Constant-scalar curvature metric - Revision history

Vipul: 4 revisions

2008-05-18T19:36:28Z

4 revisions

← Older revision	Revision as of 19:36, 18 May 2008
(No difference)

Vipul: /* Weaker properties */

2007-09-02T15:20:18Z

Weaker properties

← Older revision		Revision as of 15:20, 2 September 2007
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	* [[Constantly signed-scalar curvature metric]]		* [[Constantly signed-scalar curvature metric]]

			==Flows==

			{{stationaryflow\|Yamabe flow}}

			The [[Yamabe conjecture]] states that every [[conformal class of Riemannian metrics]] contains a constant-scalar curvature metric. The Yamabe flow is intended to help answer this conjecture positively. The Yamabe flow has, for its stationary points, the metrics of constant scalar curvature. Further, the Yamabe flow preserves the conformal class of the Riemannian metric.

			The idea is to start with an arbitrary member of the conformal class, evolve it using the Yamabe flow, and take the limit as <math>t \to \infty</math>.

Vipul: /* Stronger properties */

2007-09-02T09:41:16Z

Stronger properties

← Older revision		Revision as of 09:41, 2 September 2007
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	The following properties of Riemannian metrics are stronger than the property of being a constant-scalar curvature metric:		The following properties of Riemannian metrics are stronger than the property of being a constant-scalar curvature metric:

			* [[Flat metric]]
			* [[Conformally flat metric]]
	* [[Constant-curvature metric]]		* [[Constant-curvature metric]]
	* [[Einstein metric]]		* [[Einstein metric]]

Vipul: /* Stronger properties */

2007-04-25T08:24:50Z

Stronger properties

← Older revision		Revision as of 08:24, 25 April 2007
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	* [[Homogeneous metric]]		* [[Homogeneous metric]]
	* [[Locally homogeneous metric]]		* [[Locally homogeneous metric]]
			* [[Zero-scalar curvature metric]]

	===Weaker properties===		===Weaker properties===

	* [[Constantly signed-scalar curvature metric]]		* [[Constantly signed-scalar curvature metric]]

Vipul at 02:44, 25 April 2007

2007-04-25T02:44:37Z

New page

{{Riemannian metric property}}

{{constancyproperty|[[scalar curvature]]}}

==Definition==

===Symbol-free definition===

A [[Riemannian metric]] on a [[differential manifold]] is said to be a '''constant-scalar curvature metric''' if the scalar curvature at all points is equal.

===Definition with symbols===

{{fillin}}

==Relation with other properties==

===Stronger properties===

The following properties of Riemannian metrics are stronger than the property of being a constant-scalar curvature metric:

* [[Constant-curvature metric]]
* [[Einstein metric]]
* [[Ricci-flat metric]]
* [[Homogeneous metric]]
* [[Locally homogeneous metric]]

===Weaker properties===

* [[Constantly signed-scalar curvature metric]]