A universal Riemannian foliated space

A UNIVERSAL RIEMANNIAN FOLIATED SPACE JES´ US A. ´ ALVAREZ L ´ OPEZ, RAM´ ON BARRAL LIJ´ O, AND ALBERTO CANDEL Abstract. It is proved that the isometry classes of pointed connected complete Riemannian n-manifolds form a Polish space, M∞ ∗(n), with the topology described by the C∞convergence of manifolds. This space has a canonical partition into sets defined by varying the distinguished point into each manifold. The locally non-periodic manifolds define an open dense subspace M∞ ∗,lnp(n)⊂M∞ ∗(n), which becomes aC∞foliated space with the restriction of the canonical partition. Its leaves without holonomy form the subspace M∞ ∗,np(n)⊂M∞ ∗,lnp(n) defined by the non-periodic manifolds. Moreover the leaves have a natural Riemannian structure so that M∞ ∗,lnp(n) becomes a Riemannian foliated space, which is universal among all sequential Riemannian foliated spaces satisfying certain property called covering-determination. M∞ ∗,lnp(n) is used to characterize the realization of complete connected Riemannian manifolds as dense leaves of covering-determined compact sequential Riemannian foliated spaces. Contents 1. Introduction 1 2. Preliminaries 3 2.1. Foliated spaces 3 2.2. Riemannian geometry 5 3. Quasi-isometries 6 4. Partial quasi-isometries 11 5. The C∞topology on M∗(n) 12 6. Convergence in the C∞topology 13 7. M∞ ∗(n) is Polish 15 8. Some basic properties of M∞ ∗,lnp(n) 17 9. Canonical bundles over M∞ ∗,lnp(n) 20 10. Center of mass 24 11. Foliated structure of M∞ ∗,lnp(n) 25 12. Saturated subspaces of M∞ ∗,lnp(n) 29 Acknowledgements 30 References 30 1. Introduction For any n∈N(we adopt the convention that 0 ∈N), let M∗(n) denote the set of isometry classes, [M, x], of pointed complete connected Riemannian n-manifolds, (M, x). The cardinality of each complete connected Riemannian n-manifold is less than or equal to the cardinality of the continuum, and therefore it may be assumed that its underlying set is contained in R. With this assumption, M∗(n) is a well defined set. This set is only interesting for n≥2 because M∗(0) = {[{0},0]}and M∗(1) = {[R,0],[S1,1]}. The set M∗(n) can be considered as a subset of the Gromov space M∗of isometry classes of pointed proper metric spaces [14], [15, Chapter 3]. However it is interesting to consider a finer topology on M∗(n), taking the differentiable structure into account. For that purpose, the following notion of C∞convergence was defined on M∗(n). Key words and phrases. C∞convergence of Riemannian manifolds; locally non-periodic Riemannian manifolds; Riemannian foliated space. 1 Definition 1.1 (See e.g. [33, Chapter 10, Section 3.2]).For each m∈N, a sequence [Mi, xi]∈M∗(n) is said to be Cmconvergent to [M, x]∈M∗(n) if, for each compact domain Ω ⊂Mcontaining x, there are pointed Cm+1 embeddings φi: (Ω, x)→(Mi, xi) for large enough isuch that φ∗ igi→g|Ωas i→ ∞ with respect to the Cmtopology [22, Chapter 2]. If [Mi, xi] is Cmconvergent to [M, x] for all m, then it is said that [Mi, xi] is C∞convergent to [M, x]. Here, a domain in Mis a connected C∞submanifold, possibly with boundary, of the same dimension as M. It is admitted that C∞convergence defines a topology on M∗(n) [32]. However we are not aware of any proof in the literature showing that it satisfies the conditions to describe a topology [28], [17] (see also [26] and [27] if C∞convergence were defined with nets or filters). This is only proved on subspaces defined by manifolds of equi-bounded geometry, where the C∞convergence coincides with convergence in M∗[29] (see also [33, Chapter 10]). The first main theorem of the paper is the following. Theorem 1.2. The C∞convergence in M∗(n)describes a Polish topology. Recall that a space is called Polish if it is separable and completely metrizable. The topology given by Theorem 1.2 will be called the C∞topology on M∗(n), and the corresponding space is denoted by M∞ ∗(n). For each complete connected Riemannian n-manifold M, there is a canonical continuous map ι:M→ M∞ ∗(n) given by ι(x) = [M, x], which induces a continuous injective map ¯ι: Iso(M)\M→M∞ ∗(n), where Iso(M) denotes the isometry group of M. The more explicit notation ιMand ¯ιMmay be also used. The images of the maps ιMform a natural partition of M∞ ∗(n), denoted by F∗(n). A Riemannian manifold, M, is said to be non-periodic if Iso(M) = {idM}, and is said to be locally non-periodic if each point x∈Mhas a neighborhood Uxsuch that {h∈Iso(M)|h(x)∈Ux}={idM}. Let M∗,np(n) and M∗,lnp(n) be the F∗(n)-saturated subsets of M∗(n) defined by non-periodic and locally non-periodic manifolds, respectively. The notation M∞ ∗,np(n) and M∞ ∗,lnp(n) is used when these sets are equipped with the restriction of the C∞topology. The restrictions of F∗(n) to M∗,np(n) and M∗,lnp(n) are respectively denoted by F∗,np(n) and F∗,lnp(n). Note that M∗,np(0) = {[{0},0]}and M∗,lnp(1) = ∅. On the other hand, let M∞ ∗,c(n) (respectively, b M∞ ∗,o(n)) be the F∗(n)-saturated subspace of b M∗(n) consisting of classes [M, x] such that Mis compact (respectively, open). Observe that, if [N, y] is close enough to any [M, x]∈M∞ ∗,c(n), then Nis diffeomorphic to M. Thus M∞ ∗,c(n) is open in M∗(n), and therefore M∞ ∗,o(n) is closed. Hence these are Polish subspaces of M∗(n), as well as their intersections with any Polish subspace. The intersection of M∞ ∗,c/o(n) and M∞ ∗,(l)np(n) is denoted by M∞ ∗,(l)np,c/o(n). The restrictions of F∗(n) to M∗,c/o(n) and M∗,(l)np,c/o(n) are denoted by F∗,c/o(n) and F∗,(l)np,c/o(n), respectively. The second main theorem of the paper is the following. Theorem 1.3. The following properties hold for n≥2: (i)M∗,lnp(n)is Polish and dense in M∞ ∗(n). (ii)M∞ ∗,lnp(n)≡(M∞ ∗,lnp(n),F∗,lnp(n)) is a foliated space of dimension n. (iii)F∗,lnp,o(n)is transitive. (iv)The foliated space M∞ ∗,lnp(n)has canonical C∞and Riemannian structures such that ¯ι: Iso(M)\M→ ι(M)is an isometry for every locally non-periodic, complete, connected Riemannian manifold M. (v)For any locally non-periodic complete connected Riemannian manifold M, the quotient map M→ Iso(M)\Mcorresponds to the holonomy covering of the leaf ι(M)by ¯ι: Iso(M)\M→ι(M). In particular, the set M∗,np(n)is the union of leaves of M∞ ∗,lnp(n)with trivial holonomy groups. The following result states a universal property of M∞ ∗,lnp(n), which involves certain property called covering-determination (Definition 12.1). Theorem 1.4. Let Xbe a sequential Riemannian foliated space of dimension n≥2whose leaves are complete. Then Xis isometric to a saturated subspace of M∞ ∗,lnp(n)if and only if it is covering-determined. 2 Recall that a space Xis called sequential if a subset A⊂Xis open whenever each convergent sequence xn→x∈Ain Xeventually belongs to A. For instance, first countable spaces are sequential. This condition could be removed by using convergence of nets or filters instead of sequences. M∞ ∗,lnp(n) is used to prove the following result about realizations of Riemannian manifolds as leaves. It involves the obvious Riemannian versions of the conditions of being aperiodic or repetitive, which are standard for tilings or graphs (see e.g. [12, 16, 35]), and a weak version of aperiodicity (Definitions 12.4 and 12.6). Theorem 1.5. The following properties hold for a complete connected Riemannian manifold Mof bounded geometry and dimension n≥2: (i)Mis non-periodic and has a (repetitive)weakly aperiodic connected covering if and only if it is isometric to a dense leaf of a (minimal)covering-determined compact sequential Riemannian foliated space. (ii)If Mis aperiodic (and repetitive), then it is isometric to a dense leaf of a (minimal)covering-determined compact sequential Riemannian foliated space whose leaves have trivial holonomy groups. 