## 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

### Keywords:

singular Riemannian foliation; transnormal system

### Citations:

Zbl 0635.00010; Zbl 0524.58011