# Einstein metric

From Diffgeom

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

## 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:

where is uniform for the whole manifold.

This value of 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