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

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	==Definition==		==Definition==

Vipul at 14:03, 8 April 2007

2007-04-08T14:03:02Z

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{{Riemannian metric property}}

==Definition==

===Definition in terms of a vector field===

A [[Riemannian metric]] <math>g</math> on a [[differential manifold]] <math>M</math> is termed a '''Ricci soliton''' if it satisfies the following:

<math>L_X(g) + 2\lambda g = -2R(g)</math>

at all points <math>x</math> and all times <math>t</math>.

Here:

* <math>X</math> is a suitable [[complete vector field]] on <math>M</math> and <math>L_X</math> denotes the Lie derivative with respect to <math>X</math>
* <math>\lambda</math> is a suitable real number
* <math>R(g)</math> denotes the [[Ricci curvature tensor]] of <math>g</math>.

In general, there may be more than one <math>X</math> giving the same Ricci soliton.

===Definition in terms of the Ricci flow===

A '''Ricci soliton''' is a point in the [[space of Riemannian metrics]] whose trajectory is a [[self-similar solution]] to the [[Ricci flow]]. In other words, if we look at the Ricci flow on the moduli space obtained on quotienting both by diffeomorphism equivalence and by conformal equivalence, its image is a fixed point in the Ricci flow on that.

==Relation with other properties==

===Stronger properties===

* [[Ricci-flat metric]]: Take <math>\lambda = 0, X = 0</math>
* [[Einstein metric]]: Take <math>X = 0</math> and <math>\lambda</math> as the negative of the cosmological constant of the Einstein metric.

===Properties of solitons===

Ricci solitons are such an important class of metrics that we also talk of properties ''of'' solitons (which actually just means properties of metrics satisfied by solitons), such as:

* [[shrinking Ricci solution]]
* [[steady Ricci soliton]]
* [[expanding Ricci soliton]]
* [[gradient Ricci soliton]]

Ricci soliton - Revision history

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Vipul at 17:15, 8 April 2007

Vipul at 14:03, 8 April 2007