Torsion tensor

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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

The components of the torsion tensor T^c_ab can be derived by setting X = e_a,Y = e_b and by introducing the commutator coefficients given by . We finally obtain a component expression of the torsion tensor,

T^c_ab: = Γ^c_ab − Γ^c_ba − γ^c_ba

If the basis is coordinate induced then γ^a_bc = 0, we get T^c_ab = − T^c_ba and also T^c_ab = 2Γ^c_[ab]. Hence, while the geodesic equations determine the symmetric part of the connection, the torsion tensor determines the antisymmetric part.

In differential geometry, the torsion tensor has, just as the curvature tensor, a geometric interpretation. For simplicity, consider the torsion tensor in a coordinate induced basis. Then the torsion tensor reduces to

The geometric interpretation becomes apparent; it is the difference between the two parallel transforms. Non-vanishing torsion can be visualized as something like a screw dislocation of a crystal: traveling around a small loop causes a small translation. In Euclidean space, or more generally on any Riemannian manifold, the torsion vanishes by definition of the Levi-Civita connection.

The torsion tensor is sometimes referred to as Cartan tensor, or Cartan torsion tensor.

In Einstein's theory of gravitation the torsion is always zero, but it can be non-zero in an extension called Einstein–Cartan theory.

[edit] Further reading

• Nakahara M., Geometry, Topology and Physics, IoP, 2003.

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[edit] See also

• Curvature tensor
• Contortion tensor
• Levi-Civita connection
• Torsion of curves