2. Preliminaries 2.1. Foliated spaces. Standard references for foliated spaces are [30], [5, Chapter 11], [6, Part 1] and [13]. Let Zbe a space and let Ube an open set in Rn×Z(n∈N), with coordinates (x, z). For m∈N, a map f:U→Rp(p∈N) is of class Cmif its partial derivatives up to order mwith respect to xexist and are continuous on U. If fis of class Cmfor all m, then it is called of class C∞. Let Z0be another space, and let h:U→Rp×Z0(p∈N) be a map of the form h(x, z) = (h1(x, z), h2(z)), for maps h1:U→Rpand h2: pr2(U)→Z0, where pr2:Rn×Z→Zis the second factor projection. It will be said that his of class Cmif h1is of class Cmand h2is continuous. For m∈N∪{∞} and n∈N, a foliated structure Fof class Cmand dimension dim F=non a space Xis defined by a collection U={(Ui, φi)}, where {Ui}is an open covering of X, and each φiis a homeomorphism Ui→Bi×Zi, for a locally compact Polish space Ziand an open ball Biin Rn, such that the coordinate changes φjφ−1 i:φi(Ui∩Uj)→φj(Ui∩Uj) are locally Cmmaps of the form φjφ−1 i(x, z)=(gij(x, z), hij(z)) . These maps hij will be called the local transverse components of the changes of coordinates. Each (Ui, φi) is called a foliated chart, the sets φ−1 i(Bi×{z}) (z∈Zi) are called plaques, and the collection Uis called afoliated atlas of class Cm. Two Cmfoliated atlases on Xdefine the same Cmfoliated structure if their union is a Cmfoliated atlas. If we consider foliated atlases so that the sets Ziare open in some fixed space, then Fcan be also described as a maximal foliated atlas of class Cm. The term foliated space (of class Cm) is used for X≡(X, F). If no reference to the class Cmis indicated, then it is understood that Xis a C0(or topological) foliated space. The concept of Cmfoliated space can be extended to the case with boundary in the obvious way, and the boundary of a Cmfoliated space is a Cmfoliated space without boundary. The foliated structure of a space Xinduces a locally Euclidean topology on X, the basic open sets being the plaques of all foliated charts, which is finer than the original topology. The connected components of Xin this topology are called leaves. Each leaf is a connected Cmn-manifold with the differential structure canonically induced by F. The leaf that contains each point x∈Xis denoted by Lx. The leaves of Fform a partition of Xthat determines the topological foliated structure. The corresponding quotient space, called leaf space, is denoted by X/F. The restriction of Fto some open subset U⊂Xis the foliated structure F|Uon Udefined by the charts of Fwhose domains are contained in U. More generally, a subspace Y⊂Xis a Cmfoliated subspace when it is a subspace with a Cmfoliated structure Gso that, for each y∈Y, there is a foliated chart of Fdefined on a neighborhood Uof yin X, whose restriction to U∩Ycan be considered as a chart of Gin the obvious way; in particular, dim G≤dim F. For instance, any saturated subspace is a Cmfoliated subspace. A map between foliated spaces is said to be a foliated map if it maps leaves to leaves. A foliated map between Cmfoliated spaces is said to be of class Cmif its local representations in terms of foliated charts are of class Cm. A Cmfoliated diffeomorphism between Cmfoliated spaces is a Cmfoliated homeomorphism between them whose inverse is also a Cmfoliated map. 3 Any topological space is a foliated space whose leaves are its points. On the other hand, any connected Cmn-manifold Mis a Cmfoliated space of dimension nwith only one leaf. The Cmfoliated maps M→X can be considered as Cmmaps to the leaves of X, and may be also called Cmleafwise maps. They form a set denoted by Cm(M, F), which can be equipped with the obvious generalization of the (weak)Cmtopology. In particular, for m= 0, we get the subspace C(M, F)⊂C(M, X) with the compact-open topology. For instance, C(I, F) (I= [0,1]) is the space of leafwise paths in X. Many concepts of manifold theory readily extend to foliated spaces. In particular, if Fis of class Cmwith m≥1, there is a vector bundle TFover Xwhose fiber at each point x∈Xis the tangent space TxLx. Observe that TFis a foliated space of class Cm−1with leaves T L for leaves Lof X. Then we can consider a Cm−1Riemannian structure on TF, which is called a (leafwise)Riemannian metric on X. This is a section of the associated bundle over Xof positive definite symmetric bilinear forms on the fibers of TF, which is Cm−1as foliated map. In this paper, a Riemannian foliated space is a C∞foliated space equipped with aC∞Riemannian metric, and an isometry between Riemannian foliated spaces is a C∞diffeomorphism between them whose restrictions to the leaves are isometries; in this case, the Riemannian foliated spaces are called isomertric. A foliated space has a “transverse dynamics,” which can be described by using a pseudogroup (see [18–20]). Apseudogroup Hon a space is a maximal collection of homeomorphisms between open subsets of Zthat contains idZ, and is closed by the operations of composition, inversion, restriction to open subsets of their domains, and combination. This is a generalization of a dynamical system, and all basic dynamical concepts can be directly generalized to pseudogroups. For instance, we can consider its orbits, and the corresponding orbit space is denoted by Z/H. It is said that His generated by a subset Ewhen all of its elements can be obtained from the elements of Eby using the pseudogroup operations. Certain equivalence relation between pseudogroups was introduced [18], [19], and equivalent pseudogroups should be considered to represent the same dynamics; in particular, they have homeomorphic orbit spaces. The germ groupoid of His the topological groupoid of germs of maps in Hat all points of their domains, with the operation induced by the composite of partial maps and the ´etale topology. Its subspace of units can be canonically identified with Z. For each x∈Z, the group of elements of this groupoid whose source and range is xis called the germ group of Hat x. The germ groups at points in the same orbit are conjugated in the germ groupoid, and therefore the germ group of each orbit is defined up to isomorphisms. Under pseudogroup equivalences, corresponding orbits have isomorphic germ groups. Let U={Ui, φi}be a foliated atlas of F, with φi:Ui→Bi×Zi, and let pi= pr2φi:Ui→Zi. The local transverse components of the corresponding changes of coordinates can be considered as homeomorphisms between open subsets of Z=FiZi, which generate a pseudogroup H. The equivalence class of Hdepends only on F, and is called its holonomy pseudogroup. There is a canonical homeomorphism between the leaf space and the orbit space, X/F→Z/H, given by L7→ H(pi(x)) if x∈L∩Ui. The holonomy groups of the leaves are the germ groups of the corresponding H-orbits. The leaves with trivial holonomy groups are called leaves without holonomy. The union of leaves without holonomy is denoted by X0. If Xis second countable, then X0is a dense Gδsaturated subset of X[11,21]. Given a loop αin a leaf Lwith base point x, there is a partition 0 = t0< t1<··· < tk= 1 of Iand there are foliated charts (Ui1, φi1),...,(Uik, φik) such that α([tl−1, tl]) ⊂Uilfor l∈ {1, . . . , k}. We can assume (Uik, φik) = (Ui1, φi1) because αis a loop. Let hil−1,ilbe the local transverse component of each change of coordinates φilφ−1 il−1defined around pil−1c(tl−1) and with hil−1,ilpil−1α(tl−1) = pilα(tl). The germ the composition hik−1,ik···hi1,i0at pi0(x) = pik(x) depends only on Fand the class of αin π1(L, x), obtaining a surjective homomorphism of π1(L, x) to the holonomy group of L. This homomorphism defines a connected covering e Lhol of L, which is called its holonomy covering. Now, let Rbe an equivalence relation on a topological space X. A subset of Xis called (R-) saturated if it is a union of (R-) equivalence classes. The equivalence relation Ris said to be (topologically)transitive if there is an equivalence class that is dense in X. A subset Y⊂Xis called an (R-) minimal set if it is a minimal element of the family of nonempty saturated closed subsets of Xordered by inclusion; this is equivalent to the condition that all equivalence classes in Yare dense in Y. In particular, X(or R) is called minimal when all equivalence classes are dense in X. These concepts apply to foliated spaces with the equivalence relation whose equivalence classes are the leaves. 4 2.2. Riemannian geometry. Let Mbe a Riemannian manifold possibly with boundary or corners (in the sense of [7], [10]). Connectedness of Riemannian manifolds is not assumed in Sections 2.2, 3 and 10 because it is not relevant for the concepts of these sections, but this property is assumed in the rest of the paper: it is needed in Section 4, and it is implicit in Sections 5–9 and 11–12 because the manifolds are given by elements of M∗(n). 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 GL(n)-principal bundle of tangent frames by π:PM →M, the O(n)-principal bundle of orthonormal tangent frames by π:QM →M, the Levi-Civita connection by ∇, the curvature by R, the exponential map by exp : TM →M(if Mis complete and ∂M =∅), the open and closed balls of center x∈Mand radius r > 0 by B(x, r) and B(x, r), respectively, and the injectivity radius by inj (if ∂M =∅). The penumbra around a subset S⊂Mof radius r > 0 is the set Pen(S, r) = {x∈M|d(x, S)< r }. 