The structure of complete radiant manifolds modeled on modules over Weil algebras (Q2276602)

Product preserving functors on the category of ordinary smooth manifolds are already provided by functors of \({\mathbf A}\)-velocities \(T^{\mathbf A}M_n\), where the \({\mathbf A}\) are Weil algebras. Besides, such a functor \(T^{\mu}\) is associated with a Weil algebra's epimorphism \({\mu}: {\mathbf A} \to {\mathbf B}\) assigning to a foliated manifold \((M, \mathcal F)\) a specific product bundle which inherits an \({\mathbf A}\)-smooth foliated structure [\textit{V. V. Shurygin} and \textit{L. B. Smolyakova}, Lobachevskii J. Math. 9, 55--75 (2001; Zbl 0995.58001) and references therein]. In the present paper, \({\mathbf A}\)-smooth manifolds \(M^{\mathbf L}\) modelled on \({\mathbf A}\)-modules of the type \({\mathbf L}= {\mathbf A}^n \oplus {\mathbf B}^m\), where \({\mathbf B}= {\mathbf A}/{\mathbf I}\) is a quotient algebra of the Weil algebra \({\mathbf A}\) by an ideal \({\mathbf I} \subset {\mathbf A}\), are studied. The main result asserts that a complete radiant \({\mathbf A}\)-smooth manifold \(M^{\mathbf L}\) is isomorphic to the bundle \(T^{\mu}M\) for some foliated manifold \((M, \mathcal F)\).