# 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) |

**How to Cite This Entry:**

Torsion tensor.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Torsion_tensor&oldid=14358