Definition with symbols
A connection on a vector bundle over a differential manifold is said to be flat or integrable or curvature-free or locally flat if the curvature form vanishes identically, viz for any vector fields and :
Definition in local coordinates
In local coordinates, we require that the curvature matrix should vanish identically; in other words:
where is the matrix of connection forms.
Further information: Flat connection equals module structure over differential operators
Recall that one alternative view of a connection is as giving the space of sections the structure of a module over the connection algebra of . Equivalently, it is a way of giving the sheaf of sections the structure of a sheaf-theoretic module over the sheaf of connection algebras.
The connection is flat if and only if this descends to a module structure over the sheaf of differential operators. In other words, a flat connection is equivalent to a structure of as a module over the sheaf of differential operators.