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

JOURNAL OF GEOMETRY AND PHYSICS

卷:60

Article

Lebtahi, Leila

关键词: Foliations;    Fiber bundles;    Lie algebra;   

DOI  :  10.1016/j.geomphys.2009.09.003

来源: Elsevier

【 摘 要 】

J. Lehmann-Lejeune in [J. Lehmann-Lejeune, Cohomologies sur le fibre transverse a 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[JX, JY] + J(2) [X, Y] = 0. For every open set Omega of V, J. Lehmann-Lejeune studied the Lie Algebra L(J) (Omega) of vector fields X defined on Omega such that the Lie derivative L(X)J is equal to zero i.e., for each vector field Yon Omega: [X, JY] = J[X, Y] and showed that for every vector field X on Omega such that X is an element of KerJ, we can write X = Sigma [Y, Z] where Sigma is a finite sum and Y, Z belongs to L(J)(Omega) boolean AND (KerJ(|Omega)). In this note, we study a generalization for a decreasing family of foliations. (C) 2009 Elsevier B.V. All rights reserved.

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