aa r X i v : . [ m a t h . DG ] F e b HAUSDORFF LEAF SPACES FOR CODIM-1 FOLIATIONS
SZYMON M. WALCZAK
Abstract.
The topology of the Hausdorff leaf spaces (briefly the HLS) for acodim-1 foliation is the main topic of this paper. At the beginning, the con-nection between the Hausdorff leaf space and a warped foliations is examined.Next, the author describes the HLS for all basic constructions of foliations suchas transverse and tangential gluing, spinning, turbulization, and suspension.Finally, it is shown that the HLS for any codim-1 foliation on a compact Rie-mannian manifold is isometric to a finite connected metric graph. In addition,the author proves that for any finite connected metric graph G there exists acompact foliated Riemannian manifold ( M, F , g ) with codim-1 foliation suchthat the HLS for F is isometric to G . Finally, the necessary and sufficientcondition for warped foliations of codim-1 to converge to HLS( F ) is given. Introduction
In the 70-s M. Berger has presented the concept of modification of a Riemannianmetric of S along the fibers of the Hopf fibration. Following this concept, theauthor of this paper has introduced the notion of warped foliation [8]. Later on,the author has examined the limits of a sequence of warped compact foliations [7]and has proposed the notion of the Hausdorff leaf space (briefly the HLS) for afoliation on a compact Riemannian manifold.This paper is the continuation of the research held in [7]. At the beginning, theauthor shows that the HLS for any foliation F on a compact Riemannian manifold( M, g ) is the Gromov-Hausdorff limit of a sequence of warped foliations with warp-ing functions converging to zero on a dense subset G ⊂ M (Section 3, Theorem3.1). Next, he examines the Hausdorff leaf spaces for all natural constructions ofthe foliation listed in [2]. Namely, the HLS for tangential and transverse gluing,spinning, turbulization, and suspension are studied (Section 4).The main results of this paper are developed in Section 5 (Theorem 5.2 andTheorem 5.3), where the complete description of the Hausdorff leaf space for acodim-1 foliation on a compact Riemannian manifold is presented. It is shown thatthe HLS for a codim-1 foliation is isometric to a finite connected metric graph,while for every finite connected metric graph G there exists a foliated Riemannianmanifold ( M, F , g ) such that the Hausdorff leaf space for F is isometric to G .Finally (Theorem 5.4), the necessary and sufficient condition for the sequence ( f n )of warping function on a compact Riemannian manifold carrying foliation of codim-1 to have a sequence of warped foliations ( M f n ) converging to the Hausdorff leafspace for the foliation F is shown.For the theory of foliations we refer to [2] or [4]. Date : version November 20, 2018.
Figure 1.
The idea of ρ .2. Preliminaries
Hausdorff leaf spaces.
Let us recall the notion of Hausdorff leaf space [7]:Let ( M, F , g ) be a compact foliated manifold. Let us set ρ ( L, L ′ ) = inf { n − X i =1 dist( L i , L i +1 ) } , where the infimum is taken over all finite sequences of leaves beginning at L = L and ending at L n = L ′ (Figure 1). Let ∼ be an equivalence relation in the space ofleaves L defined by: L ∼ L ′ ⇔ ρ ( L, L ′ ) = 0 , L, L ′ ∈ L . Let ˜ L = L / ∼ . Put ˜ ρ ([ L ] , [ L ′ ]) = ρ ( L, L ′ ) , where [ L ] , [ L ′ ] ∈ ˜ L . ( ˜ L , ˜ ρ ) is a metric space. We call it the Hausdorff leaf space forthe foliation F (briefly the HLS), and we denote it by HLS( F ). Remark 2.1.
Equivalently, the Hausdorff leaf space can be defined as follows:Following [1] , one can define in a metric space ( X, d ) equipped with an equivalencerelation R the quotient pseudo-metric d R as d R ( x, y ) = inf { k X i =1 d ( p i , q i ) : p = x, q k = y, k ∈ N } . where the infimum is taken over all sequences { pi } ≤ i ≤ N , { q i } ≤ i ≤ N , N ∈ N , suchthat ( p i +1 , q i ) ∈ R. Consider a metric space ( X/R, d R ) and identify such points for which d R is equalto zero. Obtained metric space is called the quotient metric space.Let ( M, F , g ) be a compact foliated Riemannian manifold, and let R be the re-lation of belonging to the same leaf of F . Using R in M we get the alternativedefinition. Remark 2.2.
Let F be a codim-1 foliation on a compact Riemannian manifold ( M, g ) . One can define in the space of leaves a relation as follows: a leaf L isrelated to a leaf L ′ iff L is contained in a closure cl L ′ of a leaf L ′ . This relationdefines an equivalence relation ≡ in the leaf space L . One can check that equipping AUSDORFF LEAF SPACES 3 L / ≡ with quotient metric one obtains (for a foliation of codimension one) Hausdorffleaf space for the foliation F . Lemma 2.1.
For every foliation F on a compact foliated Riemannian manifoldthe 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 , → HLS( F ) such that c (0) = x , c (1) = y we have ˜ ρ ( x, y ) ≤ l ( c ). The opposite inequality follows directly from the definition of the HLS. (cid:3) Gluing metric spaces.
Following [1], we now describe how to glue lengthspaces:Let ( X α , d α ) be a family of length spaces. Set the length metric d on a disjointunion ∐ 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 bijectionbetween two subsets A ⊂ X and B ⊂ Y . Equip Z = X ∐ Y with the length metricof disjoint union. Introduce the equivalence relation ∼ as the smallest equivalencerelation containing relation generated by the relation x ∼ y iff f ( x ) = y . The resultof 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]. TheHausdorff leaf space for warped foliation will be the main topic of our interest inSection 2. Moreover, the results of Section 2 will be used as a tool in Sections 3and 4.Let ( M, F , g ) be a foliated Riemannian manifold and f : M → (0 , ∞ ) be abasic function on M , i.e. a function constant along the leaves of F . We modify theRiemannian structure g to g f in the following way: g f ( v, w ) = f g ( v, w ) while both v, w are tangent to the foliation F , but if at least one of vectors v, w is perpendicularto F then we set g f ( v, w ) = g ( v, w ). Foliated Riemannian manifold ( M, F , g f ) iscalled here the warped foliation and denoted by M f . The function f is called thewarping function .2.4. Gromov-Hausdorff convergence.
Recall the notion of Gromov-Hausdorffconvergence [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 admissible metric on disjoint union X ∐ Y , i.e. d is anextension of d X and d Y , while d H denotes the Hausdorff distance. The number d GH ( X, Y ) is called the Gromov-Hausdorff distance of metric spaces X and Y . Theorem 2.1. d GH ( X, Y ) = 0 iff ( X, d X ) is isometric to ( Y, d Y ) . (cid:3) The Gromov Lemma (below) will be used widely throughout his paper.
Lemma 2.2.
