Torsion of a linear connection - Revision history

Vipul: /* Definition part */

2012-01-06T17:57:18Z

Definition part

← Older revision		Revision as of 17:57, 6 January 2012
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	<math>\tau(\nabla)(X,Y) = \nabla_X Y- \nabla_Y X - [X,Y]</math>		<math>\tau(\nabla)(X,Y) = \nabla_X Y- \nabla_Y X - [X,Y]</math>

	A linear connection whose torsion is zero is termed a ~~'''~~torsion-free linear connection~~'''~~.		A linear connection whose torsion is zero is termed a [[torsion-free linear connection]].

	==Facts==		==Facts==

Vipul: /* Tensoriality */

2012-01-06T17:48:27Z

Tensoriality

← Older revision		Revision as of 17:48, 6 January 2012
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	The torsion map is a <math>(1,2)</math> tensor. It is tensorial in both <math>X</math> and <math>Y</math>. This means that the value of the torsion of a connection for two vector fields at a point depends only on the values of the vector fields at that point. In other words, <math>\tau(\nabla)(X,Y)</math> at <math>p</math> depends on <math>\nabla, X(p), Y(p)</math> only and does not depend on how <math>X</math> and <math>Y</math> behave elsewhere on the manifold.		The torsion map is a <math>(1,2)</math> tensor. It is tensorial in both <math>X</math> and <math>Y</math>. This means that the value of the torsion of a connection for two vector fields at a point depends only on the values of the vector fields at that point. In other words, <math>\tau(\nabla)(X,Y)</math> at <math>p</math> depends on <math>\nabla, X(p), Y(p)</math> only and does not depend on how <math>X</math> and <math>Y</math> behave elsewhere on the manifold.

			===Antisymmetry===

			{{further\|[[Torsion is antisymmetric]]}}

			We have that the torsion tensor is antisymmetric, i.e., we have:

			<math>\tau(\nabla)(Y,X) = -\tau(\nabla)(X,Y)</math>

			Equivalently, we have that:

			<math>\tau(\nabla)(X,X) = 0</math>

Vipul at 17:46, 6 January 2012

2012-01-06T17:46:55Z

← Older revision		Revision as of 17:46, 6 January 2012
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	A linear connection whose torsion is zero is termed a '''torsion-free linear connection'''.		A linear connection whose torsion is zero is termed a '''torsion-free linear connection'''.

			==Facts==

	===Tensoriality===		===Tensoriality===

	{{further\|[[Torsion is tensorial]]}}		{{further\|[[Torsion is tensorial]]}}
	The torsion map is a <math>(1,2)</math> tensor. It is tensorial in both <math>X</math> and <math>Y</math>.
			The torsion map is a <math>(1,2)</math> tensor. It is tensorial in both <math>X</math> and <math>Y</math>. This means that the value of the torsion of a connection for two vector fields at a point depends only on the values of the vector fields at that point. In other words, <math>\tau(\nabla)(X,Y)</math> at <math>p</math> depends on <math>\nabla, X(p), Y(p)</math> only and does not depend on how <math>X</math> and <math>Y</math> behave elsewhere on the manifold.

Vipul: 5 revisions

2008-05-18T20:12:02Z

5 revisions

← Older revision	Revision as of 20:12, 18 May 2008
(No difference)

Vipul at 22:58, 5 April 2008

2008-04-05T22:58:42Z

← Older revision		Revision as of 22:58, 5 April 2008
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	The torsion of <math>\nabla</math>, denoted as <math>\tau(\nabla)</math>, is defined as a map that takes as input 2 vector fields and outputs a third vector field, as follows:		The torsion of <math>\nabla</math>, denoted as <math>\tau(\nabla)</math>, is defined as a map that takes as input 2 vector fields and outputs a third vector field, as follows:

	<math>\tau(\nabla)(X,Y) = \~~nabla_YX~~ - \~~nabla_XY~~ - [X,Y]</math>		<math>\tau(\nabla)(X,Y) = \nabla_X Y- \nabla_Y X - [X,Y]</math>

	A linear connection whose torsion is zero is termed a '''torsion-free linear connection'''.		A linear connection whose torsion is zero is termed a '''torsion-free linear connection'''.

Vipul at 00:33, 5 April 2008

2008-04-05T00:33:25Z

← Older revision		Revision as of 00:33, 5 April 2008
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	The torsion of <math>\nabla</math>, denoted as <math>\tau(\nabla)</math>, is defined as a map that takes as input 2 vector fields and outputs a third vector field, as follows:		The torsion of <math>\nabla</math>, denoted as <math>\tau(\nabla)</math>, is defined as a map that takes as input 2 vector fields and outputs a third vector field, as follows:

	<math>\tau(\nabla)(X,Y) = \~~nabla_XY~~ - \~~nabla_YX~~ - [X,Y]</math>		<math>\tau(\nabla)(X,Y) = \nabla_YX - \nabla_XY - [X,Y]</math>

	A linear connection whose torsion is zero is termed a '''torsion-free linear connection'''.		A linear connection whose torsion is zero is termed a '''torsion-free linear connection'''.

Vipul at 23:53, 4 April 2008

2008-04-04T23:53:15Z

← Older revision		Revision as of 23:53, 4 April 2008
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	* A [[differential manifold]] <math>M</math>		* A [[differential manifold]] <math>M</math>
	* A [[linear connection]] <math>\nabla</math> on <math>M</math> (viz a connection on the [[tangent bundle]] <math>TM</math>).		* A [[linear connection]] <math>\nabla</math> on <math>M</math> (viz., a connection on the [[tangent bundle]] <math>TM</math>).

	===Definition part===		===Definition part===
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	===Tensoriality===		===Tensoriality===

	The torsion map is a <math>(1,2)</math> tensor. It is tensorial in both <math>X</math> and <math>Y</math>~~. This can actually be easily checked by hand~~.		{{further\|[[Torsion is tensorial]]}}
			The torsion map is a <math>(1,2)</math> tensor. It is tensorial in both <math>X</math> and <math>Y</math>.

Vipul at 18:54, 7 March 2007

2007-03-07T18:54:12Z

← Older revision		Revision as of 18:54, 7 March 2007
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			{{tensoroftype\|(1,2)}}

	==Definition==		==Definition==

Vipul: Started the page

2007-03-07T16:02:30Z

Started the page

New page

==Definition==

===Given data===

* A [[differential manifold]] <math>M</math>
* A [[linear connection]] <math>\nabla</math> on <math>M</math> (viz a connection on the [[tangent bundle]] <math>TM</math>).

===Definition part===

The torsion of <math>\nabla</math>, denoted as <math>\tau(\nabla)</math>, is defined as a map that takes as input 2 vector fields and outputs a third vector field, as follows:

<math>\tau(\nabla)(X,Y) = \nabla_XY - \nabla_YX - [X,Y]</math>

A linear connection whose torsion is zero is termed a '''torsion-free linear connection'''.

===Tensoriality===

The torsion map is a <math>(1,2)</math> tensor. It is tensorial in both <math>X</math> and <math>Y</math>. This can actually be easily checked by hand.