Hausdorff leaf spaces for foliations of codimension one

Abstract

The topology of the Hausdorff leaf spaces (briefly the HLS) for a codim-1 foliation is the main topic of this paper. At the beginning, the connection between the Hausdorff leaf space and a warped foliations is examined. Next, the author describes the HLS for all basic constructions of foliations such as transverse and tangential gluing, spinning, turbulization, and suspension. Finally, it is shown that the HLS for any codim-1 foliation on a compact Riemannian manifold is isometric to a finite connected metric graph. In addition, the author proves that for any finite connected metric graph G there exists a compact foliated Riemannian manifold (M, F , g) with codim-1 foliation such that the HLS for F is isometric to G. Finally, the necessary and sufficient condition for warped foliations of codim-1 to converge to HLS(F ) is given.L/ ≡ with quotient metric one obtains (for a foliation of codimension one) Hausdorff leaf space for the foliation F . Lemma 2.1. For every foliation F on a compact foliated Riemannian manifold the HLS(F ) is a length space. Proof. By the definition of the length metric [3], for every two points x, y ∈ HLS(F ) and any curve c : [0, 1] → HLS(F ) such that c(0) = x, c(1) = y we haveρ(x, y) ≤ l(c). The opposite inequality follows directly from the definition of the HLS. 2.2. Gluing metric spaces. Following [1], we now describe how to glue length spaces: Let (X α , d α ) be a family of length spaces. Set the length metric d on a disjoint union ∐ α X α as follows: If x, y ∈ X α , then d(x, y) = d α (x, y); Otherwise, set d(x, y) = ∞. The metric d is called the length metric of disjoint union. Now, let (X, d X ) and (Y, d Y ) be two length spaces, while f : A → B be a bijection between two subsets A ⊂ X and B ⊂ Y . Equip Z = X ∐ Y with the length metric of disjoint union. Introduce the equivalence relation ∼ as the smallest equivalence relation containing relation generated by the relation x ∼ y iff f (x) = y. The result of gluing X and Y along f is the metric space (Z/ ∼ , d ∼ ). 2.3. Warped foliations. We recall here the notion of warped foliation [7]. The Hausdorff leaf space for warped foliation will be the main topic of our interest in Section 2. Moreover, the results of Section 2 will be used as a tool in Sections 3 and 4. Let (M, F , g) be a foliated Riemannian manifold and f : M → (0, ∞) be a basic function on M , i.e. a function constant along the leaves of F . We modify the Riemannian structure g to g f in the following way: g f (v, w) = f 2 g(v, w) while both v, w are tangent to the foliation F , but if at least one of vectors v, w is perpendicular to F then we set g f (v, w) = g(v, w). Foliated Riemannian manifold (M, F , g f ) is called here the warped foliation and denoted by M f . The function f is called the warping function. 2.4. Gromov-Hausdorff convergence. Recall the notion of Gromov-Hausdorff convergence [3]. Let (X, d X ) and (Y, d Y ) be an arbitrary compact metric spaces. The distance of X and Y can be defined as d GH (X, Y ) := inf{d H (X, Y )}, where d ranges over all ad...