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Let be M a smooth manifold, A a local algebra and $M^A$ a manifold of infinitely near points on M of kind A. We build the canonical foliation on $M^A$ and we show that the canonical foliation on the tangent bundle TM is the foliation defined by its canonical field.

Let $M$ be a smooth manifold, $A$ a local algebra in sense of André Weil, $M^A$ the manifold of near points on $M$ of kind $A$ and $𝔛\left(M^A\right)$ the module of vector fields on $M^A$. We give a new definition of vector fields on $M^A$ and we show that $𝔛\left(M^A\right)$ is a Lie algebra over $A$. We study the cohomology of $A$-differential forms. Résumé. On considère $M$ une variété différentielle, $A$ une algèbre locale au sens d’André Weil, $M^A$ la variété des points proches de $M$ d’espèce $A$ et $𝔛\left(M^A\right)$ le module des champs de vecteurs sur $M^A$. On donne une nouvelle...

In this paper, $M$ denotes a smooth manifold of dimension $n$, $A$ a Weil algebra and $M^A$ the associated Weil bundle. When $(M,{\omega }_M)$ is a Poisson manifold with $2$-form ${\omega }_M$, we construct the $2$-Poisson form ${\omega }_{M^A}^A$, prolongation on $M^A$ of the $2$-Poisson form ${\omega }_M$. We give a necessary and sufficient condition for that $M^A$ be an $A$-Poisson manifold.