Torsion Tensor

In General > s.a. connection; Hyperkähler Structure; lie derivative.
* Idea: A tensor, defined by a linear connection, measuring the antisymmetric part of , or the failure of closing of infinitesimal parallelograms under parallel translation.
$ Def: The tensor T: TM × TM → TM, defined by

T(X, Y):= _X Y – _Y X – [X, Y]   or   T ^a_bc = ^a_[bc] ,   [_a,_b] f = –T_ab^c _c f .

* And curvature: Double covariant derivatives of tensor 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 .

* History: Introduced by É Cartan in his studies of geometry and gravitation.
* Properties: It satisfies the first Bianchi identity (> see curvature).
* And more structure: With a vierbein, we can define a contorsion form by

_ai^j:= _ai^j_Christoffel – K_ai^j ,   or   T_ab^c:= 2 _[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:= (D_a g_bc– D_c g_ab– D_b g_ca) ,   ^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.
@ Geodesic deviation: Iliev & Manoff gq/05-in.
@ Discrete: Aspinwall JHEP(00)ht [Vafa's and Douglas's pictures]; Sharpe PRD(03)ht/00, PLB(01)ht/00.
@ Related topics: Briggs gq/99 [conservation]; Capozziello et al AdP(01)gq [classification].
> Discussion: Hehl – Weinberg PT(07)mar.

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