torsion of affine connection

Definition

The torsion of an affine connection is a map

$$T:\mathfrak{X}(M)\times \mathfrak{X}(M)⟶\mathfrak{X}(M)$$

such that

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

This is a $(2,1)$ tensor, called the torsion tensor of $\nabla$.

A connection is called symmetric or torsion free if $T(X,Y)=0$.