Geodesic
Decomposition of a~jet bundle of differentiable mappings into a~Whitney sum of tangent bundles
Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 23 (1997), pp. 139-148

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We prove that the bundle $J_m^rM_n$ of jets of differential mappings of open neighbourhoods of $0\in\mathbb R^m$ into differential manifold $M_n$ can be decomposed into the Whitnev sum $\bigoplus\limits_{a=1}^NT_a(M_n)$, where $N=\binom{m+r}r-1$. To get such a decomposition $J_m^rM_n$ it is sufficient to take a linear connection on $M$. We use this decomposition to construct lifts of linear forms, vector fields and Riemannian metrics from the base $M_n$ into the bundle $J_m^rM_n$. 
@article{KUTGS_1997_23_a12,
     author = {A. Ya. Sultanov},
     title = {Decomposition of a~jet bundle of differentiable mappings into {a~Whitney} sum of tangent bundles},
     journal = {Trudy Geometricheskogo Seminara},
     pages = {139--148},
     publisher = {mathdoc},
     volume = {23},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/KUTGS_1997_23_a12/}
}
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A. Ya. Sultanov. Decomposition of a~jet bundle of differentiable mappings into a~Whitney sum of tangent bundles. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 23 (1997), pp. 139-148. http://geodesic.mathdoc.fr/item/KUTGS_1997_23_a12/