Torsion tensor

A tensor of type $ ( 1, 2) $ that is skew-symmetric with respect to its indices, obtained by decomposing the torsion form of a connection in terms of a local cobasis on a manifold $ M ^ {n} $. In particular, in terms of a holonomic cobasis $ dx ^ {i} $, $ i = 1 \dots n $, the components $ S _ {ij} ^ {k} $ of the torsion tensor are expressed in terms of the Christoffel symbols (cf. Christoffel symbol) $ \Gamma _ {ij} ^ {k} $ of the connection as follows:

$$ S _ {ij} ^ {k} = \Gamma _ {ij} ^ {k} - \Gamma _ {ji} ^ {k} . $$

Comments

In terms of covariant derivatives $ \nabla $ and vector fields $ X $, $ Y $ the torsion tensor $ T $ can be described as follows:

$$ T ( X, Y) = \nabla _ {X} Y - \nabla _ {Y} X - [ X, Y ] . $$

References

[a1]	N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a2]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[a3]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)