Flat connection

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Template:Connection property

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 \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) = \nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]} = 0

Definition in local coordinates

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

\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 \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.