Let ( X, d X ) and ( Y, d Y ) be arbitrary compact metric spaces, and let A = { x , . . . , x k } ⊂ X,B = { y , . . . , y k } ⊂ Y be ε -nets satisfying for all ≤ i, j ≤ k the condition | d X ( x i , x j ) − d Y ( y i , y j ) | ≤ ε. SZYMON M. WALCZAK
Then d GH ( X, Y ) ≤ ε . (cid:3) Convergence theorem
Consider a sequence ( f n ) n ∈ N , f n : M → (0 , ∞ ), of warping function on a com-pact foliated Riemannian manifold ( M, F , g ). One can ask, how does the limit inGromov-Hausdorff topology of a sequence of warped foliations ( M f n ) n ∈ N look like.Let G ⊂ M be a dense subset. Theorem 3.1.
For an arbitrary compact foliated manifold ( M, F , g ) and any se-quence ( f n ) n ∈ N , f n : M → [0 , , of warping functions on M converging to zero on G , the Gromov-Hausdorff limit of a sequence of warped foliations is isometric to HLS( F ) . Before we prove Theorem 3.1, we give a simple definition and observe an impor-tant fact.We say that two metric structures g and g ′ on a compact foliated Riemannianmanifold ( M, F ) coincide on the orthogonal bundle F ⊥ if every vector v perpendic-ular to F in g is perpendicular in g ′ and vice versa, and g ( v, w ) = g ′ ( v, w ) for anyvectors v, w perpendicular to F either in g or g ′ . Lemma 3.1.
Let g and g ′ be any Riemannian structures on M which coincide onthe orthogonal bundle F ⊥ . Then ˜ ρ = ˜ ρ ′ .Proof. Since M is compact, we can assume that g ≤ C · g ′ for a certain constant C ≥
1. Let ρ and ρ ′ be pseudometrics given by ρ ( L, L ′ ) = inf { n − X i =1 dist( L i , L i +1 ) } ,ρ ′ ( L, L ′ ) = inf { n − X i =1 dist ′ ( L i , L i +1 ) } , where dist and dist ′ denote the distance of the leaves in g and g ′ , respectively.Since the geometry of M is bounded, then for every A > ǫ > δ > γ : [0 , l ( γ )] → M parametrized naturallysatisfying(1) ˙ γ (0) is perpendicular to F ,(2) the g ′ -length l ′ ( γ ) is smaller than δ ,(3) the g ′ -geodesic curvature | k g ( γ ) | is smaller than A ,the g -length of the component tangent to F satisfies | ˙ γ ⊤ | < ǫ. Let ǫ > L, L ′ ∈ F be such that d = dist ′ ( L, L ′ ) < δ . Let γ : [0 , l ′ ( γ )] → M be acurve such that its length in g ′ satisfies d ≤ l ′ ( γ ) ≤ δ . We havedist( L, L ′ ) ≤ l ( γ ) = Z [0 ,l ′ ( γ )] | ˙ γ | ≤ Z [0 ,l ′ ( γ )] | ˙ γ ⊤ | + Z [0 ,l ′ ( γ )] | ˙ γ ⊥ |≤ C · l ′ ( γ ) · ǫ + Z [0 ,l ′ ( γ )] | ˙ γ ⊥ | ′ ≤ (1 + Cǫ ) · l ′ ( γ ) . Since γ was chosen arbitrarily, we conclude thatdist( L, L ′ ) ≤ (1 + Cǫ ) · dist ′ ( L, L ′ ) . AUSDORFF LEAF SPACES 5
Now, for every sequence of leaves L , . . . , L n such that L = L , L n = L ′ andsatisfying n − X i =1 dist ′ ( L i , L i +1 ) ≤ ρ ′ ( L, L ′ ) + ǫ, and such that dist ′ ( L i , L i +1 ) < δ for all i ∈ { , . . . , n − } , we obtain ρ ( L, L ′ ) ≤ n − X i =1 dist( L i , L i +1 ) ≤ (1 + Cǫ ) · ( n − X i =1 dist ′ ( L i , L i +1 )) ≤ (1 + Cǫ ) · ( ρ ′ ( L, L ′ ) + ǫ ) . Tending with ǫ to zero we get that ρ ≤ ρ ′ . Consequently ˜ ρ ≤ ˜ ρ ′ . Similarly, we canshow that ˜ ρ ′ ≤ ˜ ρ . (cid:3) We now turn to a proof of Theorem 3.1. Denote by π : M → HLS( F ) a naturalprojection given by π ( x ) = [ L x ] ∼ , where ∼ is the equivalence relation defined inSection 1.1. Proof.
Let ǫ > { x , . . . , x k } be an ǫ -net on M contained in G . Let i, j ∈{ , . . . , k } . Choose a family of leaves F ij = { F ij , . . . , F ijd ji } such that L x i = F ij , L x j = F ijd ji , F ijν ⊂ π − ( π ( G )) for any 0 ≤ ν ≤ d ji . Next, consider a family of curves γ ij , . . . , γ ijd ji − : [0 , → M satisfying γ ijν (0) ∈ L x ν , γ ijν (1) ∈ L x ν +1 , and(1) d ji − X ν =0 l ( γ ijν ) ≤ ˜ ρ ( π ( L x i ) , π ( L x j )) + ǫ. Let d = max { d ji } . Since f n → G , and the number of leaves involved in F ij , i, j = 1 , . . . , k , is finite, there exists N ∈ N such that for any n > N , i, j ∈ { , . . . , k } and ν ∈ { , . . . , d ji − } we have d F ijν ( γ ijν (1) , γ ijν +1 (0)) ≤ ǫd , (2) d F ijν ( x i , γ ij (0)) ≤ ǫd , and d F ijν ( γ ijd ji − (1) , x j ) ≤ ǫd . Let us pick one point in each { x , . . . , x k } ∩ π − ( π ( x i )), i = 1 , . . . , k . We obtaina set { y , . . . , y m } ( m ≤ k ) with the property π ( y i ) = π ( y j ) iff i = j .Let n > N . Direct calculation shows that the points y , . . . , y m form a 3 ǫ -net on( M, g f n ). Moreover, by (1) and (2), d n ( y i , y j ) ≤ ˜ ρ ( π ( L y i ) , π ( L y j )) + 2 ǫ. Next, by Lemma 3.1,˜ ρ ([ L y i ] , [ L y j ]) = ˜ ρ n ( π ( L y i ) , π ( L y j )) ≤ d n ( y i , y j ) . The set π ( { y , . . . , y m } ) = π ( { x , . . . , x k } ) provides an ǫ -net on HLS( F ). ByLemma 2.2, d GH ( M f n , HLS( F )) ≤ ǫ . Tending with ǫ to zero we get that d GH ( M f n , HLS( F )) = 0 . Theorem 2.1 completes the proof. (cid:3)
SZYMON M. WALCZAK Basic constructions
Studying foliations one can learn that there are several basic constructions forbuilding foliations [2]. In this chapter we examine the HLS for the following con-structions: tangential and transverse gluing, two transverse modifications - turbu-lization and spinning along a transverse boundary component, and for suspension.We only provide here a detailed proof for tangential gluing and turbulization,which are used in Section 5. For transverse gluing and for spinning we give thecharacterization of their HLS’s and we only provide an outline of a proof. Details,analogically as for tangential gluing and turbulization, are left to the reader.4.1.
