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Prolongations of vector fields to near-point manifolds. (Prolongement des champs de vecteurs à des variétés de points proches.) (French) Zbl 0663.53059

A. Weil [Théorie des points proches sur les variétés différentiables, Colloq. int. Centre nat. Rech. Sci. 52, 111–117 (1953; Zbl 0053.24903)] initiated a theory of near points on a differentiable manifold. Let \(M\) be an \(n\)-dimensional paracompact manifold of class \(C^{\infty}\) and let \(A\) be a commutative unitary real algebra of finite dimension having a maximal ideal \(m\) of codimension 1 over \({\mathbb R}\) (\(A\) is said to be a local algebra). An \(A\)-kind near point of \(x\in M\) is a morphism of algebras \(\xi\): \(C^{\infty}(M)\to A\) with the property: for each \(f\in C^{\infty}(M)\), \(\xi (f)\equiv f(x)\bmod m\). Let \(M^ A_ X\) denote the set of \(A\)-kind near points of \(x\). \(M^ A=\cup_{x\in M}M^ A_ X\) is a differentiable manifold of dimension \(n\cdot \dim A\), which has a bundle structure over \(M\). The Lie algebra of vector fields on \(M^ A\) is an \(A\)-module. Using this fact the author gives a characterization of the prolongations of the vector fields on \(M\) to \(M^ A\).
Reviewer: I.D.Albu

MSC:

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

Citations:

Zbl 0053.24903
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References:

[1] Kobayashi, S.. - Canonical forms on frame bundles of higher order contact, Proc. Sym. Diff. Geom. Tucson, t. , 1960, p. 186-193. · Zbl 0109.40601
[2] Morimoto, A.. - Prolongation of connections to bundles of infinitely near points, J. Diff. Geom., t. 11, 1976, p. 479-498. · Zbl 0358.53013
[3] Okassa, E.. - Prolongement par une algèbre locale, Congrès National des Sociétés Savantes, Grenoble, t. III, 1983, p. 47-59.
[4] Okassa, E.. - Prolongements des champs de vecteurs à des variétés de points proches, C.R. Acad. Sci.Paris Sér. I Math. t.300, t. 6, 1985, p. 173-176. · Zbl 0598.58003
[5] Weil, A.. - Théorie des points proches sur les variétés différentiables, Colloque Géom. Diff. Strasbourg, t. , 1953, p. 111-117. · Zbl 0053.24903
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