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Torsion of a linear connection

246 bytes added, 17:48, 6 January 2012
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.
{{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>
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