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 [[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===
  
A [[connection]] <math>\nabla</math> on a [[differential manifold]] <math>M</math> is said to be '''flat''' or '''integrable''' or '''curvature-free''' or '''locally flat''' if the curvature form vanishes identically, viz for any vector fields <math>X</math> and <math>Y</math>:
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A [[connection]] <math>\nabla</math> on a [[vector bundle]] <math>E</math> over a [[differential manifold]] <math>M</math> is said to be '''flat''' or '''integrable''' or '''curvature-free''' or '''locally flat''' if the curvature form vanishes identically, viz for any vector fields <math>X</math> and <math>Y</math>:
  
 
<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 [[defining ingredient::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 [[defining ingredient::matrix of connection forms]].
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===Alternative definitions===
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{{further|[[Flat connection equals module structure over differential operators]]}}
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Recall that one alternative view of a connection is as giving the space of sections <math>\Gamma(E)</math> the structure of a module over the [[connection algebra]] of <math>M</math>. Equivalently, it is a way of giving the ''sheaf'' of sections <math>\mathcal{E}</math> the structure of a sheaf-theoretic module over the [[sheaf of connection algebras]].
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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 <math>\mathcal{E}</math> as a module over the [[sheaf of differential operators]].

Latest revision as of 22:01, 24 July 2011

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 vector bundle E over 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.

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 \Gamma(E) the structure of a module over the connection algebra of M. Equivalently, it is a way of giving the sheaf of sections \mathcal{E} 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 \mathcal{E} as a module over the sheaf of differential operators.