Ricci curvature tensor - Revision history

Vipul at 02:29, 24 July 2009

2009-07-24T02:29:36Z

← Older revision		Revision as of 02:29, 24 July 2009
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	===Given data===		===Given data===

	A [[differential manifold]] <math>M</math> with a [[linear connection]] <math>\nabla</math> on it.		A [[differential manifold]] <math>M</math> with a [[defining ingredient::linear connection]] <math>\nabla</math> on it.

	===Definition part===		===Definition part===
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	The Ricci curvature tensor is a <math>(0,2)</math>-tensor that takes as input two vector fields and outputs a scalar function, as follows.		The Ricci curvature tensor is a <math>(0,2)</math>-tensor that takes as input two vector fields and outputs a scalar function, as follows.

	Let <math>X</math> and <math>Y</math> be two [[vector field]]s. Then consider the map that sends a vector field <math>Z</math> to <math>R(X,Z)Y</math> (here <math>R</math> denotes the [[Riemann curvature tensor]]).		Let <math>X</math> and <math>Y</math> be two [[vector field]]s. Then consider the map that sends a vector field <math>Z</math> to <math>R(X,Z)Y</math> (here <math>R</math> denotes the [[defining ingredient::Riemann curvature tensor]]).

	This is a linear map. The Ricci curvature function <math>Ric(X,Y)</math> is defined as the trace of this map.		This is a linear map. The Ricci curvature function <math>Ric(X,Y)</math> is defined as the trace of this map.
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	Explicitly, it is given by:		Explicitly, it is given by:

	<math>Tr(Z \mapsto \nabla_X\nabla_ZY - \nabla_Z\nabla_XY - \nabla_{[X,Z]}Y)</math>		<math>\operatorname{Tr}(Z \mapsto \nabla_X\nabla_ZY - \nabla_Z\nabla_XY - \nabla_{[X,Z]}Y)</math>

	The Ricci curvature function is a tensor because the Riemann curvature function is also a tensor.		The Ricci curvature function is a tensor because the Riemann curvature function is also a tensor.

Vipul: 5 revisions

2008-05-18T19:51:57Z

5 revisions

← Older revision	Revision as of 19:51, 18 May 2008
(No difference)

Vipul: /* Definition part */

2007-08-31T03:22:38Z

Definition part

← Older revision		Revision as of 03:22, 31 August 2007
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	Explicitly, it is given by:		Explicitly, it is given by:

	<math>Tr(Z \mapsto \nabla_X\nabla_ZY - \nabla_Z\nabla_XY - \nabla_{[X,Z]}Y</math>		<math>Tr(Z \mapsto \nabla_X\nabla_ZY - \nabla_Z\nabla_XY - \nabla_{[X,Z]}Y)</math>

	The Ricci curvature function is a tensor because the Riemann curvature function is also a tensor.		The Ricci curvature function is a tensor because the Riemann curvature function is also a tensor.

Vipul at 03:22, 31 August 2007

2007-08-31T03:22:03Z

← Older revision		Revision as of 03:22, 31 August 2007
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	===Given data===		===Given data===

	A manifold <math>M</math> with a [[linear connection]] <math>\nabla</math> ~~on it. For instance, we may take a Riemannian manifold and consider the [[Levi-Civita connection]]~~ on it.		A [[differential manifold]] <math>M</math> with a [[linear connection]] <math>\nabla</math> on it.

	===Definition part===		===Definition part===
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	The Ricci curvature tensor is a <math>(0,2)</math>-tensor that takes as input two vector fields and outputs a scalar function, as follows.		The Ricci curvature tensor is a <math>(0,2)</math>-tensor that takes as input two vector fields and outputs a scalar function, as follows.

	Let <math>X</math> and <math>Y</math> be two vector ~~fields~~. Then consider the map that sends a vector field <math>Z</math> to <math>R(X,Z)Y</math> (here <math>R</math> denotes the [[Riemann curvature tensor]]).		Let <math>X</math> and <math>Y</math> be two [[vector field]]s. Then consider the map that sends a vector field <math>Z</math> to <math>R(X,Z)Y</math> (here <math>R</math> denotes the [[Riemann curvature tensor]]).

	This is a linear map. The Ricci curvature function <math>Ric(X,Y)</math> is defined as the trace of this map.		This is a linear map. The Ricci curvature function <math>Ric(X,Y)</math> is defined as the trace of this map.

	~~The Ricci curvature function~~ is ~~a tensor because the Riemann curvature function is also a tensor.~~		Explicitly, it is given by:

	~~==Properties==~~		<math>Tr(Z \mapsto \nabla_X\nabla_ZY - \nabla_Z\nabla_XY - \nabla_{[X,Z]}Y</math>

	~~===Symmetry===~~		The Ricci curvature function is a tensor because the Riemann curvature function is also a tensor.