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” to the above notation. A covering 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 sometimes 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. When the vector space structure of TxM is emphasized, its zero element is denoted by 0x, or simply by 0, and the image of the zero section of π:TM →Mis denoted by Z⊂TM. 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 Cmmap 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, φ∗∗ if m= 2, and so on). Banach 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 TM is the unique Riemannian metric g(1) so that H⊥V and the canonical identities Hξ≡TξM≡Vξare isometries for every ξ∈TM. Continuing by induction, for m≥2, the Sasaki metric on T(m)Mis defined by 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” when a family of Riemannian manifolds Miis considered. From now on, T(m)Mis assumed to be equipped with g(m). Remark 1.The following properties hold for l < m and π:T(m)M→T(l)M: (i) g(m)|T(l)M=g(l). (ii) The submanifold T(l)M⊂T(m)Mis totally geodesic and orthogonal to the fibers of π. This follows easily by induction on m, where the case m= 1 is proved in [36, Corollary of Theorem 13]. (iii) The projection πis a Riemannian submersion with totally geodesic fibers. Again, this follows by induction on m, and the case m= 1 is proved in [36, Theorems 14 and 18]. (iv) For every ξ∈T(m)M, its projection π(ξ) is the only point ζ∈T(l)Mthat satisfies d(m)(ξ, ζ) = d(m)(ξ, T(l)M). To see this, it is enough to prove that π(ξ) is the only critical point of the distance function d(m)(·, ξ) on T(l)M. These critical points are just the points ζ∈T(l)Mwhere the shortest g(m)-geodesics γfrom ζto ξare orthogonal to T(l)Mat ζ. Hence γis a geodesic in π−1(ζ) by (iii), obtaining ζ=π(ξ). (v) For all ζ, ζ0∈T(l)M, the point ζ0is the only ξ∈π−1(ζ0) satisfying d(m)(ξ, ζ) = d(m)(ξ, π−1(ζ)). This follows like (iv), using (ii) instead of (iii). Let (U;x1, . . . , xn) be a chart of M. The corresponding metric coefficients are denoted by gij, and the Christoffel symbols of the first and second kind are denoted by Γijk and Γk ij, respectively. Using the Einstein notation, recall that Γα ijgαk = Γijk =1 2(∂igjk +∂jgik −∂kgij).(1) 5 Identify the functions xi,gij, Γijk and Γk ij with their lifts to TU. We get a chart (U(1);x1 (1), . . . , x2n (1)) of TM with U(1) =TU,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). The coefficients of the Sasaki metric g(1) with respect to (TU;x1 (1), . . . , x2n (1)) are [36, Eq. (3.5)]: g(1) ij =gij −gαγΓα µβΓβ ανvµvν g(1) n+i j = Γjµivµ g(1) n+i n+j=gij        (2) for 1 ≤i, j ≤n. Thus the metric coefficients g(1) αβ are given by universal fractional expressions of the functions gij,∂kgij and vi(1 ≤i, j, k ≤n). Using induction again, 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, and let g(m) αβ be the corresponding coefficients of g(m). Lemma 2.1. (i)The coefficients g(m) αβ are given by universal fractional expressions of the coordinates xn+1 (m), . . . , x2mn (m)and the partial derivatives up to order mof the coefficients gij . (ii)For each ρ > 0, the partial derivatives up to order mof the coefficients gij are given by universal linear expressions of the functions (σ(m) ρ,µ )∗g(m) αβ for n+ 1 ≤µ≤2mn, where σ(m) ρ,µ :U→U(m)is the section of π:U(m)→Udetermined by (σ(m) ρ,µ )∗xν (m)=ρδµν for n+ 1 ≤ν≤2mn, using Kronecker’s delta. Proof. We proceed by induction on m. For m= 1, (i) holds by (1) and (2), and (ii) holds by the second and third equalities of (2), since ∂igjk = Γijk + Γikj by (1). For arbitrary m≥2, assuming that (i) and (ii) hold for the case m−1, we get both properties for mby applying the above case to (g(m−1))(1) =g(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 by kTkCm,Ω,U,K= max amax x∈Ka∩ΩX |I|≤mX J,K  ∂|I|TK a,J ∂xI a (x), using the standard multi-index notation, 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. By taking the projective limit as m→ ∞, we get the Fr´echet space of C∞tensors of that type equipped with the C∞topology (see e.g. [22]). Observe that Uand Kare also qualified to define the norm k kCm,Ω0,U,Kfor 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 only on M,m, Ω, U,Kand g, such that 1 Ck kCm,Ω,U,K≤ k kCm,Ω,g ≤Ck kCm,Ω,U,g .(3) When ∂M =∅, it is said that Mis of bounded geometry if injM>0 and the function |∇mR|is bounded for all m∈N; in particular, Mis complete since injM>0. More precisely, given r > 0 and a sequence Cm>0, if injM≥rand |∇mR| ≤ Cmfor all m∈N, then (r, Cm) is called a geometric bound of M. A family Cof Riemannian manifolds without boundary is called of equi-bounded geometry if all of them are of bounded geometry with a common geometric bound; i.e., their disjoint union is of bounded geometry. 3. Quasi-isometries Let φ:M→Nbe a C1map between Riemannian manifolds. Recall that φis 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∈TM. This λis 6 called a dilation bound of φ. The second of the above inequalities, |φ∗(ξ)| ≤ λ|ξ|for all ξ∈TM, means that |φ∗| ≤ λ; i.e., |φ∗x| ≤ λfor all x∈M. Remark 2.(i) Every quasi-isometry is an immersion. (ii) If |φ∗| ≤ λ, then φis λ-Lipschitz; i.e., dN(φ(x), φ(y)) ≤λ dM(x, y) for all x, y ∈M. (iii) If φ:M→Nis a λ-quasi-isometry, then φis λ-bi-Lipschitz; i.e., for all x, y ∈M, 1 λdM(x, y)≤dN(φ(x), φ(y)) ≤λ dM(x, y). (iv) Let ψ:N→Lbe another C1map between Riemannian manifolds. If |φ∗| ≤ λand |ψ∗| ≤ µ, then |(ψφ)∗| ≤ λµ. (v) The composition of a λ-quasi-isometry and a µ-quasi-isometry is a λµ-quasi-isometry. (vi) The inverse of a λ-quasi-isometric diffeomorphism is a λ-quasi-isometric diffeomorphism. Consider the subbundle T≤rM={ξ∈TM | |ξ| ≤ r} ⊂ TM for each r > 0. If Mhas no boundary, then T≤rMis a manifold with boundary, being ∂T≤rM=TrM:= {ξ∈TM | |ξ|=r}; otherwise, T≤rMis 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. Note that T(m),≤rT(m0),≤rM=T(m+m0),≤rM. Definition 3.1. (i) It is said that φ:M→Nis a (λ-) quasi-isometry of order m∈N, or a (λ-) quasiisometric map of order m, if it is Cm+1 and φ(m) ∗:T(m),≤1M→T(m)Nis a (λ-) quasi-isometry. This λis called a dilation bound of order mof φ. The infimum of all dilations bounds of order mis called the dilation of order m. If φis a quasi-isometry of order mfor all m∈N, then it is called a quasi-isometry of order ∞. (ii) A collection Φ of maps between Riemannian manifolds is called a family of equi-quasi-isometries of order m∈Nif it is a family of quasi-isometries of order mwith some common dilation bound of order m, which is called an equi-dilation bound of order m. If Φ is a collection of equi-quasi-isometries of order mfor all m∈N, then it is called a family of equi-quasi-isometries of order ∞. (iii) A Riemannian manifold Mis said to be quasi-isometric with order mto another Riemannian manifold N when there is a quasi-isometric diffeomorphism of order m,M→N. With more generality, a collection {Mi}of Riemannian manifolds is called equi-quasi-isometric with order mto another collection {Ni} of Riemannian manifolds, with the same index set, when there is a collection of equi-quasi-isometric diffeomorphisms of order m,{Mi→Ni}. Remark 3.(i) The λ-quasi-isometries of order 0 are the λ-quasi-isometries. (ii) By Remark 1-(i), if φis a λ-quasi-isometry of order m≥1, then it is a λ-quasi-isometry of order m−1. (iii) For integers 0 ≤m0≤m, if φis a λ-quasi-isometry of order m, then φ(m0) ∗is a λ-quasi-isometry of order m−m0. To begin with, let us clarify the concept of quasi-isometry of order 1. Consider the splittings T(2)M= H⊕Vand T(2)N=H0⊕V0, where Hand H0are the horizontal subbundles, and Vand V0are the vertical subbundles. Fix any x∈Mand ξ∈TxM, and let x0=φ(x) and ξ0=φ∗(ξ). We have the canonical identities TξTM =Hξ⊕Vξ≡TxM⊕TxM , Tξ0TN =H0 ξ0⊕V0 ξ0≡Tx0N⊕Tx0N . (4) The pull-back Riemannian vector bundle φ∗TN is endowed with the pull-back ∇0of the Riemannian connection of N, and let φ∗:TM →φ∗TN also denote the homomorphism over idMinduced by φ. Let Xbe a C∞tangent vector field on some neighborhood of xin Mso that X(x) = ξ; thus φ∗Xis a C1local section of φ∗TN around xsatisfying (φ∗X)(x) = ξ0∈(φ∗TN)x≡Tφ(x)N. Then, for any ζ∈TxMand each C∞ function fdefined on some neighborhood of x, we have ∇0 ζ(φ∗(fX)) −φ∗(∇ζ(fX)) = f(x)∇0 ζ(φ∗X) + df(ζ)φ∗ξ−f(x)φ∗(∇ζX)−df(ζ)φ∗ξ =f(x) (∇0 ζ(φ∗X)−φ∗(∇ζX)) in (φ∗TN)x≡Tx0N. Therefore Aφ(ζ⊗ξ) := ∇0 ζ(φ∗X)−φ∗(∇ζX) depends only on ζ⊗ξ, and this expression defines a continuous section Aφof TM∗⊗T M∗⊗φ∗TN. Observe that Xcan be chosen so that ∇ζX= 