Torsion tensor

From Wikipedia, the free encyclopedia

In differential geometry, the torsion tensor is one of the tensorial invariants of a connection on the tangent bundle. It is a vector-valued 2-form given by

$\tau(X, Y) := \nabla_Y X - \nabla_X Y - [Y, X].$

The components of the torsion tensor $T^c_{ab}$ can be defined as the antisymmetric part of the affine connection $\Gamma^c_{ab}$ :

$T^c{}_{ab} := \frac{1}{2} ( \Gamma^c{}_{ab} - \Gamma^c{}_{ba} )$

From this definition, the torsion tensor is seen to be antisymmetric in its two lower indices $T c a b = - T c b a$

In differential geometry, the torsion tensor has, just as the curvature tensor, a geometric interpretation. Consider a parallelogram built of small displacement vectors and their parallel transports. In general, however, the lines of the parallelogram do not close. The failure of closure of such an infinitesimal parallelogram is proportional to the torsion tensor.

Note that the torsion tensor is sometimes referred to as Cartan tensor, or Cartan torsion tensor.