# Difference between revisions of "Torsion of a linear connection"

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The torsion map is a <math>(1,2)</math> tensor. It is tensorial in both <math>X</math> and <math>Y</math>. This means that the value of the torsion of a connection for two vector fields at a point depends only on the values of the vector fields at that point. In other words, <math>\tau(\nabla)(X,Y)</math> at <math>p</math> depends on <math>\nabla, X(p), Y(p)</math> only and does not depend on how <math>X</math> and <math>Y</math> behave elsewhere on the manifold. | The torsion map is a <math>(1,2)</math> tensor. It is tensorial in both <math>X</math> and <math>Y</math>. This means that the value of the torsion of a connection for two vector fields at a point depends only on the values of the vector fields at that point. In other words, <math>\tau(\nabla)(X,Y)</math> at <math>p</math> depends on <math>\nabla, X(p), Y(p)</math> only and does not depend on how <math>X</math> and <math>Y</math> behave elsewhere on the manifold. | ||

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+ | ===Antisymmetry=== | ||

+ | |||

+ | {{further|[[Torsion is antisymmetric]]}} | ||

+ | |||

+ | We have that the torsion tensor is antisymmetric, i.e., we have: | ||

+ | |||

+ | <math>\tau(\nabla)(Y,X) = -\tau(\nabla)(X,Y)</math> | ||

+ | |||

+ | Equivalently, we have that: | ||

+ | |||

+ | <math>\tau(\nabla)(X,X) = 0</math> |

## Revision as of 17:48, 6 January 2012

*This article defines a tensor (viz a section on a tensor bundle over the manifold) of type* (1,2)

## Definition

### Given data

- A differential manifold
- A linear connection on (viz., a connection on the tangent bundle ).

### Definition part

The torsion of , denoted as , is defined as a map that takes as input 2 vector fields and outputs a third vector field, as follows:

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

## Facts

### Tensoriality

`Further information: Torsion is tensorial`

The torsion map is a tensor. It is tensorial in both and . This means that the value of the torsion of a connection for two vector fields at a point depends only on the values of the vector fields at that point. In other words, at depends on only and does not depend on how and behave elsewhere on the manifold.

### Antisymmetry

`Further information: Torsion is antisymmetric`

We have that the torsion tensor is antisymmetric, i.e., we have:

Equivalently, we have that: