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

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 $(M,g)$ be a Riemannian manifold. Then $g$ is termed a Ricci-flat metric if $R_{ij}(g)=0$ at all points.

Relation with other properties

Stronger properties

• Flat metric

Weaker properties

• Einstein metric