Difference between revisions of "Torsion of a linear connection"

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A linear connection whose torsion is zero is termed a '''torsion-free linear connection'''.
 
A linear connection whose torsion is zero is termed a '''torsion-free linear connection'''.
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==Facts==
  
 
===Tensoriality===
 
===Tensoriality===
  
 
{{further|[[Torsion is tensorial]]}}
 
{{further|[[Torsion is tensorial]]}}
The torsion map is a <math>(1,2)</math> tensor. It is tensorial in both <math>X</math> and <math>Y</math>.
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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.

Revision as of 17:46, 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

Definition part

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

\tau(\nabla)(X,Y) = \nabla_X Y- \nabla_Y X - [X,Y]

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 (1,2) tensor. It is tensorial in both X and Y. 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, \tau(\nabla)(X,Y) at p depends on \nabla, X(p), Y(p) only and does not depend on how X and Y behave elsewhere on the manifold.