# Flat connection

## Contents

## 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 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.

## 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 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.