## Horizontal lift of tensor fields of type (1,1) from a manifold to its tangent bundle of higher order.(English)Zbl 0634.53023

Abstract analysis, Proc. 14th Winter Sch., Srnî/Czech. 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 43-59 (1987).
[For the entire collection see Zbl 0627.00012.]
Let $$E=E(M,F,G,P)$$ be the fibre bundle associated with a principal fibre bundle P(M,G) and with a standard fibre F. A connection $$\Gamma$$ in P(M,G) defines a horizontal lift of vector fields from M to E. Denoting by $$X^ H$$ the horizontal lift of $$X\in {\mathcal X}(M)$$ to E with respect to $$\Gamma$$, for a tensor field F of type (1,1) on M, a tensor field $$\tilde F$$ of type (1,1) can be defined on E such that $$\tilde F(X^ H)=(FX)^ H$$, for all $$X\in {\mathcal X}(M)$$. The authors will look for such a construction that the mapping $$F\to F$$ has “nice” algebraic properties which permit them to prolongate geometric structures from M to E. This problem has been studied for several fibre bundles.
The authors propose a solution of this problem in the case of the tangent bundle of order r, denoted by $$T^ rM$$. They study the connections of order r and the horizontal lift of vector fields to $$T^ rM$$, given a characterization of brackets of vertical and horizontal vector fields, and propose a definition of a horizontal lift of tensor fields of type (1,1) from M to $$T^ rM$$. Also their algebraic properties are studied. The theorems obtained generalize results of K. Yano and S. Ishihara [J. Math. Mech. 16, 1015-1029 (1967; Zbl 0152.204)].
Reviewer: I.D.Teodorescu

### MSC:

 53C05 Connections (general theory)

### Citations:

Zbl 0627.00012; Zbl 0152.204
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