Flat vector bundle

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title: "Flat vector bundle" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["vector-bundles"] topic_path: "general/vector-bundles" source: "https://en.wikipedia.org/wiki/Flat_vector_bundle" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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.

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