Einstein tensor

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

Definition

Given data

A manifold $M$ with a connection $\mathrm{\nabla }$ 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.