# Torsion form

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.

#### 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://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