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Feuilletages riemanniens réguliers et singuliers. (Regular and singular Riemannian foliations). (French) Zbl 0824.53028

Géométrie différentielle, Colloq. Géom. Phys., Paris/Fr. 1986, Trav. Cours 33, 173-201 (1988).
[For the entire collection see Zbl 0635.00010.]
As the basic model of a singular Riemannian foliation, the author takes the orbits of a connected Lie group acting isometrically on a Riemannian manifold \(M\). Thus, there is given a partition \(\mathcal F\) of \(M\) into connected submanifolds (the leaves), together with a Riemannian metric \(g\) adapted to \(\mathcal F\) (geodesics perpendicular to one leaf are perpendicular to all that they meet), and the module \({\mathcal I}_{\mathcal F}\) of vector fields tangent to \({\mathcal F}\) is transitive on each leaf. The condition that \(g\) be adapted to \(\mathcal F\) is also expressed by saying that \(\mathcal F\) is a transnormal system on \(M\). Many natural examples of singular Riemannian foliations exist in the literature. Given a triple \((M,{\mathcal F}, g)\) as above, with \(M\) compact, the author proves that (i) \(M\) is stratified by leaf dimension into imbedded submanifolds; (ii) the closures of the leaves form a new transnormal system \(\overline {\mathcal F}\); and (iii) there is a locally constant sheaf \({\mathfrak C} (M, {\mathcal F})\) of Lie algebras, independent of the adapted metric \(g\), which in each stratum projects to a sheaf of germs of “transverse Killing fields” whose orbits are the closures of the leaves. The author conjectures that \(\overline {{\mathcal F}}\) is itself a singular Riemannian foliation. He also remarks that the case of orbits of an isometry group is linearizable in the sense of A. D. Weinstein [J. Differ. Geom. 18, 523-557 (1983; Zbl 0524.58011)] and he conjectures that this remains true in general.

MSC:

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