# Ricci-flat 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 everywhere zero*: Ricci curvature

## Contents

## Definition

### Symbol-free definition

A Riemannian metric on a differential manifold is said to be **Ricci-flat** if the Ricci curvature is zero at all points.

### Definition with symbols

Let be a Riemannian manifold. Then is termed a Ricci-flat metric if at all points.

## Relation with other properties

### Stronger properties

### Weaker properties

## Metaproperties

### Direct product-closedness

*This property of a Riemannian metric on a differential manifold is closed under taking direct products of manifolds*

Given two Riemannian manifolds and , such that both and are Ricci-flat, the natural induced metric on is also Ricci-flat.

## Flows

### Ricci flow

*The Riemannian metrics with this property are precisely the stationary points for this flow:* Ricci flow

The Ricci-flat metrics are precisely those metrics that are invariant under the Ricci flow. Ricci flows are intended to address the question: *given a differential manifold, can we give it a Ricci-flat metric?* The approach is to start with an arbitrary Riemannian metric, and then evolve it using the Ricci flow and take the limit as .

Closely related are the notions of volume-normalized Ricci flow and Yamabe flow.

- Properties of Riemannian metrics
- Ricci flow-preserved properties
- Properties of Riemannian metrics corresponding to zero curvature
- Curvature-based properties of Riemannian metrics
- Direct product-closed properties of Riemannian metrics
- Properties of Riemannian metrics characterized by being stationary under a flow