Torsion is antisymmetric

Statement

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

Consider the torsion of $\mathrm{\nabla }$, namely:

$\tau (\mathrm{\nabla }):\mathrm{\Gamma }(TM)\times \mathrm{\Gamma }(TM)\to \mathrm{\Gamma }(TM)$

given by:

$\tau (\mathrm{\nabla })(X,Y)={\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X-[X,Y]$

Then, $\tau (\mathrm{\nabla })$ is antisymmetric, i.e.:

$\tau (\mathrm{\nabla })(Y,X)=-\tau (\mathrm{\nabla })(X,Y)$

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

$\tau (\mathrm{\nabla })(X,X)=0$.

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.