DEFORMATIONS OF LEVI FLAT STRUCTURES IN SMOOTH MANIFOLDS

摘要

We study intrinsic deformations of Levi flat structures on a smooth manifold. A Levi flat structure on a smooth manifold L is a couple (ξ, J) where ξ ? T(L) is an integrable distribution of codimension 1 and J: ξ → ξ is a bundle automorphism which defines a complex integrable structure on each leaf. A deformation of a Levi flat structure (ξ, J) is a smooth family {(ξ_t, J_t)}t∈]?ε,ε[ of Levi flat structures on L such that (ξ_0, J_0) = (ξ, J). We define a complex whose cohomology group of order 1 contains the infinitesimal deformations of a Levi flat structure. In the case of real analytic Levi flat structures, this cohomology group is H~1(Z~?(L), δ) × H~1(Λ_J~(0,?) (ξ) ? ξ, ?J) where (Z?(L), δ, {·,·}) is the differential graded Lie algebra associated to ξ.