Flat connection

Definition

Symbol-free definition

A connection on 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 $Y$:

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