This is the property of the following curvature being everywhere zero: Ricci curvature
Definition with symbols
Let be a Riemannian manifold. Then is termed a Ricci-flat metric if at all points.
Relation with other properties
Given two Riemannian manifolds and , such that both and are Ricci-flat, the natural induced metric on is also Ricci-flat.
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 .