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.
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
, and the flatness condition is equivalent to the property
In other words, the graded vector space
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.