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 = −Tabc ∇c f .
* And curvature: Double covariant derivatives of tensors are now related by, e.g.,
[∇a,∇b] Mmn = Rabmc Mcn − Rabcn Mmc − Tabc ∇c 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:= ΓaijChristoffel − Kaij , or Tabc:= 2 θ[ai Kb]ij ecj ;
With a metric, we can define a contorsion tensor Kabc, with
Kabc = −Tabc + Tbca − T cab , Vabc:= \(1\over2\)(Da gbc − Dc gab − Db 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.
> Online resources:
see Wikipedia page.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 31 may 2019