Torsion of a linear connection
This article defines a tensor (viz a section on a tensor bundle over the manifold) of type (1,2)
The torsion of , denoted as , is defined as a map that takes as input 2 vector fields and outputs a third vector field, as follows:
A linear connection whose torsion is zero is termed a torsion-free linear connection.
Further information: Torsion is tensorial
The torsion map is a tensor. It is tensorial in both and . This means that the value of the torsion of a connection for two vector fields at a point depends only on the values of the vector fields at that point. In other words, at depends on only and does not depend on how and behave elsewhere on the manifold.
Further information: Torsion is antisymmetric
We have that the torsion tensor is antisymmetric, i.e., we have:
Equivalently, we have that: