Hausdorff leaf spaces for foliations of codimension one (Q548523)

The origin of this paper is the modification of a Riemannian metric of \(S^3\) along the fibers of the Hopf fibration, and some in [the author, Ann. Pol. Math. 94, No. 3, 231--243 (2008; Zbl 1156.57023)]. Contents: basic constructions (tangential gluing, transverse gluing, turbulization), topology of Hausdorff leaf spaces (HLS), gluing metric spaces, warped foliations, HLS for codim-1 foliations. A complete description of the HLS for a 1-codimension foliation on a compact Riemannian manifold is presented. Let \({\mathcal F}\) be a foliation on a Riemannian manifold \((M,g)\) and \(\text{HLS}({\mathcal F})\) denote the Hausdorff leaf space for \({\mathcal F}\). The author proves the following theorems: 1) \(\text{HLS}({\mathcal F})\) of any codimension one foliation on a compact Riemannian manifold is isometric to a finite connected metric graph; 2) For every finite connected metric graph \(G\) there exists a compact foliated Riemannian manifold \((M,{\mathcal F},g)\) such that \(\text{HLS}({\mathcal F})\) is isometric to \(G\). Moreover, every leaf of \({\mathcal F}\) is proper. The case of \(\text{HLS}({\mathcal F}_r)\), where \({\mathcal F}_r\) is a certain foliation, is examined. Important collateral results are presented and certain examples are discussed.