Closure of singular foliations: the proof of Molino's conjecture
aa r X i v : . [ m a t h . DG ] J a n CLOSURE OF SINGULAR FOLIATIONS: THE PROOF OFMOLINO’S CONJECTURE
MARCOS M. ALEXANDRINO AND MARCO RADESCHI
Abstract.
In this paper we prove the conjecture of Molino that for everysingular Riemannian foliation ( M, F ), the partition F given by the closures ofthe leaves of F is again a singular Riemannian foliation. Introduction
Given a Riemannian manifold M , a singular Riemannian foliation F on M is,roughly speaking, a partition of M into smooth connected and locally equidistantsubmanifolds of possibly varying dimension (the leaves of F ), which is spanned bya family of smooth vector fields. The precise definition, given in Section 2, wassuggested by Molino, by combining the concepts of transnormal system of Bolton[5] and of singular foliation by Stefan and Sussmann [13].A typical example of a singular Riemannian foliation is the decomposition ofa Riemannian manifold M into the orbits of an isometric group action G on M .Such a foliation is called homogeneous . Another example of foliation is given by thepartition of an Euclidean vector bundle E → L , endowed with a metric connection,into the holonomy tubes around the zero section (cf. Example 2.7). Such a foliation,which we call holonomy foliation , will be a sort-of prototype in the structural resultsthat will appear later on. Holonomy foliations are in general not homogeneous (thezero section L is always a leaf but in general not a homogeneous manifold), howeverthey are locally homogeneous , in the sense that the infinitesimal foliation at everypoint of E is homogeneous (cf. Sections 2.3 and 2.4). This construction is relatedto other important types of foliations, like polar foliations [14] or Wilking’s dualfoliation to the Sharafutdinov projection [15], see Remark 2.8.In general, the leaves of a singular Riemannian foliation might not be closed, evenin the simple cases defined above. In the homogeneous case, consider for examplethe foliation on the flat torus T by parallel lines, of irrational slope. These arenon closed orbits, of an isometric R -action on T .Given a (regular) Riemannian foliation ( M, F ) with non-closed leaves, Molinoproved that replacing the leaves of F with their closure yields a new singular Rie-mannian foliation F . Moreover, he conjectured that the same result should holdtrue if one starts with a singular Riemannian foliation, and this has become known,in the last decades, as Molino’s Conjecture .Molino proved that the closure F of a singular Riemannian foliation ( M, F ) is atransnormal system [9], thus leaving to prove that it is a singular foliation as well. Mathematics Subject Classification.
Primary 53C12, Secondary 57R30.
Key words and phrases.
Singular Riemannian foliation, linearization, Molino’s conjecture.The first author was partially supported by FAPESP. The second author is part of the DFGproject SFB 878: Groups, Geometry & Actions.
Moreover, in [10] he suggested a strategy to prove the conjecture for the case of orbit-like foliations , i.e. foliations which, roughly speaking, are locally diffeomorphicto the orbits of some proper isometric group action around each point (cf. Section2.4). A formal alternative proof in this case can be found in [4]. Molino’s conjecturewas also proved for polar foliations and then infinitesimally polar foliations in [1]and [3], respectively.These partial results do not cover every possible foliation. Since the Eightiesthere are examples of non orbit-like foliations, and in recent years it was shown theexistence of a remarkably large class of “infinitesimal” foliations that are neitherhomogeneous nor polar, the so-called Clifford foliations [12] (these infinitesimal fo-liations have been shown, however, to have an algebraic nature, cf. [7]). Therefore,it is important to give a complete answer to the conjecture, to fully understand thesemi-local dynamic of singular Riemannian foliations.The goal of this paper is to prove the full Molino’s conjecture.
Theorem. (Molino’s Conjecture) Let ( M, F ) be a singular Riemannian foliationon a complete manifold M , and let F = { L | L ∈ F} be the partition of M into theclosures of the leaves of F . Then ( M, F ) is a singular Riemannian foliation. This result is in fact a direct consequence of the following.
Main Theorem.
Let ( M, F ) be a singular Riemannian foliation, let L be a (possi-bly not closed) leaf, and let U be an ǫ -neighbourhood around the closure of L . Thenfor ǫ small enough, there is a metric g ℓ on U and a singular foliation b F ℓ , such that:(1) ( U, g ℓ , b F ℓ ) is an orbit-like singular Riemannian foliation.(2) The foliation b F ℓ coincides with F on L .(3) The closure of b F ℓ is contained in the closure of F . In short, the foliation b F ℓ is obtained by first constructing the linearized foliation F ℓ of F in U , which is a subfoliation of F spanned by the first order approximations,around L , of the vector fields tangent to F (see Section 2.5 for a precise definition).The foliation b F ℓ is then obtained from F ℓ by taking the “local closure” of the leavesof F ℓ . The foliations F , F ℓ , b F ℓ , together with their closures, are then related bythe following inclusions: F ⊇ F ℓ ⊆ b F ℓ ⊆ ⊆ ⊆ F ⊇ F ℓ = b F ℓ Example 1.1.
