Geodesic
Riemannian structures on higher order frame bundles over Riemannian manifolds
Lobachevskii journal of mathematics, Tome 27 (2007), pp. 41-46.

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We describe all $\mathcal Mf_m$-natural operators $A\colon\mathcal Riem\to\mathcal Riem P^r$ transforming Riemannian structures $g$ on $m$-dimensional manifolds $M$ into Riemannian structures $A(g)$ on the $r$-th order frame bundle $P^rM=invJ_0^r(\mathbf R^m, M)$ over $M$.
Keywords: Riemannian structure, higher order frame bundle, natural operator. 
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     title = {Riemannian structures on higher order frame bundles over {Riemannian} manifolds},
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J. Kurek; W. M. Mikulski. Riemannian structures on higher order frame bundles over Riemannian manifolds. Lobachevskii journal of mathematics, Tome 27 (2007), pp. 41-46. http://geodesic.mathdoc.fr/item/LJM_2007_27_a3/

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[2] Kolář I., Michor P. W., Slovák J., Natural Operations in Differential Geometry, Springer Verlag, 1993 | MR