Ricci-flat metric - Revision history

Vipul: 6 revisions

2008-05-18T19:51:40Z

6 revisions

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Vipul: /* Metaproperties */

2007-09-02T15:16:27Z

Metaproperties

← Older revision		Revision as of 15:16, 2 September 2007
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	Given two Riemannian manifolds <math>(M,g_1)</math> and <math>(N,g_2)</math>, such that both <math>g_1</math> and <math>g_2</math> are Ricci-flat, the natural induced metric on <math>M \times N</math> is also Ricci-flat.		Given two Riemannian manifolds <math>(M,g_1)</math> and <math>(N,g_2)</math>, such that both <math>g_1</math> and <math>g_2</math> are Ricci-flat, the natural induced metric on <math>M \times N</math> is also Ricci-flat.

			==Flows==

			{{stationaryflow\|Ricci flow}}

			The Ricci-flat metrics are precisely those metrics that are invariant under the Ricci flow. Ricci flows are intended to address the question: ''given a [[differential manifold]], can we give it a [[Ricci-flat metric]]?'' The approach is to start with an arbitrary [[Riemannian metric]], and then evolve it using the [[Ricci flow]] and take the limit as <math>t \to \infty</math>.

			Closely related are the notions of [[volume-normalized Ricci flow]] and [[Yamabe flow]].

Vipul: /* Metaproperties */

2007-05-22T19:43:42Z

Metaproperties

← Older revision		Revision as of 19:43, 22 May 2007
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	==Metaproperties==		==Metaproperties==

	{{DP-closed ~~Riemannian~~ metric property}}		{{DP-closed metric property}}

	Given two Riemannian manifolds <math>(M,g_1)</math> and <math>(N,g_2)</math>, such that both <math>g_1</math> and <math>g_2</math> are Ricci-flat, the natural induced metric on <math>M \times N</math> is also Ricci-flat.		Given two Riemannian manifolds <math>(M,g_1)</math> and <math>(N,g_2)</math>, such that both <math>g_1</math> and <math>g_2</math> are Ricci-flat, the natural induced metric on <math>M \times N</math> is also Ricci-flat.

Vipul at 19:42, 22 May 2007

2007-05-22T19:42:40Z

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	* [[Einstein metric]]		* [[Einstein metric]]
			* [[Constant-scalar curvature metric]]
			* [[Zero-scalar curvature metric]]

			==Metaproperties==

			{{DP-closed Riemannian metric property}}

			Given two Riemannian manifolds <math>(M,g_1)</math> and <math>(N,g_2)</math>, such that both <math>g_1</math> and <math>g_2</math> are Ricci-flat, the natural induced metric on <math>M \times N</math> is also Ricci-flat.

Vipul at 19:17, 22 May 2007

2007-05-22T19:17:09Z

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	{{Ricci flow-preserved}}		{{Ricci flow-preserved}}

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

	==Definition==		==Definition==

Vipul at 17:09, 8 April 2007

2007-04-08T17:09:19Z

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	{{Riemannian metric property}}		{{Riemannian metric property}}

			{{Ricci flow-preserved}}

	==Definition==		==Definition==

Vipul: Started the page

2007-03-07T03:55:45Z

Started the page

New page

{{Riemannian metric property}}

==Definition==

===Symbol-free definition===

A [[Riemannian metric]] on a [[differential manifold]] is said to be '''Ricci-flat''' if the [[Ricci curvature]] is zero at all points.

===Definition with symbols===

Let <math>(M,g)</math> be a Riemannian manifold. Then <math>g</math> is termed a Ricci-flat metric if <math>R_{ij}(g) = 0</math> at all points.

==Relation with other properties==

===Stronger properties===

* [[Flat metric]]

===Weaker properties===

* [[Einstein metric]]