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. Let denote a flat vector bundle, and be the covariant derivative associated to the flat connection on E. Let denote the vector space (in fact a sheaf of modules over ) 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. 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. 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.