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 ^a_bc = Γ^a_[bc] ,    [∇_a,∇_b] f = −T_ab^c ∇_c f .

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

[∇_a,∇_b] M_mⁿ = R_abm^c M_cⁿ − R_abcⁿ M_m^c − T_ab^c ∇_c M_mⁿ .

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

Γ_ai^j:= Γ_ai^j_Christoffel − K_ai^j ,    or    T_ab^c:= 2 θ_[aⁱ K_b]i^j e^c_j ;

With a metric, we can define a contorsion tensor K_ab^c, with

K_ab^c = −T_ab^c + T_b^c_a − T ^c_ab ,    V_abc:= \(1\over2\)(D_a g_bc − D_c g_ab − D_b g_ca) ,    Γ^a_bc = \(\{{a\atop bc}\}\) − K_bc^a + V ^a_bc .

* 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.
> Online resources: see Wikipedia page.

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