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