Changes

Jump to: navigation, search

Torsion of a linear connection

246 bytes added, 17:48, 6 January 2012
Tensoriality
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.
 
===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>
Bureaucrats, emailconfirmed, Administrators
184
edits

Navigation menu