Torsion-free linear connection
- Its torsion is zero
- Whenever two vector fields are such that their Lie bracket is zero, then the covariant derivative of either with respect to the other are equal.
Definition with symbols
- The torsion of is a zero map, viz:
- Whenever and are vector fields on an open subset such that on , then:
Definition in local coordinates
In local coordinates, a linear connection is torsion-free if the Christoffel symbols are symmetric in and .
Set of all torsion-free linear connections
Further information: Affine space of torsion-free linear connections Recall that the set of all linear connections is an affine space, viz a translate of a linear subspace (the linear subspace being the maps that are tensorial in both variables).
The set of torsion-free linear connections is an affine subspace of this, in the sense that any affine combination of torsion-free linear connections is again a torsion-free linear connection.
The corresponding linear subspace for torsion-free linear connections are the symmetric 2-tensors.