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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., Srní/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.
Reviewer: J.Virsik

##### MSC:
 53C05 Connections (general theory) 58A20 Jets in global analysis