0, 7 giving Aφ(ζ⊗ξ) = ∇0 ζ(φ∗X) in this case. Then, from the definitions of tangent map and covariant derivative, it easily follows that, according to (4), φ∗∗ξ(ζ1, ζ2)≡(φ∗(ζ1), φ∗(ζ2) + Aφ(ζ1⊗ξ)) (5) for all ζ1, ζ2∈TxM. Remark 4.If TM were used instead of T≤1Min the definition of quasi-isometries of order 1, we would get Aφ= 0, which is too restrictive. On the other hand, it would be weaker to use T1Minstead of T≤1M. Lemma 3.2. Suppose that φ:M→Nis C2. Then the following properties hold for r > 0and µ, ν, K ≥0: (i)If |φ∗∗ξ| ≤ µfor all ξ∈T≤rM, then |φ∗| ≤ µand |Aφ| ≤ µ/r. (ii)If |φ∗| ≤ νand |Aφ| ≤ K, then |φ∗∗ξ| ≤ √2(ν+Kr)for all ξ∈T≤rM. Proof. Assume that |φ∗∗ξ| ≤ µfor all ξ∈T≤rM. We get |φ∗| ≤ µby Remark 1-(i). Furthermore, for all x∈Mand ξ, ζ ∈TxMwith |ξ|=r, according to (4) and (5), |Aφ(ζ⊗ξ)| ≤ |(φ∗x(ζ), Aφ(ζ⊗ξ))|=|φ∗∗ξ(ζ, 0)| ≤ µ|(ζ, 0)|=µ|ζ|=µ r|ζ||ξ|. Now, suppose that |φ∗| ≤ νand |Aφ| ≤ K. Fix all x∈Mand ξ, ζ1, ζ2∈TxMwith |ξ| ≤ r, according to (4) and (5), |φ∗∗ξ(ζ1, ζ2)| ≤ |φ∗(ζ1)|+|φ∗(ζ2) + Aφ(ζ1⊗ξ)| ≤ ν|ζ1|+ν|ζ2|+K|ζ1||ξ| ≤ν|ζ1|+ν|ζ2|+Kr |ζ1| ≤ (ν+Kr) (|ζ1|+|ζ2|)≤√2(ν+Kr)|(ζ1, ζ2)|. Lemma 3.3. Suppose that φ:M→Nis C2. Then the following conditions are equivalent for r > 0: (i)φ∗:T≤rM→TN is a quasi-isometry. (ii)φis a quasi-isometry and |Aφ|is uniformly bounded. In this case, the constants involved in the above properties are related in the following way: (a)If µis a dilation bound of φ∗:T≤rM→TN, then µis a dilation bound of φand |Aφ| ≤ µ/r. (b)If νis a dilation bound of φ,|Aφ| ≤ K, and 0< κ < 1with νKκr < 1, then µ= max (√2(ν+Kr),√2ν 1−νKκr ,√2ν κ) is a dilation bound of φ∗:T≤rM→TN. Proof. Assume that (i) holds, and let µbe a dilation bound of order 1 of φ. Then φis a µ-quasi-isometry by Remark 1-(i). This shows (ii) and (a) by Lemma 3.2-(i). Now, suppose that (ii) holds, and take ν,K,κand µlike in (b). For all x∈Mand ξ, ζ1, ζ2∈TxMwith |ξ| ≤ r, according to (4) and (5), |φ∗∗ξ(ζ1, ζ2)| ≥ 1 √2(|φ∗(ζ1)|+|φ∗(ζ2) + Aφ(ζ1⊗ξ)|)≥1 √2(|φ∗(ζ1)|+κ|φ∗(ζ2) + Aφ(ζ1⊗ξ)|) ≥1 √2(|φ∗(ζ1)|+κ(|φ∗(ζ2)|−|Aφ(ζ1⊗ξ)|)) ≥1 √21 ν−Kκ |ξ||ζ1|+κ ν|ζ2| ≥1 √21 ν−Kκr|ζ1|+κ ν|ζ2|≥1 µ(|ζ1|+|ζ2|)≥1 µ|(ζ1, ζ2)|. This gives (i) and (b) by Lemma 3.2-(ii).  For c > 0, let hc:TM →T M be the C∞diffeomorphism defined by hc(ξ) = cξ. Observe that hc(T≤1M) = T≤cM, and the following diagram is commutative: TM φ∗ −−−−→ TN hc  y  yhc TM φ∗ −−−−→ TN 8 For each m∈Z+, let H(m+1) and V(m+1) denote the horizontal and vertical vector subbundles of T(m+1)M over T(m)M. Thus, for ξ∈T(m−1)Mand ζ∈TξT(m−1)M, TζT(m)M=H(m+1) ζ⊕V(m+1) ζ≡TξT(m−1)M⊕TξT(m−1)M . (6) Lemma 3.4. For all m∈Z+, there is an orthogonal vector bundle decomposition, T(m+1)M=P(m+1) ⊕ Q(m+1), preserved by h(m) c∗, such that, for ξ∈T(m−1)M,ζ∈TξT(m−1)Mand ζ0=h(m) c∗(ζ), the canonical identity TζT(m)M≡Tζ0T(m)Mgiven by (6) induces identities, P(m+1) ζ≡P(m+1) ζ0and Q(m+1) ζ≡Q(m+1) ζ0, so that h(m) c∗:P(m+1) ζ→P(m+1) ζ0≡P(m+1) ζis the identity, and h(m) c∗:Q(m+1) ζ→Q(m+1) ζ0≡Q(m+1) ζis multiplication by c. Proof. The proof is by induction on m. By the definition of connection, hc∗preserves the orthogonal decomposition T(2)M=H⊕V. Moreover, for ζ∈T M and ζ0=cζ,hc∗:Hζ→Hζ0≡Hζis the identity, and hc∗:Vζ→Vζ0≡Vζis multiplication by c. Thus the statement is true in this case with P(2) =Hand Q(2) =V. Now, suppose that m≥2 and the result holds for m−1. For ξ∈T(m−1)Mand ζ∈TξT(m−1)M, we have canonical identities H(m+1) ζ≡V(m+1) ζ≡TξT(m−1)M=P(m) ξ⊕Q(m) ξ,(7) obtaining orthogonal decompositions, H(m+1) =HP(m)⊕HQ(m)and V(m+1) =VP(m)⊕VQ(m), where (HP(m))ζ≡P(m) ξ≡(VP(m))ζand (HQ(m))ζ≡Q(m) ξ≡(VQ(m))ζaccording to (7). Then the result follows with P(m+1) =HP(m)⊕VP(m)and Q(m+1) =HQ(m)⊕VQ(m). Corollary 3.5. For all m∈Z+and c, r > 0, we have h(m) c∗(T(m+1),≤rM)⊂T(m+1),≤¯crM, where ¯c= max{c, 1}, and h(m) c∗:T(m+1)M→T(m+1)Mis a ˆc-quasi-isometry, where ˆc= max{c, 1/c}. Lemma 3.6. For all m∈Z+,r, s > 0and λ≥0, there is some µ≥0such that, for any Cm+1 map between Riemannian manifolds, φ:M→N, if |(φ(m) ∗)∗ξ| ≤ λfor all ξ∈T(m),≤rM, then |(φ(m) ∗)∗ξ| ≤ µfor all ξ∈T(m),≤sM. Moreover µcan be chosen so that µs →0as s→0for fixed m,rand λ. Proof. We proceed by induction on m. For m= 1, we have |φ∗∗ξ| ≤ λfor all ξ∈T≤rM. Then |φ∗| ≤ λand |Aφ| ≤ λ/r by Lemma 3.2-(i). Using Lemma 3.2-(ii), it follows that |φ∗∗ξ| ≤ √2λ(1 + s/r) =: µfor all ξ∈T≤sM. Note that µs →0 as s→0 for fixed rand λin this case. Now, assume that m≥2 and the result holds for m−1. For c=r/s and t= min{cr, r}, the diagram T(m),≤rMφ(m) ∗ −−−−→ T(m)N h(m−1) 1/c∗x    yh(m−1) c∗ T(m−1),≤tT≤sMφ(m) ∗ −−−−→ T(m)N (8) is defined and commutative. By Corollary 3.5 and Remark 2-(iv), it follows that |(φ(m) ∗)∗ξ| ≤ ˆc2λfor all ξ∈T(m−1),≤tT≤sM, where ˆc= max{c, 1/c}. Then, by the induction hypothesis applied to the map φ∗:T≤sM→TN, there is some µ≥0, depending only on m−1, t,sand ˆc2λ, such that |(φ(m) ∗)∗ξ| ≤ µfor all ξ∈T(m−1),≤sT≤sM=T(m),≤sM, and so that µs →0 as s→0 for fixed m,tand ˆc2λ. Corollary 3.7. For all m∈Z+,r > 0and λ≥0, there is some s > 0such that, for any Cm+1 map between Riemannian manifolds, φ:M→N, if |(φ(m) ∗)∗ξ| ≤ λfor all ξ∈T(m),≤1M, then φ(m+1) ∗(T(m+1),≤sM)⊂ T(m+1),≤rN. Proof. This is also proved by induction on m. The statement is true for m= 0 because, if |φ∗| ≤ λ, then φ∗(T≤sM)⊂T≤λsNfor all s > 0, and therefore it is enough to take s=r/λ in this case. Now, assume that m≥1 and the result is true for m−1. By Remark 1-(i), if |(φ(m) ∗)∗ξ| ≤ λfor all ξ∈T(m),≤1M, then |(φ(m−1) ∗)∗ξ| ≤ λfor all ξ∈T(m−1),≤1M. Hence, by the induction hypothesis, for all r > 0, there is some s > 0, as small as desired, such that φ(m) ∗(T(m),≤sM)⊂T(m),≤rN. On the other 9 For each i, there is some λi∈(1, eri) and some (mi, Ri, λi)-pointed local quasi-isometry φi: (Mi, xi) (Mi+1, xi+1), which can be assumed to be C∞by Remark 6-(iii). Then ¯ λi:= Qj≥iλj< e¯ri<∞. For i < j, the pointed local quasi-isometry ψij =φj−1···φi: (Mi, xi)(Mj, xj) is of type (mi, Ri/¯ λi,¯ λi) by Lemma 4.3-(i). For i, m ∈N, let Bi=Bi(xi, Ri), B0 i=Bi(xi, R0 i), B00 i=Bi(xi, R00 i), B(m) i=B(m) i(xi, Ri), B0(m) i=B(m) i(xi, R0 i), B00(m) i=B(mi) i(xi, R00 i). A bar will be added to this notation when the corresponding closed balls are considered. We have φi(Bi)⊂ Bi+1 because Ri+1 > λiRi, and φ(mi) i∗(B00(mi) i)⊂B0(mi) i+1 ⊂B0(mi+1) i+1 since R0 i+1 > λiR00 iand by Remark 1-(i). Furthermore B00 i⊂dom ψij and B00(mi) i⊂dom ψ(mi) ij∗for i<jbecause R00 ≤Ri/¯ λi. Therefore ψij(Bi)⊂Bj and ψ(mi) ij∗(B00(mi) i)⊂B0(mj) j. The restrictions ψij :Bi→Bjform a direct system of spaces, whose direct limit is denoted by c M. Let ψi:Bi→c Mbe the induced maps, whose images, b Bi:= ψi(Bi), form an exhausting increasing sequence of subsets of c M. All points ψi(xi) are equal in c M, and will be denoted by ˆx. The space c Mis connected because it is the union of the connected subspaces b Biwhose intersection contains ˆx. By the definition of the direct limit and since the maps ψij are open embeddings, it follows that all maps ψiare open embeddings, and therefore c Mis a Hausdorff n-manifold. Equip each b Biwith the C∞structure that corresponds to the C∞structure of Biby ψi. These C∞structures are compatible one another because the open embeddings ψij are C∞, and therefore they define a C∞structure on c M. Moreover let ˆgibe the Riemannian metric on each b Bithat corresponds to gi|Bivia ψi. Take some compact domains, Ωiin every Miand Ω(mi) iin T(mi)Mi, such that B0 i⊂Ωi⊂Int(Ω(mi) i) and B0(mi) i⊂Ω(mi) i⊂B00(mi) i; thus Ωi⊂B00 iby Remark 1-(ii). Let b Ωi=ψi(Ωi). Claim 1.c M=Sib Ωi. This equality holds because, for each i, there is some jso that R0 j>¯ λiRi, obtaining ψij(Bi)⊂Bj(xj,¯ λiRi)⊂B0 j⊂Ωj, and therefore b Bi=ψjψij(Bi)⊂ψj(Ωj) = b Ωj. Claim 2.For all i, the restrictions ˆgj|b Ωi, with j≥i, form a convergent sequence in the space of Cmisections, Cmi(b Ωi;Tb Ω∗ iTb Ω∗ i), with the Cmitopology, and its limit, ˆgi,∞, is positive definite at every point. Clearly, Claim 2 follows by showing that the restrictions of the metrics gij := ψ∗ ijgjto Ωi, for j≥i, form a convergent sequence in Cmi(Ωi;TΩ∗ iTΩ∗ i), and its limit, gi,∞, is positive definite at every point. To begin with, let us show that gij|Ωiis a Cauchy sequence with respect to k kCmi,Ωi,gi. We have 1 ¯ λi|ξ|(mi) i≤ |ξ|(mi) ij ≤¯ λi|ξ|(mi) i(9) for all ξ∈TΩ(mi) i, where | |(mi) iand | |(mi) ij are the norms defined by g(mi) iand g(mi) ij , respectively. By Lemma 6.1-(i), it follows that kg(mi) i−g(mi) ij kC0,Ω(mi) i,g(mi) i≤¯ λ2 i−¯ λ−2 i. Then, for k≥j, kg(mi) ij −g(mi) ik kC0,Ω(mi) i,g(mi) ij =kg(mi) j−g(mi) jk kC0,ψ(mi) ij∗(Ω(mi) i),g(mi) j ≤ kg(mj) j−g(mj) jk kC0,Ω(mj) j,g(mj) j≤¯ λ2 j−¯ λ−2 j(10) because ψ(mi) ij∗(Ω(mi) i)⊂ψ(mi) ij∗(B00(mi) i)⊂B0(mj) j⊂Ω(mj) j 16 and g(mj) jk =g(mi) jk on Ω(mj) j∩B(mi) j⊃ψ(mi) ij∗(Ω(mi) i) (Remark 1-(i)). We get kg(mi) ij −g(mi) ik kC0,Ω(mi) i,g(mi) i≤¯ λ2 i(¯ λ2 j−¯ λ−2 j) (11) by (9), (10) and Lemma 6.1-(iii). Let Uibe a finite collection of charts of Miwith domains Ui,a, and let Ki={Ki,a}be a family of compact subsets of Mi, with the same index set as Ui, such that Ki,a ⊂Ui,a for all a, and B00 i⊂SaKi,a =: Ki. Thus Ωi⊂Ki. With the notation of Section 2.2, let U(mi) ibe the family of induced charts of T(mi)Miwith domains U(mi) i,a . Like in the proof of Proposition 6.4-(ii), let K(mi) ibe the family of compact subsets K(mi) i,a ={ξ∈B(mi) i|π(ξ)∈Ki,a, d(mi) i(ξ, πi(ξ)) ≤R000 i} ⊂ U(mi) i,a , for some R000 i> R00 i, where π:B(mi) i→Bi. We have B00(mi) i⊂SaK(mi) i,a =: K(mi) i. Hence Ω(mi) i⊂K(mi) i. Choose some Ci≥1 satisfying (3) with Ui,Ki, Ωiand gi, and some C(mi) i≥1 satisfying (3) with U(mi) i, K(mi) i, Ω(mi) iand g(mi). For any ρ > 0 and n+ 1 ≤µ≤2min, let σ(mi) i,a,ρ,µ :Ui,a →U(mi) i,a be the section of each projection π:U(mi) i,a →Ui,a of the type used in Lemma 2.1-(ii). Since Ωi⊂Int(Ω(mi) i), there is some ρ > 0 so that σ(mi) i,a,ρ,µ(Ki,a ∩Ωi)⊂K(mi) i,a ∩Ω(mi) ifor all aand µ. Thus, by Lemma 2.1-(ii), given any ε > 0, there is some δ > 0, depending on εand ρ, such that kg(mi) ij −g(mi) ik kC0,Ω(mi) i,U(mi) i,K(mi) i < δ =⇒ kgij −gikkCmi,Ωi,Ui,Ki< ε/Ci.(12) Since ¯ λj↓1, we have ¯ λ2 i(¯ λ2 j−¯ λ−2 j)< δ/C(mi) ifor jlarge enough, giving kg(mi) ij −g(mi) ik kC0,Ω(mi) i,g(mi) i < δ/C(mi) i=⇒ kg(mi) ij −g(mi) ik kC0,Ω(mi) i,U(mi) i,K(mi) i < δ =⇒ kgij −gikkCmi,Ωi,Ui,Ki< ε/Ci=⇒ kgij −gikkCmi,Ωi,gi< ε by (11), (12) and (3). This shows that gij|Ωiis a Cauchy sequence in the Banach space Cmi(Ωi;TΩ∗ iTΩ∗ i) with k kCmi,Ωi,gi, and therefore it has a limit gi,∞. For all nonzero ξ∈TΩi, we have gi,∞(ξ, ξ) = lim jgij(ξ, ξ)≥1 ¯ λi gi(ξ, ξ)>0, obtaining that gi,∞is positive definite. This completes the proof of Claim 2. According to Claim 2, each ˆgi,∞is a CmiRiemannian metric on b Ωi, and, obviously, ˆgj,∞|b Ωi= ˆgi,∞for j > i. Hence the metric tensors ˆgi,∞can be combined to define a C∞Riemannian metric ˆgon c Mby Claim 1. Let | |(mi) i,∞be the norm defined by g(mi) i,∞on TΩ(mi) i. By (9) and because | |(mi) i,∞= limj| |(mi) ij on TΩ(mi) i, we get 1 ¯ λi|ξ|(mi) i≤ |ξ|(mi) i,∞≤¯ λi|ξ|(mi) ifor all ξ∈TΩ(mi) i. Thus, by Remark 2-(iii), Ωicontains the gi,∞-ball of center xiand radius R0 i/¯ λibecause it contains B0 i; in particular, c Mis complete because R0 i/¯ λi→ ∞ and every Ωiis compact. Since gi,∞=ψ∗ iˆg, it also follows that ψ(mi) i∗: Ω(mi) i→T(mi)c Mis a ¯ λi-quasi-isometry. So ψi: (Mi, xi)(c M, ˆx) is an (mi, R0 i,¯ λi)-pointed local quasi-isometry, obtaining that ([Mi, xi],[c M, ˆx]) ∈ Umi R0 i,sifor any sequence si↓0 with ¯ λi< esi, and therefore [Mi, xi]→[c M, ˆx] as i→ ∞ in M∞ ∗(n).  Corollary 7.3. M∞ ∗(n)is Polish. Proof. This is the content of Propositions 7.1 and 7.2 together.  Corollaries 6.6 and 7.3 give Theorem 1.2. 8. Some basic properties of M∞ ∗,lnp(n) For each closed C∞manifold Mof dimension ≥2, the non-periodic metrics on Mform a residual subset of Met(M) with the C∞topology [3, Corollary 3.5], [38, Proposition 1]. Then, since M∞ ∗,c(n) is dense in M∞ ∗(n) (Remark 8), it follows that M∞ ∗,np(n) is dense in M∞ ∗(n), and therefore M∞ ∗,lnp(n) is dense in M∞ ∗(n) too. On the other hand, M∞ ∗,lnp(n) is Gδin M∞ ∗(n) by Lemmas 8.1 and 8.3 below, and therefore it is a Polish subspace [25, Theorem I.3.11]. This proves Theorem 1.3-(i). 17 Lemma 8.1. For every n∈Z+and [M, x]∈M∞ ∗,lnp(n), there is some r > 0such that, if {h∈Iso(M)|h(x)∈B(x, r)}={idM}, then there is some neighborhood Lof [M, x]in M∞ ∗,lnp(n)so that {h∈Iso(L)|h(y)∈B(y, r)}={idL} for all [L, y]∈L. Proof. Suppose that the statement is false. Then there is some convergent sequence, [Mi, xi]→[M, x], in M∞ ∗(n) so that, for each i, some hi∈Iso(Mi)r{idMi}satisfies hi(xi)∈Bi(xi, r). Choose any sequence of compact domains Ωqof Msuch that B(x, 2r)⊂Int(Ωq) and d(x, ∂Ωq)→ ∞ as q→ ∞. For each qand ilarge enough, there is some pointed smooth embedding φq,i : (Ωq, x)→(Mi, xi) so that φ∗ q,igi→g|Ωqas i→ ∞ with respect to the C∞topology. Thus Bi(xi,2r)⊂φq,i(Int(Ωq)) for ilarge enough. Claim 3.If ris small enough, we can assume that there is some δ > 0 such that, for ilarge enough, the maps hican be chosen so that di(zi, hi(zi)) ≥δfor some zi∈Bi(xi, r). Given any index i, suppose first that there is some k∈Z r {0}such that hk i(xi)6∈ Bi(xi, r/2). Then there is some k∈Z r {0}such that hk i(xi)6∈ Bi(xi, r/2) and h` i(xi)∈Bi(xi, r/2) if |`|<|k|. If k= 1, then di(xi, hi(xi)) ≥r/2. If k=−1, then di(xi, hi(xi)) = dih−1 i(xi), xi≥r/2 as well. If |k| ≥ 2, then there is some `∈Zsuch that |`|,|k−`|<|k|. Hence dixi, hk i(xi)≤dixi, h` i(xi)+dih` i(xi), hk i(xi)=dixi, h` i(xi)+dixi, hk−` i(xi)≤r . Therefore, by using hk iinstead of hi, we can assume that di(xi, hi(xi)) ≥r/2 in this case. Now, suppose that hk i(xi)∈Bi(xi, r/2) for all k∈Z. Consider the non-trivial abelian subgroup Ai= {hk i|k∈Z} ⊂ Iso(Mi). Since a(xi)∈Bi(xi, r/2) for any a∈Ai, it follows that Aiis compact in the C∞topology by Proposition 3.11, and thus Aiis a non-trivial compact abelian Lie subgroup of Iso(Mi). Let µibe a bi-invariant probability measure on Ai, and let fi:Ai→Mbe the mass distribution defined by fi(a) = a(xi). By the C∞convergence φ∗ q,igi→g|Ωq, we can suppose that ris so small that the ball Bi(xi,2r/3) of Misatisfies the conditions of Proposition 10.2 for ilarge enough. Then, since fi(Ai)⊂ Bi(xi, r/2) ⊂Bi(xi,2r/3), the center of mass yi=Cfiis defined in Bi(xi,2r/3). Moreover yiis a fixed point of the canonical action of Aion M[24, Section 2.1]. Since there is a neighborhood of the identity in the orthogonal group O(n) which contains no non-trivial subgroup (simply because O(n) is a Lie group), it follows that there is some K > 0 such that, for any non-trivial subgroup A⊂O(n), there is some a∈Aand some v∈Rnsuch that |v|= 1 and |a(v)−v| ≥ K. In our setting, the subgroup {a∗yi|a∈Ai}of the orthogonal group O(TyiMi)≡O(n) is non-trivial because Miis connected and Aiis non-trivial. Hence there is some ai∈Aiand some ξi∈TyiMisuch that |ξi|= 1 and |ai∗(ξi)−ξi| ≥ K. By the C∞convergence φ∗ q,igi→g|Ωq, we can also assume that ris so small that there exists some C≥1 such that expyi:B(0yi, r)→B(yi, r) is C-quasi-isometric for ilarge enough. Then, for zi= expyi(r 3ξi)∈Bi(yi, r/3) ⊂Bi(xi, r), we get di(zi, ai(zi)) ≥r 3C|ξi−h0 i∗(ξi)| ≥ rK 3C. Thus, by using aiinstead of hi, we can assume in this case that di(zi, hi(zi)) ≥rK/3C. Therefore Claim 3 follows with δ= min{r/2, rK/3C}. For each q, we can assume that B(x, diam(Ωq) + r)⊂Int(Ωq+1), obtaining Bi(xi,diam(φq,i(Ωq)) + r)⊂Int(φq+1,i(Ωq+1)) for all ilarge enough by the C∞convergence φ∗ q,igi→g|Ωq. Then h0 q,i := φ−1 q+1,i hiφq,i : Ωq→Mis well defined for each qand all ilarge enough because xi∈φq,i(Ωq) and hi(xi)∈Bi(xi, r). On the one hand, from the C∞convergence φ∗ q,igi→g|Ωqand since hi(xi)∈Bi(xi, r), we get the C∞convergence h0∗ q,ig→g|Ωq and lim supid(x, h0 q,i(x)) ≤r; in particular, for each q, the maps h0 q,i are equi-quasi-isometries of order ∞. 18 Therefore, by Proposition 3.11, some subsequence of h0 q,i is C∞convergent to some C∞map h0 q: Ωq→M, which is an isometric embedding satisfying h0 q(x)∈B(x, r). For all p≥q, the restrictions h0 p|Ωqform a sequence of isometric embeddings satisfying h0 p(x)∈B(x, r). Then, by Proposition 3.11, there is some sequence of positive integers p(q, k) for each qso that the subsequence h0 p(q,k)|Ωqof h0 p|Ωqis C∞convergent as k→ ∞ to an isometric embedding h00 q: Ωq→Msatisfying h00 q(x)∈ B(x, r). We can assume that p(q+ 1, k) is a subsequence of p(q, k) for each q, yielding h00 q+1|Ωq=h00 q. So the maps h00 qcan be combined to define an isometry h:M→Msatisfying h(x)∈B(x, r). Now, fix any qand let z0 p,i =φ−1 p,i (zi) for each p≥qand all ilarge enough. From zi∈Bi(xi, r) and the C∞convergence φ∗ p,igi→g|Ωp, it follows that z0 p,i approaches the compact set B(x, r) as i→ ∞. Then, for each p≥q, there is a sequence zp,i in B(x, r) so that d(zp,i, z0 p,i)→0. Hence, by the C∞convergence φ∗ p,igi→g|Ωpand Claim 3, we get sup{d(z, h(z)) |z∈B(x, r)}= sup{d(z, h00 q(z)) |z∈B(x, r)} ≥sup lim inf pd(z, h0 p(z)) |z∈B(x, r)≥sup lim inf plim inf id(z, h0 p,i(z)) |z∈B(x, r) ≥lim inf plim inf id(zp,i, h0 p,i(zp,i)) = lim inf plim inf id(z0 p,i, h0 p,i(z0 p,i)) ≥lim inf idi(zi, hi(zi)) ≥δ . So h6= idM, which is a contradiction because h(x)∈B(x, r).  