DIANGITUKULU NDIMBA Samuel* et MUSESA LANDA Alain

Lie-Rinehart Algebra Structure on the Leaf Space
We consider a transversally oriented foliation ℱ defined on a smooth manifold 𝑀. In this paper, wedenote by 𝑀/ℱ the leaf space of ℱ, 𝐶𝑏∞(𝑀, ℱ) its ring of basic functions, 𝑙ℱ(𝑀) its 𝐶𝑏∞(𝑀, ℱ)-module of the transverse vector fields on 𝑀/ℱ and 𝐷𝑖𝑓𝑓ℝ[𝐶𝑏∞(𝑀, ℱ)] the 𝐶𝑏∞(𝑀, ℱ)-module of differential operators of order
≤ 1 of 𝐶𝑏∞(𝑀, ℱ). We establish that if ℱ is a simple foliation on 𝑀, its leaf space 𝑀/ℱ is provided with a LieRinehart algebra structure and let us deduce from it that it is not possible to provide the leaf space 𝑀/ℱ of a Lie-Rinehart algebra structure if ℱ is not simple. Télécharger l'article