Torsion is antisymmetric

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Statement

Let M be a differential manifold and \nabla be a linear connection on M (viz., \nabla is a connection on the tangent bundle TM of M).

Consider the torsion of \nabla, namely:

\tau(\nabla): \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)

given by:

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

Then, \tau(\nabla) is antisymmetric, i.e.:

\tau(\nabla)(Y,X) = -\tau(\nabla)(X,Y)

Equivalently, \tau is alternating, i.e.:

\tau(\nabla)(X,X) = 0.

Related facts

Proof

The proof follows directly from the definition, and the fact that the Lie bracket of derivations is antisymmetric.