torsion
From Mathematics wiki
Given a manifold 
and an affine connection 
, the torsion tensor is defined by its action on two tangent vector fields 
and 
:
![T(\vec{u}, \vec{v}) = \nabla_\vec{u} \vec{v} - \nabla_\vec{v} \vec{u} - [\vec{u},\vec{v}]](/w/images/math/a/c/b/acb04e4209d0e253ba92b8fc102cf936.png)
,
where ![[\vec{u},\vec{v}]\,](/w/images/math/e/5/3/e537ca165dcbc7bb5928a08bbbe6965c.png)
is the Lie bracket.
Christoffel symbols
The definition of the Christoffel symbols is
,
so that
![\nabla_{\vec{e}_\alpha} \vec{e}_\beta - \nabla_{\vec{e}_\beta} \vec{e}_\alpha - \left[ \vec{e}_\alpha, \vec{e}_\beta \right] = \left( \Gamma^\gamma_{\beta\alpha} - \Gamma^\gamma_{\alpha\beta}\right) \vec{e}_\gamma - \left[ \vec{e}_\alpha, \vec{e}_\beta \right]\,](/w/images/math/6/e/d/6ed0d445de52f65274e5074391ad3ff1.png)
.
In particular, zero torsion implies that
![\left( \Gamma^\gamma_{\beta\alpha} - \Gamma^\gamma_{\alpha\beta}\right) = \tilde{e}^\gamma \left( \left[ \vec{e}_\alpha, \vec{e}_\beta \right] \right) = c^\gamma_{\alpha\beta}\,](/w/images/math/6/3/e/63e86bd3c17b6de83f83f5e4b150cad1.png)
,
where 
are the commutation coefficients.
See also
References
- ↑ Schutz, B. (1980). Geometrical methods of mathematical physics. Cambridge University Press. ISBN 978-0521298872.
