Einstein metric

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

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 (M,g) be a Riemannian manifold. g is teremd an Eisetin metric if:

R_{ij}(g) = \lambda g_{ij}

where \lambda is uniform for the whole manifold.

This value of \lambda is termed the cosmological constant for the manifold.

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 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