Bounded geometry and leaves

BOUNDED GEOMETRY AND LEAVES JES´ US A. ´ ALVAREZ L ´ OPEZ AND RAM´ ON BARRAL LIJ´ O Abstract. The main theorem states that any complete connected Riemannian manifold of bounded geometry can be isometrically realized as a leaf with trivial holonomy in a compact Riemannian foliated space. Contents 1. Introduction 1 2. Preliminaries 4 3. (Partial) quasi-equivalences 6 4. The C∞topology on b M∗(n) 7 5. Foliated structure of b M∞ ∗,imm(n) 11 6. Universality 17 7. Realization of manifolds of bounded geometry as leaves 18 8. Open problems 20 References 20 1. Introduction Recall that a foliated space X≡(X, F) of dimension nis a topological space Xequipped with a partition Finto connected manifolds (leaves) so that Xcan be locally described as a product B×Z, where Bis an open ball in Rnand Zany topological space (local transversal), and the slices B× {∗} correspond to open sets in the leaves. This Fis called a foliated structure or lamination. Foliated spaces are usually assumed to be Polish to get better properties. Many basic notions about foliations can be obviously extended to foliated spaces, like foliated charts, plaques, foliated atlas, holonomy pseudogroup, holonomy group and holonomy covering of the leaves, minimality, transitivity, foliated maps, etc. Some basic results can be extended as well; for instance, there is an obvious version of the Reeb local stability theorem, and the union of leaves without holonomy is a meager subset if Xis second countable. Interesting classes of foliated spaces show up in several areas of mathematics, like in dynamics, arithmetics, tessellations, graphs and foliation theory (minimal sets). AC∞foliated structure is given by a foliated atlas whose changes of coordinates are leafwise C∞, with ambient-space-continuous leafwise derivatives of arbitrary order. This gives rise to the concept of C∞foliated space. To emphasize the difference, the foliated structure underlying a C∞foliated structure may be called topological. On a C∞foliated space X≡(X, F), the concept of C∞function is defined by requiring that its local expressions, using foliated coordinates, are leafwise C∞, with ambient-space-continuous leafwise partial derivatives of arbitrary order. C∞bundles and sections also make sense on X, defined by requiring that their local descriptions are given by C∞functions in the above sense. For instance, the tangent bundle TX (or TF) is the C∞vector bundle on Xthat consists of the vectors tangent to the leaves, and a Riemannian metric on Xconsists of Riemannian metrics on the leaves fitting together nicely to form a C∞section on X. This gives rise to the concept of Riemannian foliated space. 1991 Mathematics Subject Classification. 57R30; 53C12. Key words and phrases. C∞convergence of Riemannian manifolds; bounded geometry; Riemannian foliated space. The first author is partially supported by MICINN (Spain), grant MTM2011-25656. 1 C∞foliated maps between C∞foliated spaces can be similarly defined; in particular, C∞foliated immersions, submersions, (local) diffeomorphisms and (local) embeddings between C∞foliated spaces have obvious meanings. If a homeomorphism between C∞foliated spaces is C∞and its restrictions to the leaves are diffeomorphisms, then it is a C∞diffeomorphism, as follows easily from the continuity of the inversion of C∞diffeomorphisms between C∞manifolds with respect to the C∞topology [18, p. 64, Exercise 9]. Several results about foliated spaces have obvious C∞versions, like the Reeb local stability theorem. Standard references about foliated spaces are [23], [4, Chapter 11], [5, Part 1] and [13]. See also [1, Section 2.1] for a quick summary of what is needed here. On the other hand, recall that a Riemannian manifold Mis said to be of bounded geometry when it has a positive injectivity radius, and the m-th covariant derivative of the curvature tensor has uniformly bounded norm for all order m; in particular, Mis complete by the positivity of the injectivity radius. The following are typical examples where bounded geometry holds: coverings of closed connected Riemannian manifolds, connected Lie groups with left invariant metrics, and leaves of compact Riemannian foliated spaces. More examples can be produced by using compactly supported perturbations of given Riemannian manifolds of bounded geometry. In fact, any smooth manifold admits a metric of bounded geometry [14]. We will focus in the case of leaves of compact Riemannian foliated spaces, showing that this example indeed characterizes bounded geometry. Theorem 1.1. Any connected Riemannian manifold of bounded geometry is isometric to a leaf with trivial holonomy of some compact Riemannian foliated space. It is commonly accepted that such a result should be true, and that it should follow by using the closure of the canonical embedding of the manifold into the Gromov space M∗of pointed proper metric spaces [15], [16, Chapter 3], or, better, into its smooth version, the space M∞ ∗(n) of isometry classes of pointed complete connected Riemannian n-manifolds with the topology defined by the C∞convergence [24, Chapter 10, Section 3.2], [1, Theorem 1.2]. However, to the authors knowledge, no complete proof has been given so far. A complete connected Riemannian n-manifold Mis called non-periodic (respectively, locally non-periodic) if Iso(M) = {idM}(respectively, the canonical projection M→Iso(M)\Mis a covering map), where Iso(M) denotes the isometry group of M. The non-periodic and locally non-periodic manifolds define subspaces of M∞ ∗(n) respectively denoted by M∞ ∗,np(n) and M∞ ∗,lnp(n). There is a canonical map ιM:M→M∞ ∗(n), given by ιM(x)=[M, x] (the isometry class of (M, x)), which induces a continuous injection ¯ιM: Iso(M)\M→ M∞ ∗(n). The images of all possible maps ιMform a partition F∗(n) of M∞ ∗(n). The restriction of F∗(n) to M∞ ∗,lnp(n) is denoted by F∗,lnp(n). For n≥2, M∞ ∗,lnp(n) is open and dense in M∞ ∗(n), and F∗,lnp(n) is a Riemannian foliated space of dimension nso that each map ιM:M→im ιMis a local isometry and the holonomy covering of the leaf im ιM[1, Theorem 1.3]; in particular, M∞ ∗,np(n) is the union of leaves with trivial holonomy. Moreover Cl∞(im ιM) is compact if and only if Mis of bounded geometry [1, Theorem 12.3] (see also [8], [24, Chapter 10, Sections 3 