# Torsion form

From Encyclopedia of Mathematics

The covariant differential of the vector-valued $1$-form of the displacement of an affine connection, the $2$-form $$ \Omega = D \omega = d \omega + \theta \wedge \omega $$ where $\theta$ is the connection form. The torsion form satisfies the first Bianchi identity: $$ d \Omega = \theta \wedge \Omega + \omega \wedge \Theta $$ where $\Theta$ is the curvature form of the given connection. The definition of a torsion form for reductive connections is analogous.

#### Comments

#### References

[a1] | S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963) |

**How to Cite This Entry:**

Torsion form.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Torsion_form&oldid=39365

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article