Definition with symbols
Let be a Riemannian manifold. is teremd an Eisetin metric if:
where is uniform for the whole manifold.
This value of is termed the cosmological constant for the manifold.
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 an Einstein metric with Ricci curvature constant everywhere
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