Ricci-flat metric

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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

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 (M,g_1) and (N,g_2), such that both g_1 and g_2 are Ricci-flat, the natural induced metric on M \times N 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 t \to \infty.

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