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On the Lie algebra of a transversally complete foliation. (English) Zbl 0637.57018

Given a foliation F of a manifold M, X(F) denotes the Lie algebra of all vector fields Z for which [Z,V] is tangent to F for any vector field V on M tangent to F. Theorem: If \(F_ 1\) and \(F_ 2\) are transversely complete foliations of compact manifolds and \(\phi\) : X(F\({}_ 1)\to X(F_ 2)\) is a Lie algebra isomorphism, then there exists a diffeomorphism f such that f \(*F_ 2=F_ 1\) and \(f_*=\phi\). This generalizes the analogous result for fibrations [H. Omori, Infinite dimensional Lie transformation groups, Lect. Notes Math. 427 (1970; Zbl 0328.58005)].
Reviewer: P.Walczak

MSC:

57R30 Foliations in differential topology; geometric theory
17B65 Infinite-dimensional Lie (super)algebras

Citations:

Zbl 0328.58005
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