# Flat connection

From Diffgeom

## 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 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.