	The Ricci curvature function is ~~symmetric, viz <math>Ric(X,Y) = Ric(Y,X)</math>.~~

	~~==Related notions==~~

	~~===Ricci curvature===~~

	~~The Ricci curvature is closely related to~~ the ~~Ricci~~ curvature function ~~and~~ is ~~defined as follows. It associates to each one-dimensional subspace, the value <math>Ric(X,X)</math> for any unit vector in that one-dimensional subspace.~~

	~~The following turns out to be a way of computing the Ricci curvature of~~ a one-dimensional space. Take a unit vector in that subspace, and complete it to an orthonormal basis of the tangent space at the point. Now, add up the values of the sectional curvature for the subspaces spanned by this unit vector with each other unit vector in that orthonormal basis.

	~~Note that the Ricci curvature at one-dimensional spaces determines the Ricci curvature~~ tensor ~~by means of polarization~~.

	~~===Related properties of metrics===~~

	* [[Einstein metric]] is one where the Ricci curvature tensor is a ~~constant multiple of the metric tensor.~~		===For a Riemannian or pseudo-Riemannian manifold===
	* [[Ricci-flat metric]] is one where the Ricci curvature tensor vanishes identically (or ~~equivalently, the Ricci curvature is zero for all one~~-~~dimensional subspaces)~~

	~~==Flow==~~		{{further\|[[Ricci curvature tensor of Levi-Civita connection]]}}

	~~Given any~~ [[~~flow of a metric~~]], we ~~get~~ a ~~corresponding flow~~ of ~~the~~ Ricci curvature tensor~~. Note that~~ the Ricci curvature tensor is of the ~~same type (a <math>(0,2)</math>~~-~~tensor) as the metric tensor, hence we can actually compare the metric tensor with the Ricci curvature tensor~~.		For a [[Riemannian manifold]] or [[pseudo-Riemannian manifold]], we can give the [[Levi-Civita connection]], a natural choice of [[linear connection]]. The Ricci curvature tensor of this connection, is termed the Ricci curvature tensor of the Riemannian or pseudo-Riemannian manifold.

Vipul: /* Given data */

2007-07-31T03:59:22Z

Given data

← Older revision		Revision as of 03:59, 31 July 2007
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	===Given data===		===Given data===

	A manifold <math>M</math> with a connection <math>\nabla</math> on it. For instance, we may take a Riemannian manifold and consider the [[Levi-Civita connection]] on it.		A manifold <math>M</math> with a [[linear connection]] <math>\nabla</math> on it. For instance, we may take a Riemannian manifold and consider the [[Levi-Civita connection]] on it.

	===Definition part===		===Definition part===

Vipul: Added flow

2007-03-07T18:27:02Z

Added flow

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 ==Description==
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 * [[Einstein metric]] is one where the Ricci curvature tensor is a constant multiple of the metric tensor.
 * [[Ricci-flat metric]] is one where the Ricci curvature tensor vanishes identically (or equivalently, the Ricci curvature is zero for all one-dimensional subspaces)

Vipul: Started the page

2007-03-07T14:10:07Z

Started the page

New page

==Description==

===Given data===

A manifold <math>M</math> with a connection <math>\nabla</math> on it. For instance, we may take a Riemannian manifold and consider the [[Levi-Civita connection]] on it.

===Definition part===

The Ricci curvature tensor is a <math>(0,2)</math>-tensor that takes as input two vector fields and outputs a scalar function, as follows.

Let <math>X</math> and <math>Y</math> be two vector fields. Then consider the map that sends a vector field <math>Z</math> to <math>R(X,Z)Y</math> (here <math>R</math> denotes the [[Riemann curvature tensor]]).

This is a linear map. The Ricci curvature function <math>Ric(X,Y)</math> is defined as the trace of this map.

The Ricci curvature function is a tensor because the Riemann curvature function is also a tensor.

==Properties==

===Symmetry===

The Ricci curvature function is symmetric, viz <math>Ric(X,Y) = Ric(Y,X)</math>.

==Related notions==

===Ricci curvature===

The Ricci curvature is closely related to the Ricci curvature function and is defined as follows. It associates to each one-dimensional subspace, the value <math>Ric(X,X)</math> for any unit vector in that one-dimensional subspace.

The following turns out to be a way of computing the Ricci curvature of a one-dimensional space. Take a unit vector in that subspace, and complete it to an orthonormal basis of the tangent space at the point. Now, add up the values of the sectional curvature for the subspaces spanned by this unit vector with each other unit vector in that orthonormal basis.

Note that the Ricci curvature at one-dimensional spaces determines the Ricci curvature tensor by means of polarization.

===Related properties of metrics===

* [[Einstein metric]] is one where the Ricci curvature tensor is a constant multiple of the metric tensor.
* [[Ricci-flat metric]] is one where the Ricci curvature tensor vanishes identically (or equivalently, the Ricci curvature is zero for all one-dimensional subspaces)