Torsion Tensor - The Torsion Tensor

The Torsion Tensor

Let M be a manifold with a connection ∇ on the tangent bundle. The torsion tensor (sometimes called the Cartan (torsion) tensor) is a vector-valued 2-form defined on vector fields X and Y by

where is the Lie bracket of two vector fields. By the Leibniz rule, T(fX,Y) = T(X,fY) = fT(X,Y) for any smooth function f. So T is tensorial, despite being defined in terms of the non-tensorial covariant derivative: it gives a 2-form on tangent vectors, while the covariant derivative is only defined for vector fields.