Tangential gluing.
Let us assume that ( M i , F i , g i ), i = 1 ,
2, are compactfoliated Riemannian manifolds with boundary, while F i is a foliation tangent tothe boundary. Let S i ⊂ ∂M i ( i = 1 ,
2) be a union of boundary components, and let h : S → S be an isometry mapping leaves onto leaves. According to [2], identify S with S using x ≡ h ( x ), and form the quotient foliated manifold M = M ∪ h M with foliation F = F ∪ h F defined by the leaves of F i (Figure 2).Let us assume that one can obtain a smooth Riemannian structure g on M withthe property g | M i = g i ( i = 1 , Figure 2.
Tangential gluing.Denote by π : M → HLS( F ), π i : M i → HLS( F i ) ( i = 1 ,
2) natural projections.Consider the smallest equivalence relation ∼ in disjoint unionHLS( F ) ∐ HLS( F )containing the relation defined as follows: π ( L ) ∼ π ( L ′ ) ⇔ ∃ x ∈ π − ( π ( L )) π ( h ( x )) = π ( L ′ ) . Let X = HLS( F ) ∐ HLS( F ) / ∼ endowed with quotient metric d X .Denote by Φ : HLS( F ) ∐ HLS( F ) → X , ˜ π : M ∐ M → HLS( F ) ∐ HLS( F ), and p : M ∐ M → M natural projections (see Figure 3). Theorem 4.1.
The space
HLS( F ) is isometric to ( X, d X ) . We begin the proof by a following:
Lemma 4.1.
For any x, y ∈ M ∐ M d X (Φ(˜ π ( x )) , Φ(˜ π ( y ))) = ˜ ρ ( π ( p ( x )) , π ( p ( y ))) . AUSDORFF LEAF SPACES 7 M ∐ M p / / ˜ π (cid:15) (cid:15) M π / / HLS( F )HLS( F ) ∐ HLS( F ) Φ / / X Figure 3.
The projections for Theorem 4.1.
Proof.
Let ǫ >
0. Consider points π ( p ( x )) and π ( p ( y )). By the definition ofHLS( F ), there exist points r , q , . . . , r k , q k in the disjoint union M ∐ M suchthat r ν , q ν are the points in the component M or M , and r ∈ L x , q k ∈ L y ( L x and L y are here the leaves of the appropriate foliation F or F ). Moreover, p ( q ν )and p ( r ν +1 ) lie in the same leaf of F , and k X ν =1 ¯ d ( r ν , q ν ) ≤ ˜ ρ ( π ( p ( x )) , π ( p ( y ))) + ǫ, where ¯ d is the length metric of disjoint union in M ∐ M (see Section 1.2). Denoteby ˜ d the length metric of disjoint union in HLS( F ) ∐ HLS( F ). By the definitionof X , we have k X ν =1 ¯ d ( r ν , q ν ) ≥ k X ν =1 ˜ d (˜ π ( r ν ) , ˜ π ( q ν )) ≥ d X (Φ(˜ π ( x )) , Φ(˜ π ( y ))) . Finally,(3) d X (Φ(˜ π ( x )) , Φ(˜ π ( y ))) ≤ ˜ ρ ( π ( p ( x )) , π ( p ( y ))) + ǫ Next, consider points Φ(˜ π ( x )) and Φ(˜ π ( y )). There exist points r , q , . . . , r k , q k in the disjoint union HLS( F ) ∐ HLS( F ) such that Φ( q ν ) = Φ( r ν +1 ) ( ν = 1 , . . . , k ),Φ( r ) = Φ(˜ π ( x )), Φ( q k ) = Φ(˜ π ( y )), and k X ν =1 ˜ d ( r ν , q ν ) ≤ d X (Φ(˜ π ( x )) , Φ(˜ π ( y ))) , where ˜ d denotes again the length metric of disjoint union in HLS( F ) ∐ HLS( F ).Now, for every ν = 1 , . . . , k one can find a sequence of leaves L ν , . . . , L νl ν of theappropriate foliation ( F if r ν , q ν ∈ HLS( F ) or F if r ν , q ν ∈ HLS( F )) satisfying r ν ∈ L ν , q ν ∈ L νl ν , and l ν − X µ =1 dist( L νµ , L νµ +1 ) ≤ ˜ d ( r ν , q ν ) + ǫk . Since h maps leaves onto leaves, one can consider the leaves described above asleaves of a foliation F . Moreover, ˜ π ( x ) ∈ ˜ π ( L ), and ˜ π ( y ) ∈ ˜ π ( L kl ν ). Hence we have(4) ˜ ρ ( π ( p ( x )) , π ( p ( y ))) ≤ d X (Φ(˜ π ( x )) , Φ(˜ π ( y ))) + 2 ǫ. Passing with ǫ to zero in inequalities (3) and (4), we get that˜ ρ ( π ( p ( x )) , π ( p ( y ))) ≤ d X (Φ(˜ π ( x )) , Φ(˜ π ( y ))) + 2 ǫ. SZYMON M. WALCZAK
This completes the proof. (cid:3)
Now, we turn to the proof of Theorem 4.1.
Proof.
Let us consider a sequence ( f n : M → (0 , M converging to zero. Obviously, it can be used as a sequence of warping functions.By Theorem 3.1, the limit lim GH M f n = HLS( F ). We will show, that lim GH M f n =( X, d X ).Let { x , . . . , x k } ⊂ M and { x , . . . , x k } ⊂ M be ǫ/ N ∈ N , K ≤ k , and K ≤ k such that • ˜ π ( x i ) = ˜ π ( x j ) while i = j ( i, j ≤ K ), ˜ π ( x l ) = ˜ π ( x m ) while l = m ( l, m ≤ K ); • { x , . . . , x K } is an ǫ -net in ( M ) n , while { x , . . . , x K } provides an ǫ -netin ( M ) n , n > N .Denote • y j = Φ(˜ π ( x j )), j = 1 , . . . , K ; • y K + j = Φ(˜ π ( x j )), j = 1 , . . . , K ; • x j = π ( p ( x j )), j = 1 , . . . , K ; • x K + j = π ( p ( x j )), j = 1 , . . . , K .One can easily check that { y , . . . , y K + K } and { x , . . . , x K + K } have the samenumber of elements, and y i = y j iff x i = x j . Finally we get two ǫ -nets { y , . . . , y K } and { x , . . . , x K } in X and HLS( F ) respectively. By Lemma 4.1 d X ( x i , x j ) = ˜ ρ ( y i , y j )for all i, j ≤ K . By Lemma 2.2, d GH (( X, d X ) , HLS( F )) = 0. Finally, by Theorem2.1, we get the statement. (cid:3) Remark 4.1.