and 4]), where Cl∞denotes the closure operator in M∞ ∗(n). Then, analyzing the cases where Cl∞(im ιM)⊂M∗,lnp(n), a version of Theorem 1.1 follows assuming restrictions on M[1, Theorem 1.5]. To prove Theorem 1.1 with complete generality, we refine the above arguments as follows. Fix a separable Hilbert space Eand any natural1n. Consider pairs (M, f) and triples (M, f, x), where Mis a complete connected Riemannian n-manifold, f∈C∞(M, E) and x∈M. An equivalence φ: (M, f)→(N, h) is an isometry φ:M→Nsuch that φ∗h=f. If moreover distinguished points, x∈Mand y∈N, are preserved, then φ: (M, f, x)→(N, h, y) is called a pointed equivalence. The group of self equivalences of (M, f) is denoted by Iso(M, f). If there is a pointed equivalence (M, f, x)→(N, h, y), then the triples (M, f, x) and (N, h, y) are declared to be equivalent. The equivalence class of each (M, f, x) is denoted by [M, f, x]. Let b M∗(n) denote the set2of such equivalence classes. Definition 1.2. For each m∈N, a sequence [Mi, fi, xi] in b M∗(n) is said to be Cmconvergent to [M, f, x]∈ b M∗(n) if, for each compact domain3Ω⊂Mcontaining x, there is a pointed Cm+1 embedding φi: (Ω, x)→ 1It is assumed that 0 is natural. 2Like in the cases of M∗and M∞ ∗(n), without loss of generality, it can be assumed that the underlying set of any such Mis contained in R, so that b M∗(n) becomes a well defined set. 3Here, a domain in Mis a connected C∞submanifod, possibly with boundary, of the same dimension as M. 2 (Mi, xi) for each large enough isuch that φ∗ igi→g|Ωand φ∗ ifi→f|Ωas i→ ∞ with respect to the Cm topology [18, Chapter 2]. If [Mi, fi, xi] is Cmconvergent to [M, f, x] for all m, then it is said that [Mi, fi, xi] is C∞convergent to [M, f, x]. It is not completely obvious that this C∞convergence satisfies the conditions to define a topology [21], [17]. Thus the following result is not trivial. Theorem 1.3. The C∞convergence in b M∗(n)describes a Polish topology. The topology given by Theorem 1.3 will be called the C∞topology, and the corresponding space is denoted by b M∞ ∗(n). The closure operator in this space will be denoted by c Cl∞. The following maps are canonical and continuous: a forgetful map b M∞ ∗(n)→M∞ ∗(n), [M, f, x]7→ [M, x], and an evaluation map ev : b M∞ ∗(n)→E, [M, f, x]7→ f(x). Note that ev : b M∗(0) →Eis a homeomorphism. Moreover, for each complete connected Riemannian n-manifold Mand any f∈C∞(M, E), there is a canonical continuous map ˆιM,f :M→b M∞ ∗(n), given by ˆιM,f (x) = [M, f, x], which induces a continuous injection ¯ιM,f : Iso(M, f)\M→M∞ ∗(n). The images of the maps ˆιM,f form a natural partition of b M∞ ∗(n), denoted by b F∗(n). Let C∞ imm(M, E) be the set of C∞immersions M→E, and let b M∞ ∗,imm(n) be the b F∗(n)-saturated subspace of b M∗(n) consisting of classes [M, f, x] with f∈C∞ imm(M, E). The restriction of b F∗(n) to b M∗,imm(n) is denoted by b F∗,imm(n). Observe that the canonical projection M→Iso(M, f)\Mis a covering map if f∈C∞ imm(M, E). On the other hand, let b M∞ ∗,c(n) (respectively, b M∞ ∗,o(n)) be the b F∗(n)-saturated subspace of b M∗(n) consisting of classes [M, f, x] such that Mis compact (respectively, open). Observe that, if [N, h, y] is close enough to any [M, f, x]∈b M∞ ∗,c(n), then Nis diffeomorphic to M. Thus b M∞ ∗,c(n) is open in b M∗(n), and therefore b M∞ ∗,o(n) is closed. Hence these are Polish subspaces of b M∗(n), as well as their intersections with any Polish subspace. Let b M∞ ∗,imm,c/o(n) = b M∞ ∗,c/o(n)∩b M∞ ∗,imm(n). The restrictions of b F∗(n) to b M∗,c/o(n) and b M∗,imm,c/o(n) are denoted by b F∗,c/o(n) and b F∗,imm,c/o(n), respectively. Theorem 1.4. The following properties hold: (i)b M∞ ∗,imm(n)is Polish and dense in b M∞ ∗(n). (ii)b F∗,imm(n)is a foliated structure of dimension n. (iii)b F∗,imm,o(n)is transitive. (iv)There is a unique C∞foliated structure b F∞ ∗,imm(n)on b M∞ ∗,imm(n), whose underlying topological foliated structure is b F∗,imm(n), such that ev : b M∞ ∗,imm(n)→Eis a C∞immersion. (v)There is a unique Riemannian metric on b M∞ ∗,imm(n)≡(b M∞ ∗,imm(n),b F∞ ∗,imm(n)) such that ιM,f :M→ ˆιM,f is a local isometry for all complete connected Riemannian n-manifold Mand f∈C∞ imm(M, E). (vi)For all Mand fas above, the map ˆιM,f :M→im ˆιM,f is the holonomy covering of the leaf im ˆιM,f . It is possible to give a version of Theorem 1.4 closer to [1, Theorem 1.3], using the subspace b M∞ ∗,lnp(n) consisting of the classes [M, f, x] such that M→Iso(M, f)\Mis a covering map. Such a result could be proved with the obvious adaptation of the proof of [1, Theorem 1.3], using the exponential map to define foliated charts. Instead, we have opted for studying b M∞ ∗,imm(n) because, in this case, the immersions f directly provide foliated charts. The following result states that b M∞ ∗,imm(n) is universal among the class of Polish Riemannian foliated spaces that satisfy a condition called covering-continuity (Definition 6.1). Theorem 1.5. A Polish Riemannian foliated space Xof dimension nwith complete leaves is isometric to a saturated Riemannian foliated subspace of b M∞ ∗,imm(n)if and only if Xis covering-continuous. In Theorem 1.5, when Xconsists of a single leaf M, the isometric injection of Minto b M∞ ∗,imm(n) is ˆιM,f for any C∞embedding f:M→E. If moreover Mis of bounded geometry, then fcan be chosen so that c Cl∞(imˆιM,f ) is a compact Riemannian foliated subspace of b M∞ ∗,imm(n) (Proposition 7.1). Then Theorem 1.1 follows by considering the isometric injection ˆιM,f :M→c Cl∞(imˆιM,f ). There are examples of Lie groups with left invariant metrics that are not coarsely quasi-isometric to any finitely generated group [7], [11]. Applying the above argument to those Riemannian manifolds, we get 3 compact Riemannian foliated spaces whose leaf holonomy covers are not coarsely quasi-isometric to any finitely generated group. Theorem 1.1 contrasts with the examples of connected Riemannian manifolds of bounded geometry whose quasi-isometry type cannot be realized as leaves of foliations of codimension one on closed manifolds [2], [37], [30], [31]. If the metric is not considered, any surface can be realized as a leaf of a codimension one foliation on a closed manifold [6], but this fails in higher dimension [12], [19], [2], [35], [32]. The study of this realizability problem was initiated in [34]. This work can be considered as a continuation of [1], and therefore many references to [1] are included. 