Einstein metric - Revision history

Vipul: /* Relation with other properties */

2009-02-13T00:50:42Z

Relation with other properties

← Older revision		Revision as of 00:50, 13 February 2009
Line 35:		Line 35:
	The following properties of Riemannian metrics are stronger than the property of being an Einstein metric:		The following properties of Riemannian metrics are stronger than the property of being an Einstein metric:

	* [[Ricci-flat metric]]: This is an Einstein metric with cosmological constant zero, that is, with Ricci curvature zero everywhere		* [[Weaker than::Ricci-flat metric]]: This is an Einstein metric with cosmological constant zero, that is, with Ricci curvature zero everywhere
	* [[Constant-curvature metric]]: This is a metric with the property that the sectional curvature for all 2-dimensional subspaces being equal. The implication holds because the Ricci curvature associated with a direction is a sum of sectional curvatures of planes containing that direction, and all the sectional curvatures in turn are constant.		* [[Weaker than::Constant-curvature metric]]: This is a metric with the property that the sectional curvature for all 2-dimensional subspaces being equal. The implication holds because the Ricci curvature associated with a direction is a sum of sectional curvatures of planes containing that direction, and all the sectional curvatures in turn are constant.

	===In low dimensions===		===In low dimensions===

Vipul: 9 revisions

2008-05-18T19:39:20Z

9 revisions

← Older revision	Revision as of 19:39, 18 May 2008
(No difference)

Vipul: /* Relation with other properties */

2007-09-02T15:10:50Z

Relation with other properties

← Older revision		Revision as of 15:10, 2 September 2007
Line 43:		Line 43:

	* For manifolds of dimension upto three, Einstein metrics are precisely the same as constant-curvature metrics		* For manifolds of dimension upto three, Einstein metrics are precisely the same as constant-curvature metrics

			==Flows==

			{{stationaryflow\|volume-normalized Ricci flow}}

			The volume-normalized Ricci flow is a flow on the [[space of all Riemannian metrics]] on a differential manifold, for which the stationary points are precisely the Einstein metrics.

			The interest in Ricci flows in the context of Einstein metrics arises from the following general question: ''given a [[differential manifold]], can we associate an [[Einstein metric]] to that differential manifold?'' The idea would be to start with an arbitrary [[Riemannian metric]] and then evolve it using the [[volume-normalized Ricci flow]], and take the limit as <math>t \to \infty</math>.

Vipul: /* For pseudo-Riemannian manifolds */

2007-09-02T15:06:28Z

For pseudo-Riemannian manifolds

← Older revision		Revision as of 15:06, 2 September 2007
Line 27:		Line 27:
	===For pseudo-Riemannian manifolds===		===For pseudo-Riemannian manifolds===

	We can also talk of whether a [pseudo-Riemannian metric]] is an Einstein metric. {{further\|[[Einstein pseudo-Riemannian metric]]}}		We can also talk of whether a [[pseudo-Riemannian metric]] is an Einstein metric. {{further\|[[Einstein pseudo-Riemannian metric]]}}

	==Relation with other properties==		==Relation with other properties==

Vipul at 02:24, 31 August 2007

2007-08-31T02:24:23Z

← Older revision		Revision as of 02:24, 31 August 2007
Line 24:		Line 24:

	This constant is the same <math>\lambda</math> as above -- the cosmological constant.		This constant is the same <math>\lambda</math> as above -- the cosmological constant.

			===For pseudo-Riemannian manifolds===

			We can also talk of whether a [pseudo-Riemannian metric]] is an Einstein metric. {{further\|[[Einstein pseudo-Riemannian metric]]}}

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Definition with symbols */

2007-08-27T08:18:06Z

Definition with symbols

← Older revision		Revision as of 08:18, 27 August 2007
Line 22:		Line 22:

	* <math>Ric(x,x)</math> is constant for all unit tangent vectors at all points. In other words, the Ricci curvature is constant for all one-dimensional subspaces.		* <math>Ric(x,x)</math> is constant for all unit tangent vectors at all points. In other words, the Ricci curvature is constant for all one-dimensional subspaces.

			This constant is the same <math>\lambda</math> as above -- the cosmological constant.

	==Relation with other properties==		==Relation with other properties==

Vipul at 02:37, 25 April 2007

2007-04-25T02:37:27Z

← Older revision		Revision as of 02:37, 25 April 2007
Line 3:		Line 3:
	{{Ricci flow-preserved}}		{{Ricci flow-preserved}}

			{{constancyproperty\|[[Ricci curvature]]}}

	==Definition==		==Definition==

61.12.4.26: /* Stronger properties */

2007-04-24T18:09:58Z

Stronger properties

← Older revision		Revision as of 18:09, 24 April 2007
Line 29:		Line 29:

	* [[Ricci-flat metric]]: This is an Einstein metric with cosmological constant zero, that is, with Ricci curvature zero everywhere		* [[Ricci-flat metric]]: This is an Einstein metric with cosmological constant zero, that is, with Ricci curvature zero everywhere
	* [[Constant-curvature metric]]: This is a metric with the property that the sectional curvature for all 2-dimensional subspaces being equal		* [[Constant-curvature metric]]: This is a metric with the property that the sectional curvature for all 2-dimensional subspaces being equal. The implication holds because the Ricci curvature associated with a direction is a sum of sectional curvatures of planes containing that direction, and all the sectional curvatures in turn are constant.

	===In low dimensions===		===In low dimensions===

Vipul at 06:48, 9 April 2007

2007-04-09T06:48:57Z

← Older revision		Revision as of 06:48, 9 April 2007
Line 12:		Line 12:
	===Definition with symbols===		===Definition with symbols===

	Let <math>(M,g)</math> be a Riemannian manifold. <math>g</math> is teremd an Eisetin metric if:		Let <math>(M,g)</math> be a Riemannian manifold. <math>g</math> is teremd an Eisetin metric if it satisfies the following equivalent conditions:

	<math>R_{ij}(g) = \lambda g_{ij}</math>		* <math>Ric_{ij}(g) = \lambda g_{ij}</math>

	where <math>\lambda</math> is uniform for the whole manifold.		where <math>\lambda</math> is uniform for the whole manifold.

	This value of <math>\lambda</math> is termed the '''cosmological constant''' for the manifold.		This value of <math>\lambda</math> is termed the '''cosmological constant''' for the manifold.

			* <math>Ric(x,x)</math> is constant for all unit tangent vectors at all points. In other words, the Ricci curvature is constant for all one-dimensional subspaces.

	==Relation with other properties==		==Relation with other properties==
Line 27:		Line 29:

	* [[Ricci-flat metric]]: This is an Einstein metric with cosmological constant zero, that is, with Ricci curvature zero everywhere		* [[Ricci-flat metric]]: This is an Einstein metric with cosmological constant zero, that is, with Ricci curvature zero everywhere
	* [[Constant-curvature metric]]: This is ~~an Einstein~~ metric with ~~Ricci~~ curvature ~~constant everywhere~~		* [[Constant-curvature metric]]: This is a metric with the property that the sectional curvature for all 2-dimensional subspaces being equal

	===In low dimensions===		===In low dimensions===

Vipul at 17:14, 8 April 2007

2007-04-08T17:14:11Z

← Older revision		Revision as of 17:14, 8 April 2007
Line 1:		Line 1:
	{{Riemannian metric property}}		{{Riemannian metric property}}

			{{Ricci flow-preserved}}


	==Definition==		==Definition==