Torsions and connections on some natural bundles

The natural vector valued 1-forms Q on the natural bundles associated with product preserving functors (including the tangent bundle, the bundle of first order k-velocities, the bundle of second order 1-velocities and the bundle of linear frames) and on the cotangent bundle are classified. Then, these forms Q are used to study the torsion τ = [Γ, Q] of connections Γ on the above bundles, where [.,.] is the Frölicher-Nijenhuis bracket.