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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 \(M\).
Reviewer: M.Adachi

53C99 Global differential geometry
58A99 General theory of differentiable manifolds
57R25 Vector fields, frame fields in differential topology
Zbl 0053.24903