Torsion form

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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.



[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963)
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Torsion form. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article