Torsion is antisymmetric

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

• Torsion is tensorial
• Curvature is tensorial
• Curvature is antisymmetric in first two variables

Proof

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