# Einstein tensor

From Diffgeom

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

## Definition

### Given data

A manifold with a connection 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 -tensor that takes as input two vector fields and outputs a scalar function, as follows.

where denotes the Ricci curvature tensor and denotes the scalar curvature function.