Consider an Euclidean vector bundle E over a complete Riemannianmanifold L , with a metric connection ∇ E and a connection metric g E (cf. Example2.7). Let H p denote the holonomy group of ( E, ∇ E ) at p , acting by isometries on theEuclidean fiber E p , and let ( E p , F p ) be a singular Riemannian foliation preservedby the H p -action. Finally, let K p be the maximal connected group of isometries of E p that fixes each leaf of F p as a set.Letting F the partition of E into the holonomy translates of the leaves of F p (i.e., for every leaf L ∈ F p , L L denotes the set of points in E that can be reachedvia ∇ E -parallel translation from a point in L ), then F is a singular Riemannianfoliation. In this case, the linearized foliation F ℓ is the foliation by the holonomy translates of the K p -orbits in E p , and the local closure of b F ℓ is the foliation by theholonomy translates of the K p -orbits in E p , where K p denotes the closure of K p in O ( E p ).This can be restated in the language of groupoids: defining H the holonomygroupoid of the connection ∇ E , then F = { H ( L ) | L ∈ F p } , F ℓ is given by theorbits of HK p , and its local closure ˆ F ℓ is given by the orbits of HK p .This paper is organized as follows: after a section of preliminaries (Section 2) weshow how the Molino’s Conjecture follows from the Main Theorem (Section 3). InSection 4 we fix the setup in which we work for the rest of the paper. In Section 5we define three distributions of the tangent bundle T U . We first use these to obtaininformation on the local structure of F ℓ and define the local closure b F ℓ (Section6) and then to define the metric g ℓ used in the Main Theorem (Section 7). In thisfinal section we also prove the Main Theorem. Acknowledgements
The authors thank Prof. Lytchak and Prof. Thorbergsson for consistent support.2.
Preliminaries
Given a Riemannian manifold ( M, g), a partition F of M into complete connectedsubmanifolds (the leaves of F ) is called a transnormal system if geodesics startingperpendicular to a leaf stay perpendicular to all leaves, and a singular foliation ifevery vector tangent to a leaf can be locally extended to a vector field everywheretangent to the leaves. A singular Riemannian foliation will be denoted by the triple( M, g , F ). However, if the Riemannian metric of M is understood, we will drop itand simply write ( M, F ).The following notation will be used throughout the rest of the paper. Given apoint p ∈ M , the leaf of F through p will be denoted by L p . A small relativelycompact open subset P ⊂ L is called a plaque . The tangent and normal spacesto L p at p are denoted by T p L p and ν p L p , respectively. Given some ǫ > ν ǫp L p denotes the set of vectors x ∈ ν p L p with norm < ǫ . If ǫ is small enough that thenormal exponential map exp : ν ǫp L p → M is a diffeomorphism onto the image,such image is called a slice of L p at p , and it is denoted by S p . The slice foliation F| S p denotes the partition of S p into the connected components of the intersections L ∩ S p , where L ∈ F .2.1. Vector fields of a singular Riemannian foliation.
We review here themain notations about vector fields of a singular Riemannian foliation.A vector field V is called vertical if it is tangent to the leaves at each point. Theset of smooth vertical vector fields is a Lie algebra, which is denoted by X ( M, F ).A vector field X is called foliated if its flow takes leaves to leaves or, equivalently,if [ X, V ] ∈ X ( M, F ) for every V ∈ X ( M, F ). Any vertical vector field is foliated,but there are other foliated vector fields. A vector field is called basic if it is bothfoliated and everywhere normal to the leaves.2.2. Homothetic Transformation Lemma.
One of the most fundamental re-sults in the theory of singular Riemannian foliations is the Homothetic Transfor-mation Lemma. A deeper discussion of this lemma, with proof and applications,can be found in Molino [9], Ch. 6, in particular Lemma 6.1 and Proposition 6.7.
MARCOS M. ALEXANDRINO AND MARCO RADESCHI
Let ( M, F ) be a singular foliation, let L be a leaf of F , and let P ⊂ L a plaque.Let ǫ > ν ǫ P → M is a diffeomor-phism onto its image B ǫ ( P ). For any two radii r , r = λr in (0 , ǫ ), it makes senseto define the homothetic transformation h λ : B r ( P ) → B r ( P ) , h λ (exp v ) = exp λv. The leaves of F intersect B r i ( P ), i = 1 ,
2, in plaques that foliate B r i ( P ). We call F| B ri ( P ) the foliation of B r i ( P ) into the path components of such intersections.One has then the following: Theorem 2.1 (Homothetic Transformation Lemma) . The homothetic transforma-tion h λ takes the leaves of ( B r , F| B r ( P ) ) onto the leaves of ( B r , F| B r ( P ) ) . This result still holds, more generally, if we replace the plaque P by an opensubset B of some submanifold N ⊂ M which is a union of leaves of the samedimension. In this case we consider some ǫ > ν ǫ B → M is adiffeomorphism onto the image B ǫ ( B ), and define the homothetic transformation around W , h λ : B r ( B ) → B λr ( B ), as before. In this case, an analogous version ofthe Homothetic Transformation Lemma applies.2.3. Infinitesimal foliation.
Let ( M, F ) be a singular Riemannian foliation, p ∈ M a point, and S p a slice at p . Definition 2.2 (Infinitesimal foliation at p ) . The infinitesimal foliation of F at p ,denoted by ( ν p L p , F p ) is defined as the partition of ν p L p whose leaf at v ∈ ν p L p isgiven by L v = { w ∈ ν p L p | exp p tw ∈ L exp p tv ∀ t > } , where L exp p tv denotes the leaf of ( S p , F| S p ) through exp p tv .The leaf L v is well defined because, by the Homothetic Transformation Lemma,if exp p t w belongs to the same leaf of exp p t v for some small t , then exp p tw belongs to the same leaf of exp p tv for every t ∈ (0 , t ). In the following propositionwe collect the important facts about infinitesimal foliations that we will need. Theorem 2.3.
Given a singular Riemannian foliation ( M, F ) and a point p ∈ M with infinitesimal foliation ( ν p L p , F p ) , then:(1) The foliation ( ν p L p , F p ) is a singular Riemannian foliation with respect tothe flat metric g p at p .(2) The normal exponential map exp p : ν ǫp L p → M sends the leaves of F p tothe leaves of ( S p , F| S p ) .(3) ( ν p L p , F p ) is invariant under rescalings r λ : ν p L p → ν p L p , r λ ( v ) = λv .Proof.
