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

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• 作者：Leila Lebtahi
• 关键词：Foliations ; Fiber bundles ; Lie algebra
• 刊名：Journal of Geometry and Physics
• 出版年：2010
• 出版时间：January 2010
• 年：2010
• 卷：60
• 期：1
• 页码：122-130
• 全文大小：726 K

文摘

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 J²=0 and for every pair of vector fieldsX,Y on M: [JX,JY]−J[JX,Y]−J[X,JY]+J²[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 Yon Ω: [X,JY]=J[X,Y] and showed that for every vector field X on Ω such thatXKerJ, 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.