# Difference between revisions of "Linear connection"

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===Definition part (pointwise form)=== | ===Definition part (pointwise form)=== | ||

− | A '''linear connection''' is a smooth choice <math>\nabla</math> of the following: at each point <math>p \in M</math>, there is a map <math>{}^p\nabla: T_p(M) \times \Gamma(TM) \to T_p(M)</math>, satisfying some conditions. The map is written as <math>{}^p\nabla_X(v)</math> where <math>X \in T_p(M)</math> and <math>v \in \Gamma( | + | A '''linear connection''' is a smooth choice <math>\nabla</math> of the following: at each point <math>p \in M</math>, there is a map <math>{}^p\nabla: T_p(M) \times \Gamma(TM) \to T_p(M)</math>, satisfying some conditions. The map is written as <math>{}^p\nabla_X(v)</math> where <math>X \in T_p(M)</math> and <math>v \in \Gamma(TM)</math>. |

* It is <math>\R</math>-linear in <math>X</math> (that is, in the <math>T_p(M)</math> coordinate). | * It is <math>\R</math>-linear in <math>X</math> (that is, in the <math>T_p(M)</math> coordinate). | ||

− | * It is <math>\R</math>-linear in <math>\Gamma(TM)</math> (viz the space of sections on <math> | + | * It is <math>\R</math>-linear in <math>\Gamma(TM)</math> (viz the space of sections on <math>TM</math>). |

* It satisfies the following relation called the Leibniz rule: | * It satisfies the following relation called the Leibniz rule: | ||

## Latest revision as of 17:29, 6 January 2012

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.