1) [9], Prop. 6.5.2) Follows from the definition of infinitesimal foliation, and of slice foliation.3) Via the exponential map exp : ν ǫ L p → S p , this corresponds to the HomotheticTransformation Lemma on S p . (cid:3) The following fact will come very useful.
Proposition 2.4.
Given singular Riemannian foliations ( M, F ) , ( M ′ , F ′ ) and afoliated diffeomorphism φ : U → U ′ , between open sets U, U ′ of M, M ′ respectively,sending a point p ∈ U to p ′ ∈ U ′ , the differential of φ induces a linear, foliatedisomorphism φ ∗ : ( ν p L p , F p ) → ( ν p ′ L p ′ , F ′ p ′ ) . Proof.
By substituting ( M ′ , g ′ , F ′ ) with ( M, φ ∗ g ′ , F ), the problem can be reducedto the case where M = M ′ , φ = id , p = p ′ , and F = F ′ is a singular Riemannianfoliation with respect to two metrics, g and ˜g. In the following, we will denote witha “tilde” (˜) every geometric object related to the metric ˜g, and without the tildeany geometric object related to g.Let S p (resp. e S p ) denote a slice at p with respect to g (resp. ˜g). Consider the set { X , . . . X k } ⊂ X ( M, F ), k = dim L p , of vector fields such that { X ( p ) , . . . , X k ( p ) } is a basis of T p L p . Denote by Φ ti the flow of X i , and define Φ ( t ,...t k ) = Φ t k k ◦ . . . ◦ Φ t .Around p , both S p and e S p are transverse to span( X , . . . X k ) and, up to possiblyreplacing S p and e S p with smaller open subsets, we can assume that for every q ∈ S p there exists a unique ˜ q ∈ e S p of the form ˜ q = Φ ( t ,...t k ) ( q ). This gives rise to a map H : S p → e S p , H ( q ) = ˜ q which is differentiable and, since q and ˜ q belong to thesame leaf of F , sends the leaves of F| S p to the leaves of F| e S p . In other words, thereis a foliated diffeomorphism H : ( S p , F| S p ) → ( e S p , F| e S p ).Consider the composition ψ of foliated diffeomorphisms( ν ǫp L p , F p ) exp p −→ ( S p , F| S p ) H −→ ( e S p , F| e S p ) ˜exp − p −→ (˜ ν ǫp L p , e F p ) . For any λ ∈ (0 , ψ λ : ( ν ǫ/λp L p , F p ) → (˜ ν ǫ/λp L p , e F p ) , ψ λ ( v ) = 1 λ ψ ( λv ) . As λ →
0, the maps ψ λ converge to the differential d ψ of ψ at 0. This is aninvertible linear map (in particular a diffeomorphism) and, as a limit of foliatedmaps, it is itself foliated. Therefore, the map φ ∗ := d ψ : ( ν p L, F p ) −→ (˜ ν p L, e F p )satisfies the statement of the proposition. (cid:3) Remark . Given a singular Riemannian foliation ( M, F ) and a submanifold N ⊂ M which is a union of leaves of the same dimension, the infinitesimal foliationat a point p ∈ M splits as a product ( ν p ( L p , N ) × ν p N, { pts. } × F p | ν p N ), where ν p ( L p , N ) = ν p L p ∩ T p N . In this case, the foliation ( ν p N, F p | ν p N ) is the “essentialpart” of the infinitesimal foliation ( ν p L p , F p ). By abuse of notation, we will callthe foliation ( ν p N, F p | ν p N ) infinitesimal foliation at p as well, and denote it by F p .Given a singular Riemannian foliation ( M, F ) and a point p ∈ M , the infinitesi-mal foliation ( ν p L p , F p ) at p contains the origin as a leaf of F p . Based on this fact,we make the following definition. Definition 2.6 (Infinitesimal foliation) . An infinitesimal foliation is a singularRiemannian foliation ( V, F ) on an Euclidean vector space, with the origin { } being a 0-dimensional leaf.2.4. Homogeneous and orbit like foliations.
A singular Riemannian foliation( M, F ) is called homogeneous (sometimes Riemannian homogeneous ) if there existsa connected Lie group G acting by isometries on M , whose orbits are precisely theleaves of F . Furthermore, a singular Riemannian foliation ( M, F ) is called orbit-like if at every point p ∈ M , the infinitesimal foliation ( ν p L p , F p ) is closed andhomogeneous. MARCOS M. ALEXANDRINO AND MARCO RADESCHI
Example 2.7 (Holonomy foliations) . An example of orbit like foliation, whichwill be useful to keep in mind later on, can be constructed as follows. Consider aRiemannian manifold L , and an Euclidean vector bundle E over L , that is, a vectorbundle over L with an inner product h , i p on each fiber E p , p ∈ L . Let ∇ E be ametric connection on E , i.e. a connection on E such that, for every vector field X on L and sections ξ, η of E , one has X h ξ, η i = h∇ EX ξ, η i + h ξ, ∇ EX η i . Given ( E, ∇ E ), there is an induced Riemannian metric g E on E , called connectionmetric . Moreover, ∇ E induces a parallel transport on E : given ξ ∈ E p and a curve γ : [0 , → L with γ (0) = p , there exists a unique lift ξ ( t ), t ∈ [0 ,
1] with ξ (0) = ξ such that ∇ Eγ ′ ( t ) ξ ( t ) = 0 for every t ∈ [0 , E one can now define a foliation F E , by declaring two vectors ξ, η ∈ E in the same leaf if they can be connected toone another via a composition of parallel transports. The leaves of F E are usuallyreferred to as the holonomy tubes around the zero section L ⊂ E , and they definea singular Riemannian foliation on ( E, g E ). Moreover, the infinitesimal foliationat any point of E is homogeneous: in fact, for any point p along the zero section L , one can first construct the holonomy group H p of the connection ∇ E , whichacts by isometries on the fiber E p and whose orbits are precisely the leaves of theinfinitesimal foliation of F E at p . Similarly, the infinitesimal foliation at a point ξ ∈ E p is given by the orbits in ν ξ L ξ of the stabilizer H ξ ⊂ H p of ξ . The foliation F E coincides with its own linearization with respect to the zero section (see definitionin Section 2.5). Moreover, if the leaves of F E are closed then ( E, g E , F E ) is anorbit-like foliation. Remark . When L ⊂ M is a submanifold of somewhat special geometry, theholonomy foliation on the normal bundle E of L , endowed with the Levi-Civitaconnection, induces via the normal exponential map a foliation on the whole of M . For example, if L has parallel focal structure , then the induced foliation on M is a polar foliation [14]. If M is a complete, non-compact manifold with sectionalcurvature ≥ L is a soul of M [6], then the induced foliation on M is Wilking’s dual foliation to the Sharafutdinov projection [15].Although in principle the property of being orbit-like might depend on the metric,the following proposition shows in fact that being orbit like is invariant underfoliated diffeomorphisms. Proposition 2.9.