2. Preliminaries Let Mbe a Riemannian manifold (possibly with boundary or corners). The following standard notation will be used. The metric tensor is denoted by g, the distance function on each of the connected components of Mby d, the tangent bundle by π:TM →M, the Levi-Civita connection by ∇, and the open and closed balls of center x∈Mand radius r > 0 by B(x, r) and B(x, r), respectively. If needed, “M” will be added to all of the above notation as a subindex or superindex; when a family of Riemannian manifolds Miis considered, we may add the subindex or superindex “i” instead of “Mi”. A covering space of Mis assumed to be equipped with the lift of g. For m∈Z+, let T(m)M=T· · · TM (mtimes); we also set T(0)M=M. If l < m,T(l)Mis identified with a regular submanifold of T(m)Mvia zero sections, and therefore, for each x∈M, the notation xmay be also used for the zero elements of TxM,TxTM, etc. Let π:T(m)M→T(l)Mbe the vector bundle projection given by composing the tangent bundle projections; in particular, we have π:T(m)M→M. Given any Cm map between Riemannian manifolds, φ:M→N, the induced map T(m)M→T(m)Nwill be denoted by φ(m) ∗(or simply φ∗if m= 1). Hilbert manifolds are also considered in some parts of the paper, using analogous notation. The Levi-Civita connection determines a decomposition T(2)M=H⊕V, as direct sum of the horizontal and vertical subbundles. The Sasaki metric on T M is the unique Riemannian metric g(1) so that H⊥Vand the canonical identities Hξ≡TξM≡Vξare isometries for every ξ∈TM [27]. Continuing by induction, for m≥2, the Sasaki metric on T(m)Mis g(m)= (g(m−1))(1). The notation d(m)is used for the corresponding distance function on the connected components, and the corresponding open and closed balls of center ξ∈T(m)Mand radius r > 0 are denoted by B(m)(ξ, r) and B(m)(ξ, r), respectively. We may add the subindex “M” to this notation if necessary, or the subindex “i” instead of “Mi” for a family of Riemannian manifolds Mi. From now on, T(m)Mis assumed to be equipped with g(m). For l < m,T(l)Mbecomes a totally geodesic Riemannian submanifold of T(m)Morthogonal to the fibers of π:T(m)M→T(l)M, which are also totally geodesic [1, Remark 1-(i)–(iii)] (see also [27, Corollary of Theorem 13, and Theorems 14 and 18]). Let (U;x1, . . . , xn) be a chart of M. As usual, the corresponding metric coefficients are denoted by gij, and write (gij) = (gij)−1. Identify the functions xiwith their lifts to TU. We get a chart (U(1);x1 (1), . . . , x2n (1)) of TM with U(1) =T U,xi (1) =xiand xn+i (1) =vifor 1 ≤i≤n, where the functions vigive the coordinates of tangent vectors with respect to the local frame (∂1, . . . , ∂n) of T U induced by (U;x1, . . . , xn). By induction, for m≥2, let (U(m);x1 (m), . . . , x2mn (m)) be the chart of T(m)Minduced by the chart (U(m−1);x1 (m−1), . . . , x2m−1n (m−1) ) of T(m−1)M. Let Ω ⊂Mbe a compact domain and m∈N. Fix a finite collection of charts of Mthat covers Ω, U={(Ua;x1 a, . . . , xn a)}, and a family of compact subsets of Mwith the same index set as U,K={Ka}, such that Ω ⊂SaKa, and Ka⊂Uafor all a. The corresponding Cmnorm of a Cmtensor Ton Ω is defined by4 kTkCm,Ω,U,K= max amax x∈Ka∩ΩX |I|≤mX J,K  ∂|I|TK a,J ∂xI a (x), where TK a,J are the coefficients of Ton Ua∩Ω with respect to the frame induced by (Ua;x1 a, . . . , xn a). With this norm, the Cmtensors on Ω of a fixed type form a Banach space, whose underlying topology is called 4The standard multi-index notation is used here. 4 the Cmtopology. By taking the projective limit as m→ ∞, we get the Fr´echet space of C∞tensors of that type, whose underlying topology is called the C∞topology (see e.g. [18]). We will always consider the Ck topology for Cktensors on Ω of a given type (k∈N∪ {∞}); in particular, Ck(Ω) is always assumed to be equipped with the Cktopology. Observe that Uand Kare also qualified to define the norm k kCm,Ω0,U,K for any compact subdomain Ω0⊂Ω. It is well known that k kCm,Ω,U,Kis equivalent to the norm k kCm,Ω,g defined by kTkCm,Ω,g = max 0≤l≤mmax x∈Ω|∇lT(x)|; i.e., there is some C≥1, depending on M, Ω, U,K,gand m, such that 1 Ck kCm,Ω,U,K≤ k kCm,Ω,g ≤Ck kCm,Ω,U,K.(1) In particular, for m= 0 and f∈C∞(M), kfkΩ:= kfkC0,Ω,U,K=kfkC0,Ω,g = max x∈Ω|f(x)|,(2) which is independent of the choices U,Kand g. The norms k kCm,Ω,U,Kand k kCm,Ω,g have straightforward extensions to tensors with values in a separable Hilbert space E, and satisfy the obvious versions of (1) and (2), and Ck(M, E) is assumed to be equipped with the Cktopology (k∈N∪ {∞}). For f∈C∞(M, E), recall that ∇f=df (its de Rham differential). For each m, the map f(m) ∗≡f(m),1 ∗, . . . , f(m),2m ∗:T(m)M→T(m)E≡E2m is also C∞and with values in a separable Hilbert space. In the following lemma, we consider the local representations of fand every f(m),λ ∗with respect to coordinate systems (U, x1, . . . , xn) and (U(m), x1 (m), . . . , x2mn (m)) of Mand T(m)M. Moreover each function on Mor Uis identified with its lift to T(m)Mor U(m). Lemma 2.1. The following properties hold: (i)The local representation of every f(m),λ ∗is a universal polynomial expression of xn+1 (m), . . . , x2mn (m)and the partial derivatives up to order mof the local representation of f. (ii)For each ρ > 0, the partial derivatives up to order mof the local representation of fare given by universal linear expressions of the functions (σ(m) ρ,µ )∗f(m),λ ∗for n+1 ≤µ≤2mn, where σ(m) ρ,µ :U→U(m) is the section of π:U(m)→Udetermined by5(σ(m) ρ,µ )∗xν (m)=ρδµν for n+ 1 ≤ν≤2mn. Proof. By using induction on m, the result clearly boils down to the case m= 1. But, in this case, the statement follows because f∗≡(f, df) : TM →TE≡E2. By using the supremum on Ω instead of the maximum, the definition of k kCm,Ω,g can be extended to any non-compact n-submanifold Ω ⊂M(including Ω = M), with possible infinite values. The tensors on Ω with finite norm k kCm,Ω,g are said to be uniformly Cm, or Cm b. For a given type, they form a Banach space, and the corresponding projective limit as m→ ∞ is a Fr´echet space, whose elements are said to be uniformly C∞, or C∞ b(see e.g. [26, Definition 2.7] or [28, Definition 3.15]). In particular, this gives rise to the Fr´echet spaces C∞ b(Ω) and C∞ b(Ω,E) when