Einstein operator

Definition

Given data

A Riemannian manifold $(M,g)$.

Definition part

The Einstein operator $G$ is a map that takes as input a 2-tensor over the Riemannian manifold and outputs another 2-tensor as follows:

$G(v)=v-(1/2)(Tr_{g}v)g$

Note that $G$ maps the Ricci curvature tensor to the Einstein tensor.