The following hold:(1) Given a foliated linear isomorphism ϕ : ( V, F ) → ( V ′ , F ′ ) between infini-tesimal foliations, ( V, F ) is homogeneous if and only if ( V ′ , F ′ ) is homoge-neous.(2) Given a foliated diffeomorphism φ : ( M, F ) → ( M ′ , F ′ ) between singularRiemannian foliations, ( M, F ) is orbit-like if and only if ( M ′ , F ′ ) is orbit-like.Proof.
1) By the symmetric roles of V and V ′ , it is enough to show that if ( V, F ) ishomogeneous, so is ( V ′ , F ′ ). Suppose that ( V, F ) is homogeneous, and therefore thefoliation F is spanned by Killing fields. Recall that a vector field X on an Euclideanspace ( V, g) is Killing if and only if is of the form X ( v ) = Av , where A is a skewsymmetric endomorphism of V , in the sense that g( Av, v ) = 0 for every v ∈ V .Letting { X , . . . X k } denote a set of Killing fields on V spanning the foliation F , the set { Y , . . . Y k } with Y i ( v ′ ) = ϕ ∗ (cid:0) X i (cid:0) ϕ − ( v ′ ) (cid:1)(cid:1) spans the foliation F ′ as well.Since ϕ is a linear map and the vector fields X i are linear, it follows that Y i canbe written as Y i ( v ′ ) = B i v ′ for some endomorphism B i : V ′ → V ′ , i = 1 . . . k .Since ( V ′ , g ′ , F ′ ) is a singular Riemannian foliation, the leaf L v ′ through v ′ lies ina distance sphere from the origin, and in particular g ′ ( T v ′ L v ′ , v ′ ) = 0. Since Y i ( v ′ )is tangent to L v ′ , it follows that0 = g ′ ( Y i ( v ′ ) , v ′ ) = g( B i v ′ , v ′ )In other words, B i is skew-symmetric and thus Y i is a Killing field as well. Thereforethe foliation ( V ′ , F ′ ) is spanned by Killing vector fields, hence it is homogeneousas well.2) Up to exchanging the roles of M and M ′ , it is enough to show that if ( M, F )is orbit-like, so is ( M ′ , F ′ ). Fixing a point p ∈ M , Proposition 2.4 states that thefoliated diffeomorphism φ induces a foliated linear isomorphism φ ∗ : ( ν p L p , F p ) −→ ( ν p ′ L p ′ , F p ′ ), where p ′ = φ ( p ). Since ( M, F ) is orbit-like, it follows that ( ν p L p , F p )is closed and homogeneous. From the first point above it follows that ( ν p ′ L p ′ , F p ′ )is homogeneous as well, and by the continuity of φ ∗ one has that ( ν p ′ L p ′ , F p ′ ) isclosed. Since p ′ was chosen arbitrarily, it follows that ( M ′ , F ′ ) is orbit-like. (cid:3) Linearization, and linearized foliation.
Let ( M, F ) be a singular Rie-mannian foliation, B ⊂ M a submanifold saturated by leaves, and U ∈ M an ǫ -tubular neighbourhood of B with metric projection p : U → B . Given a vectorfield V in U tangent to the leaves of F , it is possible to produce a new vector field V ℓ , called the linearization of V with respect to B , as follows: V ℓ = lim λ → ( h − λ ) ∗ ( V | h λ ( U ) )where h λ : U → U denotes the homothetic transformation around B . From [8],Prop. 5, the linearization V ℓ is a smooth vector field invariant under the homothetictransformation h λ , and it coincides with V along B .On U , consider the module X ( U, F ) ℓ given by the linearization, with respect to B , of the vector fields in X ( U, F ): X ( U, F ) ℓ = { V ℓ | V ∈ X ( U, F ) } . Let D the pseudogroup of local diffeomorphisms of U , generated by the flows oflinearized vector fields, and let ( U, F ℓ ) the partition of U into the orbits of dif-feomorphisms in D . By Sussmann [13, Thm. 4.1], such orbits are (possibly non-complete) smooth submanifolds of M . Moreover, as noted By Molino [9, Lem. 6.3],this foliation is spanned, at each point, by the vector fields in X ℓ ( U, F ).We call ( U, F ℓ ) the linearized foliation of F with respect to B . We will show,later, that the leaves of the linearized foliation are actually complete, and have aparticularly nice local structure (cf. Section 6).Given a point p ∈ B , define U p = p − ( p ) ⊂ U and let F p (resp. ( F ℓ ) p ) denote thepartition of U p into the connected components of L ∩ U p , as L ranges through theleaves of F (resp. F ℓ ). If U p is given the flat metric g p of ν p B via the exponentialmap exp p : ν ǫp B → U p , then F p corresponds to the infinitesimal foliation at p (cf.Remark 2.5) which justifies the notation of F p for this foliation. Furthermore, asnoted in [9, Sec. 6.4], ( F ℓ ) p is given by the linearization of ( U p , g p , F p ) with respectto the origin. In other words, ( F ℓ ) p = ( F p ) ℓ and it makes sense to denote this MARCOS M. ALEXANDRINO AND MARCO RADESCHI foliation simply by F ℓp . Moreover, letting O ( F p ) denote the Lie group of (linear)isometries of ( U p , g p ) sending every leaf to itself, one has: Proposition 2.10.
