Flat connection

Definition

Symbol-free definition

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 of the connection is zero everywhere.

Definition with symbols

A connection $\mathrm{\nabla }$ on a vector bundle $E$ over a differential manifold $M$ is said to be flat or integrable or curvature-free or locally flat if the curvature form vanishes identically, viz for any vector fields $X$ and $Y$:

$R(X,Y)={\mathrm{\nabla }}_X{\mathrm{\nabla }}_Y-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_X-{\mathrm{\nabla }}_{[X,Y]}=0$

Definition in local coordinates

In local coordinates, we require that the curvature matrix should vanish identically; in other words:

$\mathrm{\Omega }:=d\omega +\omega \wedge \omega =0$

where $\omega$ is the matrix of connection forms.

Alternative definitions

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 $\mathrm{\Gamma }(E)$ the structure of a module over the connection algebra of $M$. Equivalently, it is a way of giving the sheaf of sections $\mathcal{E}$ 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 $\mathcal{E}$ as a module over the sheaf of differential operators.