# 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.

- 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 .