Atiyah-Molino classes of a smooth manifold over a local algebra $\mathbb A$ as obstacles to the continuation of transversal connections to $\mathbb A$-smooth connections

Each ideal $\mathbb I$ of a local algebra $\mathbb A$ in the sense of A. Weil gives rise to the canonical $\mathbb I$-foliations on $\mathbb A$-smooth manifolds and the corresponding lifted foliations on $\mathbb A$-smooth principal bundles. An $\mathbb A$-smooth connection $\Gamma$ in an $\mathbb A$-smooth principal bundle $P^{\mathbb A}$ induces an $\mathbb A$-smooth connection $\overline\Gamma_{\overline{\mathbb A}}$ in the tranverse, with respect to the canonical $\mathbb I$-foliation, bundle $P^{\overline{\mathbb A}}$, where $\overline{\mathbb A}=\mathbb A/\mathbb I$. Given an $\mathbb A$-smooth connection $\Gamma_{\overline{\mathbb A}}$ in the bundle $P^{\overline{\mathbb A}}$ we construct the Atiyah-Molino class $a(\Gamma_{\overline{\mathbb A}})$ of the connection $\Gamma_{\overline{\mathbb A}}$, the obstruction for existence of $\mathbb A$-smooth connection in $P^{\mathbb A}$ which projects into the connection $\Gamma_{\overline{\mathbb A}}$ in $P^{\overline{\mathbb A}}$.