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* :
T*M* × T*M* → T*M*, defined by

*T*(*X*,* Y*):= ∇_{X} *Y* – ∇_{Y} *X* – [*X*,* Y*] or *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*_{m}^{n}
= *R*_{abm}^{c}* M*_{c}^{n} – *R*_{abc}^{n}* M*_{m}^{c} – *T*_{ab}^{c} ∇_{c} *M*_{m}^{n}
.

* __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*_{[a}^{i}* 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 a1804 [torsional topological invariants].

> __Discussion__: Hehl – Weinberg PT(07)mar.

> __Online resources__: see Wikipedia page.

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