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Prolongations of vector fields to near-point manifolds. (Prolongements des champs de vecteurs à des variétés de points proches.) (French) Zbl 0598.58003
Let $$A$$ be a local algebra (commutative, unitary) of finite dimension on $$\mathbb R$$ whose maximal ideal $$\mathfrak m$$ is of codimension 1. Let $$M$$ be a paracompact $$C^{\infty}$$-manifold and $$M^ A$$ be the manifold of near points of $$A$$-kind on $$M$$ [cf. A. Weil, Colloq. Int. Centre Nat. Réch. Sci. 52, 111–117 (1953; Zbl 0053.24903)]. We have a canonical fibration $$\pi^ A: M^ A\to M$$ such that for $$f\in C^{\infty}(M,{\mathbb R})$$ and for $$\xi \in M^ A$$, $$f(\pi^ A(\xi))$$ is the image of $$\xi(f)$$ by the augmentation $$A\to {\mathbb R}$$. Using the $$A$$-module structure on the Lie algebra $$\chi (M^ A)$$ of vector fields on $$M^ A$$, the author gives a characterization of the prolongations to $$M^ A$$ of the vector fields on $$M$$.