Lifts of Foliated Linear Connectionsto the Second Order Transverse Bundles

summary: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