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**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., SrnĂ®/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)].

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) |