Flat vector bundle

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 denote a flat vector bundle, and be the covariant derivative associated to the flat connection on E.
Let denote the vector space of differential forms on X with values in E. The covariant derivative defines a degree-1 endomorphism d, the differential of, and the flatness condition is equivalent to the property.
In other words, the graded vector space 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 with the connection forms 0 and. 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 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.