# Ricci curvature tensor

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

## Contents

## Description

### Given data

A differential manifold with a linear connection on it.

### Definition part

The Ricci curvature tensor is a -tensor that takes as input two vector fields and outputs a scalar function, as follows.

Let and be two vector fields. Then consider the map that sends a vector field to (here denotes the Riemann curvature tensor).

This is a linear map. The Ricci curvature function is defined as the trace of this map.

Explicitly, it is given by:

The Ricci curvature function is a tensor because the Riemann curvature function is also a tensor.

### For a Riemannian or pseudo-Riemannian manifold

`Further information: Ricci curvature tensor of Levi-Civita connection`

For a Riemannian manifold or pseudo-Riemannian manifold, we can give the Levi-Civita connection, a natural choice of linear connection. The Ricci curvature tensor of this connection, is termed the Ricci curvature tensor of the Riemannian or pseudo-Riemannian manifold.