The lifts of polyaffinor structures on the pure sections of a tensor bundle
A polyaffinor structure \(\Pi\) on a manifold \(M\) is a collection of affinor fields. A tensor field \(t\) is said to be pure with respect to \(\Pi\) if for each \(\phi \in \Pi\) there holds \( t(\phi x_1, \dots, x_q, \xi_1, \dots, \xi_p) = \dots = t(x_1, \dots, \phi x_q, \xi_1, \dots, \xi_p) = t(x_1, \dots, x_q, \phi^*\xi_1, \dots, \xi_p) = \dots = t(x_1, \dots, x_q, \xi_1, \dots, \phi^*\xi_p) \). If \(\Pi\) is rigid, i.e., in a neighborhood of each point in \(M\) there exists a frame with respect to which all affinors in \(\Pi\) have constant components, then the pure tensors of type \((p,q)\) constitute the pure tensor subbundle \(t^p_q(M)\) with respect to \(\Pi\). The author considers the vertical, horizontal and complete lifts of vector fields to the tensor bundle \(T^p_q(M)\), and finds their properties. Then along \(t^p_q(M)\) the author constructs lifts of affinors of \(\Pi\) to \(T^p_q(M)\) and studies them with the use of Tachibana operators. For example, he proves the following Theorem. Let \(\Pi\) be almost integrable. Then the complete lift and the horizontal lift of \(\Pi\) to the tensor bundle \(T^p_q(M)\) along \(t^p_q(M)\) coincide on a pure section.
scientific article; zbMATH DE number 1135537