Lifts and isomorphisms of commutation in bundles of jets (Q2639386)

Let M be a smooth real manifold, \(T_ rM\) be the fibre bundle of r-jets and \(\alpha_{rs}: T_ r(T_ sM)\to T_ s(T_ rM)\) be the isomorphisms of commutation. A linear connection (resp. a linear pseudo- connection) \(\nabla\) may be thought of as a (1,1)-tensor field F(\(\nabla)\) on \(TM=T_ 1M\). This fact suggests to define a generalized connection on a vector bundle \(\pi\) : \(E\to M\) as a (1,1)-tensor field on E. This notion is carefully studied. Let \(\nabla^*\) and \(F(\nabla)^{(r)}\) be the lifts of \(\nabla\) and F(\(\nabla)\) to r-jets, respectively. It is shown that the local coefficients of \(F(\nabla^*)\) are the composition of \(\alpha_{1r}\) and those of \(F(\nabla)^{(r)}\) (r\(\geq 1)\). In the case \(r=1\) a similar result for the complete and horizontal lifts of a linear connection is obtained.