Note that it can be impossible to construct a smooth Riemannianstructure g on M such that g | M i = g i ( i = 1 , ). But all Riemannian structures ona compact manifold are equivalent. We slightly modify the Riemannian structures g i to obtain structures with desired properties. In this case we can only prove that HLS( F ) of the glued foliation is homeomorphic to ( X, d X ) . Transverse gluing.
Following [2], let ( M , F , g ), ( M , F , g ) be smoothcompact foliated Riemannian manifolds of dimension n with nonempty boundaryand codimension q foliations. Suppose that S i ⊂ ∂M i is a union of boundarycomponents ( i = 1 ,
2) and φ : S → S is an isometry mapping leaves to leaves.Suppose further that F i is g i -orthogonal to S i . Form a manifold M = M ∪ φ M from the disjoint union M ∐ M by identifying x ∼ φ ( x ). Endow M with aninduced foliation.Consider the smallest equivalence relation ∼ in disjoint unionHLS( F ) ∐ HLS( F )containing the relation defined by˜ π ( x ) ∼ ˜ π ( φ ( x )) . Next, glue HLS( F ) with HLS( F ) along ∼ and denote the result endowed withquotient metric by ( X, d X ).Let us denote the natural projections as shown on the Figure 4. AUSDORFF LEAF SPACES 9 M ∐ M p / / ˜ π (cid:15) (cid:15) M π / / HLS( F )HLS( F ) ∐ HLS( F ) Φ / / X Figure 4.
The projections for Theorem 4.2.
Lemma 4.2.
For any two points x ∈ M ∐ M d X (Φ(˜ π ( x )) , Φ(˜ π ( y ))) = ˜ ρ ( π ( p ( x )) , π ( p ( y ))) . Proof.
Analogical to the proof of Lemma 4.1. Left to the reader. (cid:3)
Theorem 4.2.
HLS( F ) coincides with ( X, d X ) .Proof. Denote by A = { x , . . . , x k } ⊂ M \ ∂M and A = { y , . . . , y m } ⊂ M \ ∂M two ǫ -nets. One can easily check that p ( A ∪ A ) is an ǫ -net in M , π ( p ( A ∪ A ))is an ǫ -net in HLS( F ) and Φ(˜ π ( A ∪ A )) is an ǫ -net in X . Moreover, by theconstruction of X we have that ♯ (Φ( ˜ piπ ( A ∪ A ))) = ♯ ( π ( p ( A ∪ A ))) . Lemma 4.2 and Lemma 2.2 yield the statement. (cid:3)
Figure 5.
Transverse gluing.
Remark 4.2.
Of course, not every foliation transverse to the boundary componentis orthogonal to it. But one can easily modify (see [2] ) any transverse foliation toobtain a foliation orthogonal to the boundary component with the same space as theHausdorff leaf space (see Figure 5). Hence, Theorem 4.2 is true for any foliationstransverse to the boundary.
Turbulization.
Let now ( M, F , g ) be a foliated Riemannian manifold of di-mension n + 1 endowed with a codimension one foliation which is leaf-wise andtransversely orientable. Let γ : [0 , → M be a closed transversal curve and let N ( γ ) be a fixed foliated tubular neighbourhood of γ . Let us equip N ( γ ) = D n × S with cylindrical coordinates ( r, z, t ) (we take t modulo
1, and the leaves of F| N ( γ )are the sets D n × { t } ). Let ω = cos λ ( r )d r + cos λ ( r )d t, where λ : [0 , → [ − π/ , π/
2] is a smooth, strictly increasing on [0 , /
4] functionsatisfying λ (0) = − π/ λ (2 /
3) = 0, λ ( t ) = π/ t ≥ /
4, and with derivativesof all orders at zero vanishing. Since ω is integrable, it defines a foliation F γ of N ( γ ), which agrees with F near ∂N ( γ ) and has a Reeb component R inside N ( γ ).Modified foliation F γ of M is called a turbulized foliation, while this deformation iscalled turbulization [2] (Figure 6). Figure 6.
Turbulization.Denote by L x ( L γx ) the leaf of F ( F γ ) passing through x ∈ M . Next, let π : M → HLS( F ) be a natural projection, and let X be a metric space obtained fromHLS( F ) by identification π ( γ ([0 , X with thequotient metric d X . Theorem 4.3.
HLS( F γ ) is isometric with ( X, d X ) . Before we start a proof, we shall prove technical lemmas.
Lemma 4.3.
For every two leaves L , L ∈ F and every ǫ > there exists a finitesequence of leaves F , . . . , F k ∈ F γ , k ≤ , satisfying (1) F \ N ( γ ) = L \ N ( γ ) and F k \ N ( γ ) = L \ N ( γ ) , (2) P k − ν =1 dist( F ν , F ν +1 ) ≤ dist( L , L ) + ǫ .Proof. Let ǫ >
0, and x ∈ L , y ∈ L be such that d ( x, y ) ≤ dist( L , L ) + ǫ . Weshall consider three cases:(1) x, y / ∈ M \ N ( γ ). Put F = L γx and F = L γy . Then dist( F , F ) ≤ d ( x, y ) ≤ dist( L , L ) + ǫ .(2) x, y ∈ N ( γ ). By the definition of the turbulization we have that dist( L γx , L γy ) =0 ≤ dist( L , L ).[0 , γ / / ( M, F , g ) π (cid:15) (cid:15) turb. / / ( M, F γ , g ) π γ (cid:15) (cid:15) HLS( F ) p (cid:15) (cid:15) HLS( F γ )( X, d X ) f qqqqqqqqqq Figure 7.
The projections for Theorem 4.3.
AUSDORFF LEAF SPACES 11
Figure 8.
HLS for turbulized foliation.(3) x ∈ N ( γ ) and y ∈ M \ N ( γ ). Let z ∈ L \ N ( γ ). Put F = L γz , F = L γx , F = L γy . We havedist( F , F ) + dist( F , F ) ≤ d ( x, y ) ≤ dist( L , L ) + ǫ. This completes the proof. (cid:3)
Lemma 4.4.
For any x, y ∈ M the following inequality holds: d X ( p ( π ( x )) , p ( π ( y ))) ≤ dist( L γx , L γx ) . Proof.
