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