Torsion-free linear connection

Definition

Symbol-free definition

A linear connection on a differential manifold is said to be torsion-free if its torsion is zero.

Definition with symbols

A linear connection $\mathrm{\nabla }$ on a differential manifold $M$ is said to be torsion-free if the torsion of $\mathrm{\nabla }$ is a zero map, viz:

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

Facts

Set of all 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.