Torsion Tensor

In General > s.a. connection; Hyperkähler Structure; lie derivative.
* Idea: A tensor, defined by a linear connection, measuring the antisymmetric part of Γ, or the failure of closing of infinitesimal parallelograms under parallel translation.
* History: The concept was introduced in differential geometry by Élie Cartan in 1922 in his studies of geometry and gravitation.
\$ Def: The tensor T : TM × TM → TM, defined by

$$T(X,Y):= \nabla_{\!X}^~ Y - \nabla_{\!Y}^~\, X - [X, Y]\qquad{\rm or}\qquad$$ T abc = Γa[bc] ,    [∇a,∇b] f = −Tabcc f .

* And curvature: Double covariant derivatives of tensors are now related by, e.g.,

[∇a,∇b] Mmn = Rabmc McnRabcn MmcTabcc Mmn .

* Properties: It satisfies the first Bianchi identity (> see curvature).
* And more structure: If a vierbein eai is available, we can define a contorsion form by

Γaij:= ΓaijChristoffelKaij ,    or    Tabc:= 2 θ[ai Kb]ij ecj ;

With a metric, we can define a contorsion tensor Kabc, with

Kabc = −Tabc + TbcaT cab ,    Vabc:= $$1\over2$$(Da gbcDc gabDb gca) ,    Γabc = $$\{{a\atop bc}\}$$ − Kbca + V abc .

* Consequences: In a manifold with torsion, geodesics as extremal lines do not coincide with autoparallels.

References > s.a. torsion in physical theories.
@ General references: Gogala IJTP(80) [overview]; Petti GRG(86)-a1301 [interpretation]; Fabbri AFLB(07)gq/06 [completely antisymmetric torsion tensor].
@ Geodesic deviation: Iliev & Manoff in(81)gq/05.
@ Discrete: Aspinwall JHEP(00)ht [Vafa's and Douglas's pictures]; Sharpe PRD(03)ht/00, PLB(01)ht/00; Sharpe JGP(11) [in orientifolds].
@ Related topics: Briggs gq/99 [conservation]; Capozziello et al AdP(01)gq [classification]; Nieh PRD(18)-a1804 [torsional topological invariants].
> Discussion: Hehl PT(07)mar, Weinberg PT(07)mar.