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**Horizontal lifts of tensor fields and connections to the tangent bundle of higher order.**
*(English)*
Zbl 0685.53026

Proc. Winter Sch. Geom. Phys., SrnĂ/Czech. 1988, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 21, 151-178 (1989).

[For the entire collection see Zbl 0672.00006.]

The paper generalizes results obtained by A. Morimoto [Nagoya Math. J. 40, 99-120 (1970; Zbl 0208.502)] and K. Yano and S. Ishihara [Tangent and cotangent bundles (1973; Zbl 0262.53024)] as well as by the authors in their earlier papers. A connection \(\Gamma\) of order r on an n-dimensional \(C^{\infty}\)-manifold M is a connection in the principal bundle \(F^ rM\) of frames of order r. This defines a horizontal distribution on \(T^ rM\), the space of r-velocities on M, which is a fibre bundle associated with \(F^ rM\). Hence there is a lift of each vector field X on M to a horizontal field \(X^ H\) on \(T^ rM\). On the other hand, the jet structure of \(T^ rM\) allows to define so- called v-lifts of one-forms, and more generally of tensor fields, on M to \(T^ rM\), for \(v=0,1,...,n\). The combination of these two constructions gives horizontal v-lifts of tensor fields from M to \(T^ rM.\)

Next, horizontal lifts of connections are defined: Given \(\Gamma\) as above, each linear connection \(\nabla\) on M gives rise to a unique connection \(\nabla^ H\) on the manifold \(T^ rM\) satisfying certain conditions. The relation between the torsions and curvatures of these connections are studied. In particular, for \(r=1\), one obtains the horizontal lift of a linear connection \(\nabla\) on M with respect to a linear connection \(\nabla_ 0=\Gamma\) on M. For \(\nabla_ 0=\nabla\) this gives the construction by Yano and Ishihara.

The paper generalizes results obtained by A. Morimoto [Nagoya Math. J. 40, 99-120 (1970; Zbl 0208.502)] and K. Yano and S. Ishihara [Tangent and cotangent bundles (1973; Zbl 0262.53024)] as well as by the authors in their earlier papers. A connection \(\Gamma\) of order r on an n-dimensional \(C^{\infty}\)-manifold M is a connection in the principal bundle \(F^ rM\) of frames of order r. This defines a horizontal distribution on \(T^ rM\), the space of r-velocities on M, which is a fibre bundle associated with \(F^ rM\). Hence there is a lift of each vector field X on M to a horizontal field \(X^ H\) on \(T^ rM\). On the other hand, the jet structure of \(T^ rM\) allows to define so- called v-lifts of one-forms, and more generally of tensor fields, on M to \(T^ rM\), for \(v=0,1,...,n\). The combination of these two constructions gives horizontal v-lifts of tensor fields from M to \(T^ rM.\)

Next, horizontal lifts of connections are defined: Given \(\Gamma\) as above, each linear connection \(\nabla\) on M gives rise to a unique connection \(\nabla^ H\) on the manifold \(T^ rM\) satisfying certain conditions. The relation between the torsions and curvatures of these connections are studied. In particular, for \(r=1\), one obtains the horizontal lift of a linear connection \(\nabla\) on M with respect to a linear connection \(\nabla_ 0=\Gamma\) on M. For \(\nabla_ 0=\nabla\) this gives the construction by Yano and Ishihara.

Reviewer: J.Virsik