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 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.