# Curvature of a connection

This curvature is also sometimes known as theRiemann curvature tensor. However, the latter term is usually reserved for situations where we have a linear connection, in particular, the Riemann curvature tensor arising from the Levi-Civita connection for a Riemannian or pseudo-Riemannian manifold

## Contents

## Definition

### Given data

- A connected differential manifold
- A vector bundle over
- A connection for

### Definition part

The **curvature** of is defined as the map:

where

Note that itself outputs a linear map . We can thus write this as:

### In local coordinates

`Further information: curvature matrix of a connection`

Consider a system of local coordinate charts for such that the vector bundle is trivial on each chart. For any connection , we can write a matrix that, in local coordinates, describes the curvature of . This matrix is sometimes denoted as , and is defined by:

Here, is a matrix of connection forms.

### In the linear case

In the special case where (the case of a linear connection) we get that . We can thus think of this map as a (1,3)-tensor because it takes as input three vector fields and outputs one vector field.

## Properties

### Tensoriality

`Further information: Curvature is tensorial`

The curvature is tensorial in *all* three arguments. This is best proved by proving -linearity in all arguments, via a computation.

### Antisymmetry

`Further information: Curvature is antisymmetric in first two variables`

We have the following identity: