Difference between revisions of "Flat connection"

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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.
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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>
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===Definition in local coordinates===
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In local coordinates, we require that the [[curvature matrix of a connection|curvature matrix]] should vanish identically; in other words:
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<math>\Omega := d\omega + \omega \wedge \omega = 0</math>
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where <math>\omega</math> is the [[matrix of connection forms]].

Revision as of 23:43, 12 April 2008

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 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.