Einstein operator

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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.