tailieunhanh - Applications of the Tachibana operator on problems of lifts
The purpose of the present paper is to study, using the Tachibana operator, the complete lifts of affinor structures along a pure cross-section of the tensor bundle and to investigate their transfers. The results obtained are to some extent similar to results previously established for tangent (cotangent) bundles. | Turk J Math 24 (2000) , 173 – 183. ¨ ITAK ˙ c TUB Applications of the Tachibana Operator on Problems of Lifts A. Ma˘gden, E. Kadıo˘glu and . Salimov Abstract The purpose of the present paper is to study, using the Tachibana operator, the complete lifts of affinor structures along a pure cross-section of the tensor bundle and to investigate their transfers. The results obtained are to some extent similar to results previously established for tangent (cotangent) bundles [1]. However there are various important differences and it appears that the problem of lifting affinor structures to the tensor bundle on the pure cross-section presents difficulties which are not encountered in the case of the tangent (cotangent) bundle. Key words and phrases. Tensor, bundle, affinor, complete lift, pure cross-section, Tachibana operator 1. Introduction Let Mn be a differentiable manifold of class C ∞ and finite dimension n, and let Tqp (Mn ), p+ S q > 0 be the bundle over Mn of tensors of type (p, q): Tqp (Mn ) = P ∈Mn Tqp (P ), where Tqp (P ) denotes the tensor(vector) spaces of tensors of type (p, q) at P ∈ Mn . We list below notations used in this paper. i. π : Tqp (Mn ) 7→ Mn is the projection Tqp (Mn ) onto Mn . indices i, j, · · · run from 1 to n , the indices i, j, · · · from n + 1 to n + np+q = dim Tqp (Mn ) and the indices I = (i, i), J = (j, j), . from 1 to n + np+q . The so-called Einsteins summation convention is used. iii. F(M ) is the ring of real-valued C ∞ functions on Mn . Tpq (Mn ) is the module over F(M ) of C ∞ tensor fields of type (p, q). iv. Vector fields in Mn are denoted by V, W, . . The Lie derivation with respect to V is denoted by LV . Affinor fields (tensor fields of type (1, 1) ) are denoted by ϕ,ψ, . . Subject classification number: Primary 53A45, Secondary 53C55. 173 ˘ ˘ MAGDEN, KADIOGLU, SALIMOV ∼ ∼ Denoting by xj the local coordinates of P = π(P ) (P ∈ Tqp (Mn )) in a neighborhood ∼ i ···i U ⊂ Mn and if we make (xj , .
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