Weil bundles and jet spaces. (English) Zbl 1079.58500

Summary: In this paper we give a new definition of the classical contact elements of a smooth manifold \(M\) as ideals of its ring of smooth functions: they are the kernels of Weil’s near points. Ehresmann’s jets of cross-sections of a fibre bundle are obtained as a particular case. The tangent space at a point of a manifold of contact elements of \(M\) is shown to be a quotient of a space of derivations from the same ring \(C^{\infty } (M)\) into certain finite-dimensional local algebras. The prolongation of an ideal of functions from a Weil bundle to another one is the same ideal, when its functions take values into certain Weil algebras. Following the same idea, vector fields are prolonged without any considerations about local one-parameter groups. As a consequence, we give an algebraic definition of Kuranishi’s fundamental identification on Weil bundles, and study their affine structures, as a generalization of the classical results on spaces of jets of cross-sections.


58A20 Jets in global analysis
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