This is the property of the following curvature being constant: Ricci curvature
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.
- 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
Relation with other 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
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 .