Torsion tensor

The Torsionstensor is a mathematical object from the field of differential geometry. This tensor was introduced by Élie Cartan in his studies on the geometry and gravity.

Definition

Let be a differentiable manifold with an affine connection. The Torsionstensor is a tensor field, which by

Is defined. Here are two vector fields and means the Lie bracket.

Local representation

Be a local frame of the tangent bundle. These are sections in the tangent bundle, which form a vector space basis in each tangent space. Substituting, and then applies to the components of the local coordinates in Torsionstensor

The symbols denote the Christoffel symbols. As it is always possible to select the local frame, so that the Lie bracket disappears anywhere in this coordinate is valid for the components of the Tensorfelds

Properties

• The Torsionstensor is a (2,1) - tensor field, is thus in particular - linear in its three arguments.
• The Torsionstensor is skew-symmetric, that is, it is.

Balanced connection

An affine connection is called symmetric or torsion when the Torsionstensor disappears, ie when

Applies. The most important symmetric connection is the Levi- Civita connection, which is additionally metric.

For balanced connections a kind of generalization of the black for differentiable curves can be proved. Let be a differentiable manifold with a symmetric connection and a smooth homotopy of smooth curves, then applies

Simply put, can be reversed by in the case of a symmetrical relationship so the derivative with respect to the.