Rybicki, Tomasz On the Lie algebra of a transversally complete foliation. (English) Zbl 0637.57018 Publ., Secc. Mat., Univ. Autòn. Barc. 31, No. 1, 5-16 (1987). 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 Keywords:isomorphisms between Lie algebras of foliate vectorfields; foliation preserving diffeomorphisms; transversely complete foliations Citations:Zbl 0328.58005 PDFBibTeX XMLCite \textit{T. Rybicki}, Publ., Secc. Mat., Univ. Autòn. Barc. 31, No. 1, 5--16 (1987; Zbl 0637.57018)