Geodesic
Lifts of Foliated Linear Connectionsto the Second Order Transverse Bundles
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 55 (2016) no. 1, pp. 111-120.

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The second order transverse bundle $T^2_{}M$ of a foliated manifold $M$ carries a natural structure of a smooth manifold over the algebra $\mathbb {D}^2$ of truncated polynomials of degree two in one variable. Prolongations of foliated mappings to second order transverse bundles are a partial case of more general $\mathbb {D}^2$-smooth foliated mappings between second order transverse bundles. We establish necessary and sufficient conditions under which a $\mathbb {D}^2$-smooth foliated diffeomorphism between two second order transverse bundles maps the lift of a foliated linear connection into the lift of a foliated linear connection.
Classification : 53C12, 53C15, 58A20, 58A32
Keywords: Foliation; transverse bundle; second order transverse bundle; projectable linear connection; Lie derivative; Weil bundle 
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Shurygin, Vadim V.; Zubkova, Svetlana K. Lifts of Foliated Linear Connectionsto the Second Order Transverse Bundles. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 55 (2016) no. 1, pp. 111-120. http://geodesic.mathdoc.fr/item/AUPO_2016__55_1_a12/