On lifts of projectable-projectable classical linear connections to the cotangent bundle

Abstract

We describe all \(\mathcal{F}^2\mathcal{M}_{m_1,m_2,n_1,n_2}\)-natural operators \(D\colon Q^{\tau}_{proj-proj} \rightsquigarrow QT^*\) transforming projectable-projectable classical torsion-free linear connections \(\nabla\) on fibred-fibred manifolds \(Y\) into classical linear connections \(D(\nabla)\) on cotangent bundles \(T^*Y\) of \(Y\). We show that this problem can be reduced to finding \(\mathcal{F}^2 \mathcal{M}_{m_1,m_2,n_1,n_2}\)-natural operators \(D\colon Q^{\tau}_{proj-proj}\rightsquigarrow
(T^*,\otimes^pT^*\otimes\otimes^q T)\) for \(p=2\), \(q=1\) and \(p=3\), \(q=0\).