# Difference between revisions of "Flat connection"

From Diffgeom

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===Symbol-free 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. | + | 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=== | ===Definition with symbols=== | ||

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<math>R(X,Y) = \nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]} = 0</math> | <math>R(X,Y) = \nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]} = 0</math> | ||

+ | |||

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

+ | |||

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

+ | |||

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

+ | |||

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

## Revision as of 23:43, 12 April 2008

## 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.