R-valued and E-valued C∞ bfunctions are considered. Let Nbe another Riemannian manifold. Recall that a C1map φ:M→Nis called a (λ-) quasi-isometry, or (λ-) quasi-isometric, if there is some λ≥1 such that 1 λ|ξ|≤|φ∗(ξ)| ≤ λ|ξ|for every ξ∈T M; in particular, φis an immersion. To define higher order quasi-isometries, let T≤rM={ξ∈TM | |ξ| ≤ r}for each r > 0. If Mhas no boundary, then T≤rMis a manifold with boundary; otherwise, it is a manifold with corners. Also, define T(m),≤rMby induction on m∈Z+, setting T(1),≤rM=T≤rMand T(m),≤rM=T≤rT(m−1),≤rM. It is said that φ:M→Nis a (λ-) quasi-isometry of order m∈N, or a (λ-) quasi-isometric map of order m, if it is Cm+1 and φ(m) ∗:T(m),≤1M→T(m)Nis a (λ-) quasi-isometry. If φis a quasi-isometry of order mfor all m∈N, then it is called a quasi-isometry of order ∞. If there is a quasi-isometric diffeomorphism M→Nof order m∈N∪ {∞}, then Mand Nare said to be quasi-isometric with order m. The property of being a quasi-isometry of order mis preserved by the operations of composition of maps and inversion of 5Kronecker’s delta is used here. 5 diffeomorphisms [1, Proposition 3.9], and therefore it induces an equivalence relation between Riemannian manifolds. For m∈N, a partial map φ:MNis called a Cmlocal diffeomorphism6if dom φand im φare open in Mand N, respectively, and φ: dom φ→im φis a Cmdiffeomorphism. If moreover φ(x) = yfor distinguished points, x∈dom φand y∈im φ, then it is said that φ: (M, x)(N, y) is a pointed Cmlocal diffeomorphism. For m∈N,R > 0 and λ≥1, a Cm+1 pointed local diffeomorphism φ: (M, x)(N, y) is called an (m, R, λ)-pointed local quasi-isometry, or a local quasi-isometry of type (m, R, λ), if the restriction φ(m) ∗: Ω(m)→T(m)Nis a λ-quasi-isometry for some compact domain Ω(m)⊂dom φ(m) ∗with B(m) M(x, R)⊂ Ω(m)[1, Definition 4.2]. 3. (Partial) quasi-equivalences Let Mand Nbe Riemannian n-manifolds, let f∈C∞(M, E) and h∈C∞(N, E), and let x∈Mand y∈N. Recall from Section 1 the concepts of an equivalence (M, f)→(N, h), and a pointed equivalence (M, f, x)→(N, h, y). Observe that kf(m) ∗kΩ(m)makes sense for any n-submanifold Ω(m)⊂T(m)Mbecause we consider f(m) ∗:T(m)M→T(m)E≡E2m, with values in a separable Hilbert space. Note also that (φ∗h)(m) ∗=h(m) ∗◦φ(m) ∗for any Cmmap φ:M→N. Definition 3.1. Let λ≥1 and ε≥0, and let φ:M→Nbe a C1map. It is said that φ: (M, f)→(N, h) is a ((λ, ε)-) quasi-equivalence of order m∈Nif it is Cm+1,φ(m) ∗:T(m),≤1M→T(m)Nis a (λ-) quasiisometry, and kf(m) ∗−(φ∗h)(m) ∗kT(m)M≤ε. If moreover distinguished points xand yare preserved, then φ: (M, f, x)→(N, h, y) is called a pointed quasi-equivalence of order m. If there is a quasi-equivalence (M, f)→(N, h) (respectively, (M, f, x)→(N, h, y)), then (M, f) and (N, h) (respectively, (M, f, x) and (N, h, y)) are called quasi-equivalent. Remark 1.(i) Any (λ, ε)-quasi-equivalence of order m≥1 is a (λ, ε)-quasi-equivalence of order m−1. (ii) For integers 0 ≤m0≤m, if φis a (λ, ε)-quasi-equivalence of order m, then φ(m0) ∗is a (λ, ε)-quasi- equivalence of order m−m0. For a submanifold Ω ⊂Mand f∈C∞(M, E), the notation (Ω, f) is used for (Ω, f|Ω). Proposition 3.2. The following properties hold for any m∈N,λ, µ ≥1and ε, δ ≥0: (i)There is some ν≥1, depending on m,λand µ, such that, if φ: (M, f)→(N, h)is a (λ, ε)- quasi-equivalence and ψ: (N, h)→(L, u)a(µ, δ)-quasi-equivalence, both of them of order m, then ψ◦φ: (M, f)→(L, u)is a (ν, ε +δ)-quasi-equivalence of order m. (ii)There are some ν0≥1, depending on mand λ, such that, if φ: (M, f)→(N, h)is a (λ, ε)-quasi- equivalence of order mand a diffeomorphism, then φ−1: (N, h)→(M, f)is a (ν0, ε)-quasi-equivalence of order m. Proof. By [1, Proposition 3.9], we only have to check the conditions on the E-valued functions. Thus (i) follows because, for each ξ∈T(m)M, we have f(m) ∗(ξ)−((ψ◦φ)∗u)(m) ∗(ξ) ≤f(m) ∗(ξ)−(φ∗h)(m) ∗(ξ)+h(m) ∗φ(m) ∗(ξ)−(ψ∗u)(m) ∗φ(m) ∗(ξ)≤ε+δ . Similarly, (ii) follows because, for each ζ∈T(m)N, h(m) ∗(ζ)−((φ−1)∗f)(m) ∗(ζ)=(φ∗h)(m) ∗(φ−1)(m) ∗(ζ)−f(m) ∗(φ−1)(m) ∗(ζ)≤ε .  Corollary 3.3. “Being quasi-equivalent with order m” is an equivalence relation on the sets of pairs (M, f) and triples (M, f, x). Now, suppose that Mand Nare connected, complete and without boundary. 6The term “Cmlocal diffeomorfism” (m≥1) is also used in the standard sense, referring to any Cmmap M→Nwhose tangent map is an isomorphism at every point of M. The context will always clarify this ambiguity. 6 Definition 3.4. Fix m∈N,R > 0, λ≥1 and ε≥0. Let φ: (M, x)(N, y) be a Cm+1 pointed local diffeomorphism, and let f∈C∞(M, E) and h∈C∞(N, E). It is said that φ: (M, f, x)(N, h, y) is an (m, R, λ, ε)-pointed local quasi-equivalence, or a local quasi-equivalence of type (m, R, λ, ε), if there is some compact domain Ω(m)⊂dom φ(m) ∗such that B(m) M(x, R)⊂Ω(m)and φ(m) ∗: (Ω(m), f(m) ∗)→(T(m)N, h(m) ∗) is a (λ, ε)-quasi-equivalence. Remark 2.(i) Any pointed local quasi-equivalence (M, f, x)(N, h, y) of type (m, R, λ, ε) is also of type (m0, R0, λ0, ε0) for 0 ≤m0≤m, 0 < R0< R,λ0> λ and ε0> ε. (ii) Consider integers 0 ≤m0≤m, any pointed Cm+1 local diffeomorphism φ: (M, x)(N, y), and any f∈C∞(M, E) and h∈C∞(N, E). Then φ: (M, f, x)(N, h, y) is a pointed local quasi-equivalence of type (m, R, λ, ε) if and only if φ(m0) ∗: (T(m0)M, f(m0) ∗, x)(T(m0)N, h(m0) ∗, y) is a pointed local quasi-equivalence of type (m−m0, R, λ, ε). (iii) If there is an (m, R, λ, ε)-pointed local quasi-equivalence (M, f, x)(N, h, y), then, for all R0< R, λ0> λ and ε0> ε, there is a C∞(m, R0, λ0, ε0)-pointed local quasi-equivalence (M, f, x)(N, h, y) by [18, Theorem 2.7]. Lemma 3.5. The following properties hold: (i)If φ: (M, f, x)(N, h, y)and ψ: (N, h, y)(L, u, z)are pointed local quasi-equivalences of types (m, R, λ, ε)and (m, λR, λ0, ε0), respectively, then ψ◦φ: (M, f, x)(L, u, z)is an (m, R, λλ0, ε +ε0)- pointed local quasi-equivalence. (ii)If φ: (M, f, x)(N, h, y)is an (m, λR, λ, ε)-pointed local quasi-isometry, then φ−1: (N, h, y) (M, f, x)is an (m, R, λ, ε)-pointed local quasi-isometry. Proof. To prove (i), take compact domains, Ω(m)⊂T(m)Mand Ω0(m)⊂T(m)N, such that B(m) M(x, R)⊂ Ω(m),B(m) N(x, λR)⊂Ω0(m),φ(m) ∗: (Ω(m), f(m) ∗)→(T(m)N, h(m) ∗)isa(λ, ε)-quasi-equivalence, and ψ(m) ∗: (Ω0(m), h(m) ∗)→(T(m)L, u(m) ∗) is a (λ0, ε0)-quasi-equivalence. According to the proof of [1, Lemma 4.3-(i)], there is a compact domain Ω(m) 0⊂T(m)Msuch that B(m) M(x, R)⊂Ω(m) 0and φ(m) ∗(Ω(m) 0)⊂Ω0(m). Then (ψ◦φ)(m) ∗: Ω(m) 0→T(m)Lis a λλ0-quasi-isometry by [1, Remark 2-(v)]. Moreover, for each ξ∈Ω(m) 0, f(m) ∗(ξ)−((ψ◦φ)∗u)(m) ∗(ξ) ≤f(m) ∗(ξ)−(φ∗h)(m) ∗(ξ)+h(m) ∗φ(m) ∗(ξ)−(ψ∗u)(m) ∗φ(m) ∗(ξ)≤ε+ε0. So ψ◦φ: (M, f, x)(L, u, z) is an (m, R, λλ0, ε +ε0)-pointed local quasi-equivalence. To prove (ii), let Ω(m)⊂T(m)Mbe a compact domain such that B(m) M(x, R)⊂Ω(m), and φ(m) ∗: (Ω(m), f(m) ∗)→(T(m)N, h(m) ∗) is a (λ, ε)-quasi-equivalence. According to the proof of [1, Lemma 4.3-(ii)], the compact domain Ω0(m):= φ(m) ∗(Ω(m))⊂T(m)Ncontains B(m) N(y, R). Then (φ−1)(m) ∗= (φ(m) ∗)−1: Ω0(m)→ T(m)Mis a λ-quasi-isometry by [1, Remark 2-(vi)]. Moreover, for each ξ∈Ω0(m), h(m) ∗(ξ)−((φ−1)∗f)(m) ∗(ξ)≤(φ∗h)(m) ∗(φ−1)(m) ∗(ξ)−f(m) ∗(φ−1)(m) ∗(ξ)≤ε . So φ−1: (N, h, y)(M, f, x) is an (m, R, λ, ε)-pointed local quasi-equivalence.  