The foliation ( U p , F ℓp ) is homogeneous, given by the orbits ofthe identity component H p of O ( F p ) .Proof. We identify here U p with a neighbourhood of the origin in ν p B via theexponential map, and we think of F ℓp as the linearization of F p .Given a vector field V ∈ X ( U p , F p ), its linearization V ℓ is linear, in the sensethat V ℓp = A · p for some A ∈ End( U p ). Since F p is a singular Riemannian foliation,the leaves are tangent to the distance spheres around the origin and thereforeperpendicular to the radial directions from the origin: h V ℓp , p i = 0. In other words, V ℓp = A · p with A skew symmetric, which implies that the flow of V ℓ is an isometryof U p . Moreover, since V ℓ is everywhere tangent to the leaves of F ℓp , the flow of V ℓ is a 1-parameter group in H p , moving every leaf of ( F p ) ℓ to itself. In particular,the orbits of H p are contained in the leaves of ( F p ) ℓ .However, by definition of H p , the tangent space of a H p -orbit through a point q ∈ U p is given by T q ( H p · q ) = { W q | W Killing vector field tangent to the leaves of F p } and such vector fields coincide precisely with the vector fields in X ( U p , F p ) ℓ . There-fore, H p · q is the integral manifold of X ( U p , F p ) ℓ through q . (cid:3) Molino’s conjecture, assuming the Main Theorem
Before proving the Main Theorem, we show how Molino’s Conjecture followsfrom it as a corollary.
Proof of Molino’s Conjecture.
Let ( M, F ) be a singular Riemannian foliation, andlet F denote the closure of F . Molino himself proved that F is a partition into com-plete smooth closed submanifolds, and that F is a transnormal system. Therefore,in order to prove the conjecture, it is enough to show that for any leaf L ∈ F withclosure L and any vector v ∈ ν ( L, L ) := νL ∩ T L , there exists a smooth extensionof v to a vector field V everywhere tangent to the leaves of F .Let U be a tubular neighbourhood of L , and let ( U, b F ℓ ) be the foliation satisfyingthe Main Theorem. Since b F ℓ coincides with F along L , it follows that L is a leafof b F ℓ as well. Since b F ℓ is an orbit-like foliation, by Theorem 1.6 of [4], given v ∈ ν ( L, L ) there is a vector field V extending v which is tangent to the closure of b F ℓ . Since this closure is contained in F , it follows that V is also tangent to F andthis ends the proof of the conjecture. (cid:3) The setup
Fix a leaf L , and distance tube U = B ǫ ( L ) around L . Using the normal expo-nential map exp : νL → M , U can be identified with the ǫ -tube ν ǫ L around thezero section. By the Homothetic Transformation Lemma, the pull-back foliationexp − F on ν ǫ L is invariant under the rescalings r λ : ν ǫ L → ν ǫ L , r λ ( p, v ) = ( p, λv )for any λ ∈ (0 , • U is the ǫ -tube around the zero section of some Euclidean vector bundle E → B (in our case B = L ), with projection p : U → B . • g is a Riemannian metric on U with the same radial function as the Eu-clidean metric on each fiber of E . • ( U, g , F ) is a singular Riemannian foliation on U , invariant under rescal-ings r λ . In particular, the zero-section B is saturated by leaves and theprojection p sends leaves onto leaves. • The restriction F B = F| B is a regular Riemannian foliation. • For every leaf L ⊆ B and any point p ∈ L , the normal exponential map ν ǫp L → U is an embedding.5. Three distributions
Let ( U, g , F ), p : U → B be as in Section 4. In order to prove the Main Theorem,it is first needed to produce a nicer metric on U , and for this we first need to split thetangent space of U into three components. The first, K = ker p ∗ , is the distributiontangent to the fibers of p . For the remaining two notice that, since the foliation( B, F B ) is regular, the tangent bundle T B splits into a tangent and a normal partto the foliation:
T B = T F| B ⊕ ν F| B . The last two distributions will be constructedas (appropriately chosen) extensions T and N of T F| B and ν F| B respectively, tothe whole of U . BL T NK b p b q Figure 1.
The distributions at q .5.1. The distribution T . From [2] there exists a distribution b T of rank dim F| B ,which extends T F| B and is everywhere tangent to the leaves of F .The distribution T is simply defined as the linearization of b T with respect to B , as follows: consider a family of vector fields { V α } α spanning b T . Since b T | B istangent to B , the vector fields V α lie tangent to B as well and therefore it makessense to consider their linearization V ℓα with respect to B . By the properties of thelinearization, these linearized vector fields still span a smooth distribution of thesame rank as b T , which we call T . The distribution N . At each point q ∈ U with p ( q ) = p , the slice S p =exp p ( ν ǫp L p ) contains q as well as the whole p -fiber U p through p . In particular, K q lies tangent to S p . Moreover, S p comes equipped with a flat metric g p , inheritedfrom the metric on ν p L via the diffeomorphism exp p : ν ǫ L p → S p .Define b N q as the subspace of T q S p which is g p -orthogonal to K q . Finally, define N as the linearization of b N , as defined in the previous section.The distributions b N and N satisfy the following property: Proposition 5.1.
