# Difference between revisions of "Linear connection"

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===Curvature of a linear connection=== | ===Curvature of a linear connection=== | ||

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+ | {{further|[[Riemann curvature tensor]]}} | ||

The curvature of a linear connection <math>\nabla</math> is denoted as <math>R_\nabla</math>. It is defined as: | The curvature of a linear connection <math>\nabla</math> is denoted as <math>R_\nabla</math>. It is defined as: | ||

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<math>R_\nabla(X,Y) = \nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}</math> | <math>R_\nabla(X,Y) = \nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}</math> | ||

− | The notion of curvature actually makes sense for any connection | + | The notion of curvature actually makes sense for any connection. |

## Revision as of 23:34, 23 July 2009

This lives as an element of: the space of -bilinear maps

## Contents

## Definition

### Given data

- A connected differential manifold with tangent bundle denoted by

### Definition part (pointwise form)

A **linear connection** is a smooth choice of the following: at each point , there is a map , satisfying some conditions. The map is written as where and .

- It is -linear in (that is, in the coordinate).
- It is -linear in (viz the space of sections on ).
- It satisfies the following relation called the Leibniz rule:

### Definition part (global form)

A **linear connection** is a map , satisfying the following:

- It is -linear in the first
- it is -linear in the second
- It satisfies the following relation called the Leibniz rule:

where is a scalar function on the manifold and denotes scalar multiplication of by .

### Generalizations

The notion of linear connection can be generalized to the more general notion of a connection, where the second is replaced by an arbitrary vector bundle over .

## Operations on a linear connection

### Torsion of a linear connection

`Further information: torsion of a linear connection`

The torsion of a linear connection is denoted as . It is a -tensor defined as:

.

A connection whose torsion is zero is termed a torsion-free linear connection.

Note that torsion makes sense *only* for linear connections.

### Curvature of a linear connection

`Further information: Riemann curvature tensor`

The curvature of a linear connection is denoted as . It is defined as:

The notion of curvature actually makes sense for any connection.