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

\(T(X,Y):= \nabla_{\!X}^~ Y - \nabla_{\!Y}^~\, X - [X, Y]\qquad{\rm or}\qquad\)
*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*_{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 PRD(18)-a1804 [torsional topological invariants].

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

> __Online resources__:
see Wikipedia page.

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