Lemma 8.2. For n≥2and each point [M, x]∈M∞ ∗,lnp(n), there is some r > 0such that, for each ε∈(0, r), there is some neighborhood Nof [M, x]in M∞ ∗,lnp(n)so that, if an equivalence class ι(L)of M∞ ∗,lnp(n)meets Nat points [L, y]and [L, z], then either dL(y, z)< ε or dL(y, z)> r. Proof. Since Mis locally non-periodic, there is some r > 0 such that {h∈Iso(M)|d(x, h(x)) ≤r}={idM}.(13) Suppose that the statement is false for this r. Then, given any ε∈(0, r), there are sequences [Li, yi] and [Li, zi] in M∞ ∗,lnp(n) converging to [M, x] in M∞ ∗,lnp(n) such that ε≤di(yi, zi)≤rfor all i. Take a sequence of compact domains Ωqof Msuch that x∈Ωqand d(x, ∂Ωq)→ ∞ as q→ ∞. For each q, there are C∞embeddings φq,i : Ωq→Miand ψq,i : Ωq→Mifor ilarge enough so that φq,i(x) = yi, ψq,i(x) = zi, and φ∗ q,igi, ψ∗ q,igi→g|Ωqas i→ ∞ with respect to the C∞topology. We can also assume that, for each q, B(x, diam(Ωq) + r)⊂Int(Ωq+1), giving φq,i(Ωq)⊂Bi(yi,diam(φq,i(Ωq))) ⊂Bi(zi,diam(φq,i(Ωq)) + r)⊂Int(ψq+1,i(Ωq+1)) for ilarge enough by the C∞convergence φ∗ q,igi, ψ∗ q,igi→g|Ωqand since di(yi, zi)≤r. So hq,i := ψ−1 q+1,i φq: Ωq→Mis well defined for each qand all ilarge enough. From the C∞convergence φ∗ q,igi, ψ∗ q,igi→g|Ωq, we also get the C∞convergence h∗ q,ig→g|Ωq, and moreover lim inf id(x, hq,i(x)) ≥ε , lim sup i d(x, hq,i(x)) ≤r , because φq,i(x) = yi,ψq,i(x) = ziand ε≤di(yi, zi)≤r. Then, like in the proof of Lemma 8.1, an isometry h:M→Mcan be constructed so that ε≤d(x, h(x)) ≤r, which contradicts (13).  Lemma 8.3. Let n∈Nand r > 0. For any convergent sequence [Mi, xi]→[M, x]in M∞ ∗(n)and each y∈B(x, r), there are points yi∈Bi(xi, r)such that [Mi, yi]→[M, y]in M∞ ∗(n). Proof. Take a sequence of compact domains Ωqof Msuch that x, y ∈Ωqand d(x, ∂Ωq)→ ∞ as q→ ∞. For each q, there is some index iqsuch that, for each i≥iqthere is a C∞embedding φq,i : Ωq→Mi satisfying φq,i(x) = xiand φ∗ q,igi→g|Ωqas i→ ∞ with respect to the C∞topology. Let yq,i =φq,i(y) for all i≥iq. Then, for each qand every m∈Z+, there is some index iq,m ≥iqsuch that di(xi, yq,i)< r and kφ∗ q,igi−gkCm,Ωq,g <1/m for all i≥iq,m. Moreover we can assume that iq,q < iq+1,q+1 for all q. Now, let yibe any point of Bi(xi, r) for i < i0,0, and let yi=yq,i for iq,q ≤i < iq+1,q+1. Let us check that [Mi, yi]→[M, y] in M∞ ∗(n). Fix any compact domain Ω of Mcontaining y, and let m∈N. We have 19 d(y, ∂Ωq)→ ∞ as q→ ∞ because d(x, ∂Ωq)→ ∞ and d(x, y)< r. So there is some q0≥msuch that Ω⊂Ωqfor all q≥q0. For i≥iq0,q0, let φi=φq,i|Ωif iq,q ≤i<iq+1,q+1 with q≥q0. Then φi(y) = yiand kφ∗ igi−gkCm,Ωq,g ≤ kφ∗ q,igi−gkCq,Ωq,g <1 q for iq,q ≤i<iq+1,q+1, obtaining φ∗ igi→g|Ωas i→ ∞. Lemma 8.4. For n∈N, let [M, x]∈M∞ ∗(n), and let Nbe a neighborhood of [M, x]in M∞ ∗(n). Then there is some δ > 0and some neighborhood Lof [M, x]in M∞ ∗(n)such that [L, z]∈Nfor all [L, y]∈Land all z∈BL(y, δ). Proof. There are some m∈Z+and ε > 0, and a compact domain Ω of Mcontaining xsuch that, for all [L, z]∈M∞ ∗(n), if there is some C∞embedding φ: Ω →Lso that φ(x) = zand kφ∗gL−gMkCm,Ω,gM< ε, then [L, z]∈N. Take any compact domain Ω0of Mwhose interior contains Ω. There is some ε0>0 and some neighborhood Hof idMin the group of diffeomorphisms of Mwith the weak Cmtopology such that, for all h∈Hand any metric tensor g0on Ω0satisfying kg0−gMkCm,Ω0,gM< ε0, we have h(Ω) ⊂Ω0and kh∗g0−gMkCm,Ω,gM< ε. Moreover there is some δ0>0 such that, for each z0∈BM(x, δ0), there is some h∈Hso that h(x) = z0. Let Lbe the neighborhood of [M, x] in M∞ ∗(n) that consists of the points [L, y]∈M∞ ∗(n) such that there is some C∞embedding ψ: Ω0→Lso that ψ(x) = yand kψ∗gL−gMkCm,Ω0,gM< ε0. There is some δ > 0 such that BL(y, δ)⊂ψ(Ω0) and ψ−1(BL(y, δ)) ⊂BM(x, δ0) for all [L, y]∈Land ψ: Ω0→Las above. Hence z0=ψ−1(z)∈BM(x, δ0) for each z∈BL(y, δ), and therefore there is some h∈Hsuch that h(x) = z0. Then φ:= ψh is defined on Ω and satisfies φ(x) = ψ(z0) = z. Moreover kφ∗gL−gMkCm,Ω,gM=kh∗ψ∗gL−gMkCm,Ω,gM< ε because kψ∗gL−gMkCm,Ω0,gM< ε0and h∈H. 9. Canonical bundles over M∞ ∗,lnp(n) For each n∈N, consider the set of pairs (M, ξ), where Mis a complete connected Riemannian manifold without boundary of dimension n, and ξ∈TM. Like in the case of M∗(n), we can assume that the underlying set of each complete connected Riemannian n-manifold is contained in R, obtaining that these pairs (M, ξ) form a well defined set. Define an equivalence relation on this set by declaring that (M, ξ) is equivalent to (N, ζ) if there is an isometric diffeomorphism φ:M→Nsuch that φ∗(ξ) = ζ. The class of a pair (M, ξ) will be denoted by [M, ξ], and the corresponding set of equivalence classes will be denoted by T∗(n). If orthonormal tangent frames are used instead of tangent vectors in the above definition, we get a set denoted by Q∗(n). Let πT∗(n):T∗(n)→M∗(n) and πQ∗(n):Q∗(n)→M∗(n) be the maps defined by π([M, ξ]) = [M, πM(ξ)] and π([M, f]) = [M, πM(f)] for [M, ξ]∈T∗(n) and [M, f]∈Q∗(n); the simpler notation πwill be used for πT∗(n)and πQ∗(n)if there is no danger of misunderstanding. For each [M, x]∈M∗(n), there are canonical surjections TxM→π−1 T∗(n)([M, x]), ξ7→ [M, ξ], and QxM→π−1 Q∗(n)([M, x]), f7→ [M, f]. Via the canonical surjection QxM→π−1 Q∗(n)([M, x]), the canonical right action of O(n) on QxMinduces a right action on π−1 Q∗(n)([M, x]); in this way, we get a canonical action of O(n) on Q∗(n) whose orbits are the fibers of πQ∗(n). The operation of multiplication by scalars on TxMalso induces an action of Ron π−1 T∗(n)([M, x]). However the sum operation of TxMmay not induce an operation on π−1 T∗(n)([M, x]). The following definition is analogous to Definition 1.1. Definition 9.1. For each m∈N, a sequence [Mi, ξi]∈T∗(n) (respectively, [Mi, fi]∈Q∗(n)) is said to be Cmconvergent to [M, ξ]∈T∗(n) (respectively, [M, f]∈Q∗(n)) if, with the notation x=π(ξ) and xi=πi(xi) (respectively, x=π(f) and xi=πi(fi)), for each compact domain Ω ⊂Mcontaining x, there are pointed Cm+1 embeddings φi: (Ω, x)→(Mi, xi) for large enough isuch that φi∗(ξ) = ξi(respectively, φi∗(f) = fi), and φ∗ igi→g|Ωas i→ ∞ with respect to the Cmtopology. If [Mi, ξi] (respectively, [Mi, fi]) is Cmconvergent to [M, ξ] (respectively, [M, f]) for all m, then it is said that [Mi, ξi] (respectively, [Mi, fi]) is C∞convergent to [M, ξ] (respectively, [M, f]). Theorem 9.2. The C∞convergence in T∗(n)and Q∗(n)describes a Polish topology. 20 To prove Theorem 9.2, we follow the steps of Sections 5–7. Definition 9.3. For m∈Nand R, r > 0, let Vm R,r (respectively, Wm R,r) be the set of pairs ([M, ξ],[N, ζ]) ∈ T∗(n)×T∗(n) (respectively, ([M, f],[N, h]) ∈Q∗(n)×Q∗(n)) such that there is some (m, R, λ)-pointed local quasi-isometry φ: (M, x)(N, y) for some λ∈[1, er) so that φ∗(ξ) = ζ(respectively, φ∗(f) = h). The following proposition is proved like Proposition 5.2. Proposition 9.4. The following properties hold for all m, m0∈Nand R, S, r, s > 0: (i) (Vm erR,r)−1⊂Vm R,r and (Wm erR,r)−1⊂Wm R,r. (ii)Vm0 R0,r0⊂Vm R,r ∩Vm0 S,s and Wm0 R0,r0⊂Wm R,r ∩Wm0 S,s, where m0= max{m, m0},R0= max{R, S}and r0= min{r, s}. (iii) ∆ ⊂Vm R,r and ∆⊂Wm R,r. (iv)Vm er+sR,r ◦Vm er+sR,s ⊂Vm R,r+sand Wm er+sR,r ◦Wm er+sR,s ⊂Wm R,r+s. Proposition 9.5. TR,r>0Vm R,r = ∆ and TR,r>0Wm R,r = ∆ for all m∈N. Proof. We only prove the first equality because the proof of the second one is analogous. The inclusion “⊃” is obvious; thus let us prove “⊂”. Let ([M, ξ],[N, ζ]) ∈TR,r>0Vm R,r, and let x=πM(ξ) and y= πN(ζ). Then there is a sequence of pointed local quasi-isometries φi: (M, x)(N, y), with corresponding types (m, Ri, λi), such that φi∗(ξ) = ζ, and Ri↑ ∞ and λi↓1 as i→ ∞. According to the proof of Proposition 5.3, there is a pointed isometric immersion ψ: (M, x)→(N, y) so that, for any i, the restriction ψ:BM(x, Ri)→Nis the limit of the restrictions of a subsequence φk(i,l)in the weak Cmtopology. Hence ψ∗(ξ) = limlφk(i,l)∗(ξ) = ζ, obtaining [M, ξ]=[N, ζ].  By Propositions 9.4 and 9.5, the sets Vm R,r (respectively, Wm R,r) form a base of entourages of a Hausdorff uniformity on T∗(n) (respectively, Q∗(n)), which is also called the C∞uniformity. The corresponding topology is also called the C∞topology, and the corresponding space is denoted by T∞ ∗(n) (respectively, Q∞ ∗(n)). Remark 9.