Difference between revisions of "Torsion is antisymmetric"
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Latest revision as of 17:51, 6 January 2012
Statement
Let be a differential manifold and be a linear connection on (viz., is a connection on the tangent bundle of ).
Consider the torsion of , namely:
given by:
Then, is antisymmetric, i.e.:
Equivalently, is alternating, i.e.:
.
Related facts
Proof
The proof follows directly from the definition, and the fact that the Lie bracket of derivations is antisymmetric.