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Lie algebra on the transverse bundle of a decreasing family of foliations

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Lie algebra on the transverse bundle of a decreasing family of foliations

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Lebtahi, Leila

Aquest document és un/a article, creat/da en: 2010

J. Lehmann-Lejeune in [J. Lehmann-Lejeune, Cohomologies sur le fibré transverse à un feuilletage, C.R.A.S. Paris 295 (1982), 495-498] defined on the transverse bundle V to a foliation on a manifold M, a zero-deformable structure J such that J^2 = 0 and for every pair of vector fields X,Y on M: [JX,JY]−J[JX,Y]−J[X,JY]+J^2[X,Y]=0. For every open set Ω of V, J. Lehmann-Lejeune studied the Lie Algebra L_J(Ω) of vector fields X defined on Ω such that the Lie derivative L(X)J is equal to zero i.e., for each vector field Y on Ω: [X,JY]=J[X,Y] and showed that for every vector field X on Ω such that X∈KerJ, we can write X=∑[Y,Z] where ∑ is a finite sum and Y,Z belongs to L_J(Ω)∩(KerJ|Ω). In this note, we study a generalization for a decreasing family of foliations.

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