# Article

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Keywords:
Foliation; transverse bundle; second order transverse bundle; projectable linear connection; Lie derivative; Weil bundle
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.
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