Complete Lifts of Tensor Fields on a Pure Cross-section in the Tensor Bundle

Abstract.

The main purpose of the present paper is to study the behaviour on the pure cross-section of the complete lifts of tensor fields in M _n to its tensor bundle T⁰ _q (M _n ), q > 0. The results obtained are to some extent similar to results previously established for cotangent T⁰₁ (M _n ) [11] and tensor bundles T¹_q (M _n ) [7]. However there are various important differences and it appears that the problem of lifting tensor fields to the tensor bundle T⁰ _q (M _n ), q > 1 the cross-section presents difficulties connected with purity of the cross-section and γ-operator which are not encountered in the case of the cotangent bundle and tensor bundle T¹ _q (M _n ), respectively. These difficulties were solved with applications of the Tachibana operator.