An interpretation of the torsion tensor of a connection (Q705693)

Let \(\nabla \) be a linear connection on the manifold \(M\), \(T\) its torsion and \(\nabla ^{\perp }=\nabla -T\) the conjugate connection of \(\nabla \). Let \(H^{*}\) and \(H^{*\perp }\) be the horizontal distributions associated to \(\nabla \) and \(\nabla ^{\perp }\), respectively, on the cotangent manifold \(T^{*}M\). The main result of the paper is the following: \(H^{*}\) and \(H^{*\perp }\) are orthogonal distributions with respect to the canonical symplectic structure on \(T^{*}M\) and \(T=0\) if and only if \(H^{*}\) is a Lagrangian distribution.