# Torsion is antisymmetric

From Diffgeom

## 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.