4. The C∞topology on b M∗(n) Definition 4.1. For m∈Nand R, r > 0, let b Um R,r be the set of pairs ([M, f, x],[N, h, y]) ∈b M∗(n)×b M∗(n) such that there is some (m, R, λ, ε)-pointed local quasi-equivalence (M, f, x)(N, h, y) for some λ∈[1, er) and ε∈(0, r). Proposition 4.2. The following properties7hold for all m, m0∈Nand R, S, r, s > 0: 7The following standard notation is used for a set Xand relations U, V ⊂X×X: U−1={(y, x)∈X×X|(x, y)∈U}, V◦U={(x, z)∈X×X| ∃y∈Xso that (x, y)∈Uand (y, z)∈V}. Moreover the diagonal of X×Xis denoted by ∆. 7 (i) (b Um erR,r)−1⊂b Um R,r. (ii)b Um0 R0,r0⊂b Um R,r ∩b Um0 S,s, where m0= max{m, m0},R0= max{R, S}and r0= min{r, s}. (iii) ∆ ⊂b Um R,r. (iv)b Um R,r ◦b Um erR,s ⊂b Um R,r+s. Proof. Properties (ii) and (iii) are elementary, and (i) and (iv) are consequences of Lemma 3.5.  Proposition 4.3. TR,r>0b Um R,r = ∆ for all m∈N. Proof. We only have to prove “⊂” by Proposition 4.2-(iii). For ([M, f, x],[N, h, y]) ∈TR,r>0b Um R,r, there is a sequence of pointed local quasi-equivalences φi: (M, f, x)(N, h, y), with corresponding types (m, Ri, λi, εi), such that Ri↑ ∞,λi↓1 and εi↓0 as i→ ∞. According to the proof of [1, Proposition 5.3], for each i, there is some subsequence φk(i,l)whose restriction to BM(x, Ri) converges to some pointed isometric immersion ψi: (BM(x, Ri), x)→(N, y) in the weak Cmtopology, ψi+1|BM(x,Ri)=ψifor all i, and the combination of the maps ψiis a pointed isometry ψ: (M, x)→(N, y). For every x0∈M and ε > 0, there are some iand δ > 0 so that x0∈BM(x, Ri), εi≤ε/2, and kh(y0)−h(y00)k< ε/2 if dN(y0, y00)< δ for all y0, y00 ∈BM(x, Ri). Moreover there is some lsuch that dN(φk(i,l)(x0), ψi(x0)) < δ. Hence kf(x0)−h◦ψ(x0)k≤kf(x0)−h◦φk(i,l)(x0)k+kh◦φk(i,l)(x0)−h◦ψ(x0)k< εi+ε/2≤ε. Since x0and εare arbitrary, it follows that ψ: (M, f, x)→(N, h, y) is an equivalence, and therefore [M, f, x]=[N, h, y].  By Propositions 4.2 and 4.3, the sets b Um R,r form a base of entourages of a separating uniformity on b M∗(n), which is called the C∞uniformity. Definition 4.4. For R, r > 0 and m∈N, let b Dm R,r be the set of pairs ([M, f, x],[N, h, y]) ∈b M∗(n)× b M∗(n) such that there is some Cm+1 pointed local diffeomorphism φ: (M, x)(N, y) so that kgM− φ∗gNkCm,Ω,gM< r and kf−φ∗hkCm,Ω,gM< r for some compact domain Ω ⊂dom φwith BM(x, R)⊂Ω. Remark 3.By (1), and its version for E-valued functions, a sequence [Mi, fi, xi]∈b M∗(n) is C∞convergent to [M, f, x]∈b M∗(n) if and only if it is eventually in8b Dm R,r(M, f, x) for arbitrary m∈Nand R, r > 0. Proposition 4.5. The following properties hold: (i)For all R, r > 0, if 0< r0≤min{1−e−2r, e2r−1, r}, then b D0 R,r0⊂b U0 R,r. (ii)For all m∈Z+,R, r > 0and [M, f, x]∈b M∗(n), there is some r0>0such that b Dm R,r0(M, f, x)⊂ b Um R,r(M, f, x). Proof. Let us show (i). If ([M, f, x],[N, h, y]) ∈b D0 R,r0, then there is a C1pointed local diffeomorphism φ: (M, x)(N, y) such that r0 0:= kgM−φ∗gNkC0,Ω,gM< r0and ε0:= kf−φ∗hkC0,Ω,gM< r0for some compact domain Ω ⊂dom φwith BM(x, R)⊂Ω. Take some λ∈[1, er) such that r0 0≤min{1−λ−2, λ2−1}. According to the proof of [1, Proposition 6.4-(i)], φ: Ω →Nis a λ-quasi-isometry. Since moreover kf−φ∗hkΩ≤ε0, it follows that φis a (0, R, λ, r0 0, ε0)-pointed local quasi-equivalence, obtaining that ([M, f, x],[N, h, y]) ∈b U0 R,r. Let us prove (ii). Take m∈Z+,R, r > 0 and [M, f, x]∈b M∗(n). Let Ube a finite collection of charts of Mwith domains Ua, and let K={Ka}be a family of compact subsets of M, with the same index set as U, such that Ka⊂Uafor all a, and BM(x, R)⊂Int(K) for K=SaKa. Let r0>0, to be fixed later. For any [N, h, y]∈b Dm R,r0(M, x), there is a Cm+1 pointed local diffeomorphism φ: (M, x)(N, y) so that kgM−φ∗gNkCm,Ω,gM< r0and kf−φ∗hkCm,Ω,gM< r0for some compact domain Ω ⊂dom φ∩Int(K) with BM(x, R)⊂Ω. By continuity, there is another compact domain Ω0⊂dom φ∩Int(K) such that Ω ⊂Int(Ω0), kgM−φ∗gNkCm,Ω0,gM< r0and kf−φ∗hkCm,Ω0,gM< r0. According to the proof of [1, Proposition 6.4-(i)], if r0is small enough (depending on m,R,rand [M, x]), then there is some compact domain Ω(m)⊂T(m)M 8Given a set X, for U⊂X×Xand x∈X, let U(x) = {y∈Y|(x, y)∈U}. In the case of U⊂b M∗(n)×b M∗(n) and [M, f, x]∈b M∗(n), we simply write U(M, f, x). 8 such that B(m) M(x, R)⊂Ω(m)⊂π−1(Ω0), where π:T(m)M→M, and φ(m) ∗: Ω(m)→T(m)Nis a λ-quasi- isometry for some λ∈[1, er). Given ε∈(0, r), choose some C≥1 satisfying (1) for E-valued functions with U,K, Ω0and g, and, according to Lemma 2.1-(i), choose some ε0>0 such that kf−φ∗hkCm,Ω0,U,K< ε0=⇒ kf(m) ∗−(φ∗h)(m) ∗kΩ(m)<ε. Suppose that r0≤ε0/C. Then kf−φ∗hkCm,Ω0,gM< r0=⇒ kf−φ∗hkCm,Ω0,U,K< Cr0≤ε0=⇒ kf(m) ∗−(φ∗h)(m) ∗kΩ(m)<ε. Hence φis an (m, R, λ, ε)-pointed local quasi-equivalence (M, f, x)(N, h, y), and therefore [N, h, y]∈ b U(m) R,r (M, f, x).  