Let x, y ∈ M . We shall consider few cases: Case 1. L γx ∩ N ( γ ) = L γy ∩ N ( γ ) = ∅ . Then L x = L γx , L y = L γy , and d X ( p ( π ( x )) , p ( π ( y ))) ≤ dist( L γx , L γx ) . Case 2. L γx ∩ N ( γ ) = ∅ , L γx ∩ N ( γ ) = ∅ . Then, by the construction of F γ , L x ∩ N ( γ ) = ∅ , and L y ∩ N ( γ ) = ∅ . Finally, d X ( p ( π ( x )) , p ( π ( y ))) = 0 ≤ dist( L γx , L γx ) . Case 3. L γx ∩ N ( γ ) = ∅ , L γx ∩ N ( γ ) = ∅ . Let ǫ >
0, and r ∈ L γx , q ∈ L γy be suchpoints that d ( r, q ) ≤ dist( L γx , L γy ) + ǫ .Suppose first that x / ∈ N ( γ ), r / ∈ N ( γ ). Then L x = L r , L y = L q , and d X ( p ( π ( x )) , p ( π ( y ))) ≤ dist( L x , L y ) ≤ d ( r, q ) ≤ dist( L γx , L γx ) . Next, suppose that x / ∈ N ( γ ), r ∈ N ( γ ). Set L = L x , L = L r , L = L y . Recallthat L q = L y . Hence, by Case 1, d X ( p ( π ( x )) , p ( π ( y ))) ≤ d X ( p ( π ( x )) , p ( π ( r ))) + d X ( p ( π ( r )) , p ( π ( q ))) ≤ d ( r, q ) ≤ dist( L γx , L γx ) + ǫ. Analogically we show that d X ( p ( π ( x )) , p ( π ( y ))) ≤ dist( L γx , L γx ) , for x ∈ N ( γ ), r / ∈ N ( γ ), and x ∈ N ( γ ), r ∈ N ( γ ).Passing with ǫ to zero gives the desired inequality. (cid:3) Lemma 4.5.
For any x, y ∈ M we have d X ( p ( π ( x )) , p ( π ( y ))) ≤ ˜ ρ γ ( π γ ( L γx ) , π γ ( L γx )) . Proof.
Let ǫ > x, y ∈ M . There exists a sequence of leaves L γ , . . . , L γk such that L γ = L γx , L γk = L γy , and k =1 X ν =1 dist( L γν , L γν +1 ) ≤ ˜ ρ γ ( π γ ( L γx ) , π γ ( L γx )) + ǫ. Let r , q , . . . , r k − , q k − ∈ M be such that r i ∈ L γi , q i ∈ L γi +1 , and d ( r i , q i ) ≤ dist( L γν , L γν +1 ) + ǫk . Note that L γx = L γr , L γy = L γq k − , and L γr i +1 = L γq i . By Lemma 4.4, d X ( p ( π ( x )) , p ( π ( r ))) = 0 ,d X ( p ( π ( y )) , p ( π ( q k − ))) = 0 ,d X ( p ( π ( r i +1 )) , p ( π ( q i ))) = 0for all i ∈ { , . . . , k − } . By the construction of X , d X ( p ( π ( x )) , p ( π ( y ))) ≤ k − X ν =1 d X ( p ( π ( r i )) , p ( π ( q i ))) + k − X ν =1 d X ( p ( π ( r i +1 )) , p ( π ( q i )))+ d X ( p ( π ( x )) , p ( π ( r ))) + d X ( p ( π ( y )) , p ( π ( q k − ))) ≤ k − X ν =1 d ( r i , q i ) ≤ ˜ ρ γ ( π γ ( L γx ) , π γ ( L γx )) + 2 ǫ. Passing with ǫ to zero gives us the statement. (cid:3) Lemma 4.6.
For any x, y ∈ M we have ˜ ρ γ ( π γ ( L γx ) , π γ ( L γx )) ≤ d X ( p ( π ( x )) , p ( π ( y ))) . Proof.
Let x, y ∈ M , ǫ >
0. There exist points r , q , . . . , r k , q k ∈ HLS( F ) suchthat p ( q i ) = p ( r i +1 ), π ( x ) = r , π ( x ) = q k , and(5) k − X ν =1 ˜ ρ ( r i , q i ) ≤ d X ( p ( π ( x )) , p ( π ( y ))) + ǫ. For any i ∈ { , . . . , k } one can find a family of leaves L i, , . . . , L i,µ i satisfying L i +1 , = L i,µ i , x ∈ L , , y ∈ L k,µ k , and(6) µ i − X ν =1 dist( L i,ν , L i,ν +1 ) ≤ ˜ ρ ( r i , q i ) + ǫk . By (5), (6), and Lemma 4.3, one can find a finite sequence L γ , . . . , L γm of leaves of F γ such that L γ = L γx , L γm = L γy and˜ ρ γ ( π γ ( L γx ) , π γ ( L γx )) ≤ m − X ν =1 dist( L γν , L γν +1 ) ≤ d X ( p ( π ( x )) , p ( π ( y ))) + 3 ǫ. Passing with ǫ to zero gives us the statement. (cid:3) AUSDORFF LEAF SPACES 13
Let f : X → HLS( F γ ) be defined as follows:(7) f ( p ( π ( x ))) = π γ ( L γx ) , where π γ : M → HLS( F γ ) denotes the natural projection. Lemma 4.7. f is bijective.Proof. Suppose that p ( π ( x )) = p ( π ( y )), x, y ∈ M . Consider two cases: Case 1. π ( x ) = π ( y ). If π − ( π ( x )) ∩ γ ([0 , ∅ then π − ( π ( y )) ∩ γ ([0 , ∅ , and π − γ ( π γ ( x )) = π − γ ( π γ ( y )). Hence π γ ( L γx ) = π γ ( L γy ), and f ( p ( π ( x ))) = f ( p ( π ( y ))). Case 2. If π ( x ) = π ( y ), then there exist ξ x ∈ π − ( π ( x )) ∩ γ ([0 , ξ y ∈ π − ( π ( y )) ∩ γ ([0 , π γ ( ξ x ) = π γ ( ξ x ) = π γ ( R ), where R denotes the Reebcomponent. But π γ ( L γx ) = π γ ( L γξ x ). Thus f ( p ( π ( x ))) = f ( p ( π ( x ))).Finally, f is well defined. By the definition, f is ”onto” HLS( F γ ). Checking that f is one-to-one we leave to the reader. (cid:3) Now, we can turn to the proof of Theorem 4.3.
Proof.
By Lemma 4.7, f defined in (7) is a bijection from X onto HLS( F γ ). ByLemmas 4.5 and 4.6, f is an isometry. (cid:3) Spinning.
Following the definition given in [2] we recall the notion of spinning a foliation along a transverse boundary component.Let ( M, F , g ) be a compact Riemannian manifold carrying codim-1 foliationtransverse to the boundary ∂M = ∅ . Let S be a transverse connected componentof ∂M with ∂S = ∅ . Assume that F| S can be defined by a closed non-singular1-form ω ∈ A ( S ). Figure 9.
