Local and global stability of compact leaves and foliations (Q2874853)

Let \(\mathcal F\) be a foliation of codimension \(q\) on a manifold \(M\). A leaf \(L\) of \(\mathcal F\) is called locally stable if it has a family of saturated neighborhoods, \(\{\,W_k\mid k\in\mathbb N\,\}\), such that: (1) there is a submersion \(f_1:W_1\to L\) so that each \(f_k:=f_1|_{W_k}\) is a locally trivial fibration by \(q\)-disks transverse to the leaves; and, (2) for any \(x\in L\), \(\{\,f_k^{-1}(x)\mid k\in\mathbb N\,\}\) is a base of neighborhoods of \(x\) in \(f_1^{-1}(x)\). \(\mathcal F\) is called locally stable if all of its leaves satisfy this condition. This is a classical concept in foliation theory, with several characterizations, specially when \(\mathcal F\) is compact (its leaves are compact), whose study has attracted important mathematicians like Reeb, Millet, Edwards, Sullivan, Epstein, Vogt, Thurston and others. In this paper, the author gives new characterizations of local stability of \(\mathcal F\). For instance, this property holds if its holonomy pseudogroup is complete and quasi-analytic (two properties introduced by A.~Haefliger). Another characterization of local stability is given by the quasi-analiticity of the holonomy pseudogroup and the existence of an Ehresmann connection (a classical object in foliation theory). Several interesting consequences are derived from these results.