Flat connection - Revision history

Vipul: /* Definition */

2011-07-24T22:01:20Z

Definition

← Older revision		Revision as of 22:01, 24 July 2011
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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===

Vipul: 4 revisions

2008-05-18T19:40:18Z

4 revisions

← Older revision	Revision as of 19:40, 18 May 2008
(No difference)

Vipul: /* Alternative definitions= */

2008-04-13T00:32:34Z

Alternative definitions=

← Older revision		Revision as of 00:32, 13 April 2008
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	where <math>\omega</math> is the [[matrix of connection forms]].		where <math>\omega</math> is the [[matrix of connection forms]].

	==Alternative definitions===		===Alternative definitions===

	{{further\|[[Flat connection equals module structure over differential operators]]}}		{{further\|[[Flat connection equals module structure over differential operators]]}}

Vipul at 00:29, 13 April 2008

2008-04-13T00:29:14Z

← Older revision		Revision as of 00:29, 13 April 2008
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	===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>:		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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	where <math>\omega</math> is the [[matrix of connection forms]].		where <math>\omega</math> is the [[matrix of connection forms]].

			==Alternative definitions===

			{{further\|[[Flat connection equals module structure over differential operators]]}}

			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]].

			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]].

Vipul at 23:43, 12 April 2008

2008-04-12T23:43:05Z

← Older revision		Revision as of 23:43, 12 April 2008
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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]].

Vipul at 17:57, 7 March 2007

2007-03-07T17:57:38Z

New page

{{connection property}}

==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.

===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>:

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