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On Lie algebras of vector fields related to Riemannian foliations. (English) Zbl 0784.57016

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 \(\varphi_*=\Phi\).

MSC:

57R30 Foliations in differential topology; geometric theory
53C12 Foliations (differential geometric aspects)
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