For every smooth F B -basic vector field X along a plaque P in B there exists a smooth extension X to an open set of U such that(1) X is foliated and tangent to b N .(2) The linearization X ℓ of X with respect to B is tangent to N , and it isfoliated with respect to both F and F ℓ .Proof.
1) Fix a leaf L in B , a plaque P ⊂ L and a parametrization ϕ : ( − , k → P ⊂ L, where k = dim F B . We first show that there exists a small neighbourhood of P in U , on which any F B -basic vector field X along P can be extended to a foliatedvector field X ′ , whose restriction to p − ( P ) is tangent to b N .Let ∂ y , . . . , ∂ y k be coordinate vector fields on P , and let Y , . . . Y k denote vectorfields, linearized with respect to L , that extend ∂ y , . . . ∂ y k to a neighbourhood of P in U . There is a foliated diffeomorphism F : ( P × S p , P × F S p ) −→ ( U, F )( ϕ ( y , . . . y k ) , q ) Φ y k k ◦ . . . ◦ Φ y ( q )where Φ y i i is the flow of Y i , after time y i .Furthermore, the foliation ( P × S p , P × F S p ) locally splits as( P × ν p ( L, B ) × ν p B, P × { pts. } × F| ν p B )where ν ( L, B ) = νL ∩ T B . Moreover, if S p is endowed with the Euclidean metricg p on ν p L , the splitting S p = ν p ( L, B ) × ν p B is in fact Riemannian.The map F satisfies the following: • The set P × { } × ν ǫp B is sent to p − ( P ) = ν ǫ B | P . • The set P × ν ǫp ( L, B ) × { } is sent to a neighbourhood of P in B . • Since F is defined via linearized vector fields, each fiber { p ′ } × S p ⊂ P × S p is sent, via F , to the slice S p ′ , isometrically with respect to the flat metricson S p and S p ′ (cf. [8]).From the last point, it follows that the distribution of P × ν p ( L, B ) × ν p B tangentto the second factor is sent, along ν ǫ B | P , precisely to the distribution b N .Any F B -basic vector field X along P corresponds, via F , to a vector field along P × { } × { } of the form (0 , x ,
0) where x ∈ ν p ( L, B ) is a fixed vector. One canclearly extend such a vector field to the foliated vector field X ′ = F ∗ (0 , x , F is a foliated map, the vector field X ′ is a foliated vector field, whose restrictionto B is tangent to B by the second point above. Moreover, by the discussion abovethe restriction of X ′ to p − ( P ) is tangent to b N .This proves the first claim, made at the beginning of the proof. In particular,since the plaque P was chosen arbitrarily, this shows that the distribution b N is foliated : that is, given a vector y tangent to b N at a point q , there exists a foliated extension Y along a plaque containing q which is everywhere tangent to b N . It iseasy to see that K and T are foliated as well. In particular, given the foliated vectorfield X ′ , the (unique) decomposition X ′ = X ′ K + X ′ T + X ′ c N , X ′ K ∈ K , X ′ T ∈ T , X ′ c N ∈ b N produces three vector fields X ′ K , X ′ T , X ′ c N which are foliated. In particular, thevector field X = ( X ′ ) b N is foliated, everywhere tangent to b N , and it extends X =( X ) c N to an open set of U , as we needed to show.2) Since X is tangent to b N , its linearization X ℓ is tangent to the linearizationof b N , which is N . Moreover, since X is foliated and r λ : U → U is a foliated map, X ℓ = lim λ → ( r − λ ) ∗ X ◦ r λ is foliated as well. Finally, since X is foliated, for everyvector field V tangent to F one has that [ X, V ] is also tangent to F . Since r λ is adiffeomorphism, one computes[ X ℓ , V ℓ ] = lim λ → [( r − λ ) ∗ X ◦ r λ , ( r − λ ) ∗ V ◦ r λ ]= lim λ → ( r − λ ) ∗ [ X, V ] ◦ r λ = [ X, V ] ℓ . Since the linearization V ℓ are precisely the vector fields generating F ℓ , it followsfrom the equation above that [ X ℓ , V ℓ ] is tangent to F ℓ whenever V ℓ is, and therefore X ℓ is foliated with respect to F ℓ . (cid:3) Structure of F ℓ , and the local closure b F ℓ Using the extensions X ℓ defined in Proposition 5.1, one can prove the following: Proposition 6.1.
