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## Summary

In mathematics, a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection.

## de Rham cohomology of a flat vector bundle

Let $\pi :E\to X$  denote a flat vector bundle, and $\nabla :\Gamma (X,E)\to \Gamma \left(X,\Omega _{X}^{1}\otimes E\right)$  be the covariant derivative associated to the flat connection on E.

Let $\Omega _{X}^{*}(E)=\Omega _{X}^{*}\otimes E$  denote the vector space (in fact a sheaf of modules over ${\mathcal {O}}_{X}$ ) of differential forms on X with values in E. The covariant derivative defines a degree-1 endomorphism d, the differential of $\Omega _{X}^{*}(E)$ , and the flatness condition is equivalent to the property $d^{2}=0$ .

In other words, the graded vector space $\Omega _{X}^{*}(E)$  is a cochain complex. Its cohomology is called the de Rham cohomology of E, or de Rham cohomology with coefficients twisted by the local coefficient system E.

## Flat trivializations

A trivialization of a flat vector bundle is said to be flat if the connection form vanishes in this trivialization. An equivalent definition of a flat bundle is the choice of a trivializing atlas with locally constant transition maps.

## Examples

• Trivial line bundles can have several flat bundle structures. An example is the trivial bundle over $\mathbb {C} \backslash \{0\},$  with the connection forms 0 and $-{\frac {1}{2}}{\frac {dz}{z}}$ . The parallel vector fields are constant in the first case, and proportional to local determinations of the square root in the second.
• The real canonical line bundle $\Lambda ^{\mathrm {top} }M$  of a differential manifold M is a flat line bundle, called the orientation bundle. Its sections are volume forms.
• A Riemannian manifold is flat if and only if its Levi-Civita connection gives its tangent vector bundle a flat structure.