Proposition 4.6. The following properties hold: (i)For all R, r > 0, if e2r0−e−2r0≤r, then b U0 R,r0⊂b D0 R,r. (ii)For all m∈Z+,R, r > 0and [M, f, x]∈b M∗(n), there is some r0>0such that b Um R,r0(M, f, x)⊂ b Dm R,r(M, f, x). Proof. Let us show (i). If ([M, f, x],[N, h, y]) ∈b U0 R,r0, then there is a (0, R, λ, ε)-pointed local quasiequivalence φ: (M, f, x)(N, h, y) for some λ∈[1, er0) and ε∈(0, r0). Thus there is some compact domain Ω ⊂dom φsuch that BM(x, R)⊂Ω and φ: (Ω, f)→(N, h)isa(λ, ε)-quasi-equivalence. According to the proof of [1, Proposition 6.5-(i)], kgM−φ∗gNkC0,Ω,g < r. So ([M, f, x],[N, h, y]) ∈b D0 R,r. Let us prove (ii). Let m∈Z+,R, r > 0 and [M, f, x]∈b M∗(n). Take U,Kand Klike in the proof of Proposition 4.5-(ii). Let r0>0, to be fixed later. For any [N, h, y]∈b Um R,r0(M, x), there is an (m, R, λ, ε)- pointed local quasi-equivalence φ: (M, f, x)(N, h, y) for some λ∈[1, er0) and ε∈(0, r0). Thus there is a compact domain Ω(m)⊂dom φ(m) ∗∩Int(K(m)) so that B(m) M(x, R)⊂Ω(m)and φ(m) ∗: (Ω(m), f(m) ∗)→ (T(m)N, h(m) ∗) is a (λ, ε)-quasi-equivalence. According to the proof of [1, Proposition 6.5-(ii)], there are compact domains, Ω0(m)⊂dom φ(m) ∗and Ω ⊂M, such that Ω(m)⊂Int(Ω0(m)), Ω(m)∩M⊂Ω⊂Int(Ω0(m)), and kgM−φ∗gNkCm,Ω,g < r if r0is small enough; in particular, BM(x, R)⊂Ω because Mis a totally geodesic Riemannian submanifold of T(m)M. Take some C≥1 satisfying (1) for E-valued functions with U, K, Ω and gM. With the notation of Section 2, for ρ > 0 and n+ 1 ≤µ≤2mn, let σ(m) a,ρ,µ :Ua→U(m) abe the section of each π:U(m) a→Uaof the type used in Lemma 2.1-(ii). Since Ω ⊂Int(Ω0(m)), there is some ρ > 0 so that σ(m) ρ,µ (Ka∩Ω) ⊂Ω0(m)for all aand µ. Thus, by Lemma 2.1-(ii), there is some ε0>0, depending on rand ρ, such that kf(m) ∗−(φ∗h)(m) ∗kΩ0(m)< ε0=⇒ kf∗−φ∗hkCm,Ω,U,K< r/C . Suppose that moreover r0< ε0, and therefore ε<ε0. Then kf(m) ∗−(φ∗h)(m) ∗kΩ0(m)≤ε<ε0=⇒ kf−φ∗hkCm,Ω,U,K< r/C =⇒ kf−φ∗hkCm,Ω,g < r , showing that [N, h, y]∈b D(m) R,r (M, f, x).  Corollary 4.7. The C∞convergence in b M∗(n)describes the topology induced by the C∞uniformity. Proof. This is a direct consequence of Remark 3 and Propositions 4.5 and 4.6.  According to Corollary 4.7, the C∞uniformity induces what was called the C∞topology in Section 1. Recall that the corresponding space is denoted by b M∞ ∗(n), and the notation c Cl∞is used for the closure operator in b M∞ ∗(n). Proposition 4.8. b M∞ ∗(n)is separable. Proof. According to the proof of [1, Proposition 7.1], there is a countable family Cof C∞compact manifolds containing exactly one representative of every diffeomorphism class, and, for every M∈C, there is a countable 9 Mand f∈C∞ imm(M), the map ˆιM,f :M→imˆιM,f is a C∞local diffeomorphism with respect to the C∞ structure induced by Gon the leaf im ˆιM,f because ev is a C∞immersion and ev ◦ˆιM,f =f, which is a C∞local embedding. Thus the restriction of χ:N2→Bto each plaque is a C∞diffeomorphism. Using again [18, p. 64, Exercise 9], it follows that Φ : N2→B×Zis also C∞foliated diffeomorphism with respect to the restriction of Gand the C∞product foliated structure of B×Z. This shows that G=b F∞ ∗,imm(n), completing the proof of (i).  Consider a leaf im ˆιM,f of b F∞ ∗,imm(n). Every x∈Mhas an open neighborhood Uin Mso that f:U→E is an embedding, obtaining that φ(U)∩U=∅for all φ∈Iso(M, f)r{idM}. Therefore the subgroup Iso(M, f)⊂Iso(M) is discrete, the quotient projection M→Iso(M, f)\Mis a covering map, and there is a unique Riemannian structure on the manifold Iso(M, f)\Mso that M→Iso(M, f)\Mis a local isometry. Moreover ˆιM,f :M→imˆιM,f induces a diffeomorphism ¯ιM,f : Iso(M, f)\M→imˆιM,f . Thus ˆιM,f :M→ imˆιM,f is a covering map, and im ˆιM,f has a unique Riemannian metric so that ˆιM,f :M→im ˆιM,f is a local isometry, and therefore ¯ιM,f : Iso(M, f)\M→imˆιM,f becomes an isometry. Proposition 5.14. The above Riemannian metrics on the leaves of b F∞ ∗,imm(n)form a C∞Riemannian metric on (b M∞ ∗,imm(n),b F∞ ∗,imm(n)). Proof. Let Φ = (χ, Θ) : N2→B×Zbe defined by any choice of (V, e, ρ, κ, σ) as above, and let [Mi, fi, xi]→ [M, f, x] be a convergent sequence in Z. Let ¯gMand ¯gibe the metrics on Bthat correspond to gMand gi by the diffeomorphisms χM,f :P:= BM(x, 2ρ)∩χ−1 M,f (B)→B , χMi,fi:Pi:= Bi(xi,2ρ)∩χ−1 Mi,fi(B)→B , respectively (see Lemma 5.7). According to the proof of Proposition 5.13-(ii), we have to prove that ¯gi→¯gM as i→ ∞ in the weak C∞topology. Given m∈N,R, r > 0, for each ilarge enough, there is an (m, R, λi, εi)-pointed local quasi-equivalence φi: (M, f, x)(Mi, fi, xi) for some λi∈(1, er) and εi∈(0, r). Assuming R > 2erρ, we get BM(x, 2ρ)⊂ BM(x, R) and Bi(xi,2ρ)⊂φi(BM(x, R)), like in the proof of Proposition 5.12. Take a compact domain Ω(m) i⊂dom φ(m) i∗such that B(m) i(xi, R)⊂Ω(m) iand φ(m) i∗: Ω(m) i→T(m)Mis a (λi, εi)-quasi-isometry. Let Ξ⊂Bbe a compact domain, and let Ξ(m)be a compact domain contained in T(m)Bsuch that Ξ⊂Int(Ξ(m)),(χ−1 M,f )(m) ∗(Ξ(m))∩T(m)P⊂Ω(m) i,(χ−1 Mi,fi)(m) ∗(Ξ(m))∩T(m)Pi⊂φ(m) i∗(Ω(m) i). Like in (10), there is some ν≥1, independent of i, such that d(m) i(φ−1 i◦χ−1 Mi,fi)(m) ∗(ξ),(χ−1 M,f )(m) ∗(ξ)< νr , for all ξ∈Ξ(m). Since the choice of Ξ(m)is valid for all rsmall enough, it follows that φ−1 i◦χ−1 Mi,fi→χ−1 M,f in Cm(Ξ, M) by the obvious version of Lemma 2.1 for maps between manifolds. Since the choice of Ξ is valid for all m, it follows that this convergence also holds in C∞(Ξ, M). Take a compact domain Ω ⊂M such that BM(x, R)⊂Ω and φ∗ igi→gMon Ω with respect to the C∞topology. We get (φ−1 i◦χ−1 Mi,fi)∗(φ∗ igi−gM)→(χ−1 M,f )∗0 = 0 on Ξ with respect to the C∞topology. So ¯gi−¯gM= (χ−1 Mi,fi)∗gi−(χ−1 M,f )∗gM = (φ−1 i◦χ−1 Mi,fi)∗(φ∗ igi−gM)+(φ−1 i◦χ−1 Mi,fi)∗gM−(χ−1 M,f )∗gM→0 on Ξ with respect to the C∞topology. Since every point in Bbelongs to some domain Ξ as above if ris chosen small enough, it follows that ¯gi−¯gM→0 on Bwith respect to the weak C∞topology.  Proposition 5.15. The holonomy covering of any leaf im ˆιM,f of b F∗,imm(n)is ˆιM,f :M→imˆιM,f . This proposition follows directly from the obvious version of [1, Lemma 11.9] for b M∞ ∗,imm(n). 