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Desingularisation des feuilletages Riemanniens. (Desingularisation of Riemannian foliations). (French) Zbl 0576.57022

This paper is a continuation of the author’s study of the geometric structure of \(C^{\infty}\) Riemannian foliations F of compact manifolds V [Indagationes Math. 44, 45-76 (1982; Zbl 0516.57016)]. In this study, which is recalled in Section 1, one shows that the closures of the leaves of F define a foliation with singularities \(\bar F.\) If \(\bar F\) has no singularities, F is defined by a submersion on a V-manifold [I. Satake, J. Math. Soc. Japan 9, 464-492 (1957; Zbl 0080.374)].
The present paper provides a method for the resolution of the singularities of \(\bar F \)(Section 2). It consists of blowing up V along the submanifold S of the leaves of lowest dimension of \(\bar F,\) i.e., replacing S by its projective tangent bundle. After a finite number of successive blowing up operations, one arrives at the regular case. Applications and examples are given in Section 3. Particularly, one studies there foliations by orbits of groups of isometries.
Reviewer: I.Vaisman

MSC:

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