Einstein metric: Difference between revisions

Revision as of 06:48, 9 April 2007

This article defines a property that makes sense for a Riemannian metric over a differential manifold

This property of a Riemannian metric is Ricci flow-preserved, that is, it is preserved under the forward Ricci flow

A Riemannian metric on a differential manifold is said to be an Einstein metric if the Ricci curvature tensor is proportional to the metric tensor.

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

where $λ$ is uniform for the whole manifold.

This value of $λ$ is termed the cosmological constant for the manifold.

$R i c (x, x)$ is constant for all unit tangent vectors at all points. In other words, the Ricci curvature is constant for all one-dimensional subspaces.

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
Constant-curvature metric: This is a metric with the property that the sectional curvature for all 2-dimensional subspaces being equal

The following turn out to be true:

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

@@ Line 12: / Line 12: @@
 ===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.
 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==
@@ Line 27: / Line 29: @@
 * [[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===