^{m})$.
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> In this case, the 'elementwise [[exterior derivative]] $d:C^{\infty}(B; \mathbb{R}^{m})=\Omega^{0}(B) \otimes \mathbb{R}^{m} \to \Omega^{1}(B) \otimes \mathbb{R}^{m}