E) $ of $ E $-valued differential forms. A soldering form on $ E $ is a vector bundle morphism $ \vartheta: \TBundle M \to E $ (usually, $ \vartheta $ is required to be an isomorphism of vector bundles, but we do not need this assumption). The tangent bundle $ E = \TBundle M $ comes equipped with a tautological soldering form given by the identity $ \vartheta: \TBundle M \to \TBundle M $. Given a soldering form $ \vartheta \in \DiffFormSpace^1(M, E) $, the torsion of a connection $ K $ on $ E $ relative to $ \vartheta $ is defined to be $ T = \dif_K \vartheta \in \DiffFormSpace^2(M, E) $. In particular, we have $$T (X, Y)
        = \dif_K \vartheta (X, Y)
        = \nabla_X \bigl(\vartheta (Y)\bigr) - \nabla_Y \bigl(\vartheta (X)\bigr) - \vartheta (\commutator{X}{Y}),$$ which, for the tangent bundle $ E = \TBundle M $ with the tautological soldering form, reduces to the usual defining relation for the torsion. Let $ f: N \to M $ be a smooth map, and let $ E \to M $ be a vector bundle endowed with a connection $ K $. The pull-back bundle $ f^* E $ is 