# 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.:

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

## Proof

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