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Flat vector bundle

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated January 24, 2026

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.

01de 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.

02Flat 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.

03Examples

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This article is adapted from the Wikipedia article “Flat vector bundle”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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