Curvature of a connection - Revision history

Vipul: /* In local coordinates */

2008-05-22T12:24:33Z

In local coordinates

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

	Here, <math>\~~Omega~~</math> is a [[matrix of connection forms]].		Here, <math>\omega</math> is a [[matrix of connection forms]].

	===In the linear case===		===In the linear case===

2008-05-18T19:37:21Z

11 revisions

← Older revision	Revision as of 19:37, 18 May 2008
(No difference)

2008-04-11T22:41:45Z

In local coordinates

← Older revision		Revision as of 22:41, 11 April 2008
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	<math>\Omega := d\omega + \omega \wedge \omega</math>		<math>\Omega := d\omega + \omega \wedge \omega</math>

			Here, <math>\Omega</math> is a [[matrix of connection forms]].

	===In the linear case===		===In the linear case===

2008-04-11T22:40:02Z

Definition

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

			===In local coordinates===

			{{further\|[[curvature matrix of a connection]]}}

			Consider a system of local coordinate charts for <math>M</math> such that the vector bundle <math>E</math> is trivial on each chart. For any connection <math>\nabla</math>, we can write a matrix that, in local coordinates, describes the curvature of <math>\nabla</math>. This matrix is sometimes denoted as <math>\Omega</math>, and is defined by:

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

	===In the linear case===		===In the linear case===

2008-04-05T00:55:07Z

Properties

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	{{further\|[[Curvature is tensorial]]}}		{{further\|[[Curvature is tensorial]]}}

			The curvature is tensorial in ''all'' three arguments. This is best proved by proving <math>C^\infty</math>-linearity in all arguments, via a computation.

			===Antisymmetry===

			{{further\|[[Curvature is antisymmetric in first two variables]]}}

			We have the following identity:

			<math>R(X,Y) = -R(Y,X)</math>

2008-04-05T00:53:04Z

← Older revision		Revision as of 00:53, 5 April 2008
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	{{quotation\|''This curvature is also sometimes known as the '''Riemann curvature tensor'''. However, the latter term is usually reserved for situations where we have a [[linear connection]], in particular, the [[Riemann curvature tensor of Levi-Civita connection\|Riemann curvature tensor arising]] from the [[Levi-Civita connection]] for a [[Riemannian manifold\|Riemannian]] or [[pseudo-Riemannian manifold]]}}		{{quotation\|''This curvature is also sometimes known as the '''Riemann curvature tensor'''. However, the latter term is usually reserved for situations where we have a [[linear connection]], in particular, the [[Riemann curvature tensor of Levi-Civita connection\|Riemann curvature tensor arising]] from the [[Levi-Civita connection]] for a [[Riemannian manifold\|Riemannian]] or [[pseudo-Riemannian manifold]]''}}
	==Definition==		==Definition==

2008-04-05T00:52:48Z

← Older revision		Revision as of 00:52, 5 April 2008
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	{{quotation\|''This curvature is also sometimes known as the '''Riemann curvature tensor'''. However, the latter term is usually reserved for situations where we have a [[linear connection]], in particular, the [[Levi-Civita connection]] for a [[Riemannian manifold]]}}		{{quotation\|''This curvature is also sometimes known as the '''Riemann curvature tensor'''. However, the latter term is usually reserved for situations where we have a [[linear connection]], in particular, the [[Riemann curvature tensor of Levi-Civita connection\|Riemann curvature tensor arising]] from the [[Levi-Civita connection]] for a [[Riemannian manifold\|Riemannian]] or [[pseudo-Riemannian manifold]]}}
	==Definition==		==Definition==

2008-04-05T00:51:35Z

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	{{~~further~~\|[[Riemann curvature tensor]]}}		{{quotation\|''This curvature is also sometimes known as the '''Riemann curvature tensor'''. However, the latter term is usually reserved for situations where we have a [[linear connection]], in particular, the [[Levi-Civita connection]] for a [[Riemannian manifold]]}}
	==Definition==		==Definition==

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	===In the linear case===		===In the linear case===

	In the special case where <math>E = TM</math>, we ~~have~~ that <math>X,Y, Z \in \Gamma(TM)</math>. We can thus think of this map as a (1,3)-tensor because it takes as input three vector fields and outputs one vector field.		In the special case where <math>E = TM</math> (the case of a [[linear connection]]) we get that <math>X,Y, Z \in \Gamma(TM)</math>. We can thus think of this map as a (1,3)-tensor because it takes as input three vector fields and outputs one vector field.

	~~This is the famed~~ [[~~Riemann curvature tensor~~]] ~~that is important for its algebraic and differential properties.~~		==Properties==

			===Tensoriality===

			{{further\|[[Curvature is tensorial]]}}

2007-04-08T15:50:00Z

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	{{~~tensoroftype~~\|~~(1,3)~~}}		{{further\|[[Riemann curvature tensor]]}}

	==Definition==		==Definition==

2007-03-07T18:31:32Z