16 6. Universality Definition 6.1. Let Xbe a sequential Riemannian foliated space with complete leaves, and let Lxdenote the leaf through every x∈X, whose holonomy covering is denoted by e Lhol x. It is said that Xis coveringcontinuous when there is a connected pointed covering (e Lx,˜x) of (Lx, x) for all x∈Xsuch that [e Lxi,˜xi] is C∞convergent to [e Lx,˜x] if xi→xis a convergent sequence in X. When this condition is satisfied with e Lx=e Lhol xfor all x∈X, it is said that Xis holonomy-continuous. Remark 5.Observe the following: (i) Covering-continuity and holonomy-continuity are weaker than covering-determination and holonomydetermination [1, Definition 12.1], which were defined by using “if and only if” instead of “if”. (ii) The condition of being covering-continuous is hereditary (by saturated subspaces). (iii) Covering/holonomy-continuity/determination have obvious generalizations to arbitrary Riemannian foliated spaces by using nets instead of sequences. Example 6.2. The following simple examples clarify Definition 6.1: (i) The Reeb foliation on S3with the standard metric is covering-continuous, but it is not holonomycontinuous with any Riemannian metric. If the metric is modified around the compact leaf T2=S1×S1 so that the diffeomorphism (x, y)7→ (y, x) of T2is not an isometry, then this foliation becomes noncovering-continuous. (ii) The Riemannian foliated space of [22, Example 2.5] is covering-determined but not holonomy-continuous. This example can be easily realized as a saturated subspace of a Riemannian foliated space where the holonomy coverings of the leaves are isometric to R. So holonomy-continuity is not hereditary. (iii) b M∞ ∗,imm(n) is holonomy-continuous. However it is not holonomy-determined for n≥1 by [1, Remark 10- (iii)], since there are different points with isometric pointed holonomy covers of the corresponding pointed leaves. To see this, take any connected complete Riemannian n-manifold M, and some x∈M and f, f0∈C∞ imm(M, E) such that f(x)6=f0(x). Then ˆιM,f (x)6= ˆιM,f0(x), but (M, x) is isometric to the holonomy covers of the pointed leaves (imˆιM,f ,ˆιM,f (x)) and (im ˆιM,f0,ˆιM,f0(x)). Proposition 6.3 (Cf. [4, Theorem 11.4.4]).For any Polish C∞foliated space Xwith complete leaves, there is a C∞embedding X→E. Proof. This is an adaptation of the usual argument to show the existence of C∞embeddings of C∞manifolds in Euclidean spaces [18, Theorem 1.3.4]. Let n= dim X(as foliated space), and let Br=BRn(0, r) and Br=BRn(0, r) for each r > 0. Claim 3.Let Zbe a Polish space, and consider the C∞foliated structure on U:= B2×Zwith leaves B2× {∗}. Let Vand Wbe open subsets of Usuch that V⊂Wand W⊂B1×Z. Then there is some h∈C∞(U) such that h= 1 on Vand supp h⊂W. Since B1is compact, it easily follows that each z∈Zhas an open neighborhood Pzin Zsuch that, for some open subsets Gz, Hz⊂B2with Gz⊂Hzand Hz⊂B1, we have V∩(B1×Pz)⊂Gz×Pzand Hz×Pz⊂W. Let {λi}be a partition of unity of Zsubordinated to the open cover {Pz|z∈Z}; in particular, for every i, there is some zi∈Zso that supp λi⊂Pzi. Let hi∈C∞(B2) such that hi= 1 on Gziand supp hi⊂Hzi. Then hiλi∈C∞(U), hiλi=λion Gzi×Pziand supp(hiλi)⊂Hzi×Pzi. It follows that h=Pihiλisatisfies the properties stated in Claim 3. Now, let Ube a countable collection of C∞foliated charts φi:U2,i →B2×Ziof Xsuch that the open sets U1,i := φ−1 i(B1×Zi) cover X. Using the paracompactness and regularity of X, a standard argument gives locally finite open covers, V={Vi}and W={Wi}, with the same index set as U, such that Vi⊂Wi and Wi⊂U1,i. For each i, let Eibe a copy of E. Take embeddings ψi:Zi→Ei[9, Corollary IX.9.2]. Thus each composite U2,i φi −−−−→ B2×Zi id ×ψi −−−−→ B2×Ei,→Rn×Ei=: e Ei 17 is a C∞embedding with respect to the restriction of F, which will be denoted by ˜ φi. By Claim 3, there are functions hi∈C∞(U2,i) such that hi= 1 on Viand supp hi⊂Wi. Then a C∞embedding9f:X→ c Lie Ei∼ =Eis defined by f(x) = Paha(x)˜ φika.  Proof of Theorem 1.5. The Polish Riemannian foliated space b M∞ ∗,imm(n) has complete leaves and is holonomycontinuous (Example 6.2-(iii)). Thus any Polish Riemannian foliated subspace of b M∞ ∗,imm(n) is also coveringcontinuous (Remark 5-(ii)). Let Xbe any covering-continuous Polish Riemannian foliated space with complete leaves. By Proposition 6.3, there is a C∞embedding f:X→E. With the notation of Definition 6.1, suppose that the covering-continuity of Xis satisfied with the connected pointed coverings (e Lx,˜x)→(Lx, x) (x∈X). Let ˆιX,f :X→b M∞ ∗,imm(n) be defined by ˆιX,f (x) = [e Lx,˜ fx,˜x], where ˜ fxis the lift of f|Lxto e Lx. This map is well defined because the leaves of Xare complete. Moreover it is obviously foliated and continuous by the definitions of covering-continuity and the topology of b M∞ ∗,imm(n). To show that ˆιX,f is C∞, take a foliated chart Φ = (χ, Θ) : N2→B×Zof b F∞ ∗,imm(n) defined by any choice of (V, e, ρ, κ, σ) as above. Let Ube the domain of a foliated chart of Xsuch that ˆιX,f (U)⊂N2. Then the composite UˆιX,f −−−−→ N2 χ −−−−→ B is equal to ΠV◦(f−e), and therefore it is C∞. Finally, ˆιX,f is a C∞embedding because the composite XˆιX,f −−−−→ b M∞ ∗,imm(n)ev −−−−→ E equals the C∞embedding f. 7. Realization of manifolds of bounded geometry as leaves Proposition 7.1. Let Mbe any connected, complete Riemannian n-manifold of bounded geometry. Then there is a C∞embedding f:M→Esuch that c Cl∞(imˆιM,f )is a compact subspace of b M∞ ∗,imm(n). Proof. Let Br=BRn(0, r) for each r > 0. By the bounded geometry of M, there is some r > 0, smaller than the injectivity radius of M, such that the following properties hold: (i) For the normal parametrizations κx:Br→BM(x, r) (x∈M), the corresponding metric coefficients, gij and gij, as a family of C∞functions on Brparametrized by x,iand j, lie in a bounded subset of the Fr´echet space C∞ b(Br) [28, Theorem A.1], [29, Theorem 2.5] (see also [26, Proposition 2.4], [10]). (ii) There is some countable subset {xi|i∈N} ⊂ Mand some c∈Nsuch that the family of balls BM(xi, r/2) covers M, and BM(x, r) meets at most csets BM(xi, r) for all x∈M[33, A1.2 and A1.3], [29, Proposition 3.2]. Let κi=κxifor each i. Claim 4.There is a partition of Ninto finitely many sets, I1, . . . , Ic+1, such that BM(xi, r)∩BM(xj, r) = ∅ for i∈Ikand j∈Ilwith k6=l. This claim follows by considering the graph Gwhose set of vertices is N, and such that there is a unique edge connecting two different vertices, iand j, if and only if BM(xi, r)∩BM(xj, r)6=∅. Since there are at most cedges meeting at each vertex according to (ii), Gis c+ 1-colorable10; i.e., there is a partition of N into subsets, I1, . . . , Ic+1, such that there is no edge joining any pair of different vertices in any Ik. Let Sbe an isometric copy in Rn+1 of the standard n-dimensional sphere containing the origin 0. Choose some spherically symmetric C∞function ρ∈C∞(Rn) such that ρ(x)=1if|x| ≤ r/2 and ρ(x)=0if|x| ≥ r. Take also some C∞map τ:Rn→Rn+1 that restricts to a diffeomorphism Br→Sr{0}and maps RnrBr 9The notation c LiFiis used for the Hilbert space direct sum of a family of Hilbert spaces Fi; i.e., the Hilbert space completion of LiFiwith the scalar product h(vi),(wi)i=Pihvi, wii. 10This easily follows by induction, assigning to each ia color different from the colors of the previous vertices that are neighbors of i, which is possible because there are at most cof them (see [3]). 