Spinning along the boundary component.Let N ( S ) = S × [0 ,
1) be a foliated collar, i.e. the leaves of F| N ( S ) are of theform L × [0 , L is a leaf of F| S .Decompose T ( x,t ) ( N ( S )) = T x ( S ) ⊕ T t ([0 , ζ ∈ X ( N ( S )) as ζ = f v + g∂ t , where f, g ∈ C ∞ ( N ( S )), v ∈ X ( S ), and ∂ t = ∂∂t . ω extends to a closed non-singularform ω N ( S ) by ω N ( S ) ( f v + g∂ t ) = f ω ( v ) . Let h : [0 , → [0 ,
1] be a C ∞ -function such that h ( t ) = 0 for t ∈ [ , h (0) = 1,and h is decreasing strictly monotonically on [0 , ]. Moreover, let the derivativesof all orders of h vanish at t = 0. Set θ = (1 − h ( t )) ω N ( S ) + h ( t )d t.θ agrees with ω N ( S ) on S × [ ,
1) and with d t on S × { } . Moreover, θ is integrableand S becomes a leaf of a new foliation F S on S × [0 , F coincides with F S outside the collar S × [0 , ). Thus, we extend F S to a foliation F S on M which istangent to the boundary component S .Now, identify in HLS( F ) the points of π ( S ) and denote the result by X . Endow X with the quotient metric denoted by d X .Before we examine the Hausdorff leaf space for a spinned foliation we formulatetechnical lemmas. Easy proofs are omitted and left to the reader. M π / / π S (cid:15) (cid:15) HLS( F ) φ (cid:15) (cid:15) HLS( F S ) X Figure 10.
The projections for Theorem 4.4.Let π : M → HLS( F ), π S : M → HLS( F S ), and φ : HLS( F ) → X denotethe natural projections (Figure 10). Denote by L z ( L Sz ) a leaf of F ( F S ) passingthrough a point z ∈ M . Lemma 4.8.
For every two points p, q ∈ M such that L Sp = L Sq we have d X ( φ ( π ( L p )) , φ ( π ( L q ))) = 0 . (cid:3) Lemma 4.9.
For any two points p, q ∈ M such that L p = L q we have ˜ ρ ( π S ( L Sp ) , π S ( L Sq )) = 0 . (cid:3) Lemma 4.10.
For any two points x, y ∈ M we have d X ( φ ( π ( L x )) , φ ( π ( L y ))) = ˜ ρ S ( π S ( x ) , π S ( y )) . (cid:3) Theorem 4.4.
HLS( F ) coincides with ( X, d X ) .Proof. Let f n = n be a constant function on M . Let A ′ = { x , . . . , x k ′ } ⊂ M be an ǫ/ M . One can select from a subset A = { x , . . . , x k } ⊂ A ′ and N ∈ N such that π ( x i ) = π ( x j ) ( i = j ) and A an ǫ -net on M n = ( M, F , g n ) for all n > N . We may assume that the points x k − l , . . . , x k are the only ones that belongto π − ( π ( S )). Now, pick from the points x k − l , . . . , x k exactly one, let say x k − l . AUSDORFF LEAF SPACES 15
Figure 11.
HLS of a foliation spinned along the boundary com-ponent S .Observe that π S ( { x k − l , . . . , x k } ) is a single point in HLS( F S ). Hence, thereexists N ′ such that { x , . . . , x k − l } is an ǫ -net on M S n = ( M, F S , g n ) for all n > N ′ .Moreover, since x k − l , . . . , x k ∈ π − ( π ( S )), φ ( π ( x µ )) = φ ( π ( x ν )) , µ, ν ∈ { k − l, . . . , k } . Set ζ i = φ ( π ( x i )), ξ j = π S ( x j ) ( i, j = 1 , . . . , k − l ). By the construction and Lemma3.1, the sets { ζ i } and { ξ j } are 2 ǫ -nets on X and HLS( F S ), respectively. By Lemma4.10, d X ( ζ i , ζ j ) = ˜ ρ S ( ξ i , ξ j ) , for all i, j ∈ { , . . . , k } . By Lemma 2.2, d GH ( X, HLS( F S )) = 0, and by Theorem 2.1, X is isometric toHLS( F S ). (cid:3) Suspension.
Denote by B a smooth connected manifold, and by p : ˜ B → B the universal covering of B . Let x ∈ B . Recall that the covering transformationgroup Γ acts from the right on ˜ B and hence Γ ⊂ Diff( ˜ B ). Let F be a q -dimensionalmanifold. Consider a group homomorphism h : Γ → Diff( F ). Then Γ acts on ˜ B × F by γ ( x, z ) = ( γ ( x ) , h ( γ )( z )) , ( x ∈ ˜ B, z ∈ F ) . Consider a foliation ˜ F = { ˜ B × { z } , z ∈ F } . Using canonical projection one canproject ˜ F onto a foliation F of M = ( ˜ B × F ) / Γ. The foliation F is called the suspension of the homomorphism h . One can check that M is a fibre bundle over B , and F coincides with its fibre.Analogically as in Section 2.1, one can define the Hausdorff orbit space :Let G be a group acting on a metric space ( X, d X ). Denote by O the space oforbits of G -action. Set ρ ( G ( x ) , G ( y )) = inf { n − X i =1 d X ( G , G i +1 ) } , where the infimum is taken over all finite sequences of orbits beginning at G = G ( x ) and ending at L n = G ( y ), and G ( z ) denotes the orbit of z ∈ X . Define anequivalence relation ∼ in O by: G ( x ) ∼ G ( y ) ⇔ ρ ( G ( x ) , G ( y )) = 0 , x, y ∈ X. Let ˜ O = O / ∼ . Put ˜ ρ ([ G ( x )] , [ G ( y )]) = ρ ( G ( x ) , G ( y )) , where [ G ( x )] , [ G ( y )] ∈ ˜ O . ( ˜ O , ˜ ρ ) is a metric space. We call it the Hausdorff orbitspace of the G -action, and we denote it by HOS( X/G ). Theorem 4.5.
HLS( F ) is homeomorphic to HOS(
F/h (Γ)) .Proof.
By the construction of suspension, there exists a homeomorphism betweenthe space of leaves of F and the space of orbits of h (Γ). It induces a homeomorphismbetween HLS( F ) and HOS( F/h (Γ)). (cid:3) Main results - HLS for codim-1 foliations
HLS for compact I-bundles.
Let ( M, F , pr) be a foliated I -bundle, I =[0 , L the boundary leaf passing through the points 0 ∈ I of every fiber. Consider thefunction d : L → [0 ,
1] ( L denotes here the space of leaves of the foliation F ) definedby d ( L ) = ˜ ρ ( L , L ), where ˜ ρ denotes the metric in HLS( F ). Let π : M → HLS( F )again be the natural projection. Lemma 5.1.