(i) The maps π:T∞ ∗(n)→M∞ ∗(n) and π:Q∞ ∗(n)→M∞ ∗(n) are uniformly continuous and open because (π×π)(Vm R,r) = (π×π)(Wm R,r) = Um R,r for all m∈Nand R, r > 0. (ii) The canonical right O(n)-action on Q∞ ∗(n) is continuous. This follows easily by using that the composite of maps is continuous in the weak C∞topology [22, p. 64, Exercise 10], and the following property that can be easily verified: for each [M, f]∈Q∞ ∗(n) and any neighborhood Nof idMin the space of C∞ diffeomorphisms of Mwith the weak C∞topology, there is a neighborhood Oof the identity element ein O(n) such that, for all a∈O, there is some φ∈Nso that φ(x) = xand φ∗(f) = h. Definition 9.6. For R, r > 0 and m∈N, let Em R,r (respectively, Fm R,r) be the set of pairs ([M, ξ],[N, ζ]) ∈ T∗(n)×T∗(n) (respectively, ([M, f],[N, h]) ∈Q∗(n)×Q∗(n)) such that, with the notation x=πM(ξ) and y= πN(ζ), there is some Cm+1 pointed local diffeomorphism φ: (M, x)(N, y) so that φ∗(ξ) = ζ(respectively, φ∗(f) = h), and kgM−φ∗gNkCm,Ω,gM< r for some compact domain Ω ⊂dom φwith BM(x, R)⊂Ω. Like in the case of relations on M∗(n), for V⊂T∗(n)×T∗(n), W⊂Q∗(n)×Q∗(n), [M, ξ]∈T∗(n) and [M, f]∈Q∗(n), the simpler notation V(M, ξ) and W(M, f) is used instead of V([M, ξ]) and W([M, f]). Remark 10.By (3), a sequence [Mi, ξi]∈T∗(n) (respectively, [Mi, fi]∈Q∗(n)) is C∞convergent to [M, ξ]∈ T∗(n) (respectively, [M, f]∈Q∗(n)) if and only if it is eventually in Em R,r(M, ξ) (respectively, Fm R,r(M, f)) for arbitrary m∈Nand R, r > 0. Proposition 9.7. (i)For R, r > 0, if 0< ε ≤min{1−e−2r, e2r−1}, then E0 R,ε ⊂V0 R,r and F0 R,ε ⊂W0 R,r. (ii)For all m∈Z+,R, r > 0and [M, ξ]∈T∗(n) (respectively, [M, f]∈P∗(n)), there is some ε > 0such that Em R,ε(M, ξ)⊂Vm R,r(M, ξ) (respectively, Fm R,ε(M, ξ)⊂Wm R,r(M, ξ)). Proof. Let us show (i) for the case of V0 R,r, the case of W0 R,r being analogous. Let ([M, ξ],[N, ζ]) ∈E0 R,ε, and let x=πM(ξ) and y=πN(ζ). Then there is a C1pointed local diffeomorphism φ: (M, x)(N, y) such that φ∗(ξ) = ζ, and ε0:= kgM−φ∗gNkC0,Ω,gM< ε for some compact domain Ω ⊂dom φwith BM(x, R)⊂Ω. According to the proof of Proposition 6.4-(i), φis a (0, R, λ)-pointed local quasi-isometry if 1 ≤λ < erand ε0≤min{1−λ−2, λ2−1}, obtaining that ([M, ξ],[N, ζ]) ∈V0 R,r. 21 As above, let us prove (ii) only for the case of Vm R,r(M, ξ). Take m∈Z+,R, r > 0 and [M, ξ],[N, ζ]∈T∗(n), and let x=πM(ξ) and y=πN(ζ). According to the proof of Proposition 6.4-(ii), there is some ε > 0 such that, for every Cm+1 pointed local diffeomorphism φ: (M, x)(N, y), if kgM−φ∗gNkCm,Ω,gM< ε for some compact domain Ω ⊂dom φ∩Int(K) with BM(x, R)⊂Ω, then φis an (m, R, λ)-pointed local quasi-isometry (M, x)(N, y) for some λ∈[1, er). Therefore [N, ζ]∈Vm R,r(M, ξ) if [N, ζ]∈Em R,ε(M, ξ).  Proposition 9.8. (i)For all R, r > 0, if e2ε−e−2ε≤r, then V0 R,ε ⊂E0 R,r and W0 R,ε ⊂F0 R,r. (ii)For all m∈Z+,R, r > 0and [M, ξ]∈T∗(n) (respectively, [M, f]∈Q∗(n)), there is some ε > 0such that Vm R,ε(M, ξ)⊂Em R,r(M, ξ) (respectively, Wm R,ε(M, f)⊂Fm R,r(Mf)). Proof. This result follows from the proof of Proposition 6.5 in the same way as Proposition 9.7 follows from Proposition 6.4.  As a direct consequence of Remark 10, and Propositions 9.7 and 9.8, we get that the C∞convergence in T∗(n) and Q∗(n) describes the C∞topology. Proposition 9.9. T∞ ∗(n)and Q∞ ∗(n)are separable Proof. With the notation of Proposition 7.1, for every M∈C, let D0 Mand D00 Mbe countable dense subsets of TM and QM, respectively. Then the countable sets {[(M, g), ξ]|M∈C, g ∈GM, ξ ∈D0 M}and {[(M, g), f]|M∈C, g ∈GM, f ∈D00 M} are dense in T∞ ∗(n) and Q∞ ∗(n), respectively.  Proposition 9.10. T∞ ∗(n)and Q∞ ∗(n)are completely metrizable Proof. Only the case of T∞ ∗(n) is proved, the other case being similar. The C∞uniformity on T∞ ∗(n) is metrizable because it has a countable base of entourages. Thus it is enough to check that this uniformity is complete. Consider an arbitrary Cauchy sequence [Mi, ξi] in T∗(n) with respect to the C∞uniformity, and let xi=πi(ξi)∈Mi. We have to prove that [Mi, ξi] is convergent in T∞ ∗(n). By taking a subsequence if necessary, we can suppose that ([Mi, ξi],[Mi+1, ξi+1]) ∈Vmi Ri,rifor sequences mi, and Riand risatisfying the conditions of the proof of Proposition 7.2. Thus, for each i, there is some λi∈(1, eri) and some (mi, Ri, λi)-pointed local quasi-isometry φi: (Mi, xi)(Mi+1, xi+1), which can be assumed to be C∞ (Remark 6-(iii)), such that φi∗(ξi) = ξi+1. Then, with the notation of the proof of Proposition 7.2, we have ψij∗(ξi) = ξjfor i<j. Therefore there is some ˆ ξ∈Tˆxc Mso that ψi∗(ξi) = ˆ ξfor all i, obtaining that ([Mi, ξi],[c M, ˆ ξ]) ∈Umi R0 i/¯ λi,sifor all iaccording to the proof of Proposition 7.2. Hence [Mi, ξi]→[c M, ˆ ξ] as i→ ∞ in T∞ ∗(n).  Propositions 9.9 and 9.10 together mean that T∞ ∗(n) and Q∞ ∗(n) are Polish, completing the proof of Theorem 9.2. Let T∞ ∗,lnp(n)⊂T∞ ∗(n) and Q∞ ∗,lnp(n)⊂Q∞ ∗(n) be the subspaces defined by locally non-periodic manifolds. Proposition 9.11. (i)The projection π:T∞ ∗,lnp(n)→M∞ ∗,lnp(n)admits the structure of a Riemannian vector bundle of rank nso that the canonical map TxM→π−1([M, x]) is a orthogonal isomorphism for each [M, x]∈M∞ ∗,lnp(n). (ii)The projection π:Q∞ ∗,lnp(n)→M∞ ∗,lnp(n)admits the structure of a O(n)-principal bundle canonically isomorphic to the O(n)-principal bundle of orthonormal references of T∞ ∗,lnp(n). Proof. Obviously, the canonical O(n)-action on Q∞ ∗(n) preserves Q∞ ∗,lnp(n), and the O(n)-orbits in Q∞ ∗,lnp(n) are the fibers of π:Q∞ ∗,lnp(n)→M∞ ∗,lnp(n). Claim 4.For all [M, x]∈M∞ ∗,lnp(n), the canonical maps TxM→π−1 T∗(n)([M, x]) and QxM→π−1 Q∗(n)([M, x]) are bijections. Let us show the case of the first map in Claim 4, the case of the second one being similar. It was already pointed out that the canonical map TxM→π−1 T∗(n)([M, x]) is surjective, and let us to prove that it is also 22 injective. If [M, ξ]=[M, ζ] for some ξ, ζ ∈TxM, then φ∗(ξ) = ζfor some φ∈Iso(M) with φ(x) = x. But φ= idMbecause Mis locally non-periodic, obtaining ξ=ζ. Let Xbe a completely regular space with a right action of a Lie group G, and let Gx⊂Gdenote the isotropy subgroup at some point x∈X. Recall that a slice at xis a subspace S⊂Xcontaining xsuch that S·Gis open in X, and there is a G-equivariant continuous map κ:S·G→Gx\Gwith κ−1(Gx) = S[31, Definition 2.1.1]. Since Q∞ ∗,lnp(n) is completely regular and O(n) is compact, the O(n)-action on Q∞ ∗,lnp(n) has a slice Sat each point [M, f]∈Q∞ ∗,lnp(n) [31, Theorem 2.3.3] (see also [23], [34, Theorems 5.1 and 5.2] and [5, Theorems 11.3.9 and 11.3.14]). Then Θ := π(S) = π(S·O(n)) is open in M∞ ∗,lnp(n) by Remark 9-(i). Claim 5.π:S→Θ is a homeomorphism. This is the restriction of a continuous map (Remark 9-(i)), and therefore it is continuous. This map is also open because, for every open W⊂S, the set W·O(n) is open in Q∞ ∗,lnp(n) [31, Corollary of Proposition 2.1.2], and thus π(W) = π(W·O(n)) is open in M∞ ∗(n) (Remark 9-(i)). Obviously, π:S→Θ is surjective, and let us show that it is also injective. Take [N, p],[L, q]∈Ssuch that π([N, p]) = π([N, q]) =: x. Thus there is some a∈O(n) so that [L, q] = [N, p]·a. Since the isotropy group at [M, f] is trivial by Claim 4, there is an O(n)-equivariant continuous map κ:S·O(n)→O(n) so that κ−1(e) = S. It follows that e=κ([L, q]) = κ([N, p]·a) = κ([N, p]) a=a, obtaining [L, q]=[N, p], which completes the proof of Claim 5. According to Claim 5, the inverse of π:S→Θ defines a continuous local section σ: Θ →Q∞ ∗,lnp(n) of π:Q∞ ∗,lnp(n)→M∞ ∗,lnp(n). By the existence of continuous local sections, and since the O(n)-action on Q∞ ∗,lnp(n) is continuous and free (Remark 9-(ii) and Claim 4), it easily follows that π:Q∞ ∗,lnp(n)→M∞ ∗,lnp(n) admits the structure of an O(n)-principal bundle. By Claim 4, π−1 T∗(n)([M, x]) canonically becomes an orthogonal vector space for each [M, x]∈M∞ ∗,lnp(n), and we can canonically identify πQ−1 ∗(n)([M, x]) to the set of linear isometries π−1 T∗(n)([M, x]) →Rn. The continuity of the mapping ([M, f],[M, ξ]) 7→ [M, f]([M, ξ]) is easy to check. By using this identity, we get a homeomorphism θ:π−1 T∗(n)(Θ) →Rn×Θ defined by θ([M, ξ]) = (σ([M, x])([M, ξ]),[M, x]), where π([M, ξ]) = [M, x], whose inverse map is given by θ−1(v, [M, x]) = [M, σ([M, x])−1(v)]. If σ0: Θ0→Q∞ ∗,lnp(n) is another local section of π:Q∞ ∗,lnp(n)→M∞ ∗,lnp(n) defining a map θ0:π−1(Θ0)→Rn×Θ0as above, and [M, x]∈Θ∩Θ0, then the composite Rn≡Rn×{[M, x]}θ−1 −−−−→ π−1 T∗(n)([M, x]) θ0 −−−−→ Rn×{[M, x]} ≡ Rn is the orthogonal isomorphism σ0([M, x]) ◦σ([M, x])−1. It follows that π:T∞ ∗,lnp(n)→M∞ ∗,lnp(n), with these local trivializations, becomes an orthogonal vector bundle of rank nso that the canonical map TxM→ π−1([M, x]) is a orthogonal isomorphism for all [M, x]∈M∞ ∗,lnp(n). Moreover, by Claim 4, there is a canonical isomorphism between Q∞ ∗,lnp(n) and the O(n)-principal bundle of orthonormal frames of T∞ ∗,lnp(n).  