Einstein metric
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
This is the property of the following curvature being constant: Ricci curvature
Contents
Definition
Symbol-free definition
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.
Definition with symbols
Let be a Riemannian manifold.
is teremd an Eisetin metric if it satisfies the following equivalent conditions:
where is uniform for the whole manifold.
This value of is termed the cosmological constant for the manifold.
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is constant for all unit tangent vectors at all points. In other words, the Ricci curvature is constant for all one-dimensional subspaces.
This constant is the same as above -- the cosmological constant.
For pseudo-Riemannian manifolds
We can also talk of whether a pseudo-Riemannian metric is an Einstein metric. Further information: Einstein pseudo-Riemannian metric
Relation with other properties
Stronger properties
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 implication holds because the Ricci curvature associated with a direction is a sum of sectional curvatures of planes containing that direction, and all the sectional curvatures in turn are constant.
In low dimensions
The following turn out to be true:
- For manifolds of dimension upto three, Einstein metrics are precisely the same as constant-curvature metrics
Flows
volume-normalized Ricci flow
The Riemannian metrics with this property are precisely the stationary points for this flow: volume-normalized Ricci flow
The volume-normalized Ricci flow is a flow on the space of all Riemannian metrics on a differential manifold, for which the stationary points are precisely the Einstein metrics.
The interest in Ricci flows in the context of Einstein metrics arises from the following general question: given a differential manifold, can we associate an Einstein metric to that differential manifold? The idea would be to start with an arbitrary Riemannian metric and then evolve it using the volume-normalized Ricci flow, and take the limit as .
- Properties of Riemannian metrics
- Ricci flow-preserved properties
- Properties of Riemannian metrics corresponding to constancy of curvature
- Sectional curvature-based properties of Riemannian metrics
- Curvature-based properties of Riemannian metrics
- Properties of Riemannian metrics characterized by being stationary under a flow