Feuilletage singulier défini par une distribution presque régulière. (Singular foliations defined by an almost regular distribution) (Q1110869)

Two equivalent ways of defining regular foliations of codimension one on an n-manifold are (1) by a nonsingular form \(\omega \in A^ 1(M)\) that is locally of the form \(g\cdot df\); (2) by an (n-1)-plane distribution that is involutive. These lead to distinct notions of a singular foliation. For instance, the differential forms point of view leads to \(\Gamma\)-structures and to Morse foliations, the second point of view to the Sussman-Stefan singular foliations in control theory. The paper under review considers the second type of singular foliation which, transversely along each singular leaf S, is defined by a vector field having an isolated zero at S. The author proves that, under this generic restriction, various codimension one theorems of Sacksteder, Reeb, and Moussu remain true. These consist of a version of the stability theorem, the existence of minimal sets in the region of regularity and the nonexistence of exceptional minimal sets in the absence of holonomy.