# Difference between revisions of "Flat connection"

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===Symbol-free 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. | + | A [[defining ingredient::connection]] on a [[defining ingredient::vector bundle]] over a [[defining ingredient::differential manifold]] is said to be '''flat''' or '''integrable''' or '''curvature-free''' or '''locally flat''' if the [[defining ingredient::curvature of a connection|curvature of the connection]] is zero everywhere. |

===Definition with symbols=== | ===Definition with symbols=== | ||

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===Definition in local coordinates=== | ===Definition in local coordinates=== | ||

− | In local coordinates, we require that the [[curvature matrix of a connection|curvature matrix]] should vanish identically; in other words: | + | In local coordinates, we require that the [[defining ingredient::curvature matrix of a connection|curvature matrix]] should vanish identically; in other words: |

<math>\Omega := d\omega + \omega \wedge \omega = 0</math> | <math>\Omega := d\omega + \omega \wedge \omega = 0</math> | ||

− | where <math>\omega</math> is the [[matrix of connection forms]]. | + | where <math>\omega</math> is the [[defining ingredient::matrix of connection forms]]. |

===Alternative definitions=== | ===Alternative definitions=== |

## Latest revision as of 22:01, 24 July 2011

## 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.