Integrable \(p\)-almost tangent manifolds and tangent bundles of \(p^ 1\)- velocities (Q1187293)

Let \(N\) be a \((p+1)n\)-dimensional differentiable manifold endowed with \(p(1,1)\)-tensor fields \((J_ 1,\dots,J_ p)\) satisfying the conditions \(J_ aJ_ b=0\), \(\text{rank}(J_ a)=n\) and \(\text{im}(J_ a)\cap(\bigoplus_{b\neq a}\text{im}(J_ b))=0\). Then \((J_ 1,\dots,J_ p)\) is said to be a \(p\)-almost tangent structure on \(N\). When \(p=1\), then such a structure is just an almost tangent structure. The tangent bundle \(T^ 1_ pM\) of \(p^ 1\)-velocities of any \(n\)- dimensional manifold \(M\) carries an integrable canonical \(p\)-almost tangent structure. It is already proved by the authors [Rend. Circ. Mat. Palermo, Ser. II 37, No. 2, 282-294 (1988; Zbl 0672.53040)] that any integrable \(p\)-almost tangent structure is locally equivalent to the canonical \(p\)-almost tangent structure on \(T^ 1_ pM\). In the reviewed paper, the authors study the global problem of equivalence. They prove that an integrable \(p\)-almost tangent manifold which defines a fibration is, under certain additional hypotheses, an affine bundle modelled on \(T^ 1_ pM\).