18 to 0. Let ˜ρibe the extension by zero of ρ◦κ−1 ito the whole of M, and let ˜ρk=Pi∈Ik˜ρi. For each k, define fk:M→Rn+2 by fk(x) = (0 if x /∈Si∈IkBM(xi, r) ˜ρk(x)/i, ˜ρk(x)·τ◦κ−1 i(x)if x∈BM(xi, r) for some i∈Ik. So fk◦κi= (ρ/i, ρ ·τ), obtaining that, for every multi-index α, the function |∂α(fk◦κi)|is uniformly bounded over Brby a constant depending only on |α|. Let f= (f1, . . . , fc+1) : M→R(c+1)(n+2). We have supM|∇mf|<∞for each m∈Nby (i). Moreover fk◦κi= (1/i, τ) on Br/2, obtaining that fis a C∞ embedding, and infM|Vndf|>0 by (i). By taking any isometric linear embedding of R(c+1)(n+2) into E, we can consider R(c+1)(n+2)-valued functions as E-valued functions; in particular, this applies to f. Claim 5.c Cl∞(imˆιM,f )⊂b M∞ ∗,imm(n). This claim is true because, for all [N, h, y]∈c Cl∞(imˆιM,f ), it is easy to see that infN|Vndh| ≥ infM|Vndf|>0, obtaining that his an immersion. Claim 6.c Cl∞(imˆιM,f ) is compact. This assertion follows by showing that any sequence in im ˆιM,f has a subsequence that is convergent in b M∞ ∗(n). Assume first that the sequence is of the form [M, f, xip] for some sequence of indices ip. Since Cl∞(im ιM) is compact in M∞ ∗(n) by [1, Theorem 12.3], we can suppose that [M, xip] converges to some point [N, y] in M∞ ∗(n). Take a sequence of compact domains Ωqin Nsuch that BN(y, q + 1) ⊂Ωq. For each q, there are pointed local embeddings φq,p : (N, y)(M, xip), for plarge enough, such that Ωq⊂dom φq,p and φ∗ q,pgM→gNon Ωqwith respect to the C∞topology. Let hq,p =φ∗ q,pfon Ωq. It is easy to see that, for all naturals qand m, the sequence khq,pkCm,Ωq,gNis uniformly bounded. Hence the functions hq,p form a compact subset of C∞(Ωq,R(c+1)(n+2)) with the C∞topology by [1, Proposition 3.11]. So some subsequence hq,p(q,`)is convergent to some hq∈C∞(Ωq,R(c+1)(n+2)) with the C∞topology. In fact, arguing inductively on q, it is easy to see that we can assume that each hq+1,p(q+1,`)is a subsequence of hq,p(q,`), and therefore hq+1 extends hq. Thus the functions hqcan be combined to define a function h∈C∞(M, R(c+1)(n+2)). Take sequences of integers, `q↑ ∞ and mq↑ ∞, so that kh−φ∗ q,p(q,`q)fkCmq,Ωq,gN=khq−hq,p(q,`q)kCmq,Ωq,gN→0. Then, considering has an E-valued function, we get that [M, f, xip(q,`q)]→[N, h, y] in b M∞ ∗(n) as q→ ∞. Now take an arbitrary sequence [M, f, x0 p] in im ˆιM,f . By (ii), there is a sequence of naturals, ip, such that dM(x0 p, xip)< r/2. By the above case in the proof, after taking a subsequence if necessary, we can assume that [M, f, xip] is convergent to some point [N, h, y] in b M∞ ∗(n). Thus, given sequences, mj↑ ∞ in N, and Sj↑ ∞ and sj↓0 in R+, there is some sequence pj↑ ∞ in Nsuch that there exists some (mj, Sj+esjr/2, λj, εj)-pointed local quasi-equivalence φj: (N, h, y)(M, f, xipj) for some λj∈[1, esj) and εj∈(0, sj). Since y0 j:= φ−1 j(x0 pj)∈BN(y, esjr/2), it follows that φj: (N, h, y0 j)(M, f, x0 pj) is an (mj, Sj, λj, εj)-pointed local quasi-equivalence, showing that [M, f, x0 pj]∈b Umj Sj,sj(N, h, y0 j). On the other hand, since the sequence y0 jis bounded in N, we can suppose that it is convergent to some y0∈Nby taking a subsequence if necessary. Hence [N, h, y0 j]→[N, h, y0] in b M∞ ∗(n) by the continuity of ˆιN,h. Hence there are sequences, nj↑ ∞ in N, and Tj↑ ∞ and tj↓ ∞ in R+, such that [N, h, y0 j]∈b Unj esjTj,tj(N, h, y0) for jlarge enough. So [M, f, x0 pj]∈b Umj Sj,sj◦b Unj esjTj,tj(N, h, y0)⊂b Umin{mj,nj} min{Sj,Tj},sj+tj(N, h, y0) for plarge enough by Propositionn 4.2-(iv). This shows that [M, f, x0 pj]→[N, h, y0] in b M∞ ∗(n), completing the proof of Claim 6.  Proof of Theorem 1.1. Given a connected, complete Riemannian n-manifold Mof bounded geometry, by Proposition 7.1, and Theorems 1.3 and 1.4, c Cl∞(imˆιM,f ) is a compact Riemannian foliated subspace of b M∞ ∗,imm(n). Moreover ˆιM,f :M→im ˆιM,f is an isometry because fis an embedding.  19 8. Open problems Question 8.1. In Theorem 1.1, is it possible to get the Riemannian foliated space so that its leaves have trivial holonomy? Question 8.1 can be reduced to the following question, in the same way as Theorem 1.1 follows from Proposition 7.1. Question 8.2. In Proposition 7.1, is it possible to get fsuch that moreover11 Iso(N, h) = {idM}if im ˆιN,h ⊂ c Cl∞(imˆιM,f )? In turn, Question 8.2 can be reduced to the following graph version. Consider only connected graphs with a countable set of vertices, all of them with finite degree. These graphs are proper path metric spaces in a canonical way so that each edge is of length one. Thus they define a subspace G∗of the Gromov space M∗ of pointed proper metric spaces. Decorate such graphs with maps of their vertex set to N. This gives rise to a space b G∗of isomorphism classes of pointed decorated graphs, like in the case of b M∞ ∗(n). Let c Cl denote the closure operator in b G∗. For each decorated graph (G, α), let Iso(G, α) denote its group of isomorphisms. There is a canonical map ˆιG,α :G→b G∗, like the above map ˆιM,f . It is said that Gis of bounded geometry if there is a uniform upper bound for the degree of its vertices. Question 8.3. For any graph Gof bounded geometry, does there exist a finite valued decoration αso that Iso(H, β) = {id}for all decorated graph (H, β) with imˆιH,β ⊂c Cl(imˆιG,α)? There are aperiodic tilings of R(like the Fibonacci tiling), or elements of {0,1}Z, giving rise to examples of decorations of the Cayley graph of Zsatisfying the condition of Question 8.3 (see e.g. [25]). If Question 8.3 had an affirmative answer, then, in the proof of Proposition 7.1, we could take a finite valued decoration α of Gsatisfying the condition of Question 8.3, and modify the definition of fso that fk(x) = ˜ρk(x)·(α(i)+1/i),˜ρk(x)·τ◦κ−1 i(x) if x∈BM(xi, r) for some i∈Ik. 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MR 1363743 (96j:57033) Departamento de Xeometr´ ıa e Topolox´ ıa, Facultade de Matem´ aticas, Universidade de Santiago de Compostela, Campus Vida, 15782 Santiago de Compostela, Spain E-mail address:[email protected] Departamento de Xeometr´ ıa e Topolox´ ıa, Facultade de Matem´ aticas, Universidade de Santiago de Compostela, Campus Vida, 15782 Santiago de Compostela, Spain E-mail address:[email protected] 21