Rybicki, Tomasz On Lie algebras of vector fields related to Riemannian foliations. (English) Zbl 0784.57016 Ann. Pol. Math. 58, No. 2, 111-122 (1993). Let \({\mathcal F}_ i\), \(i=1,2\), be Riemannian foliations of compact, connected manifolds \(M_ i\). Let \({\mathcal I}({\mathcal F}_ i)\) be the Lie algebra of vector fields tangent to \({\mathcal F}_ i\), \({\mathcal X}({\mathcal F}_ i)\) the Lie algebra of foliated vector fields on \(M_ i\), and \(\bar{\mathcal X}({\mathcal F}_ i)={\mathcal X}({\mathcal F}_ i)/{\mathcal I}({\mathcal F}_ i)\). Finally, let \(\bar{\mathcal F}_ i\) be the (singular) foliation of \(M_ i\) by the closures of the leaves of \({\mathcal F}_ i\).The author proves that for any Lie algebra isomorphism \(\Phi:\bar{\mathcal X}({\mathcal F}_ 1)\to\bar{\mathcal X}({\mathcal F}_ 2)\) (resp., \(\Phi:{\mathcal X}({\mathcal F}_ 1)\to{\mathcal X}({\mathcal F}_ 2)\)) there exists a Satake diffeomorphism \(\varphi:M_ 1/\bar{\mathcal F}_ 1\to M_ 2/\bar{\mathcal F}_ 2\) (resp., a diffeomorphism \(\varphi:M_ 1\to M_ 2)\) such that \(\varphi_*=\Phi\). Reviewer: P.Walczak (Łódź) MSC: 57R30 Foliations in differential topology; geometric theory 53C12 Foliations (differential geometric aspects) Keywords:space of leaves; singular foliation; Riemannian foliations of compact, connected manifolds; Lie algebra of vector fields; Lie algebra of foliated vector fields; Lie algebra isomorphism; Satake diffeomorphism PDFBibTeX XMLCite \textit{T. Rybicki}, Ann. Pol. Math. 58, No. 2, 111--122 (1993; Zbl 0784.57016) Full Text: DOI