For any two leaves L = L ′ such that π ( L ) = π ( L ′ ) we have d ( L ) = d ( L ′ ) .Proof. Since π ( L ) = π ( L ′ ) then ˜ ρ ( π ( L ) , π ( L ′ )) >
0. Let ǫ >
0, and let L , . . . , L k be a family of leaves such that L k = L ′ , and k − X ν =0 dist( L ν , L ν +1 ) < ˜ ρ ( L , L ′ ) . Without losing generality we can assume that there exists j ∈ { , . . . , k − } satis-fying L j = L (if not then rename the leaf L to L ′ and L ′ to L ). Then˜ ρ ( L , L ) + ˜ ρ ( L, L ′ ) ≤ j − X ν =0 dist( L ν , L ν +1 ) + k − X ν = j dist( L ν , L ν +1 ) < ˜ ρ ( L , L ′ ) + ǫ. Hence, d ( L ) + ˜ ρ ( L, L ′ ) ≤ d ( L ′ ) . By the triangle inequality and the above, we obtain d ( L ) + ˜ ρ ( L, L ′ ) = d ( L ′ ) . But ˜ ρ ( L, L ′ ) >
0. Hence, d ( L ) < d ( L ′ ). This completes the proof. (cid:3) Theorem 5.1.
Let ( M, F , pr) be a foliated I -bundle. HLS( F ) is isometric to ametric segment.Proof. Let L denote the same leaf as in Lemma 5.1, d be a function on the spaceof leaves of F defined by d ( L ) = ˜ ρ ( L , L ), and let δ = max L ∈F d ( L ). Let π : M → HLS( F ) be a natural projection, while p : M → [0 , δ ] be the mapping defined by p ( x ) = d ( L x ). By Lemma 5.1, for any two leaves such that π ( L ) = π ( L ′ ) we have d ( L ) = d ( L ′ ) . Let ǫ >
0, and
L, L ′ ∈ F be two arbitrary leaves such that d ( L ) < d ( L ′ ). Let L , . . . , L k , L k +1 , . . . , L k + l be a family of leaves satisfying L k = L , L k + l = L ′ , and k + l − X ν =0 dist( L ν , L ν +1 ) ≤ ˜ ρ ( π ( L ) , π ( L ′ )) + ǫ. AUSDORFF LEAF SPACES 17
Since ˜ ρ ( π ( L ) , π ( L )) ≤ P k − ν =0 dist( L ν , L ν +1 ), we have˜ ρ ( π ( L ) , π ( L ′ )) ≤ k + l − X ν =0 dist( L ν , L ν +1 )(8) ≤ ˜ ρ ( π ( L ) , π ( L ′ )) + ǫ − ˜ ρ ( π ( L ) , π ( L )) = | d ( L ′ ) − d ( L ) | + ǫ. Now, let L , . . . , L k , L k +1 , . . . , L k + l be a family of leaves such that L k = L , L k + l = L ′ k − X ν =0 dist( L ν , L ν +1 ) ≤ ˜ ρ ( π ( L ) , π ( L )) + ǫ , and k + l − X ν = k dist( L ν , L ν +1 ) ≤ ˜ ρ ( π ( L ) , π ( L ′ )) + ǫ . Then d ( L ′ ) ≤ k + l − X ν =0 dist( L ν , L ν +1 ) ≤ ˜ ρ ( π ( L ) , π ( L )) + ǫ ρ ( π ( L ) , π ( L ′ )) + ǫ ≤ d ( L ) + ˜ ρ ( π ( L ) , π ( L ′ )) + ǫ. We get(9) d ( L ′ ) − d ( L ) ≤ ˜ ρ ( π ( L ) , π ( L ′ )) + ǫ. Since d ( L ) − d ( L ′ ) ≤ ≤ ˜ ρ ( π ( L ) , π ( L ′ )) we finally get, by (8) and (9),(10) || d ( L ) − d ( L ′ ) | − ˜ ρ ( π ( L ) , π ( L ′ )) | ≤ ǫ. Let A = { x , . . . , x k } be an ǫ -net on M . Then π ( A ) and p ( A ) are ǫ -nets onHLS( F ) and ([0 , d ] , | · | ), respectively. Moreover, ♯π ( A ) = ♯p ( A ). By (10), we have || p ( L i ) − p ( L j ) | − ˜ ρ ( π ( L i ) , π ( L j )) | ≤ ǫ, where L ν = L x ν . By Lemma 2.2, d GH (HLS( F ) , [0 , d ]) ≤ ǫ . Finally, d GH (HLS( F ) , [0 , d ]) = 0 , and, by Theorem 2.1, HLS( F ) is isometric to the metric segment I = ([0 , d ] , |·| ). (cid:3) HLS for codim-1 foliations.
Recall now [1] that the metric graph G is theresult of gluing of a set of a disjoint metric segments E = { E i } and points V = { v i } along an equivalence relation defined in the union of V and the set of the endpointsof the segments equipped with the length metric. A graph G is called finite if V and E are finite. Theorem 5.2.
HLS( F ) of any codimension one foliation on a compact Riemannianmanifold is isometric to a finite connected metric graph.Proof. Following the proof of the main theorem of [5], we can cover M by a finitenumber of mutually disjoint saturated neighborhoods N i ( i = 1 , . . . , k ) such thatthe HLS of the foliation restricted to N i is a singleton, and a finite number ofmutually disjoint foliated I − bundles (denoted by C , . . . , C m ) with their HLS’s,by Lemma 5.1, isometric to [0 , d j ], d j >
0, 1 ≤ j ≤ m . We can assume that N i ∩ C j ⊂ ∂N i ∩ ∂C j , 1 ≤ i ≤ k , 1 ≤ j ≤ m (Figure 12), and that the sets N i ( i = 1 , , . . . , k ) are maximal, i.e. π − ( π ( N i )) = N i , where π : M → HLS( F )denotes the natural projection. Figure 12.
The sets N i and C j .Let v i = HLS( F| N i ), and V = { v , . . . , v k } . Next, let E = { I , . . . , I m } , I j = HLS( F| C j ) = [0 , d j ] . Denote by π j : C j → [ o, d j ] natural projections.Introduce in V and in the set of the endpoints of the segments I j , 1 ≤ j ≤ m ,the smallest equivalence relation ∼ generated by the following relation:A point v i is in the relation with an endpoint a ( a can be equal to 0 or d j ) ofthe segment I j iff N i ∩ π − j ( a ) = ∅ . Figure 13.
Construction of a graph.Glue points from V and segments from E along ∼ . Obtained space endow withthe length metric. In this way we obtain a metric graph G (Figure 13). By theconstruction of G and Theorem 4.1, HLS( F ) is isometric to G . (cid:3) Remark 5.1.
One can easily check that it is possible to construct a number ofmetric graphs, not necessarily finite, isometric to
HLS( F ) , but all of them areisometric as metric spaces wit length metric. For example, consider a foliation Figure 14.
A part of a foliation by Kronecker components and circles. of T by a infinite number of Kronecker components separated by circles foliation AUSDORFF LEAF SPACES 19 (Figure 14). Then every Kronecker component can define itself a node of a graph,and every circle foliation can define an edge. One also can select only one Kroneckercomponent to be a node, and the rest of foliation to be an edge. One can check thatany metric graph constructed this way is isometric to a circle.
Example 5.1.
Recall that any compact manifold of dimension is either an inter-val I or a -dimensional sphere S . Hence, a foliated bundle of codim-1 is either I -bundle or S -bundle. One can see that Hausdorff leaf space for a codim-1 foliatedbundle is a singleton, a metric segment or a circle S . Lemma 5.2.
