Volume-normalized Ricci flow - Revision history

Vipul: 4 revisions

2008-05-18T20:13:29Z

4 revisions

← Older revision	Revision as of 20:13, 18 May 2008
(No difference)

Vipul: /* Definition */

2007-09-02T14:46:41Z

Definition

← Older revision		Revision as of 14:46, 2 September 2007
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			{{Riemannian metric flow}}
	==Definition==		==Definition==

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	<math>\frac{\partial}{\partial t} g_{ij} = 2 (\frac{r}{n} g_{ij} - R_{ij})</math>		<math>\frac{\partial}{\partial t} g_{ij} = 2 (\frac{r}{n} g_{ij} - R_{ij})</math>

	Here <math>R_{ij}</math> denotes the [[Ricci curvature tensor]] ~~and~~ <math>r</math> is the ~~average value of~~ [[scalar curvature]] ~~over~~ the manifold.		Here <math>R_{ij}</math> denotes the [[Ricci curvature tensor]], <math>r</math> is the [[average scalar curvature]] and <math>n<math> is the dimension of the manifold.

	==Solution for the Ricci flow==		==Solution for the Ricci flow==

Vipul: /* Description */

2007-09-02T09:14:54Z

Description

← Older revision		Revision as of 09:14, 2 September 2007
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	==~~Description~~==		==Definition==

	~~===Equation for the~~ volume-normalized Ricci flow~~===~~		The '''volume-normalized Ricci flow''' is a [[flow]] in the space of Riemannian metrics on a differential manifold, though it can also be viewed as a flow in the space of conformal classes of Riemannian metrics.

	The equation for the volume-normalized Ricci flow in any dimension is:		The equation for the volume-normalized Ricci flow in any dimension is:

Vipul: Started the page properly

2007-03-07T19:50:04Z

Started the page properly

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 Here <math>R_{ij}</math> denotes the [[Ricci curvature tensor]] and <math>r</math> is the average value of [[scalar curvature]] over the manifold.
-==Facts==
+==Solution for the Ricci flow==
 {{stationarypoint|[[Einstein metric]]}}
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 Conversely, the stationary points for Ricci flows are precisely the Einstein metrics.

Vipul at 19:41, 7 March 2007

2007-03-07T19:41:42Z

New page

==Description==

===Equation for the volume-normalized Ricci flow===

The equation for the volume-normalized Ricci flow in any dimension is:

<math>\frac{\partial}{\partial t} g_{ij} = 2 (\frac{r}{n} g_{ij} - R_{ij})</math>

Here <math>R_{ij}</math> denotes the [[Ricci curvature tensor]] and <math>r</math> is the average value of [[scalar curvature]] over the manifold.

==Facts==

{{stationarypoint|[[Einstein metric]]}}

If <math>g</math> is an Einstein metric then the right-hand-side of the above equation becomes identically zero. Thus, we obtain that Einstein metrics are stationary points under the Ricci flow.

Conversely, the stationary points for Ricci flows are precisely the Einstein metrics.