By the compatibility of exponential maps and isometries, a map exp : T∞ ∗(n)→M∞ ∗(n) is well defined by setting exp([M, ξ]) = [M, expM(ξ)]. For each [M, x]∈M∞ ∗(n), the restriction exp : π−1([M, x]) →M∞ ∗(n) may be denoted by exp[M,x]. Lemma 9.12. Consider convergent sequences [Mi, fi]→[M, f]and [Mi, f0 i]→[M, f0]in Q∞ ∗(n)for some n∈Z+. Let x=π(f),x0=π(f0),xi=πi(fi)and x0 i=πi(f0 i). Suppose that there is some r > 0such that h∈Iso(M)|h(x)∈B(x, 2r)={idM},(14) and d(x, x0), di(xi, x0 i)≤rfor all i. Then there is some compact domain Ωin Mwhose interior contains x and x0, and there are C∞embeddings φi: Ω →Mifor ilarge enough so that φi∗(f) = fiand limiφ−1 i∗(f0 i) = f0 in PM, and limiφ∗ igi=g|Ωwith respect to the C∞topology. Proof. Let Ωqbe a sequence of compact domains in Msuch that B(x, r)⊂Int(Ωq),Pen(Ωq,diam(Ωq)) ⊂Int(Ωq+1) ; in particular, x0∈Int(Ωq). By the convergence [Mi, fi]→[M, f] and [Mi, f0 i]→[M, f0] in Q∞ ∗(n), for each q, there are C∞embeddings φq,i, ψq,i : Ωq→Mifor ilarge enough so that φq,i∗(f) = fi,ψq,i∗(f0) = f0 i, and limiφ∗ q,igi=g|Ωqand limiψ∗ q,igi=g|Ωqwith respect to the C∞topology; in particular, φq,i(x) = xi 23 and ψq,i(x0) = x0 i. We have x0 i∈Bi(xi, r)⊂Int(φq,i(Ωq)) for ilarge enough, depending on q, and therefore φq,i(Ωq)∩ψq,i(Ωq)6=∅. Hence ψq,i(Ωq)⊂Peni(φq,i(Ωq),diam(φq,i(Ωq))) ⊂Int(φq,i(Ωq+1)) for ilarge enough, depending on q. It follows that hq,i := φ−1 q+1,iψq,i is a well defined C∞embedding Ωq→M. Observe that limih∗ q,ig=g|Ωqwith respect to the C∞topology. Moreover lim sup i d(x, hq,i(x)) = lim sup i d(x, φ−1 q+1,iψq,i(x)) = lim sup i di(xi, ψq,i(x)) ≤lim sup i di(xi, x0 i) + lim sup i di(x0 i, ψq,i(x)) ≤r+d(x0, x)≤2r . If the statement is not true, then some neighborhood Uof f0in PM contains no accumulation point of the sequence φ−1 q+1,i∗(f0 i) = φ−1 q+1,i∗ψq,i∗(f0) = hq,i∗(f0) for each q. With the arguments of the proof of Lemma 8.1, it follows that there is some h∈Iso(M) such that d(x, h(x)) ≤2rand h∗(f0)6∈ U, which contradicts (14).  10. Center of mass The main tool used to prove Theorem 1.3-(ii)–(v) is the Riemannian center of mass of a mass distribution on a Riemannian manifold M[24], [8, Section IX.7]; especially, we will use the continuous dependence of the center of mass on the mass distribution and the metric tensor. Recall that a domain Ω ⊂Mis said to be convex when, for all x, y ∈Ω, there is a unique minimizing geodesic segment from xto yin Mthat lies in Ω (see e.g. [8, Section IX.6]). For example, sufficiently small balls are convex. For a fixed convex compact domain Ω in M, let C(Ω) be the set of functions f∈C2(Ω) such that the gradient grad fis an outward pointing vector field on ∂Ω and Hess fis positive definite on the interior Int(Ω) of Ω. Notice that C(Ω) is open in the Banach space C2(Ω) with the norm k kC2,Ω,g, and thus it is a C∞Banach manifold. Moreover C(Ω) is preserved by the operations of sum and product by positive numbers. Any f∈C(Ω) attains its minimum value at a unique point m(f)∈Int(Ω), defining a function m:C(Ω) →Int(Ω). Lemma 10.1. m is continuous. Proof. Consider the map v:C(Ω) ×Int(Ω) →TΩ defined by v(f, x) = grad f(x), and let Z⊂TΩ denote the image of the zero section. Since the graph of mis equal to v−1(Z), it is enough to prove the following. Claim 6.vis C1and transverse to Z. Here, smoothness and transversality refer to vconsidered as a map between C∞Banach manifolds [1, p. 45]. Let πHand πVdenote the orthogonal projections of T(2)Ω onto Hand V, respectively. Let X1(Ω) denote the Banach space of C1vector fields over Ω with the norm k kC1,Ω,g, which is equivalent to the norm k k1 defined by kXk1= sup {|X(x)|+|∇X(x)| | x∈Ω}. The gradient map, grad : C2(Ω) →X1(Ω), is a continuous linear map between Banach spaces, and therefore it is C∞. The evaluation map, ev : X1(Ω) ×Ω→TΩ, is C1because, if X∈X1(Ω), Y∈ TXX1(Ω) ≡X1(Ω), x∈Ω and ξ∈TxΩ, then ev∗(Y, ξ)∈TξTΩ is easily seen to be determined by the conditions πH(ev∗(Y, ξ)) ≡ξin Hξ≡TxΩ and πV(ev∗(Y, ξ)) ≡Y(x) + ∇ξXin Vξ≡TxΩ. Therefore vis C1because it is the restriction to C(Ω) ×Int(Ω) of the composition C2(Ω) ×Ωgrad ×idΩ −−−−−−→ X1(Ω) ×Ωev −−−−→ TΩ. Fix any f∈C(Ω) and x∈Int(Ω) with v(f, x)∈Z; thus grad f(x)=0x. Claim 7.πV:v∗({0f}×TxΩ) →V0xis an isomorphism. For any ξ∈TxΩ, πVv∗(0f, ξ) = πV(grad f)∗(ξ)≡ ∇ξgrad f in V0x≡TxΩ. Then Claim 7 follows because the mapping ξ7→ ∇ξgrad fis an automorphism of TxΩ since Hess fis positive definite at xand Hess f(ξ, ·) = g(∇ξgrad f, ·) on TxM. 24 From Claim 7, it follows that v∗({0f} × TxΩ) is a linear complement to H0x=T0xZin T0xTΩ; in particular, it is closed in T0xTΩ because T0xTΩ is Hausdorff of finite dimension. Since v∗:TfC(Ω)×TxΩ→T0xTΩ is linear and continuous, and T0xTΩ is Hausdorff of finite dimension, we get that the space v∗(f,x)−1(T0xZ) is closed and of finite codimension in the Banach space TfC(Ω) ×TxΩ, and therefore it has a closed linear complement in TfC(Ω) ×TxΩ (see e.g. [37, p. 22]), which completes the proof of Claim 6.  Remark 11.(i) In the last part of the above proof, the space v∗(f,x)−1(T0xZ) can be described as follows. Since h7→ grad h(x) defines a continuous linear map C2(Ω) →TxΩ, we have v∗(TfC(Ω) ×{0x})⊂V0x and v∗(h, 0x)≡grad h(x) in V0x≡TxΩ for any h∈C2(Ω) ≡TfC(Ω), giving v∗(f,x)−1(T0xZ)≡ {(h, ξ)∈C2(Ω) ×TxΩ|grad h(x) + ∇ξgrad f= 0 }, which is obviously closed and of finite codimension in C2(Ω) ×TxΩ. (ii) In Lemma 10.1, the map mis Cmif the Banach space Cm+2(Ω) is used instead of C2(Ω). Suppose that the Riemannian manifold Mis connected and complete. Let (A, µ) be a probability space, Ba convex open ball of radius r > 0 in M, and f:A→Ba measurable map, which is called a mass distribution on B. Consider the C∞function Pf:B→Rdefined by Pf(x) = 1 2ZA d(x, f(a))2µ(a). Proposition 10.2 (H. Karcher [24, Theorem 1.2]).With the above notation and conditions, the following properties hold: (i) grad Pfis an outward pointing vector field on the boundary ∂B. (ii)If δ > 0is an upper bound for the sectional curvatures of Min B, and 2r < π/2√δ, then Hess Pfis positive definite on B. If the hypotheses of Proposition 10.2 are satisfied, then Pf∈C(B), and therefore Pfreaches its minimum on Bat a unique point Cf∈B, which is called the center of mass of f. It is known that Cfdepends continuously on fwith respect to the supremum distance when (A, µ) is fixed [24, Corollary 1.6]; indeed, the following result follows directly from Lemma 10.1. Corollary 10.3. (i)Cfdepends continuously on fand the metric tensor of M. (ii)If Ais the Borel σ-algebra of a metric space, then Cfdepends continuously on µin the weak-∗topology. 11. Foliated structure of M∞ ∗,lnp(n) The goal of this section is to prove Theorem 1.3-(ii)–(v). For any point [M, x]∈M∞ ∗,lnp(n), choose some r, ε > 0 and some neighborhood N0of [M, x] in M∞ ∗,lnp(n) satisfying the statement of Lemma 8.2 with ε≤r/5. Using [33, Chapter 6, Theorem 3.6], we can assume that εand N0are so small that BL(y, ε) satisfies the conditions of Proposition 10.2 in Lfor all [L, y]∈N0. Take any continuous function λ:M∞ ∗(n)→[0,1] supported in N0and with λ([M, x]) = 1, whose existence is a simple consequence of the metrizability of M∞ ∗(n) (Theorem 1.2). For [L, y]∈N0, let ωLdenote the Riemannian density of L, and let λL,y :L→[0,1] be the function defined by λL,y(z) = (λ([L, z]) if dL(y, z)≤ε 0 if dL(y, z)≥ε , which is well defined and continuous by Lemma 8.2. Take another neighborhood N⊂N0of [M, x] where λ > 0. For [L, y]∈N, we have RLλL,y ωL>0, and set ¯ λL,y =λL,y RLλL,y ωL . Then µL,y =¯ λL,y ωLis a continuous density defining a probability measure on L, and the identity map (L, µL,y)→Lis a distribution of mass on Lsatisfying the conditions of Proposition 10.2 with BL(y, ε). Thus its center of mass, CL,y, is defined in BL(y, ε). Let c:N→M∞ ∗(n) be the map given by c([L, y]) = [L, CL,y]. 25 36. S. Sasaki, On the differentiable geometry of tangent bundles of Riemannian manifolds, Tˆohoku Math. J. (2) 10 (1958), no. 3, 211–368. MR 0112152 (22 #3007) 37. H.H. Schaefer, Topological vector spaces, Graduate Texts in Mathematics, vol. 3, Springer-Verlag, New York, Heidelberg, Berlin, 1971. MR 0342978 (49 #7722) 38. T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2) 121 (1985), no. 1, 169–186. MR 782558 (86h:58141) 39. S. Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581 (41 #9173) 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] Department of Mathematics, CSUN, Northridge, CA 91330, USA E-mail address:[email protected] 32