# Riemann curvature tensor of Levi-Civita connection

{{tensoroftype|(1,3)}

## Contents

## Definition

### As a (1,3)-tensor

Let be a connected differential manifold and be a Riemannian metric or pseudo-Riemannian metric on . Let denote the Levi-Civita connection of . The **Riemann curvature tensor** for the Riemannian metric is defined as the Riemann curvature tensor of the Levi-Civita connection, viz:

Since the Levi-Civita connection is a linear connection, is a linear map from the to . Thus, we can view the Riemann curvature tensor as a map given as:

That this is a tensor follows from the fact that the Riemann curvature tensor is a tensor. *For full proof, refer: Curvature is tensorial*

### As a (0,4)-tensor

A Riemannian metric or pseudo-Riemannian metric on a differential manifold brings about an isomorphism between its tangent bundle and its cotangent bundle, sending to the element in the cotangent bundle. Thus any tensor can be transformed to a -tensor. In particular the Riemann curvature tensor can be transformed to a -tensor, explicitly described as:

## Facts

### Tensoriality in all variables

The Riemann curvature tensor of the Levi-Civita connection is tensorial in all the variables. This follows from the fact that the general Riemann curvature tensor is tensorial in all the three variables. *For full proof, refer: Tensoriality of Riemann curvature tensor*

### Antisymmetry in the first two variables

The antisymmetry in the first two variables again follows from the fact that the Riemann curvature tensor is alternating in the first two variables. *For full proof, refer: Antisymmetry of Riemann curvature tensor*

### Antisymmetry in the last two variables

This is a somewhat surprising fact:

Another way of saying this is that the adjoint of the operator is . *For full proof, refer: Curvature is antisymmetric in last two variables*

### Symmetry in the two pairs

From the fact that the Riemann curvature tensor is antisymmetric in both the first two and the last two variables, we can view it as a map:

It turns out that the map is symmetric in the two arguments. In other words, it can be viewed as an element of .

More explicitly:

Note that gets interchanged with and <amth>Y</math> with .

### First Bianchi identity

`Further information: First Bianchi identity`

Also known as the **algebraic Bianchi identity**, this identity holds not just for the Levi-Civita connection, but more generally, for any torsion-free linear connection over the differential manifold. It states that:

### Second Bianchi identity

`Further information: Second Bianchi identity`

In this case, the identity can be written as: