A new method in theory of lifts of tensor fields to a tensor bundle (Q1896746)

Following \textit{G. I. Kruchkovich} [Tr. Semin. Vektorn. Tenzorn. Anal. 16, 174-201 (1972; Zbl 0249.53020)], the concept of \(S\)-structure and almost integrable \(S\)-structure is considered. For any \(S\)-structure the set of all (with respect to the \(S\)-structure) pure tensor fields is defined and a canonical \(\Phi_S\)-operation on the set of such pure tensor fields for which their tensor product is pure is defined. After reviewing some examples of \(\Phi_S\)-operations, the author examines the so-called polyaffinor structure on \(M\) and the algebraic polyaffinor structure as well. For such structures he defines the concept of quasi-holomorphicity, holomorphicity of smooth mappings and Tachibana operators of the first and the second kind. He shows that the Tachibana operator of the second kind allows up to induce an interesting algebraic polyaffinor structure. Next, the author introduces horizontal and complete lifts and shows that for almost integrable polyaffinor structure these lifts coincide. The work is completed by the definition of the generalized Yano-Ako operator of the second kind. With the help of this operator, the definition of the tensor field \(S\) and the statement that for an almost integrable structure \(S\) the complete and horizontal lifts coincide are given.