Around any point p ∈ B there is a neighbourhood W of p in B such that ( p − ( W ) , F ℓ | p − ( W ) ) is foliated diffeomorphic to a product ( D k × D m − k × U p , D k × { pts. } × F ℓp ) where k = dim F| B and m = dim B .Proof. Let W be a coordinate neighbourhood of B around p , with a foliated dif-feomorphism ϕ : ( W, F| W ) → ( D k × D m − k , D k × { pts. } ). Let ∂∂y , . . . ∂∂y k denote abasis of vector fields in W tangent to the leaves of F| W , and let V , . . . V k denotevector fields on p − ( W ), linearized with respect to B , extending ∂∂y i , i = 1 , . . . k and spanning the foliation T . Similarly, let ∂∂x , . . . ∂∂x m − k denote a basis of basicvector fields in W normal to the leaves, and let X ℓ , . . . X ℓm − k denote linearizedvector fields in π − ( W ) defined as in Proposition 5.1, extending the vectors ∂∂x i , i = 1 , . . . m − k . Finally, define Φ ti and Ψ ti the flows of V i and X ℓi respectively, aftertime t , and let G : D k × D m − k × U p −→ p − ( W )(( t , . . . , t k ) , ( s , . . . , s m − k ) , q ) Φ t k k ◦ . . . ◦ Φ t ◦ Ψ s m − k m − k ◦ . . . ◦ Ψ s ( q ) . Since the V i and X ℓi are linearized, they take fibers of D k × D m − k × U p → D k × D m − k to fibers of p : p − ( W ) → W . Since the flows Ψ i send the leaves of F ℓ to leaves, andthe flows Φ i take leaves of F ℓ to themselves, the leaves of D k × ( D m − k , { pts. } ) × ( U p , F ℓp ) are sent into the leaves of F ℓ . Since the differential dG is invertible at (0 , , p ) ∈ D k × D m − k × U p , it is a diffeomorphism around G (0 , , p ) = p and, bydimensional reasons, the leaves of ( D k × D m − k × U p , D k × { pts. } × F ℓp ) are mappeddiffeomorphically onto the leaves of ( p − ( W ) , F ℓ | p − ( W ) ). (cid:3) The local closure of F ℓ . Even though ( U p , F ℓp ) is homogeneous for every p ∈ B ,it might be the case that its leaves are not closed, which happens when the group H p ⊆ O ( U p ) defined in Proposition 2.10 is not closed. To obviate this problem wedefine a new foliation b F ℓ , called the local closure of F ℓ , such that F ℓ ⊂ b F ℓ ⊂ F ℓ and whose restriction b F ℓp to each p -fiber U p is homogeneous and closed.Recall that F ℓ is defined by the orbits of the pseudogroup D of local diffeomor-phisms, generated by the flows of linearized vector fields. For each q ∈ U p , considerthe closure H p of H p in O ( U p ), and define the b F ℓ -leaf b L q through q to be the D -orbit of H p · q : b L q = { q ′ = Φ( h · q ) | Φ ∈ D , h ∈ H p } Let ∼ denote the relation q ∼ q ′ if and only if q ′ = Φ( h · q ) for some Φ ∈ D and h ∈ H p . In this way, the leaf of b F ℓ through q can be rewritten as { q ′ ∈ U | q ′ ∼ q } .As for the other foliations, for every p ∈ B we define ( U p , b F ℓp ) to be the partition of U p into the connected components, of the intersections of U p with the leaves in b F ℓ . Proposition 6.2.
The following hold:(1) b F ℓ is a well defined partition of U .(2) For every p ∈ B the leaves of b F ℓp are the orbits of H p on U p .Proof.
1. One must prove that the relation ∼ defined above is an equivalencerelation. For this, notice that, since any Φ ∈ D defines a foliated isometry between( U p , F ℓp ) and ( U Φ( p ) , F ℓ Φ( p ) ) for any p ∈ B , in particular it defines a foliated isometrybetween the respective closures ( U p , H p ) and ( U Φ( p ) , H Φ( p ) ). In particular, for any h ∈ H p and Φ ∈ D , one has h ′ = Φ ◦ h ◦ Φ − ∈ H Φ( p ) .- Reflexivity of ∼ : if q ′ ∼ q then q ′ = Φ( h ( q )) for some h ∈ H p and Φ ∈ D . Then q ′ = h ′ (Φ( q )), where h ′ = Φ ◦ h ◦ Φ − ∈ H Φ( p ) , and therefore q = Φ − (( h ′ ) − q ′ ),that means q ∼ q ′ .- Transitivity of ∼ : if q ′ ∼ q and q ′′ ∼ q ′ then q ′ = Φ( h ( q )) and q ′′ = Ψ( g ( q ′ ))for some h ∈ H p , g ∈ H Φ( p ) , and Φ , Ψ ∈ D . Then q ′′ = (Ψ ◦ Φ)(( g ′ ◦ h )( q )), where g ′ = Φ − ◦ g ◦ Φ ∈ H p , and therefore q ′′ ∼ q .2. Let L ′ denote a leaf of b F ℓ . From (1), the intersection of L ′ with U p is aunion of orbits of H p . On the other hand, we claim that the intersection L ′ ∩ U p consists of countably many orbits of H p , so that each connected component of suchintersection must consists of a single H p -orbit. From the definition of b F ℓ , it isenough to prove that the subgroup D p ⊂ D of diffeomorphisms fixing p moves every H p -orbit in L ′ ∩ U p to at most countably many orbits. For this, consider a piecewisesmooth loop γ : [0 , → L p with γ (0) = γ (1) = p . Using linearized vector fieldswith γ as integral curve, one can construct a continuous path Φ t : [0 , → D ofdiffeomorphisms such that Φ = id U and Φ t ( p ) = γ ( t ), as described in [8, Cor. 7].Fixing some H p -orbit O in L ′ ∩ U p , its image Φ ( O ) is again some H p -orbit, whichonly depends on the class [ γ ] ∈ π ( L p , p ) and not on the actual path γ , nor on the specific choice of Φ t . This gives a map ∂ : π ( L p , p ) → { H p -orbits in L ′ ∩ U p } This map admits a section, namely: for every orbit O ′ in L ′ ∩ U p , take a path γ in L ′ from a point in a (fixed) orbit O to a point in O ′ . Under the projection p : U → B , the composition p ◦ γ is a loop in L p . The section of ∂ sends O ′ to[ p ◦ γ ] ∈ π ( L p , p ). In particular, the map ∂ is surjective, and therefore the setof H p -orbits in L ′ ∩ U p has at most the cardinality of π ( L p , p ), which is at mostcountable since L p is a manifold. (cid:3) As a corollary of Propositions 6.2 and 6.1, one gets the following:
Corollary 6.3.
