Prolongations of \(F\)-structure to the tangent bundle of order 2 (Q1204349)

Let \(V_ n\) be an \(n\)-dimensional \(C^ \infty\) differentiable manifold, \(T(V_ n)\) its tangent bundle and \(T_ 2(V_ n)\) its second-order tangent bundle. \(F\) being a nonzero \(C^ \infty\) tensor field of type (1,1) on \(V_ n\) and \(F^{II}\) the second lift of \(F\) in \(T_ 2(V_ n)\), the author has proved the following results: 1) \(F^{II}\) defines an \(F\)-structure in \(T_ 2(V_ n)\) iff \(F\) defines an \(F\)-structure in \(V_ n\). 2) \(F^{II}\) is integrable in \(T_ 2(V_ n)\) iff \(F\) is integrable in \(V_ n\). 3) \(F^{II}\) is partially integrable in \(T_ 2(V_ n)\) iff \(F\) is integrable in \(V_ n\).