For every k ∈ N there exists a compact foliated manifold ( M, F ) such that M has exactly k boundary components and HLS( F ) is a singleton, and theholonomy mappings h of the boundary leaves satisfy h (0) = 0 , h ′ (0) = 1 , h ( n ) (0) = 0 for all n ≥ .Proof. Let ˆ M = S × Σ, where Σ is a compact surface of dimension 2, and letˆ F be the product foliation by { z } × Σ, z ∈ S . Let x , . . . , x k ∈ S . Let N i ( i = 1 , . . . , k ) be disjoint tubular neighbourhoods of γ i = S × { x i } . Turbulize ˆ F simultaneously along γ i . One can check [2] that it is possible to turbulize in suchway that the holonomy mappings h of the compact leaves of the Reeb componentssatisfy h (0) = 0, h ′ (0) = 1, h ( n ) (0) = 0 for all n ≥ M be a foliated manifold obtained from ( M, F ) by removing the inte-rior of the Reeb components of the turbulized foliation. It follows that M is com-pact, and its boundary has exactly k components homeomorphic with the torus T .Moreover, every leaf different from boundary leaves accumulate on every bound-ary component. Thus HLS( F ) is a singleton, and F is a foliation with desiredproperties. (cid:3) Remark 5.2.
One can see that all leaves of the foliation constructed in Lemma5.2 are proper.
Lemma 5.3.
For any metric segment I = [0 , d ] there exists a compact foliatedRiemannian manifold ( M, F , g ) carrying codim-1 foliation such that HLS( F ) isisometric to I .Proof. Taking M = [0 , d ] × Σ, where again Σ is a compact surface, with productfoliation { t } × Σ and the product metric we get the statement. (cid:3)
Theorem 5.3.
For every finite connected metric graph G there exists a compactfoliated Riemannian manifold ( M, F , g ) such that HLS( F ) is isometric to G . More-over, every leaf of F is proper.Proof. Let G = ( V, E ) be a finite connected metric graph with k nodes. ”Cutting”every edge in the middle we obtain k connected metric graphs G i (Figure 15).Consider a graph G i . If all nodes of G i have only one edge, then assign for G i afoliated manifold indicated in Lemma 5.3.Let v be a node having more than one edge, let say m . One can assign for v a 3-dimensional foliated Riemannian manifold ( V i , F i , g i ) indicated in Lemma 5.2with exactly m boundary components homeomorphic to the torus T , and such thatHLS for V i is a singleton, and the holonomy mappings h of the boundary leavessatisfy h (0) = 0, h ′ (0) = 1, h ( n ) (0) = 0 for all n ≥ Figure 15.
Star graphs G i .Next, for every edge assign a manifold E iν = [0 , d i ] × T (as described in Lemma5.3), 1 ≤ ν ≤ m . Note that either F i or foliations of E i are tangent to the boundarycomponents.Since the holonomy mappings h of the boundary leaves satisfy h (0) = 0, h ′ (0) =1, h ( n ) (0) = 0 for all n ≥
2, then by Theorem 4.1, one can glue manifolds V i and E i to obtain a compact foliated Riemannian manifold ( M i , F i , g i ) with HLS( F i ) iso-metric to G i (Figure 16). Moreover, the boundary components of M i ( i = 1 , . . . , m ) Figure 16.
Construction of a manifold M i for the graph G i .homeomorphic to T , foliations F i on each M i are tangent to the boundary, andholonomy mappings h of boundary leaves satisfy h ′ (0) = 1, h ( n ) (0) = 0 for all n ≥ Figure 17.
The graph G and the manifold ( M, F , g ).Again, by Theorem 4.1, one can glue manifolds M i to get a compact foliatedmanifold ( M, F , g ) such that HLS( F ) is isometric to G (Figure 17).By Remark 5.2, all leaves of F are proper. This ends our proof. (cid:3) AUSDORFF LEAF SPACES 21
Warped foliations in codim-1.
Let ( M, F , g ) be an arbitrary compact fo-liated Riemannian manifold with codim-1 foliation. Let ( f n ) n ∈ N , f n : M → (0 , M f n ) n ∈ N toconverge to the Hausdorff leaf space for the foliation F .First, note that on any connected finite metric graph G with at least two nodesthere exist a measure µ constants and β ≥ η > η < η and x ∈ G (11) 1 β η ≤ µ ( B d ( x, η )) ≤ βη, where B d ( x, η ) = { y ∈ X : d ( x, y ) < η } . Indeed, denote by E = { e , . . . , e k } the set of vertices, and by V = { I , . . . , I m } the set of all edges of the graph G .Let µ be a measure induced by the Lebesgue measure on edges I j of G and let η = min l ( I j ) and β = max { , max i =1 ,...,k n ( e i ) } , where l ( I ) denotes the lengthof an edge I , and n ( e ) denotes the number of edges in a vertex e . Such µ satisfies(11).Let ( f n ) n ∈ N , f n : M → (0 , M, F , g ),where F is a foliation of codimension one. Theorem 5.4. d GH (( M, g f n ) , HLS( F )) → if and only if for every ε > thereexists N ∈ N such that for any n > N the following is satisfied:There exists a finite family of leaves F n = { F n , . . . , F nk } such that (1) S F n is ε -dense in M , (2) f n | S F n < ε . The proof of the sufficient condition is analogical to the proof of Theorem 3.1 inSection 3. The proof of the necessary condition is the same as the proof of Theorem6.5 in [7]. We don’t here repeat them and we left them for the reader.6.
Final remarks
One can ask, what is the classification of HLS for foliations of codimensiongreater than one. This question still is open. We only present some results for anarbitrary codimension.Let ( M, F , g ) be a compact connected foliated Riemannian manifold, and againlet π : M → HLS( F ) be the natural projection. One can easily check that π iscontinuous. Moreover, for any leaf L ∈ F the set π − ( π ( L )) is a closed, nonempty,saturated subset of M .Let us recall that a subset A ⊆ M is called minimal if it is nonempty, closedand saturated and there is no proper subset of A with these properties [2]. Fromthe construction of HLS( F ) it follows that for any leaf L ∈ F the set π − ( π ( L ))contains a minimal set.As a simple consequence of the above observations we have: Theorem 6.1.
If the number of minimal sets of F is countable then the HLS( F ) is a singleton.Proof. Since the number of minimal sets is countable, then HLS( F ) is a countableset. The projection π : M → HLS( F ) is continuous, hence HLS( F ) is compact andconnected. This ends our proof. (cid:3) Theorem 6.2. If F contains a compact leaf with finite holonomy then HLS( F ) contains an open subset U homeomorphic to an open set of R q , where q is a codi-mension of F .Proof. This is a direct consequence of the Reeb Stability Theorem (see [2] or [4]). (cid:3)
References [1] D. Burago, Y. Burago, S. Ivanov, A course in metric geometry, AMS 2001.[2] A. Candel, L. Conlon, Foliations I, AMS 2000.[3] M. Gromov, Metric structures for Riemannian and Non-Riemannian spaces, Birkh¨auser,Boston, 1999.[4] G. Hector, U. Hirsch, Introduction to the Geometry of Foliations. Parts A and B, F.Vieweg & Sohn, 1981, 1983.[5] T. Inaba, P. Walczak, Transverse Hausdorff dimension of codim-1 C2