Einstein tensor

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This article defines a tensor (viz a section on a tensor bundle over the manifold) of type (0,2)


Given data

A manifold M with a connection \nabla on it. BY default, we may take a Riemannian manifold and consider the Levi-Civita connection on it.

Definition part

The Einstein tensor is a (0,2)-tensor that takes as input two vector fields and outputs a scalar function, as follows.

(X,Y) \mapsto Ric(X,Y) - (1/2)Rg(X,Y)

where Ric(X,Y) denotes the Ricci curvature tensor and R denotes the scalar curvature function.