Let ( U, F ) be a singular Riemannian foliation as in Section 4, let F ℓ be its linearized foliation and b F ℓ the local closure. Then b F ℓ is a singular foliationwith complete leaves. Moreover, around each point p ∈ B there is a neighbourhood W of p in B such that ( p − ( W ) , b F ℓ | p − ( W ) ) is foliated diffeomorphic to a product ( D k × D m − k × U p , D k × { pts. } × { orbits of H p ⊆ O ( U p ) } ) , which can be given the structure of a singular Riemannian foliation. Once it is shown that b F ℓ is also a transnormal system with respect to somemetric, then by the corollary above it is globally a singular Riemannian foliation.7. A new metric
Let T , N , K be the distributions as in the previous section. Clearly, one has T U = T ⊕ N ⊕ K .Define now the new metric g ℓ on U , as the metric defined by the followingproperties: • T ⊕ N and K are orthogonal with respect to g ℓ . • g ℓ | T ⊕N = p ∗ g B , where g B denotes the restriction of the original metric on B . In particular, T and N are also orthogonal to one another. • For any q ∈ U p , recall that K q = T q U p , and define g ℓ | K q = g p the flat metricon U p induced from exp p : ν p B → U p .These conditions characterize the metric g ℓ uniquely. The most useful propertyof this metric is the following. Proposition 7.1.
The triples ( U, g ℓ , F ℓ ) and ( U, g ℓ , b F ℓ ) are singular Riemannianfoliations.Proof. The arguments for F ℓ and b F ℓ are ideantical, therefore we will only check theProposition for ( U, g ℓ , b F ℓ ) (which is the only case we need for the Main Theoremanyway). Moreover, the statement is local in nature, therefore it is enough to provethe statement on certain open sets covering the whole of U . For any point p ∈ B ,let W denote a neighbourhood of p in B and p − ( W ) a neighbourhood of p in U . We need to check that ( p − ( W ) , g ℓ , b F ℓ ) is a singular Riemannian foliation. Toprove this, we apply Proposition 2.14 of [2] which states that it is enough to checktwo conditions:(1) ( p − ( W ) , g ′ , b F ℓ ) is a singular Riemannian foliation with respect to some Riemannian metric g ′ . (2) For every stratum Σ ⊂ p − ( W ) (i.e. union of leaves of the same dimension),the restriction of b F ℓ to Σ is a (regular) Riemannian foliation.The first condition is satisfied by Corollary 6.3. The second condition is equiva-lent to checking that, for every leaf b L q of b F ℓ | p − ( W ) and every basic vector field X along b L q tangent to the stratum through b L q , the norm k X k g ℓ is constant along b L q .By definition of the metric g ℓ , the space ν b L q is given by N | b L q ⊕ ( ν b L q ∩ K ). FromProposition 5.1, along b L q the space N is spanned by linearized vector fields X ℓi ,which are then g ℓ -basic (i.e., foliated and g ℓ -orthogonal to the leaves). In particular,any basic vector field X along b L q splits as a sum X = X + X , where X is tangentto N , X is tangent to N ′ := ν b L q ∩ K , and g ℓ ( X , X ) = 0. Therefore, it is enoughto check independently that for every basic vector field X along b L q , tangent toeither N or N ′ , the norm of X is constant along b L q .If X is tangent to N , then by the construction in Proposition 5.1 it projectsto some basic vector field X along b L p ⊂ B . Since ( B, g B , F| B ) is a Riemannianfoliation, the norm k X k g B is constant along b L p . By the construction of the metricg ℓ , one has k X k g ℓ = k X k g B and, therefore, the norm of X is constant along b L q .If X is tangent to N ′ , then it is tangent to any fiber U p ′ , p ′ ∈ b L p . The restriction X | U p ′ is a basic vector field of ( U p ′ , b F ℓp ′ ) along b L q ∩ U p ′ , and therefore the norm k X | U p ′ k g p ′ is locally constant along b L q ∩ U p ′ . By the construction of g ℓ , it followsthat k X | U p ′ k g ℓ is also locally constant along each b L q ∩ U p ′ . However, given twopoints p ′ , p ′′ ∈ b L p , and a vertical, foliated vector field V ℓ whose flow Φ moves p ′ to p ′′ , one also has that Φ moves U p ′ isometrically to U p ′′ , and X | U p ′ to X | U p ′′ . Inparticular, k X | U p ′ k g ℓ = k X | U p ′ k g p ′ does not really depend on the point p ′ ∈ b L p ,and it is actually constant along the whole leaf b L q . (cid:3) With this in place, one can finally prove the Main Theorem:
Proof of the Main Theorem.
Let U be an ǫ -tubular neighbourhood around the clo-sure L of a leaf L ∈ F . Letting B = L , we are under the assumptions of Section4. In particular, it is possible to define the linearized foliation F ℓ on U , its localclosure b F ℓ , and the metric g ℓ as in Proposition 7.1. It is clear by construction that b F ℓ | L = F| L and that the closure of b F ℓ is contained in the closure of F . More-over, by Corollary 6.3 the foliation ( U, g ℓ , b F ℓ ) is, locally around each point, foliateddiffeomorphic to the orbit like foliation ( D k × D m − k × U p , D k × { pts. } × b F ℓp ). ByProposition 2.9, the foliation ( U, g ℓ , b F ℓ ) is orbit like as well, and this concludes theproof. (cid:3) References
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E-mail address : [email protected], [email protected] Marco RadeschiMathematisches Institut, WWU M¨unster, Einsteinstr. 62, M¨unster, Germany.
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